Stringy Things

Notes on String Theory: Conformal Field Theory – Massless Scalars in Flat 2-dimensions

In past entries we familiarised ourselves very briefly with conformal transformations and the 2-dimensional conformal algebra. To progress with our study of Chapter 2 in Polchinski, we need to equip ourselves with a number of other essential tools which will assist in building toward computing operator product expansions (OPEs). In this post, we will focus on notational conventions, transforming to complex coordinates, and utilising holomorphic and antiholomorphic functions. Then, in the next post, we will focus on the path integral and operator insertions, before turning attention to the general formula for OPEs.

To start, it will be beneficial if we define a toy theory of free massless scalars in 2-dimensions. For consistency, we will use the same toy theory that Polchinski describes on p.32. (Please also note, this theory is very similar to the theory on the string WS and this is why it will be useful for us, as it will support an introductory study within an applied setting).

In the context of our toy theory, the Polyakov action takes the form,

\displaystyle  S_{P} = \frac{1}{4\pi \alpha^{\prime}} \int d^{2}\sigma [\partial_{1}X^{\mu} \partial_{1}X^{\nu} + \partial_{2}X^{\mu}\partial_{2}X^{\nu}] \ \ (1)

This is the Polyakov action with {\gamma_{ab}} being replaced by a flat Euclidean metric {\delta_{ab}} and with Wick rotation. What is the benefit of the Euclidean metric and what is meant by Wick rotation?

In general, a lot of calculation in string theory is performed on a Euclidean WS, in which case, for flat metrics, standard analytic continuation may be used to relate Euclidean and Minkowksi amplitudes. The benefit is that the Euclidean metric enables us to study ordinary geometry and to use conformal field theory on the string. But one will note that in previous constructions of the Polyakov action we used a Minkowski metric, and in past discussions we have also been using light-cone coordinates. So let’s consider transforming from a Minkowski measure to a Euclidean one. To achieve a flat Euclidean metric, the idea is simple: we use a Wick rotation to rewrite the Minkowski metric. Moreover, recall that in Euclidean space, namely the x-y plane, the infinitesimal measure is given by {ds^{2} = dx^{2} + dy^{2}}. Compare this with the Minkowski measure, which we may write in WS coordinates as {ds^{2} = -d\tau^{2} + d\sigma^{2}}. Notice that in the Euclidean picture all quantities are positive (or at least share the same sign). Now, by Wick rotation, we make a transformation on the time coordinate in the Minkowski measure such that {\tau \rightarrow -i\tau}. This means {d\tau \rightarrow -id\tau} and from this it follows {ds^{2} = - (-id\tau)^{2} + d\sigma^{2} = d\tau^{2} + d\sigma^{2}}. This is a Euclidean metric.

Hence, by Wick rotation, we are working in imaginary time signature such that the new metric in Euclidean coordinates may be written as,

\displaystyle \delta =      \begin{bmatrix}     1 & 0 \\     0 & 1 \\ \end{bmatrix}  \ \ (2)

As suggested, the goal is to end up with a Euclidean theory of massless scalars in flat 2-dimensions. Note, also, that as a result of Wick rotation,

\displaystyle  (\sigma_{0}, \sigma_{1}) \rightarrow (\sigma_{2}, \sigma_{1}) \ \ (3)

Where we define {\tau \equiv \sigma_{0}} and {\sigma \equiv \sigma_{1}}, and where {\sigma_{2} \equiv i\sigma_{0}}.

Moving forward, one should note that after Wick rotation from LC coordinates {(+, -)}, we enter into the use of complex coordinates {(z, \bar{z})}. We first observed these coordinates in the last section on the 2-dimensions conformal algebra. Further clarification may be offered. Most notably, the description of the WS is now performed using complex variables by defining these complex coordinates {(z, \bar{z}} that are, in fact, a function of the variables {(\tau, \sigma)} with which we have already grown accustomed. Hence, {z =  \tau + i\sigma} and {\bar{z} = \tau - i\sigma}. The benefit of setting up complex coordinates is that it enables us to employ holomorphic (left-moving) and antiholomorphic (right-moving) indices, where holomorphic = {z} and antiholomorphic = {\bar{z}} as also observed in our discussion on the conformal generators.

Now that our field theory has been sketched, and complex coordinates have been formally established, to understand how to transform these coordinates we must understand how to compute the derivatives. The first step is to invert the coordinates and then we will differentiate,

\displaystyle  \tau = \frac{z + \bar{z}}{2}, \ \ \sigma = \frac{z - \bar{z}}{2i}  \ \ (4)

Differentiating with respect to {z} and {\bar{z}} coordinates we obtain the following,

\displaystyle  \frac{\partial \tau}{\partial z} = \frac{\partial \tau}{\partial \bar{z}} = \frac{1}{2} \ \ (5)

And,

\displaystyle  \frac{\partial \sigma}{\partial z} = \frac{1}{2i}, \ \ \ \frac{\partial \sigma}{\partial \bar{z}} = -\frac{1}{2i} \ \ (6)

With these results we can then compute for the holomorphic coordinates,

\displaystyle  \frac{\partial}{\partial z} = \frac{\partial \tau}{\partial z}\frac{\partial}{\partial \tau} + \frac{\partial \sigma}{\partial}\frac{\partial}{\partial \sigma} = \frac{1}{2}\frac{\partial}{\partial \tau} + \frac{1}{2i}\frac{\partial}{\partial \sigma} = \frac{1}{2}(\frac{\partial}{\partial \tau} - i\frac{\partial}{\partial \sigma}) \ \ (7)

One can also repeat the same steps for the antiholomorphic case {\bar{z}},

\displaystyle  \frac{\partial}{\partial \bar{z}} = \frac{\partial \tau}{\partial \bar{z}}\frac{\partial}{\partial\tau} + \frac{\partial \sigma}{\partial \bar{z}}\frac{\partial}{\partial \sigma} = \frac{1}{2}\frac{\partial}{\partial \tau} - \frac{1}{2i}\frac{\partial}{\partial \sigma} = \frac{1}{2}(\partial_{\tau} + i\partial_{\sigma}) \ \ (8)

Hence, the shorthand notation as read in Polchinksi and which we will use from this point forward,

\displaystyle  \partial \equiv \partial_{z} = \frac{1}{2}(\partial_1 - i \partial_2) \ \ (9)

\displaystyle  \bar{\partial} \equiv \partial_{\bar{z}} = \frac{1}{2}(\partial_1 + i\partial_2) \ \ (10)

Where {\partial_zz = 1} and {\partial_{\bar{z}}z = 0}.

To continue setting things up, we must now also register that we may set {\sigma = (\sigma^{1},\sigma^{2})} and {\sigma^{z} = \sigma^{1} + i\sigma^{2}} and {\sigma^{\bar{z}} = \sigma^{1} - i\sigma^{2}}. The reason for this will become clear in a moment. For the metric, given the above {\gamma_{ab} \rightarrow \delta_{ab} \rightarrow g_{ab}},

\displaystyle g_{ab} =     \begin{bmatrix}     g_{zz} & g_{z\bar{z}} \\     g_{\bar{z}z} & g_{\bar{z}\bar{z}} \\ \end{bmatrix} (11)

From this we can also say that {\det g = \sqrt{g} = \frac{1}{2}}, which is true for Minkowski and indeed {\delta_{ab}} since we have Wick rotated. When we raise indices a factor of {2} is returned: {g^{z\bar{z}} = g^{\bar{z}z} = 2}. Lastly, the area element transforms as {d\sigma^{1}d\sigma^{2} \equiv d^{2}\sigma \equiv 2 d\sigma^{1}d\sigma^{2}}. So, we see, {d^{2}z \sqrt{g} \equiv d^{2}\sigma}.

We now need to study how the delta function transforms. Given {\int d^{2}\sigma \delta^{2}(\sigma_{1},\sigma_{2}) = \int d^{2}\sigma \delta(\sigma_{1})\delta(\sigma_{2}) = 1}, we find that in our new coordinates:

\displaystyle  \int d^{2}z \delta^{2}(z,\bar{z}) = 1 \implies \delta^{2}(z,\bar{z}) = \frac{1}{2} \delta^{2}(\sigma_{1},\sigma_{2}) \ \ (12)

We can continue establishing relevant notation by focusing on how we may rewrite the Polyakov action. In the notation that we’ve constructed we find,

\displaystyle  S_{P} = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial X^{\mu} \bar{\partial}X_{\mu} \ \ (13)

Where {d^{2}z = dzd\bar{z}}. Using identities constructed throughout this post, the tools are available to see how we arrive at this simpler form of the action. The task now is to see what returns when we vary (13).

Proposition: We vary the action (13) and find the EoM to be {\partial\bar{\partial}X^{\mu} = 0}.

Proof: The string coordinate field, if not obvious, is now {X(z, \bar{z}) = X(z) + \bar{X}(\bar{z})}. It will become clear in the following discussion that we want to compute a quantity without linear dependence on {\tau}. To that end we use the derivative of the coordinate field {\partial X(z)} and {\bar{\partial}\bar{X}(\bar{z})}.

Now, when we vary the simplified action we find,

\displaystyle  0 = \frac{\delta S}{\delta X^{\mu}} = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial X^{\mu}\bar{\partial}(X_{\mu} + \delta X_{\mu})

\displaystyle  = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial X^{\mu} (\bar{\partial}X_{\mu} + \bar{\partial}\delta X_{\mu})

\displaystyle = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial X^{\mu}\bar{\partial}X_{\mu} + \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial X^{\mu}(\bar{\partial}\delta X_{\mu}) \ \ (14)

Continuing with the conventional procedure, where we now integrate by parts (and for convenience discard the boundary terms), we find the EoM to be

\displaystyle \delta S = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial X^{\mu}(\bar{\partial}\delta X_{\mu})

\displaystyle = - \frac{1}{2\pi \alpha^{\prime}} \int d^{2}z \partial\bar{\partial}X^{\mu} (\delta X_{\mu}) = 0

\displaystyle  \implies \partial\bar{\partial}X^{\mu} (z, \bar{z}) = 0 \ \ (15)

\Box

Using the fact that partial derivatives commute. This completes the proof. The classical solution can be solved by, or in other words it decomposes as, {X(z) + \bar{X}(z)}. And we should also note, for pedagogical purposes, that we may write the EoM as \partial (\bar{\partial} X^{\mu}) = \bar{\partial} (\partial X^{\mu}) = 0 such that

\displaystyle  \partial X^{\mu} = \partial X^{\mu}(z) \ \ \ \text{holomorphic function}

\displaystyle  \bar{\partial}X^{\mu} = \bar{\partial}X^{\mu} (\bar{z}) \ \ \ \text{antiholomorphic function}

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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Fiducial metric
Stringy Things

Notes on string theory: Fiducial metric

[latexpage]
In Chapter 3 of Polchinski’s book, he references the fiducial metric. It is denoted as,

[ g_{ab} rightarrow hat{g}_{ab} ]

While this topic will be referenced in the future when exploring a study of the Polyakov path integral, and especially in a discussion about the symmetries used in the Faddeev-Popov method, we can at this point explore the notion of the fiducial metric by relating it to concepts already covered.

In short, the fiducial metric comes from the gauge freedom we’re afforded from the symmetries of the action, enabling us “to eliminate the integration over the metric, fixing it at some specific functional form” (p. 85). The important point to note is that we have our two gauge symmetries: diffeomorphisms and Weyl transformations. David Tong (p.109) puts it concisely, “We will schematically denote both of these by $zeta$. The change of the metric under a general gauge transformation is $g_{ab}(sigma) rightarrow g_{ab}^{zeta}(sigma^{prime})$.” The notation used here is shorthand for the following,

[ g_{ab}(sigma) rightarrow g_{ab}^{zeta}(sigma) = e^{2w(sigma)} frac{partial sigma^{gamma}}{partial sigma^{prime a}} frac{partial sigma^{lambda}}{partial sigma^{prime b}}g_{gamma lambda}(sigma) ]

Where, under the general gauge transformation $g rightarrow g^{zeta}$ we have a change of metric.

Due to gauge freedom we can rewrite the metric in a simpler form, commonly as $hat{g}$. This is the fiducial metric; it denotes our particular choice of gauge fixing. The common choice in the literature is the flat metric $hat{g}_{ab}(sigma) = delta_{ab}$.

Again, while the topic is usually introduced at a more advanced stage, an additional now may prove beneficial for when we eventually indulge in conformal field theories. For instance, upon equipping the tools provided to us by utilising stringy CFTs, we may consider the WS as it relates to a cylinder (as the Euler characteristic vanishes).

For our present discussion, we shall maintain our usual $(tau, sigma)$ coordinates and recall the metric $gamma$. We also note the symmetric nature of this $2×2$ metric with respect to its off-diagonal components.

[ gamma_{alpha beta} = (begin{bmatrix}
gamma_{00} & gamma_{01} \
gamma_{10} & gamma_{11} \
end{bmatrix}) ]

In that we have three independent components for the WS metric, we note that for the symmetry $gamma_{01} = gamma_{10}$. We can use the local symmetries already described to specify these three components. 

So, for reparameterisation invariance, we may perform a coordinate transformation which results in transforming the metric into a flat Minkowski metric.

[ gamma_{alpha beta} = e^{phi(tau, sigma)}eta_{alpha beta} = e^{phi(tau, sigma)}(begin{bmatrix}
-1 & 0\
0 & 1 \
end{bmatrix})
]

As Polchinski notes, “One sometimes wishes to consider the effect of the diff group alone. In this case, one can bring an arbitrary metric to within a Weyl transformation of the unit form” (p.85). This is also known as the conformal group, which will be discussed again later. What is important for now, without jumping ahead, is that we can perform a Weyl transformation to remove the exponential factor. Thus, we are simply left with a transformation to a flat Minkowski metric. 

What does this mean? Well, for the $S_{P}$ action that we’ve already grown familiar with, we already know that we can write is as,

[ S_{P} = frac{1}{4pi alpha^{prime}} int d^{2}sigma sqrt{- gamma} gamma^{alpha beta}partial_{alpha}X^{mu}partial_{beta}X^{nu}g_{mu nu} ]

But now the determinate is just $-1$. As for the $gamma^{alpha beta}$ term, this is:

[ gamma^{alpha beta} = begin{bmatrix}
-1 & 0 \
0 & 1 \
end{bmatrix} ]

So, notice that we can compute for the remaining terms of the action,

[ gamma^{alpha beta}partial_{alpha}X^{mu}partial_{beta}X^{nu}g_{mu nu} = gamma^{tau tau} partial_{tau}X^{mu}partial_{tau}X^{nu}g_{mu nu} + gamma^{sigma sigma}partial_{sigma}X^{mu}partial_{sigma}X^{nu}g_{mu nu} ]

[= – partial_{tau}X^{mu}partial_{tau}X^{nu}g_{mu nu} + partial_{sigma}X^{mu}partial_{sigma}X^{nu}g_{mu nu} ]

[ – partial_{tau}X^{mu}partial_{tau}X_{mu} + partial_{sigma}X^{mu}partial_{sigma}X_{mu} ]

Here, we have the Lagrangian density for a set of massless free scalar fields. Substituting into $S_{P}$, as well as simplifying the notation by using the shorthand $frac{partial X^{mu}}{partial tau} = dot{X}^{mu}$ and $frac{partial X^{mu}}{partial sigma} = X^{prime mu}$,

[ S_{P} = frac{1}{4pi alpha^{prime}} int d^{2}sigma (partial_{tau}X^{mu}partial_{tau}X_{mu} + partial_{sigma}X^{mu}partial_{sigma}X_{mu}) ]

[ = frac{1}{4pi alpha^{prime}} int d^{2}sigma (dot{X}^{mu 2} – X^{prime 2}) ]

One may notice the expression in the integrand as a Virasoro condition. But the main focus here is a continued study of the EM tensor. In flat space $gamma_{alpha beta} = eta_{alpha beta}$ as described. This means we can also write,

[ T_{alpha beta} = partial_{alpha}X^{mu}partial_{beta}X_{mu} – frac{1}{2}eta_{alpha beta}(eta^{lambda rho} partial_{lambda}X^{mu}partial_{rho}X_{mu}) ]

And so, we may now look at each component of the EM tensor. We find,

[ T_{tau tau} = partial_{tau}X^{mu}partial_{tau}X_{mu} – frac{1}{2}eta_{tau tau}(eta^{tau tau}partial_{tau}X^{mu}partial_{tau}X_{mu} + eta^{sigma sigma} partial_{sigma}X^{mu}partial_{sigma}X_{mu}) ]

[= partial_{tau}X^{mu}partial_{tau}X_{mu} + frac{1}{2}(-partial_{tau}X^{mu}partial_{tau}X_{mu} + partial_{sigma}X^{mu}partial_{sigma}X_{mu}) ]

[= frac{1}{2}(partial_{tau}X^{mu}partial_{tau}X_{mu} + partial_{sigma}X^{mu}partial_{sigma}X_{mu}) ]

[ = frac{1}{2}(dot{X}^{mu}dot{X}_{mu} + X^{prime mu}X_{mu}^{prime}) ]

Similarly, we can compute for $T_{sigma sigma}$ which gives us the result,

[ T_{sigma sigma} = frac{1}{2}(dot{X}^{mu}dot{X}_{mu} + X^{prime mu}X_{mu}^{prime}) ]

For the off-diagonal terms,

[ T_{tau sigma} = partial_{tau}X^{mu}partial_{sigma}X_{mu} – frac{1}{2}eta_{tau sigma}(eta^{tau tau}partial_{tau}X^{mu}partial_{tau}X_{mu} eta^{sigma sigma}partial_{sigma}X^{mu}partial_{sigma}X_{mu}) ]

[= partial_{tau}X^{mu}partial_{sigma}X_{mu} = dot{X}^{mu}X_{mu}^{prime} ]

Similarly, we find,

[ T_{sigma tau} = X^{prime mu}dot{X}_{mu} ]

Therefore, the EM tensor takes the matrix form,

[ T_{alpha beta} = begin{bmatrix}
frac{1}{2}(dot{X}^{mu}dot{X}_{mu} + X^{prime mu}X_{mu}^{prime}) & dot{X}^{mu}X_{mu}^{prime} \
X^{prime mu}dot{X}_{mu} & frac{1}{2}(dot{X}^{mu}dot{X}_{mu} + X^{prime mu}X_{mu}^{prime}) \
end{bmatrix} ]

We already know the EM tensor has zero trace. This of course can be computed. It also follows that the EM tensor appears in the equations of motion for $S_{P}$ as before. But I leave it to the reader to explore these topics further.

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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Stringy Things

Notes on string theory: Hamiltonian formalism and Poisson brackets

[latexpage]

So far the picture of the relativistic string has been built in terms of the Lagrangian formalism. We can now also spend a few minutes engaging in a short review of the Hamiltonian description in the flat gauge (before formally moving to study the spectrum of the string).

Recall the $S_{P}$ action in the flat gauge, restoring explicit notation for the string tension,

[S_{P} = -frac{T}{2} int d^{2}sigma partial_{a}X partial_{b}Xeta^{ab} ] Where $mathcal{L} = -frac{T}{2} int d^{2}sigma partial_{a}X partial_{b}Xeta^{ab}$.

For the Hamiltonian we may write what is below, recalling the conjugate momenta previously defined:

[ mathcal{H} = int_{0}^{sigma_1} dsigma(dot{X}_{mu}P_{mu}^{tau} – mathcal{L}_{P}) = frac{T}{2} int_{0}^{sigma_1}dsigma (dot{X}^2 + X^{prime 2}) ]

Where we have canonical fields $X^{mu}(tau, sigma)$ and conjugate momenta,

[ prod^{mu}(tau, sigma) = frac{partial mathcal{L}}{partial dot{X}_{mu}(tau,sigma)} =Tdot{X}^{mu}(tau, sigma) ]

Finally, in anticipating what is to come in the next few posts, a few brief comments on the Poisson brackets is necessary. Firstly, in string theory, our concern is with Poisson brackets for all fields at time $tau$ (Weigand, p. 21). This will be our starting point when we finally begin discussing the quantisation of the modes of the string. Moreover, as stated in (Weigand, p.21), we can define the Poisson brackets in terms of two arbitrary fields $F(tau, sigma)$ and $G(tau, sigma^{prime})$.

[ left{F, Gright } = int dtilde{sigma} (frac{partial F(tau,sigma)}{partial X^{mu}(tau, tilde{sigma})} frac{partial F(tau, sigma^{prime})}{partialprod_{mu}(tau, tilde{sigma})} – frac{partial G(tau, sigma^{prime})}{partial X^{mu}(tau, tilde{sigma})} frac{partial F(tau, sigma)}{partialprod_{mu}(tau, tilde{sigma})}) ]

From this we arrive at canonical equal time Poisson relations,

[ {X^{mu}(sigma), prod^{nu}(sigma^{prime})} = eta^{mu nu}delta(sigma – sigma^{prime}) ] [ {X, X} = 0 = { prod, prod } ]

It also follows that one can use the Fourier expansion to derive Poisson brackets for the modes along the classical string,

[ {alpha_{m}^{mu}, alpha_{n}^{nu} } = -imdelta_{m+n, 0} eta^{mu nu} ]

An identical result is also found for the case of $tilde{a}$, while the mixed brackets equal zero. It can also be stated that the commutation relations for ${x^{mu}, p^{mu} } = eta^{mu nu}$.

References

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

David Tong. (2009). “String Theory” [lecture notes].

Timo Weigand. (2015/16). “Introduction to String Theory” [lecture notes].

Standard
Stringy Things

Notes on string theory: Energy-momentum tensor

In general in Quantum Field Theory (QFT), provided we are working in flat space, we have the following formula:

[ T^{ab} = frac{2}{sqrt{- gamma}} frac{delta S}{delta gamma_{ab}} ]

In String Theory (ST), we vary $S_{P}$ slightly with respect to the background metric and we arrive at the following,

[ T^{ab} = frac{-4 pi}{sqrt{- gamma}} frac{delta S_P}{delta gamma_{a b}} ]

In sum, when it comes to the constraints on $T^{ab}$ for physical fluctuations, I think two important comments are necessary:

1. Diff invariance on the worldsheet implies $nabla_{a} T^{ab} = 0$.

2. Weyl invariance implies $T_{a}^{b} = gamma^{ab}T_{ab} = 0$.

One will find that the tracelessness of the EM tensor is a direct consequence of the symmetries imposed, such that: $T_{a}^{a} = 0$.

Due to diff invariance, it should also be highlighted that the EM tensor is covariantly conserved,

[ nabla_{a} T^{ab} rvert_{x^mu satisfies EoM} = 0 ]

Although, in a past entry, we have already found the equations of motion by varying $S_P$ with respect to our dynamical fields $gamma$ and $X^{mu}$, I think it is worthwhile, for pedagogical purposes, to pursue the same steps but this time focusing explicitly on computing the EM tensor with slightly different notation. In fact, I think what follows is quite a useful exercise and the end result is also nice to look at.

Invoking the Polyakov action we have, written in a slightly different way than in previous discussions,

[ S_{P} = – frac{1}{4 pi alpha^{prime}} int dtau dsigma sqrt{- gamma} gamma^{ab} partial_{a} X^{mu} partial_{b} X_{mu} ]

Where, in what follows, we will set $gamma^{ab} partial_{a} X^{mu} partial_{b} X_{mu} = partial X^{2}$.

Notice, also, that what we’re actually working with is a $1+1$ QFT. We also have our metric, $gamma$, so you can think of this action describing a (1+1)-dim theory of scalar fields coupled to gravity.

We now compute the EM tensor in this theory, recalling $delta gamma^{ab} = – delta gamma_{cd} gamma^{ca} gamma^{db}$ and $delta gamma = gamma cdot gamma^{ab} delta gamma_{ab}$.

[ T^{ab} = frac{4 pi}{sqrt{- gamma}} frac{1}{-4 pi alpha^{prime}} frac{delta}{delta gamma_{ab}} int [frac{1}{2} (-gamma)^{frac{1}{2}} -partial gamma^{ab} delta gamma_{ab} (partial X^{2}) – sqrt{gamma} partial^{a} X^{mu} partial^{b} X_{mu} delta gamma_{ab}] ]

[ implies -frac{1}{alpha^{prime}}[partial^{a}X^{mu}partial^{b}X_{mu} – frac{1}{2} gamma^{ab}partial_{c}X^{mu}partial^{c}X_{mu}] ]

Which gives the EM tensor. From this configuration, the tracelessness can easily be found:

[ T_{a}^{a} = -frac{1}{alpha^{prime}} [ (partial X)^{2} – frac{1}{2} gamma^{ab} gamma_{ab} (partial X)^2] = 0 ]

This reflects the fact that when we do a Weyl transformation, it does not effect the coordinates and matter fields of our theory.

What is nice about varying our action in this theory in this way is that we discover the explicit result,

[ 0 = frac{delta S}{delta gamma_{ab}} = T^{ab}= 0 ]

What is this saying? We see, and can communicate directly, that the EoM is, or represents, the vanishing of the EM tensor.

Additionally, I want to emphasis one more result. Again, using slightly different notation, let’s go back and think about varying $S_P$ with respect to $X^{mu}$.

[ delta S_P = – frac{2}{4 pi alpha^{prime}} int dtau dsigma sqrt{- gamma} gamma^{ab} nabla_{a} X^{mu} cdot nabla_{b} delta X_{mu} ]

Now, we integrate by parts such that the covariant derivative $nabla_{b}delta X_{mu}$ acts on the rest of the action,

[ = frac{1}{2 pi alpha^{prime}} int dtau dsigma nabla_{b}[sqrt{-gamma}gamma^{ab} nabla_{a} X^{mu}] delta X_{mu} + boundary terms ]

[ implies nabla_{b}[sqrt{-gamma}gamma^{ab}nabla_{a}X^{mu}] = 0 ]

Notice, approached and written this way, we see an explicit expression for the d’Alembertian,

[ sqrt{-gamma} Box X^{mu} = 0 ]

Which makes sense, as our fields satisfy the massless Klein-Gordon equation in 2-dimensions on a curved background. Perhaps this is obvious, and perhaps the result foreseeable, but I think it is still nice to look at.

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Stringy Things

Notes on string theory: Symmetries of the Polyakov action (additional comments)

[latexpage]

Let’s engage in further discussion about the symmetries of the Polyakov action. As it is an important topic, it will serve as a useful exercise to go through things a bit more deeply.

As I cited in a previous post, Polchinski describes the symmetries of the $S_{P}$ action early on by noting on p.13,

  1. D-dimensional invariance under the Poincaré group. [X^{prime mu} (tau, sigma) = Lambda^{mu}_{nu} X^{nu}(tau, sigma) + a^{mu} ] [gamma^{prime}_{alpha, beta} (tau, sigma) = gamma_{alpha beta} (tau, sigma) ]
  2. Diffeomorphism invariance. [X^{prime mu} (tau^{prime}, sigma^{prime}) = X^{mu} (tau, sigma) ] [ frac{partial sigma^{prime c}}{partial sigma^a} frac{partial sigma^{prime d}}{partial sigma^b} gamma^{prime}_{cd} (tau^{prime}, sigma^{prime}) = gamma_{alpha beta} (tau, sigma) ]
  3. Two-dimensional Weyl invariance. [ X^{prime mu} (tau, sigma) = X^{mu} (tau, sigma) ] [gamma^{prime}_{alpha beta} (tau, sigma) = exp (2 omega (tau, sigma)) gamma_{alpha beta} (tau, sigma) ]

I think it is useful to add some elaborating comments. There is also a nice point of discussion that broaches the introduction of conformal Killing vectors as related to the metric transformation under local Weyl rescaling and diffeomorphism. From a purely pedagogical point of view, a few comments can also be made directly relating the above symmetries with conserved quantities via Noether’s theorem.

Local and global symmetries

To begin, let’s return our attention to the symmetries of the Polyakov action. I think it is important to highlight that $S_{P}$ has both local and global symmetries. Moreover, $S_{P}$ has symmetries of the worldsheet and of the background spacetime. Furthermore, it is important to reemphasise that the local and global symmetries are considered and discussed from the perspective of the theory of the 2-dim worldsheet.  Thus, in terms of spacetime symmetries, we know $S_{P}$ is manifestly Poincaré invariant due to the very nature of its construction. This is a benefit of how $S_{P}$ was derived, and it can be said that the the Poincaré group is a global internal symmetry from the perspective of 2-dim field theory. As a point of emphasis, however, one should note that $X^{mu}$ can take on the interpretation as either spacetime coordinates or dynamical fields on the WS. This means that when considering Lorentz transformations, for example, while local spacetime symmetries, they are in fact global symmetries of the WS. In other words, the Poincaré transformations are global symmetries; but reparameterisations and Weyl transformations are local symmetries, with the latter used when making a decision on a gauge (Becker, Becker and Schwarz, p.30).

Proof of d-dimensional invariance under the Poincaré group can be found by noting,

[X^{prime mu} (tau, sigma) = Lambda^{mu}_{nu} X^{nu}(tau, sigma) + a^{mu} ]

Where, $Lambda_{mu nu} = – Lambda_{nu mu}$ and $a^{mu}$ is a constant. It follows,

[ S^{prime} [tau, sigma] = frac{T}{2} int d^2 sigma sqrt{- gamma} gamma^{alpha beta} g_{mu nu} partial_{a}(X^{mu} + Lambda^{mu}_{delta}X^{delta}) partial_{b}(X^{nu} + Lambda^{nu}_{delta}X^{delta}) ]

[= S + frac{T}{2} int d^2 sigma sqrt{- gamma} gamma^{alpha beta}(Lambda_{mu delta} partial_{a} X^{mu}partial_{b}X^{delta} + Lambda_{nu delta} partial_{a}X^{delta} partial_{b}X^{nu}) + mathcal{O}(Lambda^2) ]

[ = S + frac{T}{2} int d^2 sigma sqrt{- gamma} gamma^{alpha beta} (Lambda_{mu delta} + Lambda_{delta mu})partial_{a}X^{mu}partial_{b}X^{delta} + mathcal{O}(Lambda^2) = S + mathcal{O}(Lambda^2) ]

As for the WS symmetries, it is noted above that we have local diffeomorphism invariance and 2-dim Weyl invariance. Regarding the former,  if from the perspective of the WS we have a 2-dim field theory $X^{mu}$ coupled to 2-dim gravity, it has been noted in a past entry that $S_{P}$ is invariant under any WS reparameterisation. For any diffeomorphism, $X^{mu} rightarrow X^{prime^{mu}}$ the scalar field transforms more completely as,

[ X^{mu} (sigma) rightarrow X^{prime^{mu}} (sigma^{prime})= X^{mu} (sigma(sigma^{prime})) = X^{mu}(sigma) + epsilon^{c}partial_{c}X^{mu}(sigma) ]

[ equiv X^{mu} + delta X^{mu} + mathcal{O}(epsilon^2) ]

[ implies delta X^{mu} = epsilon^{c}partial_{c} X^{mu} ]

It follows that the metric $gamma_{alpha beta}$ defined in a previous post transforms as,

[ gamma_{alpha beta} (sigma) rightarrow tilde{gamma_{alpha beta}} ]

Which is commonly expanded and written  in the form (although, as with our entire discussion, notation may vary superficially),

[ frac{partial sigma^{prime c}}{partial sigma^{a}} frac{partial sigma^{prime d}}{partial sigma^{b}} gamma_{c d}^{prime} (sigma^{prime}) = gamma_{alpha beta} + delta gamma_{alpha beta} + mathcal{O}(epsilon^2) ]

[ delta gamma_{alpha beta} = epsilon^{c} partial_{c} gamma_{alpha beta} + (partial_{a} epsilon^{c}) gamma_{c b} + (partial_b epsilon^c) gamma_{ac} ]

[ = nabla_a epsilon_b + nabla_b epsilon_a ] 

We also have Weyl invariance – or local conformal invariance. 

[ delta X^{mu} = 0 ]

[ gamma_{alpha beta} rightarrow exp(2 Lambda(sigma)) gamma_{alpha beta} = gamma_{alpha beta} + delta gamma_{alpha beta} + mathcal{O}(Lambda^2) ]

[ delta gamma_{alpha beta} = 2 Lambda(sigma) gamma_{alpha beta} ]

This is an extra symmetry that we’ve been bestowed due to the simple fact that, as frequently mentioned, what we’re working with now is a 2-dim WS theory. Moreover, “the appearance of Weyl invariance for 2-dimensional worldsheets identifies String Theory as a very special generalisation of the point particle theory” (Weigand, p. 17). This special quality of ST will also become explicate when we broach the topic of the quantisation of the string and the lovely marriage that takes place with the path integral formalism. Indeed, there is a lovely approach to deriving the Nambu-Goto and Polyakov actions by way of the path integral formalism that reveals the many symmetries we’ve been discussing (I will perhaps dedicate a separate post to this study). This is a point worth emphasising, I think, because as we will see in the future the stringy generalisation is absolutely lovely in this regard.

Metric transformation and conformal Killing vectors

Following Weigand’s approach to bosonic string theory in his lecture notes, I think it is good to supplement reading Polchinski by highlighting how with the combination of diffeomorphism and Weyl rescaling, the metric undergoes the following transformation:

[ delta gamma_{alpha beta} = nabla_{alpha} epsilon_{beta} +nabla_{beta} epsilon_{alpha} + 2Lambda gamma_{alpha beta} ]

[ =  nabla_{alpha} epsilon_{beta} + nabla_{beta} epsilon_{alpha} – nabla^{c} epsilon_c gamma_{alpha beta} + 2(Lambda + frac{1}{2} nabla^{alpha} epsilon_{alpha}) gamma_{alpha beta} ]

[ implies delta gamma_{alpha beta} equiv (P cdot epsilon)_{alpha beta} 2 Lambda^{prime}gamma_{alpha beta} ]

Here, as Weigand points out, this “linear operator P maps vectors symmetric traceless 2-tensors” (p. 17). And, what is nice is how, for any transformation $epsilon_{alpha}$ where $(P cdot epsilon)_{alpha beta} = 0$, it can be said that the resultant effect can be undone by local Weyl rescaling. These $epsilon_{alpha}$ terms are conformal Killing vectors (CKV).

Now, the presence of conformal Killing vectors in our discussion anticipates deeper engagement with conformal geometry, conformal symmetry and, one may have guessed it, Conformal Field Theory (CFT). 

But let us bracket these last few comments for a future post.

In the meantime, on the presence of conformal Killing vectors, let’s magnify in on this. To do so, as is common in most texts, I should first follow a discussion on the symmetries of the $S_{P}$ action by noting that reparameterisation invariance implies the existence of conserved currents (the currents vanish on-shell, such that $nabla^{alpha}T_{alpha beta} = 0$ on-shell for $X$). We can write the conservation of the energy-momentum current in the following way,

[ mathcal{P}_{mu}^{a} = -T sqrt{- det gamma} gamma^{alpha beta} partial_{beta} X{mu} ]

[partial_{alpha} mathcal{P}_{mu}^{a} = nabla_{alpha} mathcal{P}_{mu}^{a} = 0 ]

Moreover, we can denote the conserved currents as $j_{b}^{f} = f^{a}(sigma)T_{ab}$ for any function $f^{a}(sigma)$. It was noted in a past post that the energy-momentum tensor is traceless – that is to say it is zero – and this is a consequence of Weyl invariance, such that $T_{a}^{a} = 0$ as $delta S = 0$. (For the reader comfortable with QFT, you will know or might recall that a local quantum field theory will have a conserved stress tensor). This can be shown by simply noting that, with conformal invariance of the action under reparameterisation, 

[ gamma_{alpha beta} rightarrow e^{phi} gamma_{alpha beta} ]

If we were to apply this in the context of an infinitesimal $phi$ notice,

[ delta S = frac{delta S}{delta gamma^{alpha beta}} ]

Wherein,

[ delta gamma^{alpha beta} = frac{sqrt{-gamma}}{4 pi} T_{alpha beta}(e^{- phi} – 1)gamma^{alpha beta} ]

[ implies gamma^{alpha beta}T_{alpha beta} = 0 ]

Additionally, if reparameterisation implies conserved currents, we can determine that under global Poincaré transformations – this time invariance under global Lorentz transformation – we also find the conservation of angular momentum currents,

[ J_{mu nu}^{alpha} = – T sqrt{- gamma} gamma^{alpha beta}(X_{mu}partial_{beta}X_{nu} – X_{nu}partial_{beta}X_{mu}) ]

[ = X_{mu}P_{nu}^{alpha} –  X_{nu}P_{mu}^{alpha} ]

[ nabla_{alpha} J_{mu nu}^{alpha} = 0 ]

The observations are, again, quite standard in the literature. Following Weigand (p.18) in particular, the following generalisation seems rather apt at this stage of our study. Returning to the brief mentioning of conformal Killing vectors, if every $epsilon_{alpha}$ which satisfies $P cdot epsilon_{alpha beta}$ results in or produces a conserved current, we can say:

[ J_{epsilon}^{alpha} = T^{alpha beta} epsilon_{beta} ]

Where, as noted, $nabla_{alpha} J_{epsilon}^{alpha} = 0$. The proof for this is quite simple. You can find it on p. 18 in Weigand’s lecture notes, or, just recall from GR a study of Killing vector fields.

References

Katrin Becker, Melanie Becker, John H. Schwarz. (2006). “String Theory and M-Theory: A Modern Introduction“.

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

Timo Weigand. (2015/16). “Introduction to String Theory” [lecture notes].

 

Standard
Stringy Things

Notes on string theory: Polyakov action – Symmetries, classical equivalence, equations of motion, and Virasoro constraints

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We know from a past note that the Nambu-Goto action can be written as,

[ S_{NG}= – frac{1}{2 pi alpha^{prime}} int dtau dsigma sqrt{- h} ]

Where I’ve used slightly different notation than in the past in order to align with Polchinski, and so it should be restated $h = det(h_{alpha beta})$. This is just a superficial point. (I prefer to reserve $gamma$ for the auxilliary worldsheet metric described below).

One more thing we might recall. It was mentioned in the last post that, as in the case of the point particle, when studying the action $S_{NG}$ problems arise in the quantum theory that are associated primarily with the presence of the square root. So, for the study of the quantum physics of strings, we should like for an action of another form. What form may that be? Recall that in the case of the point particle, we modified the action by introducing an auxiliary field. This enabled the elimination of the pesky square root. By complete analogy, we can take the same approach for $S_{NG}$. In fact, in majority of textbooks and certainly in most lecture series I’ve reviewed, introduction to the Nambu-Goto action is followed almost immediately by the standard explanation of how $S_{NG}$ can be modified with some sort of auxiliary field.

However, it is worth pointing out that this new auxiliary field, which we will define as $gamma_{alpha beta} (tau, sigma)$, is actually an “intrinsic” metric on the worldsheet (WS). It should be understood as completely independent of the pullback metric. In other words, it is a dynamic variable in the action and, inasmuch that one should think of $gamma_{alpha beta}$ as an auxiliary WS metric, another notable implication is that it also leads to its own field equations. So, moving forward, $gamma_{alpha beta}$ will represent the WS metric and $h_{mu nu}$ will represent the induced metric. It may also be noted,

[ gamma = det gamma_{alpha beta} ]

[ gamma^{alpha beta} = (gamma^{-1})_{alpha beta} ]

From this, the Polyakov action, which we may also think of in terms of the string sigma model action, takes the form:

[ S_{P} [X, gamma] =-frac{1}{2}T int_{sum} dtau dsigma sqrt{-gamma} gamma^{alpha beta} h_{alpha beta} ] [ =-frac{T}{2} int_{sum} dtau dsigma sqrt{-gamma} gamma^{alpha beta} partial _{alpha} X^{mu} cdot partial _{beta} X^{mu} ] [ = -frac{1}{4 pi alpha^{prime}} int_{sum} dtau dsigma sqrt{-gamma} gamma^{alpha beta} partial _{alpha} X^{mu} partial _{beta} X^{nu} g_{mu nu} ]

In that this is the Polyakov action, it was discovered independently by Brink, Di Vecchia, and Howe and by Deser and Zumino. Here, $X^{mu}(tau, sigma)$ are scalar fields on the WS. Another way to put this: the spacetime coordinates $X^{mu}$ of previous are now promoted to dynamical fields. These fields are the centre of focus when it comes to the 2-dim field theory on the WS. From the point of view of the WS, the action describes the way in which these fields are coupled to 2D gravity. The indices $mu$ and $nu$ correspond to the target space while $alpha$ and $beta$ correspond to the 2D WS.

There are also several other points worth emphasising:

First, the Polyakov action indeed looks very much like a 2D sigma model. Compare, for example, with a 4D sigma model in QFT where $g_{mu nu}$ is the metric and where we’re working on a general spacetime background:

[ S_{sigma} = int d^{4} x sqrt{g} g^{mu nu} partial_{mu} phi^{M} partial_{nu} phi^{N} G_{MN} ]

This connection makes a lot of sense, given the relation between Quantum Field Theory and String Theory.

Second, $S_{P}$ is classically equivalent to the NG action. We will prove this in just a moment. Additionally, what is also nice is how the variation of the action with respect to the metric $gamma_{alpha beta}$ gives the energy-momentum tensor (again, see below).

Third, in that one of the primary advantages of $S_{P}$ is how it provides the correct quantum theory, my understanding here is that $S_{P}$ has the advantage of bilinearity with respect to the $X(sigma)$ fields. As Polchinski writes, “Its virtues” are “especially for path integral quantization” (p.34) in that it enables or allows for direct access to quantum procedures in Fock space. Moreover, it is not that $S_{NG}$ cannot be quantised. Indeed, there’s a lot that can be accomplished with it in general. It is just that things can and do become untidy with the $S_{NG}$ form of the action – we end up with square roots everywhere and I think it is generally more difficult to derive the wave equations, momenta identities, and other things. It seems to me that $S_{P}$ is easier to generalise when considering curved or arbitrary backgrounds, given, indeed, that the metric $gamma_{alpha beta}$ is generally curved. Another terrific quality is that it provides the Levi-Civita connection. These advantages are emphasised as early as Chapter 2 in Polchinski, where we begin to learn how, when generalised, $S_{P}$ takes the form of an interacting field theory. Hence, the treatment of the string sigma model as a synonym.

Fourth, the WS metric here has a lovely quality in that its components, in a sense, play the role of Lagrange multipliers. These Lagrange multipliers impose the Virasoro constraints, which prove incredibly important moving forward. This is something we will talk a lot about (for example, when we get into the quantum theory of strings among other topics). The Virasoro constraints can first be derived (a nice derivation, in fact) from the NG action.

Symmetries of the Polyakov action

Fifth, as Polchinski notes (p.13), a number of other advantages include the symmetries that come with $S_{P}$:

  1. D-dimensional invariance under the Poincaré group. [X^{prime mu} (tau, sigma) = Lambda^{mu}_{nu} X^{nu}(tau, sigma) + a^{mu} ] [gamma^{prime}_{alpha, beta} (tau, sigma) = gamma_{alpha beta} (tau, sigma) ]
  2. Diffeomorphism invariance. [X^{prime mu} (tau^{prime}, sigma^{prime}) = X^{mu} (tau, sigma) ] [ frac{partial sigma^{prime c}}{partial sigma^a} frac{partial sigma^{prime d}}{partial sigma^b} gamma^{prime}_{cd} (tau^{prime}, sigma^{prime}) = gamma_{alpha beta} (tau, sigma) ]
  3. Two-dimensional Weyl invariance. [ X^{prime mu} (tau, sigma) = X^{mu} (tau, sigma) ] [gamma^{prime}_{alpha beta} (tau, sigma) = exp (2 omega (tau, sigma)) gamma_{alpha beta} (tau, sigma) ]

But to maintain a clear picture of what is being developed, we should be careful to delineate the origin of our symmetries (Weigand, 2015/16). I will save the reader a detailed description of the transformations, instead offering only a few brief comments (perhaps I will write a separate post on these matters so as to limit the length of the current discussion). In terms of spacetime symmetries, $S_{P}$ is manifestly Poincaré invariant. From the perspective of 2-dim field theory, this is a global internal symmetry. As for the WS symmetries, we have local diffeomorphism invariance under $xi^{a} rightarrow tilde{xi}^{a} (xi) = xi^{a} – epsilon^{a} (xi)$. From this we can describe the manner in which the fields transform, such as in how a scalar field from the WS perspective transforms or how the metric $gamma_{alpha beta}$ transforms as a WS 2-tensor (Weigand, 2015/16).

We also have Weyl invariance – or local conformal invariance. On the introduction of Weyl invariance that follows the introduction of the auxiliary metric embedded on the WS, Polchinski writes, “The Weyl invariance, a local rescaling of the worldsheet metric, has no analog in the Nambu–Goto form” (p.13). This is an extra symmetry that we’ve been granted on the basis of the fact that what we’re working with now is a 2-dimensional WS theory. It also turns out that Weyl invariance will prove crucial, as we will see in future notes.

More can be said on these matters, but I should like to keep focused on the main aims of this post.

Varying the action with respect to the metric

[ S_{P} [X, gamma] = – frac{1}{4 pi alpha^{prime}} int_{sum} dtau dsigma sqrt{-gamma} gamma^{alpha beta} partial_{alpha} X^{mu} partial_{beta} X^{nu} g_{mu nu} ]

To simplify things, let:

[ sigma^{0} = tau ]

[ sigma^{1} = sigma ]

And set $T = frac{1}{2 pi alpha^{prime}}$. So,

[ S_{P} [X, gamma] = – frac{T}{2} int_{sum} d^2 sigma sqrt{-gamma} gamma^{alpha beta} partial_{alpha} X^{mu} partial_{beta} X^{nu} g_{mu nu} ]

Varying the action such that $delta gamma = gamma + delta gamma$,

[ delta S_{P} [X, gamma] = – frac{T}{2} int_{sum} d^2 sigma delta (sqrt{-gamma} gamma^{alpha beta}) partial_{alpha} X^{mu} partial_{beta} X^{nu} g_{mu nu} ]

[ = – frac{T}{2} int_{sum} d^2 sigma sqrt{-gamma} delta gamma^{alpha beta} partial_{alpha} X^{mu} partial_{beta} X^{nu} g_{mu nu} – int_{sum} d^2 sigma gamma^{alpha beta} delta sqrt{- gamma} partial_{alpha} X^{mu} partial_{beta} X^{nu} g_{mu nu} ]

Now, two things to note before proceeding further. To vary the metric $gamma^{alpha beta}$, notice that if $A$ is some two-by-two matrix, and $delta A$ is its variation,

[ A = begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \ end{pmatrix} ]

And, [ delta A = begin{pmatrix} delta a_{11} & delta a_{12} \ delta a_{21} & delta a_{22} \ end{pmatrix} ]

It should be known and understood that the variation of $det A$ can be found,

[ delta det A = (det A) Tr(A^{-1} delta A) ]

Where $A^{-1}$ is the inverse of $A$.

With that said, and with the previous working in mind, let’s now think of $delta gamma^{alpha beta}$. We can use the identity above to vary $gamma$:

[ delta gamma = delta det(gamma_{alpha beta}) = gamma (gamma^{alpha beta} delta gamma_{beta alpha}) ]

But we must write the variation in terms of $gamma^{alpha beta}$, not $gamma_{alpha beta}$. What to do? Note, $gamma^{alpha beta} gamma_{beta alpha} = delta_{lambda}^{alpha}$ and $gamma^{alpha beta} gamma_{alpha beta} = 2$.

So, we can see how we obtain the correct expression if we vary $gamma^{alpha beta} gamma_{alpha beta} = 2 $.

[ delta gamma^{alpha beta} gamma_{alpha beta} +
gamma^{alpha beta} gamma_{alpha beta} = 0 ]

[ implies gamma^{alpha beta} delta gamma_{beta alpha} = -delta gamma^{alpha beta} gamma_{alpha beta} ]

From this, we can determine that the variation of $gamma$ can be written more concisely as

[ delta gamma = – gamma delta gamma^{alpha beta} gamma_{alpha beta} ]

As an aside, one will find this exact expression in Polchinski (1.2.15), as we have invoked the general relation for the variation of the determinant (p.12). What is also being implied here, however little we have yet to truly explicate it, is how, if $delta_{gamma}S_{P} = 0$, it follows the induced metric $h_{alpha beta} = frac{1}{2} gamma_{alpha beta} gamma^{cd}h_{cd}$. This, again, can be found on p.12 in Polchinski, as he sets himself the task of showing $S_{P}$ and $S_{NG}$ are classically equivalent. As we’re currently undertaking a different angle of attack, we’ll return to this point in a moment.

With that temporary departure from the main line of logic, let’s now ask on the basis of our previous result: what about the variation of $sqrt{-gamma}$? It remains to be said that we can show,

[ delta(sqrt{-gamma}) = – frac{1}{2} frac{delta gamma}{sqrt{-gamma}} = – frac{1}{2} frac{(-gamma) delta gamma^{alpha beta} gamma_{alpha beta}}{sqrt{-gamma}} = -frac{1}{2} sqrt{-gamma} delta gamma^{alpha beta} gamma_{alpha beta} ]

Therefore, we can complete the variation of the action with respect to the metric, which was our goal. Carrying on from above,

[ delta S_{P} [X, gamma] = -frac{T}{2} int d^2 sigma sqrt{-gamma} delta gamma^{alpha beta} partial_{alpha} X^{mu} partial_{beta} X^{nu} g_{mu nu} – frac{T}{2} int d^2 sigma – frac{1}{2} sqrt{-gamma} gamma_{alpha beta} gamma^{c d} partial_{c} X^{mu} partial_{d} X^{nu} g_{mu nu} delta gamma^{alpha beta} ]

[ = int d^{2} sigma sqrt{- gamma} delta gamma^{alpha beta} (partial_{alpha} X^{mu} partial_{beta} X^{nu} – frac{1}{2} gamma_{alpha beta} (gamma^{c d} partial_{c} X^{mu} partial_{d} X^{nu}))g_{mu nu} ]

This means that the EoM as a result of the variation $delta gamma^{alpha beta}$ is found to be,

[ partial_{alpha} X partial_{beta} X – frac{1}{2} gamma_{alpha beta} (gamma^{c d} partial_{c} X partial_{d} X) = 0 ]

Energy-momentum tensor of 2-dim field theory

One key point of emphasis is the relation to 2-dim scalar field theory, in that the WS is just the space where the field theory lives. On this note, recall a recent post introducing $S_{P}$ by way of GR. It is a pertinent pedagogical link. One reason I wrote that post is because I like the way in which there is a certain equivalence between the study of 2-dim gravity coupled to scalar fields and string theory (of the bosonic sort). It should make a lot of sense because, as I believe I have described, “the study of bosonic string theory is equivalent to the study of 2-dim gravity coupled to scalars” (Weigand, 2015/16, p. 14), as precisely described in the post linked above. Recall, also, that in that post the energy momentum tensor was written,

[ T_{m n} : = -frac{1}{T} frac{1}{sqrt{h}} frac{delta S_{P}}{delta h^{m n}} ]

[ = partial_{m} X^{mu} partial_{n} X_{nu} – frac{1}{2} h_{m n} h^{p q} partial_{p} X^{mu} partial_{q} X_{nu} = 0 ]

Notice, this is precisely what we have found when varying the action with respect to the metric. Indeed, when performing the above variation we arrive at the energy-momentum tensor. This is a nice connection, because we know all sorts of useful properties when it comes to the EM tensor. For example, we know it is traceless. It also follows that $T_{mn}$ is conserved in the case of the EoM for $X^{mu}$ (see below). And, indeed, these are important properties or features of $T_{mn}$, which I’ll discuss in a separate post.

Classical equivalence

Earlier I mentioned $S_{P}$ and $S_{NG}$ are classically equivalent. I also referenced Polchinski’s engagement with the topic. We can now show this classical equivalence to be true.

To begin, we return to the EoM. Rearranging, we get:

[ partial_{alpha} X partial_{beta} X – frac{1}{2} gamma_{alpha beta} gamma^{c d} partial_{alpha}X partial_{beta}X= 0 ]

Let’s write $partial_{alpha}X partial_{beta}X$ as $G_{alpha beta}$. Then what we want to do is take the square root of the determinant of both sides.

[ G_{alpha beta} – frac{1}{2} gamma_{alpha beta} gamma^{c d} G_{alpha beta} = 0 ]

[ G_{alpha beta} = frac{1}{2} gamma_{alpha beta} gamma^{c d} G_{cd} ]

[ sqrt{- det(G_{alpha beta})} = frac{1}{2} sqrt{- gamma} gamma^{c d} G_{c d} ]

[ implies sqrt{- gamma} = frac{2 sqrt{-G}}{gamma^{cd} G_{}cd} ]

Substitute for $sqrt{- gamma}$ in $S_{P}$:

[ S_{P} = frac{T}{2} int d^2 sigma sqrt{- gamma} gamma^{alpha beta} G_{alpha beta} ]

[= frac{T}{2} int d^2 sigma frac{2 sqrt{-G}}{gamma^{cd} G_{}cd} gamma^{alpha beta} G_{alpha beta} ]

[ = T int d^2 sigma sqrt{-G} = S_{NG} ]

Wave equation

This time varying the $S_{P}$ action with respect to $X^{mu}$, such that $X^{mu} rightarrow X^{mu} + delta X^{mu}$.

[ delta S[X, gamma] = – frac{T}{2} int d^2 sigma sqrt{- gamma} gamma^{alpha beta} partial_{alpha} (delta X^{mu}) partial_{beta} X^{nu} g_{mu nu} ]

[ = frac{T}{2} int d^2 sigma delta X^{mu} partial_{alpha}(sqrt{- gamma} gamma^{alpha beta} partial_{beta} X^{nu} g_{mu nu}) ]

Ignoring, once again, the boundary terms. The EoM is,

[ partial_{alpha}(sqrt{- gamma} gamma^{alpha beta} partial_{beta} X^{mu})=0 ]

The wave equation follows after a few important steps. We need to make a gauge choice. In particular, we shall work in the conformal gauge, such that $sqrt{- gamma} gamma^{alpha beta} equiv eta^{alpha beta}$, where $eta^{alpha beta}$ is the 2-dim Minkowski metric. These manoeuvres are also principled on reparameterisation invariance for $gamma_{alpha beta}$ (Zwiebach, 2009, p. 476).

The EoM becomes,

[ partial_{alpha} (eta^{alpha beta} partial_{beta} X^{mu}) = eta^{alpha beta} partial_{alpha}partial_{beta}X^{mu} = 0 ]

This is the wave equation.

Virasoro constraints

To conclude this entry, I think it is worthwhile spending a few minutes thinking about the Virasoro constraints. They will appear later in the quantum theory for $S_{P}$. What follows can be found in Zwiebach, 2009, pp. 583, 585-87.

For the Virasoro constraints, we once again invoke the conformal gauge. So, if we return to,

[ partial_{alpha} X partial_{beta} X – frac{1}{2} gamma_{alpha beta} (gamma^{cd} partial_{c} X partial_{d} X) = 0 ]

The gauge condition $gamma_{alpha beta} = rho^{2}(xi) eta_{alpha beta}$ gives,

[ partial_{alpha} X partial_{beta} X – frac{1}{2} eta_{alpha beta}(eta^{cd} partial_{c}X partial_{d} X) = 0 ]

But we know that the expression in the parenthesis can be expanded. We’ve seen it before. So,

[ partial_{alpha} X partial_{beta} X – frac{1}{2} eta_{alpha beta} (- dot{X}^2 + X^{prime^2}) = 0 ]

Setting $alpha=beta=1$,

[ dot{X}^2 + frac{1}{2} (-dot{X}^2 + X^{prime^2}) = 0 rightarrow
dot{X}^2 + X^{prime^2} = 0 ]

This gives one of the constraints. The second constraint can be found by setting $alpha=1$ and $beta=2$,

[ dot{X} cdot X^{prime} = 0 ]

The third constraint, then, comes from setting $alpha=beta=2$,

[ X^{prime^2} – frac{1}{2} (- dot{X}^2 + X^{prime^2})=0 rightarrow dot{X}^2 + X^{prime^2} = 0 ]

However, we have already arrived at this constraint! So, in that this last result is clearly redundant (Zwiebach, p.587), what is interesting about this is that we ought to expect three constraints. Given that we have only two conditions, this can easily be reconciled. For example, see p. 583 in Zwiebach.

References

David Tong. (2009). “String Theory” [lecture notes].

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

Timo Weigand. (2015/16). “Introduction to String Theory” [lecture notes].

Barton Zwiebach. (2009). “A First Course in String Theory”, 2nd ed.

Standard
Stringy Things

Notes on string theory: Polyakov action (by way of General Relativity in m-dimensions)

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As has been discussed, the Nambu-Goto action can be interpreted as providing the area of the worldsheet mapped by the string in spacetime. For what it is, I think this action is quite nice. There is a lot we can do with it, and there are a lot of ways in which one can play around with it and produce some fairly neat results. But there is also a problem. Similar to the action for the point particle, with the NG action we ended up with a square root unfriendly to quantisation. In other words, it is generally not adequate for the study of the quantum physics of strings (though it is possible to quantise the NG action). It may also be added that, for the NG action, it takes quite a bit of work to arrive at the wave equations, momenta identities, and other useful things. So we want an action that does not contain a square root – that is polynomial – and that is generally easier to work with.

This means that we need to come up with another action – the Polyakov action.

Currently, I do not know of a forward derivation that works directly from the Nambu-Goto action to the Polyakov action. It would seem a highly non-trivial feat. But there are a number of things one can do. As is common in majority of textbooks and leading literature, the standard procedure is to leverage the point particle analogue as a guiding principle, such that one may think of whether the NG action can be modified with some sort of auxiliary field. The answer is that it can indeed be modified in such a way. The result is the Polyakov action. Most texts describe this and show the result – it is taken more or less as a given in Polchinski, for example. Emphasis is then placed on showing how one can work backward from the Polyakov action to the NG action, showing they are classically equivalent with the former still providing the correct quantum theory.

In the next post, we will take this generally textbook approach to the Polyakov action. We will also begin to study the equations of motion and other interesting things.

But since I don’t find that approach entirely satisfying, in this entry I want to introduce the Polyakov action slightly differently. I want to take a more extended approach through General Relativity (this is one of a number of ways that one can arrive at the Polyakov action).

***

We start by invoking GR in m-dimensions, with the inclusion of scalar fields and the cosmological constant. As this is just a simple scalar field theory the action is given as follows,

[ S[g, phi] = int d^{4}x sqrt{-g} (mathcal{R} – frac{1}{2} g^{mn} partial_{m} phi partial_{n} phi – Lambda) ]

Recall that this action is diffeomorphism invariant. It also has global symmetry, such that $phi rightarrow phi + a$.

What we want to do is replace $partial_{m} phi partial_{n} phi$ with many scalar fields (a linear combination with the products of many scalar fields). So we’re going to take the conventional route of employing $i$ and $j$ indices, with the inclusion of $M_{ij}$ in the action. Here, $M_{ij}$ is not a function of $phi$ – it is just a matrix of numbers. In other words, $M_{ij} neq f(phi)$. This is what we get,

[ S[g, phi] = int d^{4}x sqrt{-g} (mathcal{R} – frac{1}{2} g^{mn} partial_{m} phi^{i} partial_{n} phi^{j} M_{ij} – Lambda) ]

Several comments: first, notice the $i$ and $j$ indices – $phi^i$ and $phi^j$ – represent the internal space of our theory where $phi$ can take values. Secondly, the global symmetry that we had before remains: $phi^i rightarrow phi^i + a^i$. But there is also another global symmetry that we can write, because if we transform the scalars by any matrix $phi^i rightarrow Lambda_{j}^{i} phi^i$ such that $Lambda^{T}MLambda = M$, we have another global symmetry for the action.

In addition to the above, if we choose a particular $M$ such that it equals identity, $M = mathbb{I}$, where $M$ is a $D times D$ matrix and where $i = 1, … , D$, the global symmetry group can be written: $M =
mathbb{I} rightarrow SO(D)$.

Another matrix one might consider is where,

[ M = begin{pmatrix}
-1 & … & 0 \
0 & … & mathbb{I}_{(D-1)+(D-1)} end{pmatrix} ]

Now the global symmetry group can be written as $SO(1, D-1)$. (In QFT, this choice for $M$ is often avoided due to the desire for positive energy. If this symmetry group is used, one of the fields will have the incorrect sign for the kinetic term). What is nice about this, I think, is that if we combine $SO(1, D-1)$ with the previous expression for the global symmetry $phi^i rightarrow Lambda^i + a^i$, it looks like a translation. So the result is translations plus the Lorentz group together to form the Poincaré group.

But even more interestingly, if we maintain this choice of $M$ and we choose $phi^i rightarrow phi^i + a^i$, what we’re looking at for the internal space is something that particularly resembles the Minkowski space.

From this, we can return to the action written above. When we vary the metric, the equations of motion can be found as:

[ frac{delta S}{delta g_{mn}} = 0 rightarrow mathcal{R}_{m} – frac{1}{2} g_{mn} mathcal{R} + Lambda g_{mn} = T_{mn} ]

Where, $mathcal{R}_{m} – frac{1}{2} g_{mn} mathcal{R}$ is the Einstein tensor. On the right-hand side, $T_{mn}$ is the energy-momentum tensor of the matter theory. The energy-momentum tensor can be written as,

[ T_{mn} = M_{ij} (partial_{m} phi^{i} partial_{n} phi^{j} – frac{1}{2} g_{mn} g^{rs} partial_{r} phi^{i} partial_{s} phi^{j}) ]

M = 1

One of the interesting and fun things to think about with respect to the present theory is for different cases of $M$. In the case for $M=1$ – if there is only 1-dimension, as illustrated in a past post on the point particle, the space only has time. So, $X^0 rightarrow tau$. We can also establish that the metric $M_{ij} rightarrow eta_{mu nu}$, as we’ll be working in a sort of Minkowski configuration. So we will also replace the previous notion with notation more appropriate for the context, as our scalar fields no longer carry $i$ indices: $phi_(tau)^{i} rightarrow X_(tau)^{mu}$.

As for the metric $g_{mn}$, in the case of $M=1$ it only has one degree of freedom, and the convention I’ve employed (used for convenience) is simply to define $g = – e^{2}$. As there is only a time component, the idea then is to define $e$ as positive with a minus sign in front. This $e$ term is of course also a field – in fact, one can think of it as an auxiliary field – and it should be clear that $X$ is also a field. This is quite explicit in the new action,

[ S[e, X] = int dtau e(-frac{1}{2} (frac{1}{-e^2} dot{X}^{mu} dot{X}^{nu} eta_{mu nu} – Lambda) ]

Where $partial_{tau} X^{mu} equiv dot{X}^{mu}$.

We can tidy things up a bit. The action begins to look a bit more familiar,

[ S[e, X] = int frac{1}{2} int dtau (frac{1}{-e^2}
dot{X}^{mu} dot{X}^{nu} eta_{mu nu} – 2eLambda) ]

Now, the goal here in the $M=1$ case is to compute the equations of motion for the field $e$, with the hope that something interesting might ‘pop out’. We could, indeed, use the energy-momentum tensor instead; however, the above action is quite simple. So, we can compute the EoM directly:

[ frac{delta S}{delta e} = 0 rightarrow (-e^{-2} dot{X}^{mu} dot{X}^{nu} eta_{mu nu} – 2Lambda) = 0 ]

The key realisation is that $e$ can be integrated out since it does not have a kinetic term. This also means we can try to find the EoM, solving for $e$, and then substitute this value back into the action. Let’s do that.

Setting $dot{X}^{mu} dot{X}^{nu} eta_{mu nu}$ to be $dot{X}^2$ for simplification purposes,

[ -e^{-2} = frac{2 Lambda}{dot{X}^2 } ]

[ e^{2} = frac{- dot{X}^2}{2 Lambda} ]

[ implies e = frac{sqrt{- dot{X}^2}}{sqrt{2 Lambda}} ]

Thus, the effective action with just $X$ remaining (as $e$ has been integrated out):

[ S_{eff}[X] = frac{1}{2} int dtau -sqrt{2 Lambda} sqrt{- dot{X}^2} – sqrt{2 Lambda} sqrt{- dot{X}^2} ]

[ = sqrt{2 lambda} int dtau sqrt{dot{X}^2} ]

This should begin to look quite familiar. Preserving this result and putting it to one side for the moment, if we consider the non-relativistic limit, we can arrive at an expression for $sqrt{2 Lambda}$ that simplifies the above a bit more. Recall, as a first step, that the metric $eta_{mu nu}$ is mostly plus. So, we can write,

[ dot{X}^2 = -(frac{dtau}{dtau})^2 + (frac{dvec{x}}{d tau})^2 ]

We can now write the action,

[ S = – sqrt{2 Lambda} int dtau sqrt{1- (frac{dvec{x}}{d tau})^2} ]

Approximating the term under the square root, we get: $~ 1 – frac{1}{2}
(frac{dvec{x}}{d tau})^2$. So,

[ S = int dtau (-sqrt{2 Lambda} + frac{sqrt{2 Lambda}}{2}
(frac{dvec{x}}{d tau})^2 + …) ]

Ignoring the linear term, the more interesting question concerns the term $
frac{sqrt{2 Lambda}}{2} (frac{dvec{x}}{d tau})^2$. What is this? What does this term resemble? The answer is that it resembles the action of a particle – that is, the kinetic term.

If, in other words, we interpret $sqrt{2 Lambda} = m$ and plug this back into the action, look what happens:

[ S_{eff}[x] = -m int dtau sqrt{dot{X}^2} ]

This is the action of a particle propagating in flat spacetime.

Replacing the metric $eta_{mu nu}$ with a general metric $G_{mu nu}(X)$ that depends on the field $X$,

[ S[X] = -m int dtau sqrt{- dot{X}^{mu} dot{X}^{nu} G_{mu nu}} ]

This, as we know from a previous post, is the action for a particle moving in a curved spacetime. It is a nice result, given it connects with past entries and considerations, allowing us also to take a slightly different approach. There is also more we can do with this, before we completely exhaust the $M=1$ case. But we want to derive the Polyakov action for a string! All is not lost, with all of this hard work being important to complete just such a task.

M=2

The next idea is to attempt to translate the general field theory described above to the case of $M=2$. Consider, also, that we now have the coordinates $(X^0, X^1) rightarrow (tau, sigma)$. This should already be familiar, given past discussions. Additionally, the scalar fields are still $phi^i rightarrow X^{mu}(tau, sigma)$. Likewise, we should note $M_{ij} rightarrow eta_{mu nu}$.

Now, even though we can more or less translate everything we’ve already accomplished to the $M=2$ case, we do have one problem. We need to update our equations of motion, and this also entails that we study what the Riemann tensor is in the updated context. This requires that we recall our knowledge about the Riemann tensor in m-dimensions, including its symmetry relations and components, where the number of degrees of freedom is given by:

[ R_{mu rightarrow sigma}^{lambda} = frac{1}{12} M^2 (M^2 – 1) ]

So, when $M=2 rightarrow frac{1}{12} (2)^2 (2^2 – 1) = 1$. This means that we only have one parameter or degree of freedom.

Since we’re working in the case of $M=2$, we can write out what is below, where we have included the correct symmetry properties for the metric such that $g_{mn} rightarrow h_{mn}$:

[ R_{mnpq} = alpha (g_{mp}g{qn} – g_{mq}g_{pn}) ]

Where $alpha$ is the degree of freedom or, one might say, the constant of proportionality. We need to find this constant of proportionality, if we’re to move forward. The key is this: if $alpha$ is proportional to the Ricci scalar – the only other scalar in our theory – we can try to produce the Ricci scalar on the left-hand side and then solve for $alpha$. The first step is to take the Ricci tensor, contract indices, and then simplify:

[ g^{mp} mathcal{R}_{mnpq} = mathcal{R}_{nq} ]

[ implies mathcal{R}_{nq} = alpha (2 g_{qn} – g_{qn}) = alpha g_{qn} ]

[ implies mathcal{R} = g^{nq} mathcal{R}_{nq} = 2alpha ]

[ therefore alpha = frac{mathcal{R}}{2} ]

We can now substitute this result back into $mathcal{R}_{nq} = alpha g_{qn}$,

[ mathcal{R}_{nq} = frac{mathcal{R}}{2} g_{qn} ]

With this, we have have to use a bit of logic. Let’s return all the way back to our earlier formula,

[ frac{delta S}{delta g_{mn}} = 0 rightarrow mathcal{R}_{m} – frac{1}{2} g_{mn} mathcal{R} + Lambda g_{mn} = T_{mn} ]

What we find, now, is that the Einstein tensor vanishes. Moreover, we just discovered that,

[ mathcal{R}_{nq} = frac{mathcal{R}}{2} g_{qn} = 0 ]

The Einstein tensor is equal to zero. Therefore, the equations of motion from earlier suggest,

[ Lambda g_{mn} = T_{mn} ]

The pressing question, then, is what do we make of the energy-momentum tensor. Well, we already know that it is. When $M=2$,

[ T_{mn} = eta_{mu nu} (partial_{m} X^{mu} partial_{n} X^{nu} – frac{1}{2} h_{mn} h^{pq} partial_{p} X^{mu} partial_{q} X^{mu}) ]

At this point, we can also note that, as we take the trace of the EM tensor,

[ T_{m}^{m} = h^{mn}T_{mn} = 0 ]

But if $T_{mn} = 0$ then,

[ Lambda g_{mn} = T_{mn} rightarrow Lambda = 0 ]

Thus, the cosmological constant vanishes.

Here is the important conclusion. If we travel back to the action once more, the lingering question is what happens to gravity such that we see the Einstein tensor equal zero. And the answer is rather pleasant, as, when we translate the old formula using updated notation for the present context,

[ S = int dtau dsigma sqrt{-h} mathcal{R} ]

This integrand is just a constant. The number it churns out, in 2-dimensions, describes the topology in the space. If we denote the space as $sum$, $dtau dsigma$ is therefore being integrated over in this space. The $sqrt{-h} mathcal{R}$ is just some medium as $tau rightarrow tau_{i}$. It turns out, the total object $int_{sum} dtau dsigma sqrt{-h} mathcal{R}$ produces the Euler characteristic of this particular 2-dim space. So, we finally arrive at an action for our metric and fields (where we set $tau = sigma^{0}, sigma=sigma^{1})$,

[ S_{p} [h_{mn}, X^{mu}] = int d^2 sigma sqrt{-h} (- frac{1}{2} h^{mn} partial_{m} X^{mu} partial_{n} X^{nu} eta_{mu nu}) ]

This is the Polyakov action (without the inclusion of the slope parameter).

Standard