Stringy Things

Double Field Theory: The Courant Bracket

1. Introduction

In this post we are going to briefly and somewhat schematically discuss the appearance of the Courant bracket in Double Field Theory (DFT), following [1]. The point here is mainly to set the stage, so we jump straight into motivating the Courant bracket. In the next post, we will then study the B-transformations from the maths side and the C-bracket, following and expanding from [1] and others, with an emphasis in the end on how all of this relates to T-duality and strings.

What follows is primarily based on a larger collection of study notes, which I will upload in time.

2. Motivating the Courant Bracket

To understand the appearance of the Courant bracket in DFT, one way to start is by considering some general theory with a metric {g_{ij}(X)} and a Kalb-Ramond field (i.e., an antisymmetric tensor field) {b_{ij}(X)}, where {X \in M}. The symmetries and diffeomorphisms of {g_{ij}(X)} are generated by vector fields {V^{i}(X)}, where {V \in T(M)} with {T(M)} being the tangent bundle. As for {b_{ij}(X)}, the transformations are generated by one-forms {\xi_{i}(X)}, where {\xi^{i}(X) \subset T^{\star}(M)} with {T^{\star}(M)} being the cotangent bundle. We may combine {V^{i}(X)} and {\xi^{i}(X)} as a sum of bundles, such that (dropping indices) {V + \xi \in T(M) \oplus T^{\star}(M)}.

With these definitions, the opening question now is to ask, ‘what are the gauge transformations?’ To make sense of this, consider the following gauge parameters,

\displaystyle \delta_{V + \xi} g = \mathcal{L}_{V} g

\displaystyle \delta_{V + \xi} b = \mathcal{L}_{V} b + d\xi \ \ \ (1)

Here {\mathcal{L}} is the Lie derivative. Furthermore, note given that {V} generates diffeomorphisms, in (1) we get the Lie derivative in the direction of {V}. Also notice that {\xi} does not enter the gauge transformation of {g}; however, for the gauge transformation of {b}, we do have a one-form {\xi} and so we can take the exterior derivative. We should also note the following important properties of {\mathcal{L}}. For instance, when acting on forms the Lie derivative is,

\displaystyle \mathcal{L}_{V} = i_{V}d + di_{V} \ \ \ (2)

Where {iV} is a contraction with {V}. We’re just following the principle of a contraction with a vector times the exterior derivative. It is also worth pointing out that {\mathcal{L}} and the exterior derivative commutate such that,

\displaystyle \mathcal{L}_{V}d = d\mathcal{L}_{V} \ \ \ (3)

There are also some other useful identities that we are going to need. For instance, for the Lie algebra,

\displaystyle  [\mathcal{L}_{V_1}, \mathcal{L}_{V_2}] = \mathcal{L}_{[V_1,V_2]} \ \ (4)

Where {[V_1,V_2]} is just another vector such that {[V_1,V_2]^{k} = V_{1}^{p}\partial_{p}V_{2}^{k} - (1,2)}.

And, finally, we have,

\displaystyle  [\mathcal{L}_{X}, i_{Y}] = i_{[X,Y]} \ \ (5)

Now follows the fun part. Given the transformation laws provided in (1), we want to determine the gauge algebra. To do this, we must compute in reverse order the gauge transformations on the metric {g} and the {b}-field. For the metric we evaluate the bracket,

\displaystyle  [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]g = \delta_{V_2 + \xi_2} \mathcal{L}_{V_1}g - (1,2)

\displaystyle = \mathcal{L}_{V_1}\mathcal{L}_{V_2}g - (1,2)

Using the identity (4) we find,

\displaystyle  [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]g = \mathcal{L}_{[V_1, V_2]} g \ \ \ (6)

For the {b}-field we have,

\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]b = \delta_{V_2 + \xi_2} (\mathcal{L}_{V_1}b + d\xi_{1}) - (1,2)

\displaystyle = \mathcal{L}_{V_1}(\mathcal{L}_{V_2}b + d\xi_2) - (1,2)

\displaystyle = \mathcal{L}_{[V_1, V_2]} + d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}V_{2}\xi_{1} \ \ \ (7)

It turns out that this bracket satisfies the Jacobi identity, although it is not without its problems because, as we will see, there is a naive assumption present in the above calculations. In the meantime, putting this aside until later, the idea now is to compare the above with (1) and see what ‘pops out’. Notice that we find,

\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}] = \delta_{[V_1,V_2]} + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} \ \ \ (8)

In which we have discovered a bracket defined on {T(M) \oplus T^{\star}(M)},

\displaystyle  [V_{1} + \xi_{1}, V_{2} + \xi_{2}] = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2} \xi_{1} \ \ \ (9)

On the right-hand side of the equality we see a vector field in the first term and a one-form given by the final two terms. This Lie bracket is reasonable and, on inspection, we seem to have a definite gauge algebra. Here comes the problem allude a moment ago: there is a deep ambiguity in (9) in that we cannot, however much we try, determine unique parameters in our theory. Notice,

\displaystyle \delta_{V + \xi}b = \mathcal{L}_{V}b + d\xi

\displaystyle = \mathcal{L}_{V + (\xi + d \sigma)} \ \ \ (10)

The point being that the ambiguity of the one-form {\xi} is so up to some exact {d\sigma}. To put it another way, if we change {\xi} by {d\sigma}, we’re not actually changing anything at all. We would just get {\mathcal{L}_{V}b + d(\xi + d\sigma)} where, when the exterior derivative hits {d\sigma} we simply get nothing. So, given that {\xi} is ambiguous up to some exact {d\sigma}, in a sense what we have is a symmetry for a symmetry. In other words, the present construction is not sufficient.

What we can do to correct the situation is analyse the mistake in (7). Let us, for instance, look at the right-hand side of the summation sign in this equation,

\displaystyle d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}V_{2}\xi_{1}) = d(di_{V_1}\xi_{2} + i_{V_1}d\xi_{2}) - (1,2)

The logic follows that the first term {di_{V_1}\xi_{2}} is killed by {d}. It doesn’t make any contribution and its coefficient is just {1}. The trick then is to see, without loss of generality, that we may change the implicit coefficient {1} in front of {di_{V_1}\xi_{2}}. It turns out, the coefficient that we can use is {1 - \frac{\beta}{2}},

\displaystyle  = d((1-\frac{\beta}{2} di_{V_1}\xi_{2} + i_{V_1}d\xi_{2}) - (1,2)

\displaystyle = d(\mathcal{L}_{V_1}\xi_{2} - \frac{1}{2}\beta di_{V_1}\xi_{2}) - (1,2)

\displaystyle = d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2}\beta d [i_{V_1}\xi_{2} - i_{V_2}\xi_{1}]) \ \ \ (11)

What we end up achieving is the construction of a much more general bracket,

\displaystyle  [V_{1} + \xi_{1}, V_{2} + \xi_{2}]_{\beta} = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2}\beta d(iV_{1}\xi_{2} - iV_{2}\xi_{2}) \ \ \ (12)

What is so lovely about this result might at first seem counterintuitive. It turns out, as one can verify, for {\beta \neq 0}, we do not satisfy the Jacobi identity. So at first (12) may not seem lovely at all! But it makes perfect sense to consider cases of non-vanishing {\beta}. In mathematics, the case for {\beta = 1} was introduced by Theodore James Courant in his 1990 doctoral dissertation [5], where he studied the bridge between Poisson geometry and pre-symplectic geometry. The idea here is to forget about the Jacobi identity – consider its loss an artefact of field theory with anti-symmetric tensors and gravity – and impose {\beta = 1}. When we do this what we obtain is indeed the famous Courant bracket. That is, given {\beta = 1}, the case of maximal symmetry is described by,

\displaystyle  [V_{1} + \xi_{1}, V_{2} + \xi_{2}]_{\beta=1} = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2} d(iV_{1}\xi_{2} - iV_{2}\xi_{2} \ \ \ (13)

Although the Jacobi identity does not hold, one can show that for {Z_{i} = V_{i} + \xi_{i}, i = 1,2,3}, the Jacobiator assumes the form,

\displaystyle [Z_1, [Z_2,Z_3]] + \text{cyclic} = dN(Z_1, Z_2, Z_3)

Which is an exact one-form. This gives us a first hint that the unsatisfied Jacobi identity does not provide inconsistencies, because exact one-forms do not generate gauge transformations.

But why {\beta = 1}? Courant argued that the correct value of {\beta} is in fact {1} because, as he discovered, there is an automorphism of the bracket. This means that if do an operation on the elements, it respects the bracket. This automorphism is, moreover, an extra symmetry known in mathematics as a B-transformation. What follows from this is, I think, actually quite special. Given the Courant bracket is a generalisation of the Lie bracket, particularly in terms of an operation on the tangent bundle {T(M)} to an operation on the direct sum of {T(M)} and the p-forms of the vector bundle, what we will discuss is how the B-transformation in mathematics relates in a deep way to what in physics, especially string theory, we call T-duality (target- space duality). This is actually one of the finer points where mathematics and physics intersect so wonderfully in DFT.

In the next post we’ll carry on with a discussion of the B-transformation and then also the C-bracket, finally showing how everything relates.

References

[1] Zwiebach, B. (2010). ‘Double Field Theory, T-Duality, and Courant Brackets’ [lecture notes]. Available from [arXiv:1109.1782v1 [hep-th]].

[2] Hohm, O., Hull, C., and Zwiebach, B. (2010). ‘Generalized metric formulation of double field theory’. Available from [arXiv:1006.4823v2 [hep-th]].

[3] Hull, C. and Zwiebach, B. (2009). ‘Double Field Theory’. Available from [arXiv:0904.4664v2 [hep-th]].

[4] Hull, C. and Zwiebach, B. (2009). ‘The Gauge Algebra of Double Field Theory and Courant Brackets’. Available from [arXiv:0908.1792v1 [hep-th]].

[5] Courant, T. (1990). ‘Dirac manifolds’. Trans. Amer. Math. Soc. 319: 631–661. Available from [https://www.ams.org/journals/tran/1990-319-02/S0002-9947-1990-0998124-1/].

Standard
Stringy Things

Notes on String Theory: Ward Identities, Noether’s Theorem, and OPEs

1.1. Example 1

In the last entry we derived both the quantum version of Noether’s theorem and the Ward identity given in Polchinski’s textbook. This means we obtained the idea of the existence of conserved currents and how Ward identities in general constrain the operator products of these same currents. Let us now elaborate on some examples. The solutions to these examples are given in Polchinski (p.43); however, a more detailed review of the computation and of some of the key concepts will be provided below.

We start with the simplest example, where we once again invoke the theory of free massless scalars. Following Polchinski, the idea is that we want to perform a simple spacetime translation {\delta X^{\mu} = \epsilon a^{\mu}}. The action will be left invariant under worldsheet symmetry. But as what we want to derive is the current, given what we have been discussing, this means we should pay special attention to the fact that we are required to add {\rho (\sigma)} to the above translation. Recall that we defined {\rho(\sigma)} in our derivation of the Ward identity. The important point to note is that, again, the action is still invariant and from past discussions we already understand {\rho(\sigma)} has a compact or finite support. From this set-up, let us now rewrite the action for massless scalars,

\displaystyle  S_{P} = \frac{1}{4\pi \alpha^{\prime}} \int d^2\sigma \partial X^{\mu}\partial X_{\mu} \ \ (1)

When we vary (1) we obtain the following,

\displaystyle  \delta S = \frac{\epsilon a_{\mu}}{2\pi \alpha^{\prime}} \int d^2 \sigma \partial^{a}X^{\mu} \partial_{a}\rho \ \ (2)

There is a factor of 2 from varying {\partial X} that gives us a reduced denominator. We have also used the identity stated in Polchinski’s textbook, namely {\delta X^{\mu}(\sigma) = \epsilon \rho(\sigma) a^{\mu}}, where we can treat {\epsilon} and {a^{\mu}} as constants and therefore pull them in front of the integral.

Before we can move forward, there is something we have to remember. Recall the path integral formulation from our last discussion, where we found the variation to be proportional to the gradient. The result is written again below for convenience,

\displaystyle  [d\phi^{\prime}]e^{-S[\phi^{\prime}]} = [d\phi]e^{-S[\phi]}[1 + \frac{i\epsilon}{2\pi} \int d^2\sigma J^{a}\partial_{a}\rho + \mathcal{O}(\epsilon)^2] \ \ (3)

If there is no contribution from the metric, then the measure in brackets becomes {- \delta S}. What this tells us is that the variation must be something like,

\displaystyle  \delta S = -\frac{i\epsilon}{2\pi} \int d^2\sigma J^{a}\partial_{a}\rho \ \ (4)

Now notice, in computing both (2) and (4) we may establish the following interesting relation,

\displaystyle  \partial^{a}X_{a} \partial_{a} \rho \frac{\epsilon a_{\mu}}{2\pi \alpha^{\prime}} = -\frac{i\epsilon}{2\pi}J^{a} \partial_{a}\rho \ \ (5)

The first step is to simplify. Immediately, we can see that we can cancel the {\partial_{a}\rho} terms on both sides,

\displaystyle \partial_{a}X^{\mu} \frac{\epsilon a_{\mu}}{2\pi \alpha^{\prime}} = -\frac{i\epsilon}{2\pi}J_{a} \ \ (6)

This still leaves us with a bit of a mess. What we need to do is recall another useful fact. In the last section we studied the invariance of the path integral under change of variables, which, at the time, enabled us to obtain Noether’s theorem as an operator equation. Explicitly put, we had something of the general form { \frac{\epsilon}{2\pi i} \int d^{d} \sigma \sqrt{g} \rho(\sigma) \langle \nabla_{a}J^{a}(\sigma) ... \rangle}. Notice that we have all of the ingredients. Given the Noether current is,

\displaystyle J_{a} = a_{\mu}J_{a}^{\mu}

We may substitute for {J_{a}} in (6) and then work through the obvious cancellations that appear, including a cancellation of signs. Once this is done, we go on to obtain the following expression for the current,

\displaystyle J_{a}^{\mu} = \frac{i}{\alpha^{\prime}} \partial_{a}X^{\mu} \ \ (7)

Which is precisely what Polchinski gives in eqn. (2.3.13) on p.43 of his textbook. Automatically, we can see our currents are conserved. And, of course, we are free to switch to holomorphic and antiholomorphic indices and we can do so with relative ease,

\displaystyle J_{a}^{\mu} = \frac{1}{\alpha^{\prime}}\partial X^{\mu}

\displaystyle  \bar{J}_{a}^{\mu} = \frac{1}{\alpha^{\prime}}\bar{\partial} X^{\mu} \ \ (8)

In the manner indicated above, we have successfully constructed the current following a spacetime translation. For this example the goal is to now use an operator to check the Ward identity and see if the overall logic is sound. What we require in the process are the appropriate residues, and to find these we will need to compute the OPEs. So to test some of the ideas from earlier discussions in the context of the given example.

Recall the formula for OPEs given the reverse of the sum of subtractions, namely the sum of contractions, as described in (11) of this post. In this formula recall that we have two operators which are normal ordered, : \mathcal{F}: and : \mathcal{G}: . These are arbitrary functionals of X and typically the range of X is non-compact.

Now, in Polchinski’s first example we consider the case where {\mathcal{F} = J_{a}^{\mu}} and {\mathcal{G} = e^{ikX(z, \bar{z})}}. In other words, we want to compute the product of the current and the exponential operator. As the product is normal ordered, there are no singularities and the classical equations of motion are satisfied. Instead, the singularities are produced from the contractions, or, in this case, the cross-contractions as {z \rightarrow z_{0}}. Furthermore, as a sort of empirical rule, it can be said that the most singular term in {\frac{1}{z - z_{0}}} comes from the most cross-contractions. And we should recall that we compute the cross-contractions by hitting our operators with {\delta / \delta X^{\mu}_{\mathcal{F}}} and {\delta / \delta X^{\mu}_{\mathcal{G}}}, respectively. Hence, from the master formula for cross-contractions,

\displaystyle  : \frac{i}{\alpha^{\prime}} :\partial X^\mu(z): :e^{ik  X(z_0,{\bar z_{0}})}: = \exp [- \frac{\alpha^{\prime}}{2} \int d^2 z_1 d^2 z_2 \ln \mid z_{12}\mid^2 \frac{\delta}{\delta X_{\mathcal{F}}^{\mu}(z_1, \bar{z}_1)} \frac{\delta}{\delta X_{\mathcal{G} \mu}(z_2, \bar {z}_2)}] \ \ :\frac{i}{\alpha^{\prime}} \partial X^\mu(z) e^{i k X(z_{0},\bar{z}_{0})}:

\displaystyle  = : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z)  e^{i k X(z_{0},\bar{z}_{0})} : - \frac{i}{2} : \int d^2 z_1 d^2 z_2 \ln \mid z_{12} \mid^2 \frac{\delta(\partial X^{\mu}(z))}{\delta X_{\mathcal{F}}^{\mu}(z_1, \bar{z}_1)} \frac{\delta  ( e^{i k X(z_0, \bar{z}_0)})}{\delta X_{\mathcal{G} \mu}(z_2, \bar{z}_2)} \ \ (9)

Note that for {\mathcal{G}}, which, in this case is {e^{ikX}}, it is an eigenfunctional of {\delta / \delta X_{\mathcal{G}} (z_{2}, \bar{z}_{2})}. Likewise, for for {\delta / \delta X_{\mathcal{F}}} we will end up with a delta function,

\displaystyle  = : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z)  e^{i k X(z_{0},\bar{z}_{0})} : - \frac{i}{2} : \int d^2 z_1 d^2 z_2 \ln \mid z_{12} \mid^2 \partial (\delta^{\mu}_{\alpha} \delta^2(z_1, z)) i k^{\alpha} \delta^2(z_2, z_0)  e^{i k X(z_{0}, \bar{z}_0)} : \ \ (10)

Now, we can pull out the ik^{\mu} which flips the sign,

\displaystyle = : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z)  e^{i k X(z_{0},\bar{z}_{0})}: + \frac{k^{\mu}}{2} : \partial  (\int d^2 z_1 d^2 z_2 \ln \mid z_{12} \mid^2 \delta^2(z_1, z) \delta^2(z_2, z_{0})  e^{i k X(z_{0},\bar{z}_0)}) : \ \ (11)

Notice that we have delta functions inside the integrand, so we are left with,

\displaystyle  : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z)  e^{i k X(z_{0},\bar{z}_{0})} : + \frac{k^{\mu}}{2} : \partial (  \ln \mid z - z_{0} \mid^2 e^{i k X(z_0,\bar{z}_0)}) :

\displaystyle = : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z)  e^{i k X(z_{0},\bar{z}_{0})} :  + \frac{k^{\mu}}{2 (z - z_{0})}  e^{i k X(z_0,\bar{z}_0)} \ \ (12)

And so we obtain the following result,

\displaystyle \frac{i}{\alpha^{\prime}} : \partial_{a}X^{\mu} : :e^{ikX}: \sim \frac{k^{\mu}}{2 (z - z_{0})} : e^{ikX}: \ \ (13)

Where {\sim} means the most singular pieces. We can also perform the same calculations for the antiholomorphic term,

\displaystyle \frac{i}{\alpha^{\prime}} : \bar{\partial}X^{\mu}: : e^{ikX}: \sim \frac{k^{\mu}}{2(\bar{z} - \bar{z}_{0})} :e^{ikX}: \ \ (14)

As Polchinski notes and as we see, the OPE is in agreement with the Ward identity. But we can still carry on a bit further. To conclude this example, recall explicitly to mind the Ward identity and our residues. Switching back to the holomorphic case and evaluating the LHS of (14) notice we find, picking out only the residues,

\displaystyle  \frac{i}{\alpha^{\prime}} : \partial X^{\mu} e^{ikX}: = (\frac{k^{\mu}}{2} + \frac{k^{\mu}}{2}) G = k^{\mu} G \ \ (15)

So, we have that it must be equal to {k^{\mu}} times the operator {\mathcal{G}} from above. Now, given that {\mathcal{G} = \mathcal{A}}, the right-hand side of the Ward identity tells us that,

\displaystyle  k^{\mu} \mathcal{A} = \frac{1}{i \epsilon} \delta \mathcal{A} \ \ (16)

And, again, from the Ward identity we can see in (15) that with a bit of algebra the variation of the operator must be,

\displaystyle  \delta \mathcal{A} = ik^{\mu}\epsilon \mathcal{A} \ \ (17)

Where we are assuming the variation is only in one direction. Interestingly, as an aside, what is actually happening are the following transformation properties,

\displaystyle  \mathcal{A} = e^{ikX} \rightarrow e^{ikX + ik^{\mu}\epsilon}

\displaystyle  X^{\mu} \rightarrow X^{\mu} + \epsilon \ \ (18)

1.2. Example 2

In the first example we considered a spacetime translation. We can now look to the second example in Polchinski’s textbook, where we want to consider a worldsheet translation, particularly how the {a} of the {\sigma} coordinates transforms as {\delta \sigma^{a} = \epsilon v^{a}}. Here {v^{a}} is a constant vector. It follows that from the action for free massless scalars is invariant under this transformation, with the above symmetry clearly understood given {X} is a scalar and how {\delta \sigma^{a}} does not change the measure of integration. And so, just as in the first example, what we want to do is investigate the construction of the conserved current as a result of this worldsheet symmetry transformation and then test the Ward identity.

The first step is to note that because we are dealing with a scalar theory we may write explicitly,

\displaystyle  \sigma^{a} \rightarrow \sigma^{\prime a} = \sigma + \epsilon v^{a} \ \ (19)

Where, for any worldsheet symmetry transformation, the scalar fields simply transform as follows,

\displaystyle X^{\prime \mu}(\sigma^{\prime}) = X^{\mu}(\sigma) \ \ (20)

From which it also follows that,

\displaystyle  X^{\prime \mu}(\sigma + \delta \sigma) = X^{\mu}(\sigma) \implies X^{\prime \mu}(\sigma) = X^{\mu}(\sigma - \delta \sigma) \ \ (21)

Where we should recognise that in brackets on the left-hand side of the first equality, {\sigma + \delta \sigma = \sigma^{\prime}}.

Of course, like the first example, we’re interested in how operators transform. And so we want to consider,

\displaystyle  \delta X^{\mu}(\sigma) = X^{\prime \mu}(\sigma) - X^{\mu}(\sigma) = X^{\prime \mu} (\sigma^{a} - \epsilon v^{a}) - X^{\mu}(\sigma) \ \ (22)

Expanding and only keep the 1st terms, what we end up with is precisely an expression for how our operators transform,

\displaystyle  \delta X^{\mu}(\sigma) = -\epsilon(\sigma) v^{a}\partial_{a}X^{\mu} \ \ (23)

Now, what we want to do is check with the Ward identity. So, like before, let’s start by varying the action and then build from there,

\displaystyle  \delta S = \delta [\frac{1}{4\pi \alpha^{\prime}}\int d^2\sigma \partial^{a}X^{\mu}\partial_{a}X_{\mu}]

\displaystyle = \frac{1}{2\pi \alpha^{\prime}} \int d^2\sigma \partial^{a}X^{\mu}\partial_{a}\delta X_{\mu} \ \ (24)

Where {\delta X_{\mu} = -\epsilon(\sigma)v^{a}\partial_{a}X_{\mu}}. The implication is as follows. From (24) we can substitute for {\delta X_{\mu}},

\displaystyle \delta S = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}\sigma \partial^{a}X^{\mu}\partial_{a}(-\epsilon(\sigma)v^{a}\partial_{a}X_{\mu})

\displaystyle  = -\frac{\epsilon}{2\pi \alpha^{\prime}} \int d^2\sigma \partial^{a}X^{\mu}\partial_{a} v^{b}\partial_{b}X_{\mu} +  \partial^{c}X^{\mu}\partial_{c}v^{d}\partial_{d}X_{\mu}

\displaystyle = -\frac{\epsilon}{2\pi \alpha^{\prime}} \int d^2\sigma \partial^{a}X^{\mu}\partial_{a} v^{b}\partial_{b}X_{\mu} \ [1] + \partial_{d}(\frac{1}{2}v^{d} \partial^{c}X^{\mu}\partial_{c}X^{\mu}) \ [2] \ \ (25)

Where, for pedagogical purposes, the first and second integrands have been labelled [1] and [2] respectively. The reason is because it will be useful to recall these pieces separately in order to highlight some necessary computational logical and procedure. Before that, however, we should think of the conserved current. It follows, as we have already learned,

\displaystyle -\delta S = \frac{i}{2\pi} \int d^2\sigma \sqrt{-g}J^{a}\partial_{a}\epsilon \ \ (26)

Remember, looking at (2) in a previous entry, we can see clearly that {J^{a}(\sigma)} is the coefficient of {\partial_{a}\rho (\sigma)}. In the first example we become more familiar with this fact. And what Polchinski is referencing in the single passing sentence that he provides prior to eqns. (2.3.15a) and (2.3.15b) is that we need to make contact with this formalism. It is convenient to now reassert the {\rho(\sigma)} term,

\displaystyle = -\frac{\epsilon}{2\pi \alpha^{\prime}} \int d^2\sigma [ (\partial^{a}X^{\mu}\partial_{b} v^{b}X_{\mu}) \partial_{b}(\rho(\sigma)) \ [1] + (\rho(\sigma)) \partial_{c}(\frac{1}{2}v^{c} \partial^{d}X^{\mu}\partial_{d}X^{\mu})] \ [2] \ \ (27)

Now, let’s look at both pieces of (27). Piece [1] above looks fine and, on inspection, seems quite manageable. Piece [2], on the other hand, is not very nice. In taking one step forward, what we can do is integrate the second piece by parts. This has the benefit that we can eliminate the total derivative that arises and eliminate the surface terms. To save space, the result is given below,

\displaystyle  \delta S = \frac{\epsilon}{2 \pi \alpha'}\int d^2\sigma v^b\partial^a X^{\mu} \partial_b X_{\mu} \partial_a - \partial_b (\frac{1}{2}v^b \partial^a X^{\mu} \partial_a  X_\mu)

\displaystyle  =\frac{\epsilon}{2 \pi \alpha'}\int d^2\sigma [v^b(\partial^a X^\mu  \partial_b X_{\mu} -\frac{1}{2}\delta^{a}_{b} \partial_b X^\mu \partial^b  X_\mu) \partial_a] \ \ (28)

How to interpret (28)? Notice something very interesting. We have the stress-energy tensor plus some additional terms outside the small brackets. If we make the appropriate substitution for the stress-energy tensor we therefore obtain,

\displaystyle  \delta S = -\dfrac{\epsilon}{2 \pi}\int d^2\sigma \, (v^c\,T_c^a)\partial_a \ \ (29)

If we bring the constant epsilon back into the integrand, we have an integral over the worldsheet times a derivative of the parameter of an infinitesimal transformation. Whatever is left can be interpreted as a conserved current. Hence, then, if we go back and inspect (2) in this post we come to establish what Polchinski states in eqn. (2.3.15a). Our indices are slightly different up to this point, but this is merely superficial and when we rearrange things we find,

\displaystyle  J^{a} = iv^{b}T_{b}^{a}

And then lowering the index on {J},

\displaystyle  J_{a} = iv^{b}T_{ab} \ \ (30)

This is our conserved current. In certain words, it is natural to anticipate a conserved current on the string worldsheet and also for this current to be related to the stress-energy tensor. Just thinking of the physical picture gives some idea as to why this is a natural expectation. But we are not quite done.

What we want to do, ultimately, is define the stress-energy tensor as an operator with full quantum corrections. But, as we are working in conformal field theory, there is an ambiguity about how we might define it related to normal ordering. Let’s explore this for a moment.

We should think of stringy CFTs by way of how we will define a set of basic operators, and then from this show what is the stress-energy tensor. Moreover, it is a property of the stress-energy tensor and the basic operators we utilise that will give definition to the CFT. In CFT language, it is given that the stress-energy tensor can be written as,

\displaystyle  T_{ab} = \frac{1}{\alpha^{\prime}} : \partial_{a}X^{\mu}\partial_{b}X_{\mu} - \frac{1}{2}\delta_{ab}(\partial X)^2 : \ \ (31)

This is what Polchinski cites in eqn. (2.3.15b). We can still go a step further and discuss the topic of conformal invariance in relation to this definition. For instance, from the principles of conformal invariance, it remains the case that as discussed much earlier in these notes,

\displaystyle  T_{a}^{a} = 0

Which is to say, as we should remember, the stress-energy tensor is traceless. This condition of tracelessness tells us how, if we were to go to holomorphic and antiholomorphic coordinates,

\displaystyle  T_{a}^{a} = 0 \rightarrow T_{z\bar{z}} = 0 = T_{\bar{z}z} \ \ (32)

Where one may recall, also, the non-vanishing parts {T_{zz}} and {T_{\bar{z}\bar{z}}} from an earlier discussion in this collection of notes. It follows that if the stress-energy tensor is, in fact, traceless, we may invoke the conservation of the current such that,

\displaystyle  \nabla^{a}J_{a} = 0 = \nabla^{a} T_{ab} = 0 \ \ (33)

Which is to say that we have full conservation for the full stress-energy tensor. We can write this in terms of holomorphic and antiholomorphic coordinates as expected,

\displaystyle  \bar{\partial}T_{zb} + \partial T_{\bar{z}b} = 0  \ \ (34)

This gives us two choices:

\displaystyle  b = z \implies \partial T_{zz} = 0

\displaystyle  b = \bar{z} \implies \partial T_{\bar{z}\bar{z}} = 0 \ \ (35)

Where, as it was discussed some time ago, {T_{zz} = T(z)} is a holomorphic function and {T_{\bar{z}\bar{z}} = \bar{T}(\bar{z})} is an antiholomorphic function.

It is perhaps quite obvious at this point that we may also write,

\displaystyle  T(z) = -\frac{1}{\alpha^{\prime}} : \partial X^{\mu}\partial X_{\mu}:

\displaystyle  \bar{T}(\bar{z}) = -\frac{1}{\alpha^{\prime}} : \bar{\partial} X^{\mu}\bar{\partial} X_{\mu}: \ \ (36)

Now, returning to our current (30), we can be completely general in our study of the current,

\displaystyle  J_{z} = iv(z)T(z)

\displaystyle  \bar{J}_{\bar{z}} = i\bar{v(z)}\bar{T}(\bar{z}) \ \ (37)

If we have conservation of the current, then the above is the same as,

\displaystyle  \nabla_{a}J^{a} = \bar{\partial}J_{z} + \partial J_{\bar{z}} = 0 \ \ (38)

Which is to say that the new currents are conserved provided {v(z)}, previously considered a constant vector, is holomorophic. Additionally, the current is of course associated with symmetries; but what are these symmetries? They are the conformal transformations.

If, in the bigger picture, what we want to do is find {\delta X} due to symmetries {J_{z} = ivT(z)}, to proceed recall the Ward identity {\mathcal{A} = X^{\mu}}. It follows we need to compute an OPE for the stress-energy tensor with our scalar field (complete computation is given in the Appendix of this chapter, along with other important and useful OPEs),

\displaystyle  :T(z) : :X^{\mu}(z_{0}, \bar{z}_{0}): = -\frac{1}{\alpha^{\prime}} : \partial X\partial X: : X^{\mu}(z_{0}, \bar{z}_{0}) :

\displaystyle  \sim -(\frac{2}{\alpha^{\prime}}) \cdot (-\frac{\alpha^{\prime}}{2}\partial_{z}\ln \mid z - z_{0}\mid^2) : \partial X(z_{0}) :

\displaystyle  T X^{\mu} \sim \frac{1}{z - z_{0}}\partial X(z_{0})

\displaystyle  \bar{T}X^{\mu} \sim \frac{1}{\bar{z} - \bar{z}_{0}}\bar{\partial}X(\bar{z}_{0}) \ \ (39)

Which is what Polchinski states in eqn. (2.4.6). And now we can use the Ward identity and take the residue of the current with the coefficients of the OPE for the holomorphic and antiholomorphic pieces,

\displaystyle  iv(z_{0}) \partial X(z_{0}) + i \bar{v}(z_{0} \bar{\partial} X(\bar{z}_{0}) = \frac{1}{i\epsilon} \delta X \ \ (40)

And so what we find is that, for the current we have constructed, we have a symmetry transformation of the following form,

\displaystyle  \delta X = -iv(z_{0})\partial X(z_{0}) - i\bar{v}(z_{0} \bar{\partial}X(\bar{z}_{0}) \ \ (41)

For {z_{0} \rightarrow z_{0} + \epsilon v(z_{0})}. If we drop {z_{0}} and generalise,

\displaystyle  \delta X^{\mu} = -\epsilon v(z)\partial X - \epsilon\bar{v}(z)\bar{\partial}X \ \ (42)

For {z \rightarrow z + \epsilon v(z)} which is an infinitesimal transformation, where the only constraint is that {z \rightarrow z^{\prime} = f(z)} is holomorphic.

The reason the transformation is so simple,

\displaystyle  \delta X^{\mu} = X^{\prime \mu}(z^{\prime}, \bar{z}^{\prime}) - X^{\mu}(z, \bar{z}) = X^{\mu}(z - \epsilon v, z - \epsilon\bar{v}) - X^{\mu} \ \ (43)

Where, after Taylor expansion,

\displaystyle  \delta X^{\mu} = - \epsilon v\partial X - \epsilon\bar{v}\bar{\partial}X \ \ (44)

It is important to point out that {f(z) = \xi(z)} represents a global rescaling (but can also represent a local rescaling). If {\mid \xi \mid = 1} then we have a simple rotation, and in general no scaling.

To conclude, from the very outset of this chapter, we may also recall to mind that in the context of the conformal group we are working in 2-dimensions. When we ask, ‘what is the analogue of this symmetry in higher dimensions?’, the answer is that in higher dimensions we can construct scale invariance as well. Indeed, in D-dimensions, if you have {y^{\mu} \rightarrow \lambda y^{\mu}} you have additional special conformal transformations.

References

Joseph Polchinski. (2005). ‘String Theory: An Introduction to the Bosonic String’, Vol. 1.

Blumenhagen, R. and Plauschinn. (2009). ‘Introduction to Conformal Field Theory’.

Standard
Stringy Things

Notes on the Swampland (4): The Distance Conjecture for Arbitrary Calabi-Yau Manifolds, the Emergence Proposal, and the de Sitter Conjecture

The following collection of notes is based on a series of lectures that I attended by Eran Palti at SiftS 2019 at Universidad Autonoma de Madrid. The theme of the lecture series was ‘String Theory and the Swampland’. Palti’s five lectures were supported by his most recent and impressive 200 page review paper on the Swampland, which includes over 600 references [arXiv: 1903.06239 [hep-th]]. The reader is directed to this paper in addition to supplementary references that I also provide at the end of each set of notes.

In the following entry, the notes presented follow the fourth and fifth lectures of Palti’s series.

1. Introduction

In this final entry, we approach the conclusion of this collection of notes by focusing on the fourth and fifth of Palti’s lectures. Due to lack of space, we will not cover every topic in lectures four and five. Instead, we shall focus our energy on paying particular attention and detail to one of the most important and interesting subjects of study presented by Palti (from lecture four): namely, the study of the Distance Conjecture in the context of arbitrary Calabi-Yau (CY) manifolds. Then we will conclude these notes by briefly thinking about a few choice cosmological implications of the Swampland (the topic of Palti’s fifth lecture), particularly the de Sitter conjecture in the context of Type IIA string theory on CY with flux and in the context of 11-dimensional supergravity.

But before all this, we spend a short amount of time reflecting on the ‘Emergence Proposal’ (a concept introduced at the end of lecture four) and on some timely issues facing the Swampland programme.

2. A House of Cards? Emergence and the Swampland

As a summary review, let us quickly recall what we have so far emphasised in this series of notes. One of the featured viewpoints to be highlighted in Palti’s lectures is the observation that, among the growing list of Swampland Conjectures, there is ample reason to suggest that the Distance Conjecture and the Weak Gravity Conjecture are among two of the most established in terms of evidence. That is to say, short of complete and formal proof, the amount of evidence supporting these two conjectures in particular is very solid. We have already spent quite a bit of time exploring a number of tests and we have already begun to develop a deeper understanding for why the DC and the WGC are supported by significant evidence.

This emphasis on mathematical testability is important. In general, it does not seem too egregious to admit that the Swampland programme as a while has being experiencing both internal and external controversy. This controversy would seem as much scientific as philosophical. For example, consider the most recent Swampland conference (also held at IFT) that followed a couple months after the summer school from which these notes were originally written. One description of the situation at the September conference is this: the list of conjectures has experienced unrelenting growth but at the result of questionable rigour. In a moment of hyperbole issued for exaggerating effect, we might say that the programme itself is developing analogously as an infinite tower of conjectures. It is a well-established concern among portions of the Swampland research community that we are not proving/disproving conjectures faster than the rate in which new conjectures are being introduced. And from the perspective of this humble student, the situation has reached a point where proof and disproof are imperative.

In listening to and monitoring debates about the programme, I have come to be of the mind that we should proceed with particular caution. The caution is this: there is a genuine concern growing about a lack of mathematical rigour, which would seem verified by the observation that the list of conjectures is growing at a much faster rate than formal proof/disproof. This concern gains further urgency when considering the surrounding sociology, where calls for systematic evidence have been matched with what seems a more generally developing narrative against String Theory / M-theory writ large. It is understandable that the Swampland programme is compelled to react with a mission to make predictions and provide such evidence. But these sorts of commitments may still be premature. Predictions are key, but if we do not even know the theory to be accurate, any predictive claims or evidence would seem to put not just the Swampland but the entire reputation of String Theory at risk. In other words, it would seem reckless to begin contemplating demands for evidence without exhaustive rigour and confirmation of the theory at hand. In the very least, and in the best possible scenario, we have a theory not completely understood. But in either case, to then make predictions on these grounds – on a tower of conjectures, which, at the end of the day could very well be a house of cards – is risky.

But perhaps it is in this context that the moral tone of Palti’s lecture series earlier in the summer (prior to the Swampland conference in September) might be seen to be profoundly insightful and of timely inclination. In Palti’s fourth lecture, for example, the message becomes much more pronounced – that we may take the view that the DC and the WGC are in fact two fundamental pillars of the Swampland. Let me state this slightly differently. In reflecting on Palti’s lectures, to take the view that the entire programme depends on the DC and the WGC, and to map the relation of all the other conjectures from this foundation, it provides clarified view on a programme of proof/disproof.

Furthermore, if what we have done so far is focused on studying and reviewing examples of why the DC and the WGC can be trusted – why, short of complete formal proof – the evidence for these two conjectures is both substantial and inescapable, what we are coming to learn is precisely why both the DC and the WGC are an example of two first-class constraints. Taking this view has consequences. If the DC and the WGC are first-class constraints, it follows that if one understands these two conjectures they may then go on to understand all of the other conjectures. If we can disprove any number of the second-class conjectures, the Swampland programme would not collapse. If, on other hand, we should disprove the DC or the WGC, it is likely the entire programme collapses in on itself. The picture is illustrated quite explicitly in the above image.

The above picture describes what is called the ‘Emergence Proposal’ [1], based, in a sense, on the idea that Swampland Conjectures are consequences of the emergent nature of dynamic fields in quantum gravity. In lecture 4, we learned that if a coherent picture is emerging that outlines the relations between the growing conjectural assertions of the Swampland, a related internal programme of proof/disproof may also most effectively work from the bottom-up. But with the Emergence Proposal (as I currently understand it), not only is there the idea of first and second-class constraints – an idea for how we may perhaps pursue a foundational line of enquiry – another deep idea also comes to the fore: namely, the Swampland constraints are rooted in some underlying microscopic physics to be discovered. We don’t know what defines this microscopic physics, if it exists at all. That is a subject for another time. But we know, currently placed just above it in an overall web of constraints, the DC and the WGC may still offer some direct insight.

On that end, we now turn our attention to one of the deepest tests yet of the DC, beginning with a brief discussion of the refined version of the conjecture.

3. The Distance Conjecture (Refined)

We begin with the following message in mind: already we have seen several tests of the WGC and the DC. Each time, we have focused on increasing the complexity of the test and each time we have found strong evidence that both the WGC and the DC are deeply general. What we want to do now is proceed to review more tests of the DC, this time for even more complex geometry: namely, arbitrary Calabi-Yau manifolds.

Formally, the DC can be understood as follows [2]. As Palti put it in lecture four, consider how if we have a scalar field that is canonically normalised then we have already come to expect that there should be an infinite tower of states that goes something like,

\displaystyle  (\partial \phi)^{2}, \ m \sim e^{- \alpha \phi} \ \ (1)

Indeed, we are starting to understand that the behaviour in (1) would seem a general property of string theory. But we might ask, following Palti, what if the scalar is not canonically normalised? Consider, for instance, the scenario where we have some complicated function {f (\phi)} in front of the kinetic term,

\displaystyle  f(\phi) (\partial \phi)^{2} \ \ (2)

Moreover, let us consider for a moment a theory with a moduli space, {\mathcal{M}} (remember: a moduli space is a space parameterised by the value of some scalar fields). We will make it so {\mathcal{M}} is parameterised by {\phi^{i}}, and we should note that {\phi^{i}} has no potential (typically, this implies that there should be some supersymmetry in the theory). Now, take any point {P \in \mathcal{M}}, where a point in Moduli space is given by the expectation value for the scalar fields {\phi^{i}}. We define another point {Q \in \mathcal{M}} such that, in this set-up, the geodesic proper distance (i.e., the distance is equivalent to the vacuum expectation value in field space) between {P} and {Q} may be denoted as {d(P, D)} (note: we measure the distance using the field space metric in front of the kinetic terms). Crucially, the first statement of the DC says this geodesic distance is infinite, which is to say the scalar field obtains an infinite vacuum expectation value. The second statement describes the behaviour at this infinity. That is to say, the second state describes that there exists an infinite tower of states with mass scale m , such that m(Q) \sim m(P)e^{-\alpha d(P,Q) / M_{P}} as d(P,Q) \rightarrow \infty and where \alpha \sim \mathcal{O}(1) .

This is the key idea. Given two points of great distance in field space – at least greater than the Planck scale – we obtain an infinite tower of exponentially light states.

We have of course already started to become familiar with this statement. The point that ought to be highlighted here, however, is that if we have some basic canonically normalised scalar field {(\partial \phi)^{2}}, then all that we get is the familiar {m \sim e^{-\alpha \phi}}. In more complicated situations, such as when the scalar field is not canonically normalised, the refined DC tells us that we can apply it also to such completely general situations.

In these notes, we will not explore any further an example of the trivial canonical case. Instead, having discussed is the refined distance conjecture [1], what we want do is review whether it holds in the case of arbitrary complex extended structures.

4. Type IIB on Calabi-Yau C3-fold

4.1. Supergravity Set-up

In this example, we invoke Type IIB string theory on a Calabi-Yau C3-fold (i.e., we have a 6-dimensional CY space). In the construction we are about to study, the geometry we will be working with is about as complicated as it gets, so we start with some basics.

We should first note that Type IIB string theory on CY gives {\mathcal{N} = 2} supergravity (SUGRA) in 4-dimensions. Due to limited space, we are not going to establish the supergravity formalism in these notes. The reader is instead directed to ref. [1, 3-5] for an introduction, where, for these notes, we are of course following Palti in ref. [1] quite strictly. Another very important paper, which we will cover in some depth is ref. [6] on infinite distances in field space. In fact, majority of what follows is based on this paper.

Meanwhile, to continue establishing the basics, the general supergravity set-up is this: we have {n_{V}} vector multiplets with bosonic content of a complex scalar field. Similar in a sense to past discussion about the presence of scalar fields with regards to the radius of the circle, in the present case the scalar fields we are interested in studying are complex structure moduli, {t^{i}}, where {i = 1, ..., n_{V}}. These complex structure moduli parameterise the geometry of the CY.

We also have gauge fields, {A^{i}}. These gauge fields are quite interesting, as we will elaborate. For now, note that there is a gravity multiplet which contains a (bosonic fields) graviton and graviphoton, {A^{0}}. All of the gauge fields can be combined such that {I = \{0, i \}} for {A^{I}}.

The number of fields, {t^{i}} and {A^{I}}, is counted by the number of 3-cycles in the CY, which, for a typical CY, is {\sim \mathcal{O}(100)}. This means that for the field space in the effective field theory we find a space with {\sim 100} complex dimensions (and so we have a 200 dimensional field space in total).

Based on previous discussions, one might wonder whether there are charged states under the gauge field {A^{I}}. The answer is that there are charged states, they are BPS states which are {D3-branes} wrapping 3-cycles in the CY. Schematically, the moduli describe the size of the 3-cycle and then they describe the mass of the D3-branes that are wrapping the 3-cycles, behaving like particles in the external dimensions.

Generally, in this set-up, we find an action of the form,

\displaystyle  S_{\mathcal{N} = 2} = \int d^{4}x \sqrt{-g} [\frac{R}{2} - g_{ij} \partial_{\mu} t^{i} \partial^{\mu} \bar{t}^{j} -h_{\sigma \lambda} \partial_{\mu} l^{\sigma} \partial^{\mu} l^{\lambda} + \mathcal{I}_{IJ}\mathcal{F}^{I}_{\mu \nu}\mathcal{F}^{J, \mu \nu} + \mathcal{R}_{IJ}\mathcal{F}^{I}_{\mu \nu} (\star \mathcal{F})^{J, \mu \nu}]  \ \ (3)

The structure of which can be read off beginning with metrics, {g_{ij}} and {h_{\sigma \lambda}}. In totality, the moduli space is split into vector multiplets and hypermultiplets, {\mathcal{M} = \mathcal{M}_{V} \times \mathcal{M}_{H}}. And so, as one would expect even notationally, these two metrics describe two separate manifolds. We are going to focus on the vector multiplets which span a special Kähler manifold, from which we can generalise for the hypermultiplets on the quaternionic Kähler manifold. What is important to note is the periodicity {\{X^{I}, F_{I} \}} for the multiplet field space, in which we are dealing with holomorphic functions of {t^{i}}.

Notice also the gauge kinetic functions, {\mathcal{R}} and {\mathcal{I}}. These both contain real and imaginary parts of a complex matrix.

4.2. Charge Vector and Kähler Potential

It was mentioned that we have D3-branes wrapping 3-cycles. When a certain D3-brane wraps the 3-cycles in the CY, this is labelled by a charge vector {q \in \mathbb{Z}} (of {\mathcal{O}(100)}). This charge vector is in fact a 100-dimensional vector, where each entry is some holomorphic function of the 100’s of scalar fields in our theory. The basic idea, to give some more intuition, is that once we know the charge vector we know the mass of the BPS state, which, again, are the charged states under the gauge fields. Study (4) below,

\displaystyle  m(\underline{q}) = \mid z(\underline{q}) \mid = \mid \frac{\underline{q \eta \underline{\prod}}(t)}{[i \underline{\prod}^{T}(t) \eta \bar{\prod}(t)]^{1/2}} \mid \ \ (4)

Where {\prod} is the period vector. Notice that in the denominator we have complex conjugation as the object {\underline{\prod}^{T}(t)} must be real. Furthermore, all of the geometry of the CY is captured in the period vector {\underline{\prod}(t)}. One can see that it is a function of {t^{i}}. This is because it is a 100-dimensional vector that is an arbitrary function of the complex structure moduli. We should also highlight, for pedagogical purposes, that the expressions for {\eta} and {\prod} are a local choice of basis on the moduli space. Without going into all of the details, the period vector {\prod} can be defined on a local coordinate basis such that,

\displaystyle \prod = \begin{bmatrix} X^{0} \\ x^{i} \\ F_{j} \\ F_{0} \\ \end{bmatrix} (5)

So that the electric index increases down the vector and the magnetic index increases from the bottom-up. The {eta} term in (4) is the natural symplectic form of this multiplet vector space, and so we may indeed construct the appropriate symplectic inner products.

It is not difficult to understand that the field space that we are working with is very complicated. In [6], the metric is given by the derivative of the Kähler potential (also note, much of the same notation and general construction is in this paper, which as with other points discussed can also read in ref. [1]),

\displaystyle  g_{t^{i} \bar{t}^{j}} = \partial_{t^{i}} \partial_{\bar{t}^{j}} K \ \ (6)

Where the {K} is the Kähler potential, {K = -log [i \prod^{T} \eta \bar{\prod}]}. In other words, we have the log of the period vector. This potential is actually very interesting, and one can derive it by considering in general a Kähler potential for some CY manifold, {Y_{D}}, of complex dimension {D} where the complex structure moduli is given by a {h^{D-1, 1} (Y_{D})}-dimensional Kähler manifold. The potential is then generally written as K = -log [-i^{D} \int_{Y_{D}} \omega \wedge \bar{\omega}] in which one finds metric components of the form above. Once one finds the appropriate integral basis, the potential above is found.

4.3. Studying the Field Space

The discussion in this section is based almost entirely on [6], as well as parts of Palti’s summary in lecture 4 and his review in [1]. Additionally, we will be working with a number of very powerful mathematical theorems offered by Wilfried Schmid [7] building on Deligne’s work [8] in Hodge theory. (Please note, while we will not explore a detailed study / re-derivation of some of the theorems found in [7], I am very interested in this work and also in [6] which leverages Schmid’s nilpotent orbit theorem, so I will offer a detailed review in a future post).

In a schematic way, what we want to do is consider some point of infinite distance on this field space. Following Palti in his lecture, we shall label this point by the parameter {t} going to {+i \infty}. We now invoke the theorem that tells us that for such a point the period vector has a monodromy around it. In other words, if we send the real part of {t} to infinity, {\text{Re} t \rightarrow \text{Re} t + 1}, which, in a sense, is like encircling the point at infinity, we have a transformation of the period vector. In fact, we see that the period vector transforms by the action of a monodromy matrix, \prod (t) \rightarrow T_{i} \prod(t) . Then, due to properties studied in [6], we see that each {T_{i}} can be decomposed and, with the monodromy matrix massaged in a way that it only gives its infinite order part, we can define the log of this {T_{i}} in the form of a matrix equation,

\displaystyle  N_{i} = \log T^{u}_{i} = \sum_{k = 1}^{\infty} (-1)^{k + 1} \frac{1}{k} (T^{(u)}_{i} - Id)^{k} = \frac{1}{m_{i}} \log T^{mi}_{i} \ \ (7)

From this, we invoke the nilpotent orbit theorem [7]. With space limited the essentially idea may be summarised in the result that {N} is nilpotent. This means that if we take a high enough power we will get zero, {N^{n+1} = 0, \ n \leq 3}. Moreover, remember that we have sent {t} an infinite distance, and as things are currently constructed we need to know what this point looks like. What Schmid’s theorem in ref. [7] tells us is precisely what the period vector looks like around any point at infinite distance. In fact, it says that the period vector must look like,

\displaystyle  \prod (t) = \exp [t N](a_{0} (S) + \mathcal{O}(E^{2\pi i t})) \ \ (8)

What is this telling us exactly? It says that we have a parameter {t}, and as {t \rightarrow i \infty} we get exponentially small corrections. In other words, because {N} is nilpotent we see in (8) that we get some polynomial in {t}. The vector {a_{0}} depends on the other moduli, but not {t}, and as the exponential term may be neglected we see that we can know the form of the period vector around any point.

There is another theorem in [7], as Palti cites in his lecture, which, using again the nilpotent theorem, tells us if this point is indeed an infinite distance then it must be the matrix {[t N]} does not annihilate the vector {a_{0} (S)}. And so what we have, to be terse, is the following,

\displaystyle  \text{Infinite distance} \longleftrightarrow N^{d + 1} a_{0} \neq 0, \ d > 0 \ \ (9)

Now, all we need to do is take the period vector and this form {[t N]} and plug it into the formulae for the mass of the BPS states and for the metric on the moduli space. What we find is that we must have some local expression near any infinite locus in the moduli space. Schematically, from section 3.2 in ref. [6] we may write,

\displaystyle  g_{t \bar{t}} = \partial_{t} \partial_{\bar{t}} K = \frac{1}{4} \frac{d}{\text{Im} t^{2}} \ \ (10)

Where we have dropped the subleading terms. With the universal leading term only depending on degree {d}, quadratic {1 / \text{Im} t} it is found that the proper field distance is logarithmic when we send {t} to infinity,

\displaystyle  d_{\gamma}(P, Q) = \int_{Q}^{P} \sqrt{g_{t \bar{t}}} \mid dt \mid \sim \frac{\sqrt{d}}{2} \log (\text{Im} t) \ \ (11)

From which it is found that, in the case of a CY compactification that preserves {\mathcal{N} = 2} supersymmetry the BPS states become massless at the singularity point. More technically, in the paper these singular points have to do with what the author’s study as infinite quotient monodromy orbits. But for our purposes we note in particular for the mass,

\displaystyle  M_{q} \sim \frac{\sum_{j}\frac{1}{j!}(\text{Im} t)^{j} S_{j}(q, a_{0})}{(2^{d} / d!)^{1/2} (\text{Im} t)^{d/2}} \ \ (12)

In other words, as Palti motivates it, we see that the D3-branes become massless as the imaginary part goes to infinity. The behaviour of the mass is argued to be universal for any massless BPS states. Furthermore, what is observed is the presence of a power law in {t} whilst the proper distance is logarithmic in {t}. If we consider some path, {\gamma}, as implied in (11), the effective theory at two points (P, Q) in the moduli space approach singularity. The mass of the BPS states decreases exponentially fast in the proper distance. And so, in a schematic way in these notes, we may describe this in the form of {\Delta \phi \sim \log t} and M \sim \frac{1}{t^{\alpha}} \sim e^{-\alpha \Delta \phi}, which is just the Distance Conjecture and the Weak Gravity Conjecture at work.

We have of course been crude in our description, and there is a subtlety about the state not necessarily being confirmed in the theory, with the need remaining that one must show the BPS states being in the spectrum. Perhaps a detailed individual post would be beneficial in the future. For now, we can say that in [6] the case is shown for when {d = 3}. For our current purposes, the result is notable it shows that the DC and WGC hold for any CY compactification for Type IIB string theory. And this result should not in any way be understated. Altgough we are dealing with a very complicated 100-dimensional field space, the fact the it can be proven mathematically that both of these first-class Swampland conjectures hold for any CY compactification – and that very powerful mathematical theorems tell us this is necessarily true – we are driven directly toward the suggestion of some deeply general physics.

5. de Sitter Conjecture

5.1. Introduction

To conclude this series of notes, and to celebrate what has been a fairly lengthy and detailed engagement with Palti’s lectures at IFT this past summer, we turn our attention to a brief discussion on some of the cosmological implications of the Swampland. We will not discuss things like tensors modes in inflation or other topics covered in the lectures, which can be easily reviewed in [1]. Instead, we begin with a brief review of the de Sitter Conjecture, which states that the gradient of the potential is bounded,

\displaystyle  \mid \nabla V \mid \geq \frac{c}{M_{P}} V \ \ (13)

In other words, the scalar potential of the theory must satisfy (13) or the refined version below,

\displaystyle  \text{min} (\nabla_{i} \nabla_{j} V) \leq - \frac{c^{\prime}}{M_{P}^{2}}V \ \ (14)

Where this second condition is based on or motivated by entropy arguments. There are a number of connections between the de Sitter conjectures and ongoing experiments, including dark energy constraints and constraints from inflation. Interaction with experimental observation is quite active here, as Palti summarises. What we shall focus on is what motivates the de Sitter conjecture from string theory.

5.2. Evidence of the de Sitter Conjecture – Type IIA on CY with Flux

What follows is based on a simplified version of the more general study in ref. [8], where flux compactifications of Type IIA string theory are considered and the author’s study the classical stabilisation of geometric moduli. The main idea that we consider in general is that we want to switch on the fluxes for the background CY and then we study them from the perspective of the 4-dimensional effective theory. That is to say, we study the potential from the fluxes in the 4-dimensional theory. In the referenced study there are two fields in the low-energy effective theory. More precisely, there are two moduli fields that parameterise the geometry of the CY, \rho = (vol)^{1/3} , which is the volume of the CY and another field, \tau = e^{-\phi} (vol)^{1/2} , which is the string coupling times the volume of the CY. As a result of the flux being switched on, these two fields will have some potential.

Now let us consider the canonically normalised fields,

\displaystyle  \hat{\rho} = \sqrt{\frac{3}{2}} M_{P} \ln e \ \ (15)

\displaystyle  \hat{\tau} = \sqrt{2} M_{P} \ln \tau \ \ (16)

As these fields are canonically normalised, we may write the following Lagrangian in the Einstein frame,

\displaystyle \mathcal{L} = \frac{M_{P}^{2}}{2} R - \frac{1}{2} (\partial \hat{\rho})^{2} - \frac{1}{2} (\partial \hat{\tau}) + ... V(\rho, \tau) \ \ (17)

Now, the featured point here is that the potential is of course quite complicated. We can include any number of things to generate the potential – for example, we can turn off and on certain RR-fluxes or a combination of fluxes. What is interesting is that, in playing with different scenarios, a number of general properties are found. For instance, consider the case of turning on only certain RR-fluxes, where we have an expectation value for the p-form field strength, and also the H-flux which is the field strength of the NS sector,

\displaystyle \text{RR-flux:} \ V_{p} \sim \rho^{3 - p} \tau^{-4} \ \ (18)

\displaystyle \text{H-flux:} \ V_{3} \sim \rho^{-3} \tau^{-2} \ \ (19)

And with these contributions, we can also have in this case D6-branes and 06-branes that contribute to the potential,

\displaystyle V_{D6} \sim \tau^{-3} \sim V_{06} \ \ (20)

It turns out that, completely generally (regardless of the fluxes we switch on or off, their combination, and the branes we choose), the potential always takes the form,

\displaystyle V = \frac{A_{3} (\phi^{i})}{\rho^{3} \tau^{2}} + \sum_{p} \frac{A_{p} (\phi^{i})}{\rho^{3 - p} \tau^{4}} + \frac{A_{}}{\tau^{3}} \ \ (21)

Where in the first two terms in the equality we have in the numerator some function of the other fields included in our theory over the contribution from the H-flux and RR-flux, respectively. In the last term, there is a contribution from localised sources in the numerator over the brane contribution. This is the most general form the potential can take, even when we consider the inclusion of hundreds of other fields.

Inspecting the general form of the potential (21), we may consider the following combination of derivatives,

\displaystyle -\rho \frac{\partial V}{\partial \rho} - 3\tau \frac{\partial V}{\partial \tau} \ \ (22)

It turns out that, in fact,

\displaystyle -\rho \frac{\partial V}{\partial \rho} - 3\tau \frac{\partial V}{\partial \tau} = 9V + \sum_{p} pVp \ \ (23)

Where {pVp} are positive components of the potential and so the following statement is made that, {9V + \sum_{p} pVp \geq 9V}. But what does this mean? Well, if we write this in terms of the canonically normalised fields,

\displaystyle M_{P} \mid \sqrt{\frac{3}{2}} \frac{\partial V}{\partial \hat{p}} + 3\sqrt{2} \frac{\partial V}{\partial \hat{\tau}} \mid \ \geq 9V \ \ (24)

We notice something striking. If, moreover, we consider the gradient of the potential as it also pertains to the statement made in the de Sitter Conjecture, notice that after some work we can go from a completely general statement to the below,

\displaystyle M_{P} \mid \nabla V \mid \geq M_{P} \mid \frac{\partial V}{\partial \hat{p}} +\frac{\partial V}{\partial \hat{\tau}} \mid \geq \frac{27}{13} V, \ \ \nabla V > 0 \ \ (25)

Where we see that the de Sitter Conjecture has been satisfied. As it is a completely general result for any choice of fluxes and any choice of branes for the given compactification, this result is quite striking. In other words, regardless of the complexity of the potential, there is also a lower bound to it.

6. 11-dimensional Supergravity

But what about other scenarios? Let us consider one last example, namely 11-dimensional supergravity and quickly think about what sort of potentials may be generated.

We start by noting the Maldecena-Nunez no-go theorem, which tells us that there is no de Sitter vacua in compactifications of 11-dimensional SUGRA down to any dimension. Moreover, it is shown in [10] that for 11-dimensional SUGRA on a smooth manifold compactified down to d-dimensions there is once again a lower bound which may be written as follows,

\displaystyle \frac{\mid \nabla \mid}{V} \geq \frac{6}{\sqrt{(d-2)(11-d)}} \ \ (26)

This is consistent with the de Sitter conjecture. But there are caveats, such as when orientifolds are present, as once again summarised [1]. The main point, with (13), (14), and (26) in mind, is that it is very difficult, if not somewhat extraordinary, to evade these constraints. The statement here is not that it is impossible, but that it is very difficult. Most notably, one is required to use stringy ingredients. For instance to violate these constraints you can include,

* Orientifolds (without D-branes and so where charges cannot be cancelled locally) – i.e., `naked’.

* Higher derivative corrections

* Type IIA with orientifolds / something not CY

* Quantum corrections – i.e., quantum vacuum (large, like KKLT)

But these all imply a level of great difficulty, pertaining to the use of stringy ingredients of which we do not yet have a great understanding. So this is one problem, which already requires great consideration. But there is another, which refers to the Dine-Seiberg problem [11], and when combined with the first means one has to work doubly hard. The basic idea with the latter is that the source of the potential vanishes when {g_{s} \rightarrow 0}. Moreover, it says in the weakly coupled regime there is a non-interacting theory, and so any fluxes etc. vanish. This is a very generic statement; it applies to any point in the Hilbert space where many possible light tower of states may dominate. Consider, for example, a potential subject to the above statement regarding the string coupling in some expansion,

\displaystyle V \sim g_{s}^{n} + \sum_{k=1}^{\infty} g_{s}^{n+k}C_{k}

Now, imagine the expansion is controlled. To leading order,

\displaystyle V \sim g_{s}^{n} \sim e^{-n\phi} + \text{small corrections} \ \ (27)

With only small corrections in the well controlled limit such that {g_{s} << 1}. If the potential looks like {e^{-n\phi}} then one can quickly work out,

\displaystyle  \mid \partial_{\phi} V \mid \sim nV \ \ (28)

Which satisfies the conjecture. But as Palti points out, one can always fight this with coefficients, say, for instance, with some potential,

\displaystyle  V = Ag_{s} + B g_{s}^{2} + cg_{s}^{2} + ..., \ \ g_{s} << 1 \ \ (29)

Which is what people do when performing flux compactifications. As we know, we can always play with the fluxes and other things which corresponds in the above expansion to playing with the coefficients. So we can consider A and B and chose that {\frac{B}{A} > \frac{1}{g_{s}}} for which it is possible to then have these fields in minimum balance against each other. But then what of the C coefficient? One must ensure that this doesn’t takeover, so we could say {c \sim B}. But what the Dine-Seiberg argument says that if {A \sim B \sim C \sim O(1)} then we will never find the minimum to the potential, because {Ag_{s}} must be the leading term and we end up with a runaway direction in the field space. That is why for flux compactifications a general approach is to balance the coefficients by playing with the fluxes so that we can get a minimum for the potential.

We can see clearly that the situation is one where we have to overcome both problems, the no-go and the Dine-Seiberg problem, in order to show a de Sitter vacuum in string theory. One interpretation is that both the Maldecena-Nunez no-go theorem and the Dine-Sieberg problem motivates the de Sitter conjecture: i.e., string theory does not foster or does not like de Sitter vacua. But another, perfectly legitimate interpretation is that all that these two accounts are saying is that we just have to work very hard to obtain a de Sitter vacuum in string theory. For the no-go theorem, for example, to evade it requires working with stringy ingredients that we do not yet have much understanding of – such as working with naked orientifolds or in the case of higher derivative corrections. And so maybe the reality of the situation is not best described by the de Sitter Conjecture but instead motivates the need for even deeper thinking in string theory. In time, which of these interpretations is correct will likely clarify.

References

[1] E. Palti, `The Swampland: Introduction and Review’, [arXiv:1903.06239v3[hep-th]].

[2] H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl. Phys. B766 (2007) 21–33, [hep-th/0605264 [hep-th]].

[3] A. Ceresole, R. D’Auria, and S. Ferrara, The Symplectic structure of N=2 supergravity and its central extension, Nucl. Phys. Proc. Suppl. 46 (1996) 67–74, [hep-th/9509160 [hep-th]].

[4] L. Andrianopoli, M. Bertolini, A. Ceresole, R. D’Auria, S. Ferrara, P. Fre, and T. Magri, N=2 supergravity and N=2 superYang-Mills theory on general scalar manifolds: Symplectic covariance, gaugings and the momentum map, J. Geom. Phys. 23 (1997) 111–189, [hep-th/9605032 [hep-th]].

[5] J. Polchinski, ‘String Theory: Superstring Theory and Beyond’, Vol. 2. (2005).

[6] T. W. Grimm, E. Palti, and I. Valenzuela, ‘Infinite Distances in Field Space and Massless Towers of States’, JHEP 08 (2018) 143, [arXiv:1802.08264 [hep-th]].

[7] W. Schmid, ‘Variation of Hodge structure: the singularities of the period mapping’, Invent.
Math. , 22:211–319, 1973.

[8] P. Deligne, Theorie de Hodge: III, Publications Mathematiques de l’IHES´ 44 (1974) 5–77.

[9] O. DeWolfe, A. Giryavets, S. Kachru, and W. Taylor, ‘Type IIA moduli stabilization’, JHEP 07 (2005) 066, [hep-th/0505160 [hep-th]].

[10] J. M. Maldacena and C. Nunez, ‘Supergravity description of field theories on curved manifolds and a no go theorem’, Int. J. Mod. Phys. A16 (2001) 822–855, [hep-th/0007018 [hep-th]].

[11] M. Dine and N. Seiberg, ‘Is the Superstring Weakly Coupled?’, Phys. Lett. 162B (1985) 299–302.

Standard
Stringy Things

Notes on String Theory: Conformal Field Theory – Ward Identities and Noether’s Theorem

Introduction

We now turn our attention to an introduction to Ward identities, which extends the ideas of Noether’s theorem in quantum field theory. Polchinski notes (p.41), `A continuous symmetry in field theory implies the existence of a conserved current (Noether’s theorem) and also Ward identities, which constrain the operator products of the current’.

In this post we want to derive a particular form of the Ward identity, coinciding with Section 2.3 in Polchinski’s textbook. And we shall proceed with the following discussion by emphasising again the perspective we have been building for some time, which goes all the way back to the definition of local operators. Moreover, Ward identities are in fact operator equations generally satisfied by the correlation functions, which, of course, are tied to the symmetry of the theory. So we take this as a starting point. As Polchinski comments, symmetries of the string worldsheet play a very important role in string theory. A large part of our study here is to consider some general field theory and derive similarly general consequences of symmetry in that field theory, extracting what we may learn as a result. It turns out that what we learn is how, among other things, we may derive Ward identities through the functional integral of the correlation functions, utilising the method of a change of variables.

1. Ward Identities and Noether Currents

We start by taking the path integral. Now, suppose we have a general field theory. For an arbitrary infinitesimal transformation of the form {\phi_{\alpha}^{\prime}(\sigma) = \phi_{\alpha}(\sigma) + \epsilon \cdot \delta\phi_{a}(\sigma)}, where {\epsilon} is the infinitesimal parameter,

\displaystyle \int [d\phi^{\prime}]e^{-S[\phi^{\prime}]} = \int [d\phi]e^{-S[\phi]} \ \ (1)

What we have done is considered the symmetry {\phi_{\alpha}^{\prime}(\sigma) = \phi_{\alpha}(\sigma) + \epsilon \cdot \delta\phi_{a}(\sigma)} of our general field theory and found that both the measure and the action are left invariant (1). They are invariant because what we have is in fact an exact or continuous symmetry of our field theory. A continuous symmetry implies the existence of a conserved current, which, of course, infers Noether’s Theorem and also Ward identities. So, from this basic premise, we want to consider another transformation of the form {\phi_{a} \rightarrow \phi_{a}^{\prime}(\sigma) = \phi_{a}(\sigma) +\rho(\sigma)\delta\phi_{a}(\sigma)}, where {\rho(\sigma)} is an arbitrary function. Consider the following comments for clarity: in this change of variables what we are doing is basically promoting {\epsilon} to be {\epsilon(\sigma)}. In that this transformation is not a symmetry of the theory, because one will notice that the action and the measure are no longer invariant, what we find is that to leading order of {\epsilon} the variation of the path integral actually becomes proportional to the gradient {\partial_{a} \rho}. Notice,

\displaystyle \int [d\phi^{\prime}]e^{-S[\phi^{\prime}]} = \int [d\phi]e^{-S[\phi]}[1 + \frac{i\epsilon}{2\pi} \int d^{d}\sigma \sqrt{-g} J^{a}(\sigma) \partial_{a}\rho(\sigma) + \mathcal{O}(\epsilon^2)] \ \ (2)

Where {J^{a}(\sigma)} is a local function that comes from the variation of the measure and the action. Indeed, it should be emphasised, both the measure and the action are local (p.41). The picture one should have in their mind is the same we have been building for some time: namely, we are working in some localised region within which all the operators we’re considering reside. This is one of the big ideas at this point in our study of CFTs.

Now, the idea from (2) is that, whilst we have technically changed the integrand, the partition function has actually remained the same. Why? In the change of variables, we have simply redefined the dummy integration variable {\phi}. This invariance of the path integral under change of variables gives the quantum version of Noether’s theorem {\frac{\epsilon}{2\pi i} \int d^{d}\sigma\sqrt{g} \rho(\sigma) \langle \nabla_{a}J^{a}(\sigma)... \rangle = 0}, where `…’ are arbitrary additional insertions outside of the small local region in which {\rho} is taken to be zero. This is precisely why Polchinski comments that, when we take `the function {\rho} to be nonzero only in a small region’, it follows we may consider `a path integral with general insertions `…’ outside this region’ (p.41). In other words, as {\rho} is taken to be nonzero in a small region, insertions outside this region are invariant under the change of variables.

From this clever logic, where we have {\nabla_{a}J^{a} = 0} as an operator statement (p.42), we want to proceed to derive the Ward identity. It follows that, as motivated by Polchinski, given (2) we now want to insert new operators into the path integral, noting {\rho(\sigma)} has finite support. Therefore, we may write,

\displaystyle  \int [d\phi^{\prime}] e^{-S[\phi^{\prime}]} A^{\prime}(\sigma_{0}) = \int [d\phi]e^{-S}[A(\sigma_{0}) + \delta A + \frac{i\epsilon}{2\pi} \int d^2\sigma\sqrt{-g} J^{a}(\sigma)A(\sigma_{0})\partial_{a}\rho + \mathcal{O}(\epsilon)^2] \ \ (3)

Where, again,

\displaystyle  \phi_{a} \rightarrow \phi^{\prime}_{a} = \phi_{a} + \epsilon\cdot \rho(\sigma) \cdot \delta \phi_{a}(\sigma)

And, now,

\displaystyle  A(\sigma) \rightarrow A^{\prime}(\sigma) = A(\sigma) + \delta(A) \ \ (4)

Then, we may use {\int d\phi^{\prime}e^{-S^{\prime}}A^{\prime} = \int d\phi e^{-S}A} to show,

\displaystyle  0 = -\delta A(\sigma_{0}) - \frac{i\epsilon}{2\pi} \int d^2 \sqrt{-g} J^{a}\partial_{a}\rho

\displaystyle 0 = - \delta A(\sigma_{0}) + \frac{i\epsilon}{2\pi} \int d^2 \sqrt{-g} \nabla_{a}J^{a}\rho \ \ (5)

Notice, at this point, that while we now have an integral equation, we can write it without the integral. This implies the following,

\displaystyle \nabla_{a}J^{a}A(\sigma_{0}) = \frac{1}{\sqrt{-g}}\delta^{d}(\sigma - \sigma_{0}) \frac{2\pi}{i\epsilon} \delta A(\sigma_{0}) + \text{total derivative} \ \ (6)

Where we have a total {\sigma}-derivative. But this statement is equivalent to, more generally,

\displaystyle \delta A(\sigma_{0}) + \frac{\epsilon}{2\pi i} \int_{R} d^{d}\sigma \sqrt{-g}\nabla_{a}J^{a}(\sigma)A(\sigma_{0}) = 0 \ \ (7)

Which is precisely the operator relation Polchinski gives in eqn. (2.3.7). In (7) above, what we have done is let {\rho(\sigma) = 1} in some region R and {0} outside that region. In the context of our present theory, the divergence theorem may then be invoked to give,

\displaystyle  \int_{R} d^2 \sigma \sqrt{-g} \nabla_{a}[J^{a}A(\sigma_{0})] = \int_{\partial R}dA n_{a}J^{a} A(\sigma_{0}) = \frac{2\pi}{i \epsilon} \delta A(\sigma_{0}) \ \ (8)

Where the area element is {dA} and {n^{a}} the outward normal. As Polchinski explains, what we have is a relation between the integral of the current around the operator and the variation of that same operator (p.42). We can see this in the structure of the above equation.

If the current is divergenceless, then the surface interior should give zero – i.e., it should vanish. One might say, more simply, there should therefore be a conservation current. But that would be prior to the insertion of the operator. In other words, we are assuming the symmetry transformation acts on the operator.

The next thing we want to do is convert to holomorphic and antiholomorphic coordinates, instead of {(\sigma)} coordinates. To do this we may rewrite (8) in flat 2-dimensions as,

\displaystyle  \oint_{\partial R} (J_{z}dz - \bar{J}_{z}d\bar{z})A(z_{0}, \bar{z}_{0}) = \frac{2\pi}{\epsilon}\delta A(z_{0}, \bar{z}_{0}) \ \ (9)

In general, it is difficult to evaluate this integral exactly. We can evaluate it in cases, for example, where the LHS simplifies. It simplifies when, {J_z} is holomorphic, meaning {\partial J_{z} = 0}. Therefore, also, {J_{\bar{z}}} is antiholomorphic, meaning {\partial J_{\bar{z}} = 0}. In these cases we use residue theorem,

\displaystyle  2\pi i [Res J_{z}A(z_{0}, \bar{z}_{0}) + Res J_{\bar{z}}A(z_{0}, \bar{z}_{0})] = \frac{2\pi}{\epsilon}\delta A(z_{0}, \bar{z}_{0}) \ \ (10)

Another way to put it is that the integral (9) selects and gathers the residues in the OPE. And what we find is the Ward identity,

\displaystyle  Res_{z \rightarrow z_{0}} J_{z} A(z_{0}, \bar{z}_{0}) + \bar{Res}_{\bar{z} \rightarrow \bar{z}_{0}} J_{\bar{z}}A(z_{0}, \bar{z}_{0}) = \frac{1}{i\epsilon}\delta A(z_{0}, \bar{z}_{0}) \ \ (11)

Where {\text{Res}} and {\bar{\text{Res}}} are the coefficients of {(z - z_{0})^{-1}} and {(\bar{z} - \bar{z}_{0})^{-1}}.

Now, it should be stated that this Ward identity is extremely powerful. It tells us the variation of any operator in terms of currents. One will see it in action quite a bit in bosonic string theory. Moving forward, we will also be using all the tools that we so far defined or studied. For example, we will eventually look at the OPEs to extract {\frac{1}{z}} like dependence and {\frac{1}{z} - z_{0}} like dependence. And in this way we will learn how operators transform.

All of this is to say, once we find out what is our conformal symmetry group, we will see there is a very close relation between OPEs in the CFT and the singular path of the transformations of the operators. And this will lead us to some rather deep insights.

It should be mentioned, again, that from these introductory notes we will go on to compute numerous detailed examples. For now, the focus is very much on introducing key concepts and familiarising ourselves with some of the deeper ideas in relation to stringy CFTS.

References

Joseph Polchinski. (2005). ‘String Theory: An Introduction to the Bosonic String’, Vol. 1.

Standard
Swampland Conjectures
Stringy Things

Notes on the Swampland (3): Testing the Weak Gravity Conjecture – Gauge Fields, Dp-branes, Type II Strings, and F-Theory-Heterotic Duality

The following collection of notes is based on a series of lectures that I attended by Eran Palti at SiftS 2019 at Universidad Autonoma de Madrid. The theme of the lecture series was ‘String Theory and the Swampland’. Palti’s five lectures were supported by his most recent and impressive 200 page review paper on the Swampland, which includes over 600 references [arXiv: 1903.06239 [hep-th]]. The reader is directed to this paper in addition to supplementary references that I also provide at the end of each set of notes.

In the following entry, the notes presented follow the third lecture of Palti’s series.

1. Introduction

In this collection of notes, we look to review some more basic tests of the Weak Gravity Conjecture. In the last entry, recall that we reviewed a basic relation between the WGC and the Distance Conjecture. We then considered a first test of the Distance Conjecture having compactified our theory on a circle. Additionally, we reviewed evidence for the DC where we found that if we have large expectation values for the scalar fields in string theory, we obtain an infinite tower of exponentially light states. In this sense, we also reviewed the extreme parameter regime for weak and strong coupling. Finally, we reviewed a number of lessons about the DC and T-duality, concluding with a brief review of the parameter space of M-theory.

In the present entry – the third in this series of notes – we continue to expand on past discussions, turning particular attention to another basic test of the WGC. In further testing of the WGC we will also focus on a number of related topics ranging from gauge fields to Dp-branes and Type II strings, ending with a few brief comments on F-theory {\longleftrightarrow} Heterotic duality. This will then lead us directly into the fourth and second-last entry of the series, where we will begin to review more advanced tests of the DC and WGC, using for instance arbitrary Calabi-Yau manifolds.

2. Weak Gravity Conjecture

In this section we return to the WGC, which we have already grown to understand as being closely related to the DC. Following Palti’s lecture series, although the WGC is studied quite extensively from the infrared point of view, we shall instead be studying it from the ultraviolet and maximally stringy perspective.

Proceeding directly from the last entry we return to the simple example of string compactification on a circle and consider some of the physics in [3] as discussed in [1]. This time, in compactifying on {S^{1}}, we are going to instead consider a more general solution for the metric. The reason for this is because we want to study in particular the case of compactification with gauge fields. The metric may be written as follows,

\displaystyle  ds^{2} = e^{2 \alpha \phi}g_{\mu \nu}dX^{\mu}dX^{\nu} + e^{2 \beta \phi}(dX^{d} + A_{\mu}dX^{\mu})^{2} \ \ (1)

A few comments are necessary before proceeding. First, remember that we are working in perturbative superstring theory, so this metric is very similar to the one before, where the first term in the equality is a 9-dimensional object. Second, also remember from the last entry that our original metric encoded the parameter {\phi} such that it became a dynamical field in the lower d-dimensional theory. But, as Palti notes, there is also an additional degree of freedom in the metric. What does this mean? This additional degree of freedom becomes a U(1) gauge field {A_{\mu}} in the d-dimensional theory, as opposed to a scalar field, which will also have a coupling {g}. Furthermore, in that we have added another component to the metric, namely the 9-dimensional {A_{\mu}} term on the right-hand side, this is in fact the graviproton. Altogether, it follows that this is the most general solution for stringy compactification on a circle.

Now, what is of present interest is the Ricci scalar. So let’s look at what dimensional reduction now gives for the Ricci scalar,

\displaystyle  \int d^{D}X \sqrt{-G}R^{D} = \int d^{d}X\sqrt{-g} [R^{d} - \frac{1}{2}(\partial \phi)^{2} - \frac{1}{4}e^{-2(d - 1)\alpha \phi}F_{(A), \mu \nu} F^{\\mu \nu}_{(A)}] \ \ (2)

Where {F_{(A), \mu \nu} =\frac{1}{2} \partial_{[\mu}A_{\nu]}} is the gauge field kinetic term or, in other words, the field strength of the gauge field. Recall, also, from before that the {\phi} in the exponential is related to the radius in the extra dimensions. So from (2) we can read off the gauge coupling for the U(1) gauge field as follows,

\displaystyle  g_{(A)} = e^{d - 1}\alpha \phi = \frac{1}{2 \pi R} (\frac{1}{2 \pi R})^{\frac{1}{d - 2}} \ \ (3)

Which is telling us, similar to the last entry, that if we make the circle very large the theory becomes weakly coupled. But what is the symmetry of the U(1) gauge field? How do we know that symmetry of the gauge field? Consider a general U(1) gauge symmetry transformation of the form (i.e., the circle isometry),

\displaystyle  A_{\mu} \rightarrow A_{\mu} - \partial_{\mu} \lambda (X^{\nu}), \ \ X^{d} \rightarrow X^{d} + \lambda (X^{\nu}) \ \ (4)

Where {\lambda (X^{\nu})} is a local gauge parameter. Notice that the metric remains invariant, and from this we can indeed see how lower d-dimensional theory has a U(1) gauge field with the above gauge coupling.

Now, just like in the past entry, we want to look at the Kaluza-Klein expansion. Moreover, recalling the KK expansion for the higher D-dimensional field {\Psi (X^{\mu}) = \sum_{n = -\infty}^{\infty} \psi_{n} (X^{\mu})e^{2\pi i n X^{d}}}, notice that the gauge transformation (4) reveals that the KK modes {\psi_{n}} obtain a charge under the U(1) gauge field. This charge is quantised, as anticipated, and for the nth KK mode it may be given as,

\displaystyle  q_{n}^{A} = 2\pi n \ \ (5)

But what is the relation between the charge and the KK modes? Note, firstly, that the charge of {\psi_{n}} are just the phases of these objects. Secondly, the emphasis at this point in Palti’s talk is to remember that the mass of the KK states calculated in a past entry in the Einstein frame, {M^{2}_{\text{n kk mode}} = (\frac{n}{R})^{2} (\frac{1}{2 \pi R})^{2 \ d - 2}}, is related to the charge. More pointedly, we are already familiar with how, for the KK modes, there is an infinite tower of states. We see that the mass increases along this tower, and so too does the charge. In other words, it is argued that we have a charge-mass relation for the infinite tower of states. Here it is for arbitrary {n},

\displaystyle  g_{(A)} q_{(n)}^{(A)} = M_{n, 0} \ \ (6)

This relation between the charge, mass, and couping may have already been anticipated. Since all we’ve considered here is really just a reduction of Einstein gravity, let us consider the effective string action from a past set of notes, written below for convenience,

\displaystyle  S_{D} = 2\pi M_{s}^{D - 2} \int d^{D} X \sqrt{-G}e^{-2 \phi} (R - \frac{1}{12} H_{\mu \nu \rho} H^{\mu \nu \rho} + 4\partial_{\mu} \Phi \partial^{\mu} \Phi) \ \ (7)

If we compactify this action on a circle, as we are so inclined, there is a gauge field obtained from the gravitational sector. This is similar to before, and is nothing new. What is new is that we also now obtain a second gauge field, {V_{\mu}}, which comes from the Kalb-Ramond B-field with a single index in the {X^{d}} direction. For this Kalb-Ramond field we may write,

\displaystyle  V_{\mu} \equiv B_{[\mu d]} \ \ (8)

Where we note that, generally, {B_{[mn]}} is an antisymmetric 2-form. If we also reduce {B_{[mn]}}, this also leads to a gauge field. Additionally, look at {V_{\mu}} in (8). The kinetic terms for this additional gauge field are produced by the dimensional reduction of the kinetic terms from the Kalb-Ramond field. In other words, we can compute the kinetic term for the gauge field, {V_{\mu}}, as it comes from the strength of the 2-form in 10-dimensions,

\displaystyle  \int d^{d}X \sqrt{-g} [R^{d} - \frac{1}{4}e^{-2(\alpha + \beta)\phi}F_{(V), \mu \nu}F^{\mu \nu}_{(V)}] \ \ (9)

The factor in front of the kinetic terms is produced when we reduce {\sqrt{-G}H_{\mu \nu \rho}H^{\mu \nu \rho}}. From (9) one can again read off the gauge coupling,

\displaystyle  g (v) = e^{(\alpha + \beta)\phi} = 2\pi R (\frac{1}{2 \pi R})^{\frac{1}{d - 2}} \ \ (10)

What is different here? Notice, if we now make the circle of radius {R} very large, we obtain a strongly coupled theory. So, in taking from what we reviewed in the last entry, we know that charges under this gauge field are the winding modes of the string. That is, we have stringy or indeed quantum gravity states. Moreover, think about how if we take the basic Polyakov action for a string wrapping in the {X^{d}} direction {w} times in the Einstein frame, which means that we can set {\sigma = \frac{2\pi}{w}X^{d}}, then notice we have

\displaystyle S_{P} = -\frac{T}{2} \int_{\sum} d\tau d\sigma [2i V_{\mu} \partial_{\tau} X^{\mu} \partial_{\sigma} (\frac{w\sigma}{2 \pi})]

\displaystyle  = -i\frac{w}{2 \pi \alpha^{\prime}} \int_{\gamma} d\tau (\partial_{\tau} X^{\mu})V_{\mu} \ \ (11)

Which is the worldline action for a charged particle,

\displaystyle  q_{w}^{(V)} = \frac{w}{2 \pi \alpha^{\prime}} (2\pi R)^{\frac{2}{d - 2}} \ \ (12)

Or we can think of this in another way by remembering that if we have some antisymmetric form of rank {n}, there is going to be some object coupling to it. Hence, we may notice that, if we integrate some Kalb-Ramond 2-form on the string worldsheet, where the 2-form has one leg along the 9th direction and one leg along the extra dimension, and if we consider a string winding around the extra dimension, we find the string worldsheet is just a worldline in the 9th direction times a circle. If we then perform the integral along the extra direction, we obtain the coupling {V_{\mu}}. And so, we may write,

\displaystyle  \int_{\sum = C \times S^{1}} B_{[\mu d]} dX^{\mu} \wedge dX^{d} \sim \int_{C} V_{\mu} \ \ (13)

Where a worldline coupled to a gauge field means that, as in (4.11), we have a particle in the lower d-dimensional theory. What this is telling us is that winding modes in the d-dimensional theory produce charged particles that are gauge fields under the Kalb-Ramond field. Consider again (4.12), we find once again a relation between the coupling, charge, and mass, except this time it is for the winding modes. These are interesting relations,

\displaystyle  g_{(V)}q_{w}^{(V)} = M_{0, w} \ \ (14)

Which are strictly stringy – or quantum gravitational – in nature. Moreover, what we are discovering are what appear to be deeply general relations, where there is always some particle with a relation between its charge and its mass. And if these relations are, in fact, deeply general, then this means they are also intrinsic properties of quantum gravity. We will investigate this idea more deeply in the context of the Swampland in a moment.

In the meantime, also notice something else that is interesting. If we send the gauge coupling to zero (either by making the circle small or large), {g \rightarrow 0}, we obtain an infinite tower of light states. But this is just a special case of the DC, emphasising again the relation between the DC and the WGC. Furthermore, notice that the gauge coupling depends on the scalar field. So should we want to go to weak coupling, we must give the scalar a large expectation value that directly implies an infinite tower of states.

Also notice that, in the context of our wider discussion in these notes, there is a noticeable symmetry in the theory, which until now has been left implicit; because we can exchange the two gauge fields and also the KK and winding modes. This is T-duality.

3. Quick Review: Type IIA String Theory

Let us quickly review another example and think about Type IIA string theory (from the last entry). Remember, Type IIA in the strongly coupled regime is just 11-dimensional supergravity reduced on a circle. Also remember, in thinking of the Type IIA string we have a massive Ramond-Ramond 1-form, {C^{(1)}}, which is just a gauge coupling that is the graviphoton. The gauge group is U(1) and, it follows,

\displaystyle  g_{C^{(1)}} \sim \frac{1}{g_{s}^{3/4}} \ \ (15)

The states charged under this gauge field? A D0-brane, with a D6-brane representing the magnetic dual. Again, we find the following mass-charge relation,

\displaystyle  M_{D0} = g_{c^{(1)}} q_{D0} \ \ (16)

So, as Palti summarises, we have another piece of evidence that the mass-charge-coupling relation is indeed general. And, in fact, the more we search the more we become convinced this relationship is a property of quantum gravity.

4. Weak Gravity Conjecture (d-dimensions)

These considerations bring us to a more formal definition of the WGC than what we have so far previously offered. Consider the following: take a theory coupled to gravity with a U(1) gauge coupling, {g},

\displaystyle  S = \int d^{d}X \sqrt{-g} [] (\frac{M_{p}^{d}}{2})^{d-2}R^{d} - \frac{1}{4g_{s}^{2}} F^{2} + ... ] \ \ (17)

For the Electric WGC, there exists a particle with mass {m} and charge {q} satisfying,

\displaystyle  M \leq \sqrt{\frac{d - 2}{d - 3}} gq (M_{p}^{d})^{\frac{d - 2}{2}} \ \ (18)

And for the Magnetic WGC, the cutoff scale of the effective theory is bounded from above by the gauge coupling, such that we have the general statement,

\displaystyle  \Lambda \lesssim g(M_{p}^{d})^{\frac{d - 2}{2}} \ \ (19)

Where the cutoff, as we understand, should correspond to the mass scale of an infinite tower of charged states. It is argued to be completely general.

5. Testing the WGC: The Heterotic String

Following Palti, let’s now consider testing the WGC even more than what we have done previously. For example, a leading question might be: Is the WGC true for the Heterotic string? The first formal test of the WGC was for the Heterotic string on a {T^{6}} [3]. Again, much of the following discussion also echoes [1], where a summary with additional pedagogical references can be found.

One of the first things we must consider is that we have the non-abelian gauge group {SO(32)}. This is important to note because compactifying on a {T^{6}} yields the following 4-dimensional gauge fields: {U(1)^{28}}. To understand why there are 28 U(1) gauge fields, simply remember that a {T^{6}} may be thought of as a product of 6 circles. In 4-dimensions we obtain 12 gauge fields from the metric and the Kalb-Ramond field. We may break these up into 6 {B_{[mn]}} yielding 6 U(1)’s and 6 graviphotons. Additionally, particular to the Heterotic string is a 10-dimensional gauge group. This gauge group may be broken by Wilson lines on a circle to its Cartan subalgebra. That is to say, if we have a circle and take a gauge field on that circle, this will give us a Wilson line to which we can then give an expectation value. The Wilson line will break the non-abelian group to its Cartan subalgebra. For these reasons, one can see what the Cartan subalgebra gives {U(1)^{16}}.

Let us focus on these last 16 U(1) gauge fields that come from breaking the {SO(32)} gauge group. The states charged under these are string oscillators {\underbar{q} = (q_{1}, ..., q_{16})} from which we once again obtain an infinite tower of states. The first massive excitation is the {SO(32)} spinor with mass,

\displaystyle  m^{2} = \frac{4}{\alpha^{\prime}} \ \ (20)

When we compactify on a {T^{6}} we obtain charged states that correspond to the 16-dimensional charge vectors,

\displaystyle  \textbf{q} = (\pm \frac{1}{2}, ..., \pm \frac{1}{2}) \ \ (21)

The idea now is to consider how, in the Einstein frame, and working in Planck units, we have the following gauge coupling for any of the U(1) gauge fields,

\displaystyle  g^{2} = g_{s}^{2} = \frac{2}{\alpha^{\prime}} \ \ (22)

In which the gauge coupling is equal to the string coupling, and where {\alpha^{\prime}} depends on the expectation value of the dilaton. To put it explicitly, we have a dilatonic coupling. And, so, in terms of the bound set by the WGC for the mass the following inequality is satisfied,

\displaystyle  m^{2} \leq g^{2} \mid \textbf{q} \mid^{2} = \frac{8}{\alpha^{\prime}} \ \ (23)

Which is the limit of the expectation values of the small Wilson lines. As Palti notes, an interesting further test would be for arbitrary Wilson lines, but what he focuses on in his presentation is the way in which the entire analysis may be generalised for the complete {U(1)^{28}} gauge fields in which the U(1)’s from the {T^{6}} are included. So now we consider the mass of the higher oscillator modes,

\displaystyle  m^{2} = \frac{2}{\alpha^{\prime}} (\mid \underbar{q} \mid^{2} - 2) \ \ (24)

For which, in his talk, Palti gives the possible charges,

\displaystyle  \textbf{q} = (q_{1} + \frac{c}{2}, ..., q_{16} + \frac{c}{2}) \ \ (25)

Where {q_{i} \in \mathbb{Z}} and {c = 0,1}. In that the charges should be integer, they must satisfy the lattice condition {\mid \underbar{q} \mid^{2} \in 2N}.

Now, the whole point of the analysis up to the present is to consider the mass-charge relation. And, in fact, what we find is the following mass-to-charge ratio,

\displaystyle  \mid \textbf{z} \mid^{2} = \frac{\mid \textbf{q} \mid^{2}}{\mid \textbf{q} \mid^{2} - 2} \ \ (26)

Or, to put the matter differently, notice in (24) the {\frac{2}{\alpha^{\prime}}} factor is just {g_{s}^{2}}, and {g_{s}^{2} = \frac{m^{2}}{M_{P}^{2}}}. And so,

\displaystyle  \frac{m^{2}}{g^{2} \mid \textbf{q} \mid^{2}} = \frac{\mid \underbar{q} \mid^{2} - 2}{\mid \underbar{q} \mid^{2}} < 1 \ \ (27)

Where we find quite explicitly that the mass is bounded by the charge for all of the states. This again satisfies the WGC, where, for all the U(1)’s, the mass is less than the charge. We also find that there is an infinite tower of states charging at {g}, and as we go further up the tower (so to speak) the bound in (27) becomes saturated but never violated. So all of our results so far are consistent, and the WGC indeed proves true for the Heterotic string.

6. What About Other Gauge Fields?

The following question we might now ask, as Palti motivates it: what other gauge fields might we consider? So far we have consider some fairly straightforward or simple examples. Can we continue to generalise?

6.1. Testing the Electric WGC: Open String U(1)’s

Another U(1) we get in string theory is an open string U(1), which, considering again Dp-branes, it is a U(1) gauge field on the world-volume. D-branes of course live in Type II string theory, so we could in general consider Type IIA/IIB on {\mathbb{R}^{1, (q - n)} \times T^{6}}, where there is equal radius for the torus. The D-brane can be thought of as filling the non-compact spacetime. In considering string theory on this background, take in particular a Type IIB on a {T^{6}} with 6 circles of radius R as an example. We therefore have some 4-dimensional {M_{1,3} \times T^{6}}, and what we want to do is specifically put a D3-brane with its 4-dimensional world-volume completely in the {M_{1,3}} external spacetime. The D3-brane of course carries U(1), so we therefore now have a U(1) gauge symmetry in our 4-dimensional theory.

Now, with the scenario partly constructed, notice we only have one spacetime filling D-brane, which, impliedly, means that we have some fundamental open string with its endpoints ending on this brane. But this is not consistent. Why? The gauge symmetry we have included is an open string gauge symmetry, and so it is a gauge symmetry being carried by the non-perturbative D3-brane. But if we have just the single D3-brane, it will source the charge inside the 6-dimensional torus, and, one way to put it is that this scenario is akin to inserting a charged particle in a confined space in which there is nowhere for the field lines to propagate. In other words, we have a U(1) neutral state; but D-branes also source R-R fields. This is one of the great facts about D-branes, because insofar that they carry R-R charges, this gives string theory its power of being able to have a source for every gauge field [8]. In our current construction, however, the presence of the D-brane means that it will provide a source in the compact {T^{6}} whilst we lack an appropriate sink for the R-R field lines. This is obviously a problem because the field lines must end somewhere. This is why Palti points in another direction in his talk.

One option is that we could add an anti-brane; but means that the branes will then annihilate one another and, as this is an unstable configuration, it doesn’t really remedy the situation. Instead, the solution is based on a well known fact that orientifold planes are sinks for R-R charge. We might therefore instead introduce the needed negative charges by way of invoking orientifold planes. In doing so, this implies that the spectrum now also contains unoriented strings. These unoriented strings have charge 2 under the U(1), as, under orientifold involution, they stretch between the D-brane and its image. With this configuration, we have a consistent construction, which, with the presence of the orientifold, then means we have a second D3-brane as illustrated below.

In considering the scenario we have constructed, the actual states being charged under the U(1) are open strings whose endpoints end on the D3-branes with a charge {+1}.

Now let us think more deeply about the scenario in relation to the WGC. Is it not possible to violate the WGC? For instance, if the state has charge {+1}, what if we pull the D3-branes apart (i.e., moving away from the orientfold)? The string that is already stretched between the D3-branes would stretch even more over some spatial distance. This would make it massive. But what of the charge? Well, the charge would remain constant. On first inspection, this would seem to violate the WGC. Let us quantify these ideas as follows.

In {D=10}, the relation between the string scale and the Planck scale can be found as (from dimensional reduction and re-writing everything in Planck units),

\displaystyle  M_{s}^{2}g_{s}^{-2} (RM_{s})^{6} \sim M_{P}^{2} \ \ (28)

And the gauge coupling on the D3-brane is simply,

\displaystyle  g \equiv \sqrt{g_{s}} \ \ (29)

Now, for the stretched string, the mass is given as

\displaystyle  m^{2} \sim (RM_{s})^{2}M_{s}^{2} \sim \frac{g_{s}^{2}M_{P}^{2}}{(RM_{s})^{4}} \ \ (30)

Rearranging (30) it can be found that,

\displaystyle  \frac{m^{2}}{g^{2}_{s} M_{P}^{2}} \sim \frac{g_{s}}{(RM_{s})^{4}} \ \ (31)

If the main task was to try and violate the WGC by stretching the string to great length, as we pull the D3-branes away from the orientfold, the question is: have we succeeded? More precisely, to violate the WGC (31) would have to be greater than 1. Is this the case? No, it is not! The reason is because, if we’re working in the perturbative string description – i.e., the controlled weak-coupling regime – than the coupling {g_{s}  1}. So, in fact, the WGC is satisfied. That is,

\displaystyle  \frac{m^{2}}{g^{2}_{s} M_{P}^{2}} \sim \frac{g_{s}}{(RM_{s})^{4}} < 1 \ \ (32)

As we stretch the string and make it massive, with the orientfold growing very large, the gauge coupling does not change. What we are doing, in effect, is diluting gravity. What’s more, we are diluting gravity faster than the mass can increase. And, it turns out, when {M_{P} \rightarrow \infty} we obtain a weakly coupled theory.

6.2. In General for different cases of {n}

Notice that, in general, the scenario constructed above may be considered in terms of compactification of Type IIA/B string theory on {4\mathcal{R}^{1, 9-1} \times T^{n}}. We considered the case for {n >2} when we compactified on a {T^{6}}. But other subtleties arise when considering the case of {n = 2} and especially {n < 2}, particularly due to backreaction on the space. In all cases, it can be seen that the Electric WGC holds for open string U(1)s [1].

7. Testing the Magnetic Weak Gravity Conjecture: Type IIB String Theory in 6d F-theory

In the last example we considered a test of the Electric WGC for open string U(1)s. What about the Magnetic WGC? Does the MWGC likewise hold for open string U(1)s? Recall from earlier in our discussion the MWGC is not making a statement about a single charged state but about an infinite tower of charged states. Where is the infinite tower of charged states in our scenario? The answer is rather non-trivial and can be reviewed in a series of incredibly interesting and mathematically rich papers [5, 6, 7], which display some lovely stringy physics.

We will save a detailed review of these papers for a separate entry (following the formal conclusion of this series of notes on Palti’s lectures). In the meantime, looking at [5] in particular, a brief if not altogether terse description may be considered. What the authors find is that, for the infinite tower of states, they turn out to be non-perturbative states of the theory.

To see these non-perturbative states is difficult. The set-up is this: consider Type IIB string theory on a 4-dimensional manifold, meaning compactification down to 6-dimensions. A powerful method to study non-perturbative type IIB string theory is by way of uplifting to F-theory (or, for Type IIA, uplifting to M-theory). So the framework is 6-dimensional F-theory. The 6-dimensional Planck mass is defined by the volume of the F-theory compactification space, which is a complex Kähler surface {B_{2}} at the base of a Calabi-Yau 3-fold. In these notes, we have not yet considered such complex extended objects. But the idea is that we then consider a D7-brane filling the 6 external dimensions and wrapping a holomorphic curves on the Kähler surface in the 4-dimensional space. In the uncompactified 6 dimensions, the D3-brane wrapping the 2-cycle produces a solitonic ring. Associated strings on the curve {C_{0}} contained in {B_{2}} are sourced under the D7-brane gauge group.

From this construction, however roughly described, the idea is to uplift to a strong coupling (using F-theory). From this, if the goal is {g_{D7} \rightarrow 0}, where the tower of states become light according the WGC, then the 2-cycle must become very large. But, if the 2-cycle becomes big, the volume of the 4-dimensional manifold changes and, impliedly, the values of {M_{P}} and the string scale also change. So one approach is to keep the volume fixed. However, fixing the volume while making the 2-cycle big means that another 2-cycle needs to be small!

\displaystyle  volume \ fixed \rightarrow small \ 2-cycle

Now consider the following. If a D3-brane wrapped in internal dimensions gives a string in external dimensions, impliedly, in the above construction, it seems a D3-brane wrapped on the small 2-cycle is found to produce a string in the 6 external dimensions. But this string propagating in the 6-dimensions is tensionless as the volume of the curve {C_{0}} contained in B_{2} goes to zero, \text{vol}_{j}(C_{0}) \rightarrow 0 . Moreover, as the tension of the string is actually the size of the cycle, the string itself asymptotically describes an open Heterotic string. And so we observe,

\displaystyle  F-theory \longleftrightarrow Heterotic \ duality

And, as it is found that the string is charged under U(1), to finalise what is an incredible piece of evidence, the oscillator modes become massless and again what is found is an infinite tower of light states.

This concludes the summary. In a separate future entry we will study the technicalities in detail.

In the next collection of notes from Palti’s lecture series, we will continue our study by considering more complex manifolds – that is, arbitrary Calabi-Yau manifolds – to see if the WGC still holds! We will also looks to some more advanced tests of the DC, particularly in the context of Type IIB string theory.

Reference

[1] E. Palti, `The Swampland: Introduction and Review’, [arXiv:1903.06239v3[hep-th]]

[2] B. Heidenreich, M. Reece, and T. Rudelius, Sharpening the Weak Gravity Conjecture with Dimensional Reduction, JHEP 02 (2016) 140, [arXiv:1509.06374 [hep-th]].

[3] N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, The String landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060, [hep-th/0601001].

[4] B. Heidenreich, M. Reece, and T. Rudelius, Evidence for a sublattice weak gravity conjecture, JHEP 08 (2017) 025, [arXiv:1606.08437].

[5] S.-J. Lee, W. Lerche, and T. Weigand, Tensionless Strings and the Weak Gravity Conjecture, JHEP 10 (2018) 164, [arXiv:1808.05958].

[6] S.-J. Lee, W. Lerche, and T. Weigand, Modular Fluxes, Elliptic Genera, and Weak Gravity Conjectures in Four Dimensions, [arXiv:1901.08065].

[7] S.-J. Lee, W. Lerche, and T. Weigand, A Stringy Test of the Scalar Weak Gravity Conjecture, Nucl. Phys. B938 (2019) 321–350, [arXiv:1810.05169].

[8] J. Polchinski, String theory. Vol. 2: Superstring theory and beyond. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2007.

Standard
Stringy Things

Notes on String Theory – Further Introduction to Operator Product Expansions

1. Generalising the Formula for OPEs

In the last post we continued a review of Chapter 2 in Polchinski, focusing on building understanding of conformal field theories from the perspective of local operator insertions. We finally also arrived at the basic formula for operator product expansions (OPEs). What follows in this post is a continuation of that discussion. That is to say, the following review will also necessarily reference equations in the previous entry. To avoid confusion, equation numbers from the last post will be explicitly stated.

Recall that, in an introduction to the basic formula for OPEs, it was mentioned that because it is an operator statement this means it holds inside a general expectation value. It follows that the operator equation of the form that we considered can have additional operator insertions. This implies that we may write the formula for OPEs in a more general way,

\displaystyle  \langle \mathcal{O}_{i}(z, \bar{z})\mathcal{O}_{j}(z^{\prime}, \bar{z}^{\prime}) ... \rangle = \sum_{k} C_{ij}^{k}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) \langle \mathcal{O}_{k}(z^{\prime}, \bar{z}^{\prime}) ... \rangle \ \ (1)

Where ‘…’ again denotes additional insertions and is often left implicit. One can also work out quite simply the equivalent description in the path integral formalism for {n-1} fields.

1.1. OPEs – Generalise for an Infinite Set of Operators

There are a number of other caveats and subtleties about OPEs that we have not yet explored. It will be our aim to do so in this section by reviewing the remaining contents of section 2.2 in Polchinski’s textbook, before progressing toward more advanced topics that will then aid in our understanding of stringy CFTs and the procedure for how to compute OPEs.

Moreover, at this point in Polchinski’s introduction to OPEs, a number of results and definitions are given which may not make complete sense until later. This is because there are a number of key interrelated concepts that have not yet been formally introduced, such as radial ordering, Wick’s theorem, conformal invariance, and the necessary mode expansions that we must consider. These are important conceptual tools in establishing a wider understanding of CFTs and how we may think of OPEs in string theory. So what follows in this section may be considered more in the way of definition, introducing some ideas that relate to OPEs as we work toward more advanced topics that will clarify and enrich some of these ideas.

For instance, let us recall that in the last entry we discussed a normal ordered product that was defined in such a way that it satisfies the naive equation of motion [equation (17) from previous post]. What it is telling us is how the operator product is a harmonic function of {(z_{1}, \bar{z}_{1})}. This statement already offers a hint of what is to come both in this section and other future parts of our study on CFTs, particularly when we more explicitly discuss Wick’s theorem and mode expansions in relation to computing OPEs. For now, we may maintain an introductory tone and say that this statement leads us to an important insight early in Polchinski’s discussion in Section 2.2 of his textbook: notably that from the theory of complex variables a harmonic function may be decomposed locally as the sum of holomorphic and antiholomorphic functions. To begin to explain what this means, and to explain Polchinski’s discussion on pp.37-38 let us consider more deeply (17) from the last post. We can think of it this way,

\displaystyle  \bar{\partial}_{1} [\partial_{1} :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):] = 0

\displaystyle \bar{\partial}_{1} [:\partial_{1} X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):] = 0 \ \ (2)

The point of (2) is to show that we now have a holomorphic derivative inside the normal ordering. But notice also that this holomorphic derivative will get annihilated by the antiholomorphic derivative acting on it. In other words, by the equation of motion mixed {\partial \bar{\partial}} derivatives vanish. This is telling us something we may perhaps already know or suspect, namely as we continue to think in terms of operators {:\partial_{1} X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):} is in fact a holomorphic function. Now, as Polchinski explains, from the theory of complex analysis it is within the rules that we can Taylor expand such holomorphic (and antiholomorphic) functions. This use of Taylor expansion may be considered one of the first tools in understanding how to compute OPEs. Consider, for example, only the holormorphic case. When we proceed with Taylor expansion in {z_{12}} it is implied that we have nonsingularity as {z_{1} \rightarrow z_{2}} and we obtain the following infinite series,

\displaystyle  :\partial_{1 \xi} X^{\mu}(z_{1} + \xi, \bar{z}_{1} + \xi)X^{\nu}(z_{2}, \bar{z}_{2}): = \sum_{k=1}^{\infty} \frac{\xi^{k}}{k!} :X^{\nu} \partial^{k}X^{\mu}: \ \ (3)

Where {\xi = z_{12}}. We can rewrite (3) as follows, including also the antiholomorphic series,

\displaystyle  = \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}(z_{2}, \bar{z}_{2}): + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\bar{\partial}^{k}X^{\mu}(z_{2}, \bar{z}_{2}):] \ \ (4)

Which is now written only as a function of {z_{2}}. What this is telling us is that if we have some normal ordered product, we may write more generally for this product,

\displaystyle  :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):

\displaystyle = :X^{\mu}(z_{2}, \bar{z}_{2})X^{\nu}(z_{2}, \bar{z}_{2}): + \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}: + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\bar{\partial}^{k}X^{\mu}:] \ \ (5)

This is exactly the result that Polchinski describes in equation (2.2.4), with the exception that we have simplified the equation by dropping the {\alpha^{\prime}} term. Keeping the {\alpha^{\prime}} term explicit we arrive precisely at Polchinski’s equation,

\displaystyle  :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):

\displaystyle = - \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{12} \mid^{2} + :X^{\mu}(z_{2}, \bar{z}_{2})X^{\nu}(z_{2}, \bar{z}_{2}) + \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}: + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}:] \ \ (6)

In which {- \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{12} \mid^{2}} is the regular part of the OPE that one may remember from the two-point function {\langle X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime}, \bar{z}^{\prime}) \rangle}. Again, this is something we will become more familiar with as we progress. Furthermore, notice in general that (6) looks very much like an OPE as given in (1). In fact, it will become increasingly clear, especially toward the end of our present study, that we may think of this as the free field OPE hence the inclusion of the regular piece. Later, we will show explicitly the computation to achieve this result. In the meantime, since it is simply given in Polchinski’s textbook, it has also been stated here with addition of a few more comments as follows.

Note that like its equation of motion, (6) is an operator statement. Secondly, as previously alluded, OPEs in quantum field theory are very much like the analogue of Taylor expansions in calculus. When Taylor expanding some general function {\mathcal{G}(z_{1}, \bar{z}_{1}; z_{2}, \bar{z}_{2}) = :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):} as above, note that one will obtain terms of the form {\partial^{k}:X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):} in which the derivative is outside the normal ordering as opposed to inside the normal ordering. But differentiation and normal ordering commute, which can be proven using some basic identities of functional derivatives, hence the structure of the normal ordering in the OPE (6). Also, for any arbitrary expectation value that involves some product {X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2})} multiplied by a number of fields at other points, we have been building (and will continue to build) the intuition to understand exactly why the OPE describes the behaviour for when {z_{1} \rightarrow z_{2}} as an infinite series. In the case of (6), as we deepen our study of CFTs we will come to understand more clearly why it has `a radius of convergence in any given expectation value which is equal to the distance to the nearest other insertion in the path integral’ and why `The operator product is harmonic except at the positions of operators’ (p.38).

Although how we arrive at (6) may not yet make complete sense, the key idea at this point in Polchinski’s discussion is simply that we have a product of two operators and we have described this product as an infinite sum of some coefficients {C_{k}} of some basis operators {A_{k}}. As asymptotic expansions, we will come to write OPEs up to nonsingular terms.

1.2. Subtractions and Cross-contractions

To conclude a review of Section 2.2 in Polchinski, let us consider another example where we have an arbitrary number of fields. As we discussed earlier, the sum then runs over all of the different ways we might choose pairs of fields from the product. We then replace each pair with the expectation value as mentioned in the description of the definition (16) in the last post – i.e., what we have also termed to be the regular part of the OPE. So, if for instance we have three fields, the computation generally takes the following form,

\displaystyle  :X^{\mu_{1}}(z_{1}, \bar{z}_{1})X^{\mu_{2}}(z_{2}, \bar{z}_{2})X^{\mu_{3}}(z_{3}, \bar{z}_{3}):

\displaystyle =X^{\mu_{1}}(z_{1}, \bar{z}_{1})X^{\mu_{2}}(z_{2}, \bar{z}_{2})X^{\mu_{3}}(z_{3}, \bar{z}_{3}) + (\frac{\alpha^{\prime}}{2} \eta^{\mu_{1} \mu_{2}} \ln \mid z_{12} \mid^{2} X^{\mu_{3}}(z_{3}, \bar{z}_{3}) + 2 \ \text{permutations}) \ \ (7)

Now, consider again (16) from the previous entry. It can now be seen how we may write this definition in a more compact and general way. Consider, for instance, the arbitrary functional {\mathcal{F} = \mathcal{F}[\partial X^{\mu_{1} ... \mu_{n}}]}. The terms in brackets represent a combination of an arbitrary number of fields. If, as before, we Taylor expand and make this expression an expansion of polynomials of {X}, it follows that we may then write the normal ordering for each monomial. This leads directly to the equation (2.2.7) in Polchinski,

\displaystyle :\mathcal{F}: = \exp [ \frac{\alpha^{\prime}}{4} \int d^{2}z_{1}d^{2}z_{2} \ln \mid z_{12} \mid^{2} \frac{\delta}{\delta X^{\mu}(z_{1}, \bar{z}_{1})} \frac{\delta}{\delta X_{\mu}(z_{2}, \bar{z}_{2})}] \mathcal{F} \ \ (8)

Where {\mathcal{F}} is any functional of {X}. It can be shown that (8) is equivalent to (16) from the previous post. Again, this may not yet make complete sense. But for now notice that there is a double derivative in the exponent. This double derivative contracts each pair of fields. What this means is that, every time we compute the expansion we will effectively kill two {X} terms. Instead of these {X} terms, we then insert {\ln \mid z_{12} \mid^{2}} which is, of course, the subtraction. Now, reversely, if we act with the inverse exponential, we obtain the opposite of a sum of subtractions in the form of a sum of contractions,

\displaystyle  \mathcal{F} = \exp [-\frac{\alpha^{\prime}}{4} \int d^{2}z_{1}d^{2}z_{2} \ln \mid z_{12} \mid^{2} \frac{\delta}{\delta X^{\mu}()z_{1}, \bar{z}_{1}} \frac{\delta}{\delta X_{\mu}(z_{2}, \bar{z}_{2})}] :\mathcal{F}:

\displaystyle = :\mathcal{F}: + \ \text{contractions} \ \ (9)

As it will become increasingly clear when we compute some detailed examples, this means we are now summing over all of the ways of choosing pairs of fields from {:\mathcal{F}:} instead of {\mathcal{F}}. We then replace each pair with the contraction {-\frac{1}{2} \alpha^{\prime}\eta^{\mu_{i} \mu_{j}} \ln \mid z_{ij} \mid^{2}}. It follows that for any pair of operators, we can generate the respective OPE

\displaystyle :\mathcal{F}: :\mathcal{G}: = :\mathcal{F} \mathcal{G}: + \sum \ \text{cross-contractions} \ \ (10)

What (10) is saying is that we are now summing over all of contracting pairs with one field in {\mathcal{F}} and one field in {\mathcal{G}}, where, again, {\mathcal{F}} and {\mathcal{G}} are arbitrary functionals of {X}. It is this construction of the cross-contractions that enables the following formal expression,

\displaystyle : \mathcal{F}: :\mathcal{G}: = \exp [-\frac{\alpha^{\prime}}{2} \int d^{2}z_{1}d^{2}z_{2} \ln \mid z_{12} \mid^{2} \frac{\delta}{\delta X_{F}^{\mu}(z_{1}, \bar{z}_{1})} \frac{\delta}{\delta X_{G \mu}(z_{2}, \bar{z}_{2})}] : \mathcal{F} \mathcal{G}: \ \ (11)

In which the entire operation now acts on the normal ordering {: \mathcal{F} \mathcal{G}:}.

This concludes the opening discussion on OPEs in Polchinski’s textbook, from which he goes on to consider two examples of computing normal ordering (p.40) before focusing on the important study of Ward identities and Noether’s theorem. It will prove beneficial to review in the future the computation of the two examples that Polchinski offers (see the Appendix of this chapter). In the meantime, it may aid one’s understanding if we instead pause and first explore other concepts integral to stringy CFTs and their OPEs. This will enable us to introduce more notation and more deeply explicate mathematical procedure. Taking such an approach has its obvious advantages, but it also has its disadvantages. The way in which Chapter 2 is structured in Polchinski’s textbook means that, in a few instances, it will be required that we advance our study of CFTs to include a number of other key concepts before making better sense of what we have already discussed, particular in why OPEs have the structure that they do and how we may think about their computational procedure in a more exemplified way. So at this point we bracket the definitions given above to discuss other related topics, before ultimately returning specifically to the subject of OPEs and computing a number of different examples step by step.

References

Joseph Polchinski. (2005). ‘String Theory: An Introduction to the Bosonic String’, Vol. 1.

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Stringy Things

Notes on String Theory: Conformal Field Theory – Local Operators, the String Propagator, and Operator Product Expansions

1. Local Operators

In the last entry we introduced a theory of free massless scalars in flat 2-dimensions (i.e., a free X-CFT). From this we also introduced new terms and established notation relevant to our ongoing study of CFTs in string theory (Chapter 2 in Polchinski). What we now want to do is proceed with a review of a number of interrelated topics at the heart of stringy CFTs: namely, local operators, techniques with path integrals, string propagators, and finally operator product expansions. Each of these topics has a number of parts, and so we shall need to work piece by piece and then stitch everything together.

To begin, we note that in string perturbation theory, one of the main objects of interest is the expectation value of the path integral of a product of local operators (Polchinski, p.36). This interest is our entry point, and it represents a primary theme for much of the following discussion. So our first step should be to define what we mean by local operators. These objects may also be described as fields; however, in the context of CFTs, the notion of a field carries a different meaning than, for instance, the definition of a field in quantum field theory. In our case, a field may be viewed generally as a local expression, which may be the generic field {\phi} that enters the path integral in QFT, or as a composite operator {e^{i\phi}} or as a derivative {\partial^{n}\phi} (Tong, p.69). These are all different types of fields or local operators in the CFT dictionary.

With a definition of local operators in mind, we opened the discussion by mentioning the expectation value of the path integral as a primary object of interest. Let us now consider some general expectation value. Consider, for instance, {\mathcal{A}_{i}} that is some basis for a set of local operators. We may write the general expectation value as follows,

\displaystyle \langle \mathcal{A}_{i_{1}}(z_{1}, \bar{z}_{1}) \mathcal{A}_{i_{2}}(z_{2}, \bar{z}_{2}) ... \mathcal{A}_{i_{n}}(z_{n}, \bar{z}_{n}) \rangle \ \ (1)

If the basic idea, as mentioned, is to compute the expectation value of the path integral, a more technical or detailed description of our overarching interest is to understand the behaviour of this expectation value (1) in the limit of two operators taken to approach one another (Polchinski, p. 36). The tool that we use for such analysis is the operator product expansion (OPE). Understanding the definition of OPEs and how to compute them is one of the ultimate aims of studying stringy CFTs, important for more advanced topics that we will consider throughout the remainder of this paper. But before formally defining OPEs, it is useful to first build a deeper sense of intuition about their meaning. To do this, let as briefly review some more basics.

2. The Path Integral and Arbitrary Operator Insertions

What do we mean by path integral? And how do we understand this idea of local operator insertions? Additionally, how do we construct the important operator equations required to build a picture of OPEs? Polchinski offers several valuable contributions to a definition of the path integral, including a lengthy treatment in the Appendix of Volume 1. For our purposes, we might first emphasise the QFT view of the path integral as an integral over fields,

\displaystyle Z = \int [dX]e^{-S} \ \ (2)

We may describe this as a partition function. Now, if what we want to know is the expectation value given some operator, this implies that we want to employ the path integral representation to derive operator equations. For instance, as we read in Polchinski (p.34), given some operator we may compute,

\displaystyle \left\langle \mathcal{F}[X] \right\rangle = \int [dX]e^{-S}\mathcal{F}[X] \ \ (3)

Where {\mathcal{F}[X]} is some functional of X, typically a product of two operators, and where \langle \mathcal{F}[X] \rangle = \langle 0 \mid \mathcal{F} \mid 0 \rangle  . For multiple entries in the form,

\displaystyle  \mathcal{F}_{1}[X(z_{1}, \bar{z}_{1})] \mathcal{F}[X(z, \bar{z})] \mathcal{F}_{2}[X(z_{2}, \bar{z}_{2})]

We may write,

\displaystyle  \langle 0 \mid \mathcal{F}_{1} \mathcal{F} \mathcal{F}_{2} \mid 0 \rangle =\int [dX]e^{-S} \mathcal{F}_{1}[X(z_{1}, \bar{z}_{1})] \mathcal{F}[X(z, \bar{z})] \mathcal{F}_{2}[X(z_{2}, \bar{z}_{2})] \ \ (4)

There is a notion of time-ordering present in (4), which we will discuss later. For now, we should note that the path integral of a total derivative is always zero. This fact will prove useful in just a moment and on many other occasions in the future. As Polchinski reflects, `This is true for ordinary bosonic path integrals, which can be regarded as the limit of an infinite number of ordinary integrals, as well as for more formal path integrals as with Grassmann variables’ (Polchinski, pp. 34-35). Hence eq.(2.1.15) in Polchinski (p.35), where he considers the path integral with the inclusion of Grassmann variables,

\displaystyle  0 = \int [dX] \frac{\delta}{\delta X_{\mu}(z, \bar{z})} \exp (-S)

\displaystyle  = - \int [dX] \exp (-S) \frac{\delta S}{\delta X_{\mu} (z, \bar{z})}

\displaystyle = - \int \bigg \langle \frac{\delta S}{\delta X_{\mu} (z, \bar{z})} \bigg \rangle

\displaystyle = \frac{1}{\pi \alpha^{\prime}} \langle \partial \bar{\partial} X^{\mu}(z, \bar{z}) \rangle \ \ (5)

There is something interesting with this result. If we recall the action for the free X-CFT in the last post, remember that we found the classical EoM to be {\partial \bar{\partial} X^{\mu}(z, \bar{z}) = 0}. Notice, then, that the result (5) is the analogue statement in the quantum theory for the classical equations of motion. What is this telling us? Let us dig a bit deeper.

First, consider how the same calculation in (5) holds if we have arbitrary additional insertions `…’ in the path integral. We already considered what multiple entries in the path integral in (4). But there is a caveat: namely, these additional insertions cannot also be at {z} (something we will elaborate below). Second, in the case of multiple entries in the path integral, which implies that we may write something of the form {\int [dX] \frac{\delta}{\delta X^{\mu}(z, \bar{z})}[e^{-S}\mathcal{F}(z, \bar{z})]}, one can think of some of the insertions as preparing a state in the theory. In other words, we should note that these insertions prepare arbitrary initial and final states in the theory (Polchinski, p.35). These arbitrary initial and final states perform a similar role should we instead consider boundary conditions, except with the offered convenience that we may now write the following path integral statement,

\displaystyle \left\langle \partial\bar{\partial}X^{\mu}(z, \bar{z}) ... \right\rangle = 0 \ \ (6)

Now, if (5) is the analogous statement in the quantum theory for the classical equations of motion, look at (6). Notice, as an operator statement, it is the same as in the Hilbert space formalism,

\displaystyle \partial\bar{\partial}\hat{X}(z, \bar{z}) = 0 \ \ (7)

Polchinski describes (7) as holding for all matrix elements of the operator {\hat{X}(z, \bar{z})}, with all relations that hold (6) being operator equations (Polchinski, p.34). These two points are important. It should also be noted that (7) is Ehrenfest’s theorem, which makes a lot of sense because it is telling us something that we already know or suspect: namely, the expectation values of the operators obey the classical equations of motion. But, again, this proves true only when the additional insertions `…’ in the path integral are located away from {z}. So let us now look into this subtlety. If, for example, addition insertions cannot be coincident at {z}, then let us consider what happens when we do indeed have coincident points at {z}! It follows,

\displaystyle  0 = \int [dX] \frac{\delta}{\delta X_{\mu}(z, \bar{z})}[exp(-S)X^{\nu}(z^{\prime}, \bar{z^{\prime}})]

\displaystyle = \int d[X] exp(-S) [\eta^{\mu \nu}\delta^{2}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) + \frac{1}{\pi\alpha^{\prime}}\partial_{z}\partial_{\bar{z}}X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime}, \bar{z}^{\prime})]

\displaystyle = \eta^{\mu \nu} \langle \delta^{2}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) \rangle + \frac{1}{\pi\alpha^{\prime}}\partial_{z}\partial_{\bar{z}}\langle X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime}, \bar{z}^{\prime})\rangle \ \ (8)

Where the {\delta^{2} (z^{\prime} - z, \bar{z}^{\prime} - \bar{z})} term comes from differentiating {\frac{\delta X^{\mu}(z^{\prime}, \bar{z}^{\prime})}{\delta X^{\mu}(z, \bar{z})}} that appears in the computation. What we see in (8) is that at coincident points the classical equations of motion do not hold at the quantum level. This implies a few things. First, the good news is that we obtain our previous result that the EoM agrees as an operator statement of the ground state specifically under the conditions {z \neq z^{\prime}}. Second, the implication is clearly that with arbitrary additional insertions `…’ in the path integral, so long that these are far away from {z} and {z^{\prime}}, we may can rewrite (8) as,

\displaystyle \frac{1}{\pi \alpha^{\prime}} \partial_{z}\partial_{\bar{z}} \langle X^{\mu}(z, \bar{z}) X^{\nu}(z^{\prime}, \bar{z}^{\prime}) \ ... \ \rangle = -\eta^{\mu \nu} \langle \delta^{2}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) \ ... \ \rangle \ \ (9)

Where the ellipses are, again, the additional fields. Importantly, we may note that the following holds as an operator equation,

\displaystyle \frac{1}{\pi \alpha^{\prime}} \partial_{z}\partial_{\bar{z}} X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime},\bar{z}^{\prime}) = - \eta^{\mu \nu} \delta^{2}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) \ \ (10)

We are going to want to solve this equation in the future, because the solution will prove useful when computing OPEs. In the meantime, what should be understood is that what we have accomplished here is that we’ve modified the EoM to take into account that there is a collision between points at {z} and {z^{\prime}}. And we have also found that this behaviour can be derived as an operator statement. The purpose and greater logic for such an exercise will become increasingly clear. Meanwhile, notice that we now have a product of operators. Although it will not be proven here, it follows that in the Hilbert space formalism this product in the path integral becomes time-ordered (Polchinski, p.36). We also see that the delta function appears when the derivatives act on the time-ordering.

To summarise, these last results signal what has already been alluded (however vaguely) about the definition of OPEs in the final paragraph of Section 1. If, moreover, the general theme is so far one of path integrals and local operator insertions, the picture we are ultimately constructing is one of such insertions inside time-ordered correlation functions. These correlation functions can then be held as operator statements.

3. Time-ordered Correlation Functions, Normal Ordering, and the String Propagator

Before formally introducing and defining OPEs, we should spend a few more moments developing the picture and building intuition. For example, when it comes to the idea of time-ordered correlation functions, we will learn that solving the operator equation (10) gives us,

\displaystyle \langle X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime}, \bar{z}^{\prime}) \rangle = - \eta^{\mu \nu} \frac{\alpha^{\prime}}{2} \ln \mid z - z^{\prime} \mid^{2} \ \ (11)

The computation required to arrive at this result may not yet have much meaning and may be too forward thinking. We will come to understand it soon. What can be understood at this juncture are some of the pieces of this equation. The most important note is that (11) is the propagator of the theory of massless scalars that we have been working with in our study of CFTs (i.e., the free X-CFT). Notice, on the left-hand side of the equality, we a two-point correlation function. As it has been stated, correlation functions are time-ordered. Let us focus on this notion of time-ordering. For instance, consider a Wick expansion for {X^{\mu}(z, \bar{z})},

Where we have indicated the use of contraction notation that will be defined later. The first observation is that we have a two-point correlation function, and we have some term {T}. We also have colons on the right-hand side. For the {T} term, it indicates that the expression is time-ordered in the same way one will find in basic QFT (Polchinski, p.36). Writing {T} in full we find,

\displaystyle  T (X^{\mu} (z, \bar{z}), X^{\nu} (z^{\prime}, \bar{z}^{\prime}))

\displaystyle = X^{\mu} (z, \bar{z}) X^{\nu} (z^{\prime}, \bar{z}^{\prime}) \theta(z - z^{\prime}) + X^{\nu}(z^{\prime}, \bar{z}^{\prime})X^{\mu}(z, \bar{z})\theta(z^{\prime} - z) \ \ (13)

Now, looking again at (13), it is worth pointing out a few other things. Firstly, what we will learn in the future, particularly as we advance our discussion on CFTs, is that this time-ordering will prove very useful. Eventually we are going to want to make conformal transformations from an infinite cylinder to the complex plane, and we will learn that time-ordering on the cylinder corresponds to radially ordering on the complex plane. Reversely, we will see that radial ordering on the complex plane corresponds with time-ordering in the path integral. This is a featured point of study in Section 2.6 of Polchinski and it is something we will discuss later. Secondly, for the colons on the right-hand side, they indicate normal ordering. We saw normal ordering in the past discussion on the free string string spectrum using light-cone gauge quantisation. Notice, then, that on the far right-hand side we have a normal ordered product. The definition of normal ordered operators follows as (Polchinski, p.36),

\displaystyle :X^{\mu}(z, \bar{z}): = X^{\mu}(z, \bar{z}) \ \ (14)

And for the normal ordered product we have,

\displaystyle :X^{\mu}(z_{1}, \bar{z}_{1}), X^{\nu}(z_{2}, \bar{z}_{2}): = X^{\mu}(z_{1}, \bar{z}_{1}) X^{\nu}(z_{2}, \bar{z}_{2}) + \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{12} \mid^{2} \ \ (15)

Where {z_{ij} = z_{i} - z_{j}}. Furthermore, for arbitrary numbers of fields, the normal ordered product may be written as,

\displaystyle :X^{\mu_{1}} (z_{1}, \bar{z}_{1}) ... X^{\mu_{n}}(z_{n}, \bar{z}_{n}): = X^{\mu_{1}}(z_{1}, \bar{z}_{1}) ... X^{\mu_{n}}(z_{n}, \bar{z}_{n}) + \sum \text{subtractions} \ \ (16)

Where, for the subtractions, we sum the pairs of fields from the product and then replace each pair with its expectation value {\frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{ij} \mid^{2}}. We will elaborate more on (16) later. Meanwhile, consider again the operator equation (10). If what we want to do is define a product of operators that would satisfy the classical EoM, then from (16) and using (10) we can compute,

\displaystyle \partial_{z} \partial_{\bar{z}} :X^{\mu}(z_{1}, \bar{z}_{1}) X^{\nu}(z_{2}, \bar{z}_{2}): = \partial_{z} \partial_{\bar{z}} X^{\mu}(z_{1}, \bar{z}_{1}) X^{\nu}(z_{2}, \bar{z}_{2}) + \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \partial_{z} \partial_{\bar{z}} \ln \mid z_{12} \mid^{2}

\displaystyle = - \pi \alpha^{\prime} \eta^{\mu \nu} \delta^{2}(z_{1} - z_{2}, \bar{z}_{1} - \bar{z}_{2}) + \frac{\alpha^{\prime}}{2} \eta^{\mu \nu} \partial_{z}\partial_{\bar{z}} \ln \mid z_{12} \mid^2

\displaystyle = - \pi \alpha^{\prime} \eta^{\mu \nu} \delta^{2}(z_{1} - z_{2}, \bar{z}_{1} - \bar{z}_{2}) + \pi \alpha^{\prime} \eta^{\mu \nu} \delta^{2} (z_{1} - z_{2}, \bar{z}_{1} - \bar{z}_{2}) = 0 \ \ (17)

Where, for the last line in the computation, we used the standard result,

\displaystyle  \partial \bar{\partial} \ln \mid z \mid^{2} = 2\pi \delta^{2}(z, \bar{z}) \ \ (18)

Which is derived from an application of Stokes’ theorem.

Importantly, (17) is precisely the property that Polchinski highlights in equation (2.1.23) on p.36 of his textbook. What (17) is telling us is that, on the last line, {- \pi \alpha^{\prime} \eta^{\mu \nu} \delta^{2}(z_{1} - z_{2}, \bar{z}_{1} - \bar{z}_{2})} are the quantum corrections to the classical EoM. It is also telling us that, as we want to define a product of operators that satisfy the classical EoM, we must necessarily induce normal ordering.

So what does this all mean? In order to further extend the picture being developed here, we are lead directly to a definition of OPEs.

4. Operator Product Expansions

We may now define operator product expansions. The definition follows (pp. 37-38) directly from the intuition and logic that we have so far established, notably that OPEs may be considered a direct statement about the behaviour of local operators as they approach one another. The formula for OPEs is as follows,

\displaystyle \langle \mathcal{O}_{i}(z, \bar{z})\mathcal{O}_{j}(z^{\prime}, \bar{z}^{\prime}) \rangle = \sum_{k} C_{ij}^{k}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime})\langle \mathcal{O}_{k}(z^{\prime}, \bar{z}^{\prime}) \rangle \ \ (19)

Which is, again, an operator statement. This means that it also holds inside a general expectation value. Saving the general formula for OPEs until later, note that in (19) the {C_{ij}^{k}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime})} should be considered as a set of functions that depend only on the separation between the two operators (i.e., there is translational invariance).

To summarise, if OPEs describe what happens when local operators approach one another other, we have already developed a sense of technical intuition for why the key idea is one of having two local operators inserted in such a way that they are situated close to one another but not at coincident points. As we have already discussed, upon insertion of local operators at {z_{1}} and {z_{2}} for example, we obtain some normal ordered product. Then, what we can do is compute their approximation by way of a string of operators at only one of the insertion points (Tong, p.69). There can be any number of operator insertions, which is of course why we have included `…’ in the general formula for the OPE (19); it denotes insertions that are not coincident at {z}. (From this point forward, the ellipse will be removed and the following statement will be implied). This leads us directly to an illustration of OPEs as provided in Polchinski’s textbook.

In figure 4.1, we see that we have a number of local operator insertions, {z_{1}} to {z_{4}}, hence what we would be computing is the expectation of 4 local operators. Given that the OPE describes the limiting behaviour of {z_{1} \rightarrow z_{2}} as a series, where the pair of operators are replaced by a single operator at {z_{2}}, one way to think about this is analogous to the Taylor series in calculus (i.e., the OPE plays a similar role in quantum field theory). In fact, the analogue of computing a Taylor series is apt, as we will see when we start computing OPEs.

Another thing to note is that the circle in the picture illustrates the radius of convergence, such that this radius is computed as the distance to the nearest other operator positioned on the circle. In CFTs, OPEs have a finite radius of convergence.

Now, from our previous discussions, and from the formal definition of OPEs, we can see quite clearly why they are always to be understood as statements which hold as operator insertions inside time-ordered correlation functions. Should one ask, ‘what are the observables in string theory?’, the answer is that we compute a set of correlation functions of local/composite operators at their insertion points. So, should we take for example the Polyakov action, {S_{P}}, and compute the correlation functions for the CFT, one motivation is to show the correlation function to be related to the scattering amplitude in 26-dimensional spacetime (in the case of the bosonic string). So, in perturbative string theory, we look at the critical theory – that is, the critical coefficients and components of the correlation function,

\displaystyle \langle A_{ij}(z_{i}\bar{z}_{j}) ... A_{ij}(z_{n}, \bar{z}_{n})) \rangle \ \ (20)

Where we are interested in the singular behaviour. Moreover, recall the definition of the normal ordered product (15). Notice that we have very interesting log behaviour. If what we want to know exactly is what will happen with the product of the two operators as {z_{1} \rightarrow z_{2}}, this implies that we have an operator singularity. As we start computing OPEs and moving forward in our study of string CFTs, it will become very clear why this singular behaviour is actually the only thing we care about.

In the next post, we will extend our discussion of OPEs. Following that, we will look to derive the Ward Identities and then turn our attention to the Virasoro algebra among other important topics in Chapter 2 of Polchinski’s textbook.

References

Joseph Polchinski. (2005). ‘String Theory: An Introduction to the Bosonic String’, Vol. 1.

David Tong. (2009). ‘String Theory’ [lecture notes].

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