Propagators for the dual symmetric string and a familiar identity

If I could count all the moments so far when faced with a puzzle or question and it was appropriate to say, ‘it’s in my Polchinski!’ or ‘I should double-check my Polchinski!’. Use of the possessive here should be taken as an endearing reference to the role Joe Polchinski’s textbooks have played in one’s life. They’re like a trusty companion.

A great example comes from the other day. Actually, the story begins with Tseytlin’s first principle construction of the dual symmetric string, which serves as the basis of some work I am doing more generally in the doubled formalism. In one of the papers I have been reading that generalises from Tseytlin there are references made about the propagators in this dual symmetric formulation of string theory, followed by an assortment of assumptions including one about z \rightarrow 0 regularisation such that \bar{\partial} z^{-1} = \pi \delta^{(2)}(z). When I first read this over, it wasn’t immediately obvious to me from where this identity originated; I was much more focused on the actual propagators and some of the important generalisations of the construction, so I sort of left it as something to be returned to. Then in the last week I was reminded of it again, so I went back quickly and I realised how silly it was of me to not immediately recognise why this identity is true. The equation from above just comes from eqn. (2.1.24) in Polchinski, \partial \bar{\partial} \ln \mid z \mid^2 = 2\pi \delta^{(2)} (z,\bar{z}). Furthermore, one may have also thought equivalently of eqn. (2.5.8) in the context of bc CFT, which was what I first recalled (it also happens to be the subject of problem 2.1 at the end of the chapter).

There are a few ways to verify \partial \bar{\partial} \ln \mid z \mid^{2} = \partial \bar{z}^{-1} = \bar{\partial} z^{-1} = 2 \pi \delta^{2} (z, \bar{z}). One direct way is to take the terms to the left of the first equality, noting that \partial \bar{\partial} \ln \mid z \mid^{2} = 0 if z \neq 0. What we want to do is integrate this over some region R in the complex plane, using divergence theorem given in eqn. (2.1.9) which states \int_R d^{2}z (\partial_z v^z + \partial_{\bar{z}} v^{\bar{z}}) = i \oint_{\partial R} (v^{z} d\bar{z} - v^{\bar{z}} dz), where the contour integral circles R counterclockwise.

For the holomorphic case, using the test function f(z),

\int_{R} d^{2}z \ \partial \bar{\partial} \ \ln \mid z \mid^{2} \ f(z) \ (1)

From derivative properties we see \partial \bar{\partial} \ln \mid z \mid^{2} = \partial \bar{\partial} (\ln z + \ln \bar{z}) = \bar{\partial} z^{1}. Taking this fact into account and then also finally invoking divergence theorem,

= \int_{R} d^{2}z \ \bar{\partial} \ z^{-1} \ f(z)
=-i \ \oint_{\partial R} \ dz \ z^{-1}
= 2 \pi f(0) \ (2)

Where we have used the fundamental result in complex analysis that the contour integral of z^{-1} is 2\pi i. The same procedure can also be used for the antiholomorphic case. Hence, \int \ d^2z \ \partial \bar{\partial} \ln \mid z \mid^{2} \ f(z, \bar{z}) = 2\pi f(0,0), which therefore gives us \partial \bar{\partial} \ln \mid z \mid^{2} = 2 \pi \delta^{2} (z, \bar{z}).

As an aside, thinking of this reminds me of how I’ve been wanting to go back and update whatever notes I have so far uploaded to this blog as part of my ‘Reading Polchinski’ series, which I started writing in my first undergraduate year. I still like the idea of uploading my hundreds of pages of notes on Polchinski’s textbooks and formatting them into a pedagogical blog series, because there are so many subtleties and nuances that are fun to think through. I think I now also have a better sense of how I want to continue formatting the online version of the notes and communicate them, so after my thesis I intend to return to the project :).

Literature: Duality Symmetric String and the Doubled Formalism

When it comes to a T-duality invariant formulation of string theory, there are two primary actions that are useful to study as a point of entry. The first is Tseytlin’s non-covariant action. It is found in his formulation of the duality symmetric string, which presents a stringy extension of the Floreanini-Jackiw Lagrangians for chiral fields. In fact, for the sigma model action in this formulation, one can directly reproduce the Floreanini-Jackiw Lagrangians for antichiral and chiral scalar fields. The caveat is that, although we have explicit O(D,D) invariance, which is important because ultimately we want T-duality to be a manifest symmetry, we lose manifest Lorentz covariance on the string worldsheet. What one finds is that we must impose local Lorentz invariance on-shell, and from this there are some interesting things to observe about the constraints imposed at the operator level.

The main papers to study are Tseytlin’s 1990/91 works listed below. Unfortunately there is no pre-print available, so these now classic string papers remain buried behind a paywall:
1) Tseytlin, ‘Duality Symmetric Formulation of String World Sheet Dynamics
2) Tseytlin, ‘Duality Symmetric Closed String Theory and Interacting Chiral Scalars

For Hull’s doubled formalism, on the other hand, we have manifest 2-dimensional invariance. In both cases the worldsheet action is formulated such that both the string coordinates and their duals are on equal footing, hence one thinks of the coordinates being doubled. However, one advantage in Hull’s formulation is that there is a priori doubling of the string coordinates in the target space. Here, O(D,D) invariance is effectively built in as a principle of construction. This is because for the covariant double sigma model action, the target space may be written as R^{1, d-1} \otimes T^{2D}, in which we have a non-compact spacetime and a doubled torus. From the torus identifications we have manifest GL(2D; Z) symmetry. Then after imposing what we define as the self-duality constraint of the theory, which contains an O(D,D) metric, invariance of the theory reduces directly to O(D,D; \mathbb{Z}).

    1. Hull, ‘Doubled Geometry and T-Folds
    2. Hull, ‘Geometry for Non-geometric Backgrounds
    3. Hull and Reid-Edwards, ‘Non-geometric backgrounds, doubled geometry and generalised T-duality

What is neat about the two formulations is that, turning off interactions, they are found to be equivalent on a classical and quantum level. It is quite fun to work through them both and prove their equivalence, as it comes down to the constraints we must impose in both formulations.

I think the doubled formalism (following Hull) for sigma models is most interesting on a general level. I’m still not comfortable with different subtleties in the construction, for example the doubled torus fibration background or choice of polarisation from T-duality. The latter is especially curious. But, in the course of the last two weeks, things are finally beginning to clarify and I look forward to writing more about it in time.

Related to the above, I thought I’d share three other supplementary papers that I’ve found to be generally helpful:

1) Berman, Blair, Malek, and Perry, ‘O(D,D) Geometry of String Theory
2) Berman and Thompson, ‘Duality Symmetric String and M-theory
3) Thompson, ‘T-duality Invariant Approaches to String Theory

There are of course many other papers, including stuff I’ve been studying on general double sigma models and relatedly the Pasti, Sorokin and Tonin method. But those listed above should be a good start for anyone with an itch of curiosity.

Thinking About the Strong Constraint in Double Field Theory

I’ve been thinking a lot lately about the strong (or section) constraint in Double Field Theory. In this post, I want to talk a bit about this constraint.

Before doing so, perhaps a lightning review of some other aspects of DFT might be beneficial, particularly in contextualising why the condition appears in the process of developing the formalism.

One of the important facets of DFT is the unification of B-field gauge transformations and diffeomorphisms acting on the spacetime manifold {M}. The result is a generalisation of diffeomorphisms acting on the doubled space {P} [1]. This doubled space is not too difficult to conceptualise from the outset. Think, for instance, how from the perspective of a fully constructed closed string theory, the closed string field theory on a torus is naturally doubled. But in DFT, as we advance the formalism, things of course become more complicated.

Simply put, from a first principle construction of DFT, two motivations are present from the start: 1) to make T-duality manifest and 2), extend the spacetime action for massless fields. For 2), the low-energy effective action that we want to extend is famously,

\displaystyle  S_{MS} =  \int d^{D}x \sqrt{-g}e^{-2\phi}[R + 4(\partial \phi)^{2} - \frac{1}{12}H^{ijk}H_{ijk} + \frac{1}{4} \alpha^{\prime} R^{ijkl}R_{ijkl} + ...] \ (1)

In reformulating the low-energy effective action, the tools we use begin with the fact that the coordinates in DFT are doubled such that {X^{M} = (\tilde{x}_{i}, x^{i})}. Given that the full closed string theory is rather complicated – i.e., the field arguments are doubled and we would have infinite fields, so Lagrangian wouldn’t be trivial – this motivates from the start to restrict our focus to a subset of fields. Naturally, we choose the massless sector, with the motivation to obtain a description of spacetime where the gravity-field {g_{ij}}, Kalb-Ramond field {b_{ij}} and the dilaton {\phi} are manifest. We also work with a generalised metric which rediscovers the Buscher rules such that we write, mixing {g} and {b}-fields,

\displaystyle \mathcal{H}_{MN}(E) = \begin{pmatrix} g^{ij} & -g^{ik}b_{kj} \\ b_{ik}g^{kj} & g_{ij}-b_{ik}g^{kl}b_{lj} \\ \end{pmatrix} \in O(D,D) \ (2)

Here we define {E = g_{ij} + b_{ij}}.

Most importantly, to find the analogue for {S_{MS}} formulated in an O(D,D) covariant fashion (O(D,D) is the T-duality group), a lot of the problems we need to solve and the issues we generally face are deeply suggestive of constructions in generalised geometry. We will talk a little bit about generalised geometry later.

1. The Strong Constraint

Where does the strong constraint enter into this picture? From the cursory introduction provided above, one of the quickest and most direct ways to reaching a discussion about the strong constraint follows [1]. Relative to the discussion in this paper, beginning with the standard sigma model action, where the general background metric {g_{ij}} and the Kalb-Ramond field {b_{ij}} are manifest, our entry point proceeds from the author’s review of the first quantised theory of the string and the obtaining of the oscillator expansion for the zero modes. Important for this post is what turns out to be the definition of the derivatives from the oscillator expansions,

\displaystyle  D_{i} = \partial_{i} - E_{ik}\bar{\partial}^{k}, \\ D \equiv g^{ij}D_{j}

\displaystyle  \bar{D}_{i} = \partial_{i} + E_{ki}\bar{\partial}^{k}, \\ \bar{D} \equiv g^{ij}\bar{D}_{j} \ (3)

Famously, as can be reviewed in any standard string theory textbook (for instance, see [2]), in the first quantised theory we find the Virasoro operators with zero mode quantum numbers,

\displaystyle  L_{0} = \frac{1}{2}\alpha^{i}_{0}g_{ij} \alpha^{j}_{0} + (N-1)

\displaystyle \bar{L}_{0} = \frac{1}{2}\bar{\alpha}^{i}_{0}g_{ij}\bar{\alpha}^{j}_{0} + (\bar{N} - 1) \ (4)

In (2) {N} and {\bar{N}} are the number operators. Importantly, in string theory, from the Virasoro operators we come to find the level matching constraint that matches left and right-moving excitations. This is an unavoidable constraint in closed string theory that demands the following,

\displaystyle  L_{0} - \bar{L}_{0} = 0 \ (5)

As this is one of the fundamental constraints of string theory, it follows that all states in the closed string spectrum must satisfy the condition defined in (5). The complete derivation can be found in [2].

So far everything discussed can be reviewed from the view of standard string theory. What we now want to do is use the definition of the derivatives in (3) and express (5) as,

\displaystyle  L_{0} - \bar{L}_{0} = N - \bar{N} - \frac{1}{4}(D^{i}G_{ij}D^{j} - \bar{D}^{i}G_{ij}\bar{D}^{j})

\displaystyle  = N - \bar{N} - \frac{1}{4}(D^{i}D_{i} - \bar{D}^{i}\bar{D}_{i}) \ (6)

After some working, using in particular the derivative definitions and the definitions of the background fields (see an earlier discussion in [1]), one can show that

\displaystyle  \frac{1}{2}(D^{i}D_{i} - \bar{D}^{i}\bar{D}_{i}) = -2\partial^{i}\tilde{\partial}_{i} \ (7)

In (7) we use the convention as established in [1] to denote ~ as relating to the dual coordinates. Notice, then, that what remains is a relatively simple contraction between normal and dual derivatives. What is significant about (7) is that we can now express the fundamental string theory constraint (5) as a constraint on the number operators. Since {L_{0} - \bar{L}_{0} = 0} for all states of the theory we find,

\displaystyle  N - \bar{N} = \frac{1}{2}(-2\partial_{i}\tilde{\partial}^{i} = -\partial^{i}\tilde{\partial}_{i} \equiv \partial \cdot \bar{\partial} = p_{i}\omega^{i} \ (8)

So we see that from the number operators we have constraints involving differential operators. But what is this telling us? In short, it basically depends on the fields we use for the closed string theory. The fields that are arguably most natural to include are of a first quantised state expressed in the sum,

\displaystyle  \sum_{p,\omega} e_{ij}(p,\omega) \alpha^{i}_{-1}\bar{\alpha}^{j}_{-1}c_{1}\bar{c}_{1} |p, \omega \rangle

\displaystyle  \sum_{p,\omega} d (p,\omega) (c_{1}C_{-1} - \bar{c}_{1}\bar{c}_{-1}) |p, \omega \rangle \ (9)

Where we have momentum space wavefunctions {e_{ij}} and {d (p,\omega)}. Furthermore, in the first line, {e_{ij}} denotes the fluctuating field {h_{ij} + b_{ij}}. The {c} terms are ghosts. So what we observe in (9) is matter and ghost fields acting on a a vacuum with momentum and winding.

Here comes the crucial part: given {N = \bar{N} = 0} it follows that the fields, which, to make explicit depend on normal and dual coordinates, {e_{ij}(x, \tilde{x})} and {d(x,\tilde{x})} are required to satisfy,

\displaystyle  \partial \cdot \tilde{\partial} e_{ij} (x, \tilde{x}) = \partial \cdot \tilde{\partial} d(x, \tilde{x}) = 0 \ (10)

This is weak version of the strong or section constraint, a fundamental constraint in DFT for which we can go on to define an action. What it says is that every field of the massless sector must be annihilated by the differential operator {\partial \cdot \tilde{\partial}}.

This constraint (10), when developed further, turns out to actually be very strong. When we proceed to further generalise in our first principle construction of DFT, first with the study of {O(D,D)} transformations and then eventually the construction of {O(D,D)} invariant actions, we come to see that not only all fields and gauge parameters must satisfy the constraint {\partial \cdot \tilde{\partial}}. But this constraint is deepened, in a sense, to an even stronger version that includes the product of two fields.

The argument is detailed and something we’ll discuss in length another time, with the updated definition that {\partial \cdot \tilde{\partial}} annihilates all fields and all products of fields. That is, if we let {A_{i}(x, \tilde{x})} be in general fields or gauge parameters annihilated by the constraint {\partial^{M}\partial_{M}}, we now require all products {A_{i}A_{j}} are killed such that,

\displaystyle  \partial_{M}A_{i}\partial^{M}A_{j} = 0, \forall i,j \ (11)

Here {\partial_{M}A_{i}\partial^{M}} is an {O(D,D)} scalar. Formally, the result (11) is the strong {O(D,D)} constraint. What, finally, makes this condition so strong is that, from one perspective, it kills half of the fields of the theory and we in fact lose a lot of physics! In full string theory the doubled coordinates are physical. Effectively, however, the above statement ultimately implies that our fields only depend on the real space-time coordinates, due to a theorem in {O(D,D)} in which there is always some duality frame {(\tilde{x}^{\prime}_{i},\tilde{x}^{\prime})} in which the fields do not depend on {\tilde{x}^{\prime}_{i}}. So we only have dependence on half of the coordinates.

There is maybe another way to understand or motivate these statements. In the standard formulation of DFT, what we come to find is the appearance of the generalised Lie derivative. It is essentially unavoidable. The basic reason has to do with how, in pursuing the construction of the {O(D,D)} invariant action as highlighted at start, which includes the generalised metric {\mathcal{H}}, we find that the conventional Lie derivative is not applicable. It is not applicable in this set-up because, even when using trivial gauge parameters, we find that it simply does not vanish. In other words, as can be reviewed in [1], {\mathcal{L}_{\xi} \neq 0}. So the definition of the Lie derivative becomes modified using what we define as the neutral metric {\eta}. Why this is relevant has to do with how, interestingly, from the generalised Lie derivative (or Dorfman bracket) we may then define an infinitesimal transformation that, in general, does not integrate to a group action [3-5]. This means that it does not generate closed transformations.

The convention, as one may anticipate, is to place quite a strong restriction on the space of vector fields and tensors. Indeed, from the fact that DFT is formulated by way of doubling the underlying manifold, we have to use constraints on the manifold to ensure a consistent physical theory. But this restriction, perhaps as it can be viewed more deeply, ultimately demands satisfaction of what we have discussed as the strong constraint or section condition. So it is again, to word it another way, the idea that we have to restrict the space of vectors and tensors for consistency in our formulation that perhaps makes (11) more intuitive.

There is, of course, a lot more to the strong constraint and what it means [5], but as a gentle introduction we have captured some of its most basic implications.

2. Some Nuances and Subtleties

Given a very brief review of the strong constraint, there are some nuances and caveats that we might begin to think about. The first thing to note is that the strong constraint can be relaxed to some degree, and people have started researching weakly constrained versions of DFT. I’m not yet entirely familiar with these attempts and the issues faced, but an obvious example would be the full closed string field theory on a torus, because this is properly doubled from the outset and subject only to the weak level-matching constraint {\tilde{\partial} \cdot \partial = 0}.

I think a more important nuance or caveat worth mentioning is that, as discussed in [3], the strong constraint does not offer a unique solution. That is to say, from what I currently understand, there is no geometrical information that describes the remaining coordinates on which the fields depend. This contributes to, in a sense, an arbitrariness in construction because there is a freedom to choose which submanifold {P} is the base {M} for the generalised geometry {TM}.

In a future post, we’ll discuss more about this lack of uniqueness and other complexities, as well as detail more thorough considerations of the strong constraint. As related to simplified discussion above, the issue is that we can solve the basic consistency constraints that govern the theory by imposing the strong constraint (11). This is what leads to the implied view that DFT is in fact a highly constrained theory despite doubled coordinates, etc. In this approach, we have restriction on coordinate dependence such that, technically, the fields and gauge parameters may only depend on the undoubled slice of the doubled space. We haven’t discussed the technicalities of the doubled space in this post, but that can be laid out another time. The main point being that this solution is controlled. But there are also other solutions, of which I have not yet studied, but where it is understood that the coordinate dependence is no longer restricted (thus truly doubled) at the cost of the shape of geometric structure. At the heart of the matter, some argue [3] that when it comes to this problem of uniqueness the deeper issue is a lack of a bridge between DFT and generalised geometry. This is also a very interesting topic that will be saved for another time.

References

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

[2] J. Polchinski, ‘String Theory: An Introduction to the Bosonic String’, Vol. 1. 2005.

[3] L. Freidel, F. J. Rudolph, D. Svoboda, ‘Generalised Kinematics for Double Field Theory’. 2017. [arXiv:1706.07089 [hep-th]].

[4] B. Zwiebach O. Hohm. Towards an invariant geometry of double field theory. 2013. [arXiv:1212.1736v2 [hep-th]].

[5] B. Zwiebach O. Hohm, D. Lust. The Spacetime of Double Field Theory : Review, Remarks and Outlook. 2014. [arXiv:1309.2977v3 [hep-th]].

[6] K. van der Veen, ‘On the Geometry of String Theory’ [thesis]. 2018. Retrieved from [https://pdfs.semanticscholar.org/8c17/53af2fce4d174ca63a955c39ee2fedf37556.pdf]

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/].

Metastable de Sitter Solutions in 2-dimensions

To obtain stable de Sitter solutions in string theory that also avoid the Swampland is no easy task. It is difficult for many reasons. To give one example, from 10-dimensional Type IIA* and Type IIB* theories, one finds de Sitter related solutions but they come with ghosts that give a negative cosmological constant. Full de Sitter spacetime is a Lorentzian manifold that is the coset space of Lorent groups, dS^d \simeq O(d,1)/O(d-1,1), the cosmology of which, as many readers will likely be familiar, is based on a positive cosmological constant. Hence, in this example, we obtain the wrong sign. Many other examples can be cited in reference to the difficulty we currently face.

To speak generally, it has gotten to the point historically that one possibility being considered is that string theory may simply conspire against de Sitter space – that is, there is some deep incompatibility between our leading theory of quantum gravity and de Sitter vacua. The No-de Sitter Conjecture is an example of an attempt to formalise such a logical possibility, motivating beyond other things the need to understand better the structure of vacua in string theory. This conjecture is by no means rigorous, but it is supported by the fact that historically de Sitter solutions have been elusive. Having said that, one must be cautious about any claims regarding de Sitter constructions, either from the perspective of obtaining de Sitter solutions or from arguments claiming de Sitter belongs to the Swampland. Our present theory is still very much incomplete and there is a lot of work to be done, and I think it is fair to suggest that the fate of de Sitter in string theory is still uncertain. As an expression of opinion, I would currently say that it is more likely de Sitter will come from something much more fundamental than some of the current strategies being proposed. It should also be kept in mind that there may be a larger issue here. Quantum gravity in de Sitter is fundamentally difficult beyond pure string theory reasons. Indeed, it is possible that de Sitter is simply unstable in quantum gravity – that it is a space that simply cannot exist quantum mechanically. For instance, think of complications in QFT in curved space. Indeed, there is a lengthy list of authoritative papers that one may cite when it comes purely to perturbative quantum gravity (see, for example, this page for further references).

Image: de Sitter geometry courtesy of ncatlab.org

Given the sociology and history, when a new paper appears claiming to have found de Sitter solutions, one takes notice. That is precisely what happened last week when Miguel Montero, Thomas Van Riet, and Gerben Venken uploaded to the archive their recent work claiming to have found metastable de Sitter solutions in lower dimensions. More precisely, these parametrically controlled solutions appear when compactifying from 4-dimensions to 2-dimensions, particularly as a result of some clever work that invokes abelian p-1-form gauge fields to stabilise the runaway potential giving in general dS_{d-p} \times S^{p} solutions. However, not all solutions are stable with controlled saddle points. Indeed, the authors find D - p > 2 solutions to be generically unstable. On the other hand, the instability in the homogeneous mode disappears when D - p = 2. In this 2-dimensional case, the solutions relate to the near horizon dS_2 \times S^2 geometry of Nariai black holes. It is also worth pointing out that these solutions are not in string theory, but the work highlights some interesting implications and raises some important questions in quantum gravity more generally, including also when it comes to the Swampland.

If I am not mistaken, I think I remember seeing similar kinds of ideas in studies of dS / CFT where people look into cases of rotating Nariai black holes maximally large at dS_2 \times S^2. The geometry is generally interesting because in that some Nariai constructions with quintessence generalise quasi-de Sitter solutions, utilising for example decompositions from 4-dimemensions to things like 2-dimensional dilaton gravity theory (this may make one recall SYK models), one will often find descriptions of the geometric space as ds^2 = \Gamma(\theta) [-d\tau^2 +\cosh^2 \frac{\tau}{l} d\psi^2 + \alpha(\theta) d\theta^{2}] + \gamma(\theta) (d\phi + k \tanh \frac{\tau}{l} d\psi)^2 which we can think of in terms of an S^2 fibered over a dS_2 base space. I kind of see it as a bit of a playground useful to experiment with and probe. In section 2 of the de Sitter paper, one can read a short overview of Nariai black hole solutions of Einstein-Maxwell theory as a precursor to the main study.

I’ve only had time to skim through the Montero et al. paper, so I still need to give it careful attention. But on quick glance I noticed a number of interesting calculations, not least the derived inequality (p -1) \mid \frac{V^{\prime}}{V} \mid \leq \mid \frac{f^{\prime}}{f} \mid that states how, in order for a dS_{d-p} \times S^{p} saddle point to exist, this constraint that relates the gauge kinetic function of the p−1-form field and the potential must be satisfied. This is an example of one of a handful of points to which I would like to go back and think about more deeply. Additionally, it seems an open question whether, if we can obtain controlled lower-dimensional de Sitter solutions from runaway potentials, do such approaches and constructions fully escape the Swampland? It is worth reading their paper with that in mind.

As alluded at the outset, one should always approach any claims about de Sitter with a healthy dose of scepticism. At the same time, van Riet is a physicist who I admire, because he is one of a number championing the need for greater mathematical rigour in string theory and quantum gravity moving forward. So, for me, this paper immediately comes with some weight and authority. If one thing is certain today, at least from my current vantage, it is the need for thoughtful caution and careful mathematical analysis; and it appears that on a few occasions the authors also stress this point in their analysis.