Chennai String Meet day 2

So this time we did summaries instead of a live blog. Here are the summaries we have completed till now. We will put up the rest as we complete them.

The first speaker was Prasanta Tripathy. His talk was titled “The Geroch group in Einstein spaces”.

Prasanta started by explaining the Geroch group. In the 70s, Geroch had come up with a method of generating new solutions to Einstein’s equations from known solutions. The starting point is any spacetime M with a vanishing Ricci tensor and a Killing vector $\xi$ .

Two scalars which are going to be important in the story are $\lambda$ and $\omega$ :

$\displaystyle \lambda = \xi^a \xi_a$ and $\omega,_{a} = \epsilon_{abcd} \xi^b \xi^{c;d}$

The next step is to construct a coset space S as the quotient of M by the one parameter group generated by ξ. Then one can show that Einstein’s equations in M become a set of equations (we will call them the projected Einstein equations) involving only the projected metric on S which we call h, $\lambda$ and $\omega$ on S. Any triplet (h, $\lambda$ , $\omega$ ) which satisfies these equations corresponds to a solution of the Einstein’s equations. This sets up a nice one to one correspondence between solutions in M and solutions (h, $\lambda$ , $\omega$ ) in S.

The final and crucial observation by Geroch was that there exists a set of transformations of (h, $\lambda$ , $\omega$ ) which map solutions to the projected Einstein’s equations to other solutions. But once one obtains a new solution on S by such a transformation one can lift it up to M and obtain a new solution to Einstein’s equation in M. This way exact curved solutions can be obtained from flat spacetime!

Geroch had assumed Ricci tensor to vanish. But Prasanta and his collaborators have extended this construction to the more general Einstein spaces, defined as:

$\displaystyle R_{ab} = kg_{ab}$

This is a much more complicated case and one needs to introduce an additional scalar field $\kappa$ along with $\lambda$ and $\omega$ . Prasanta showed that the dynamics of the triplet ( $\kappa$ , $\lambda$ , $\omega$ ) is governed by a three dimensional sigma model subgroup and from there he showed that while the full Geroch group no longer leaves the projected equations invariant, a subgroup of it still maps solutions to new solutions for Einstein spaces. Then he used Hamilton Jacobi equations to prove that this sigma model is integrable.

Bala Sathiapalan’s talk was about the connection between Wilsonian RG and holography.

Intuitively, there is some connection between scale in CFT and the radial direction in AdS. But can it be derived or at least understood? This is the question this talk addressed.

Bala’s strategy is to start from Polchinski’s equation for a free CFT in the boundary. Polchinski equation describes the RG flow of the action. For a 0-dim field x, it looks like:

$\displaystyle \partial S_I/ \partial t = \frac{1}{2} \dot{G}(t) [\partial^2S_I/\partial x^2 -(\partial S_I/\partial x)^2]$

where $S_I$ is the interaction part of the action and G is a t-dependent function which can be thought of as the cutoff Green function. t denotes the RG parameter.

When written in terms of $\psi = e^{S_I}$ , the Polchinski equation looks like a diffusion equation:

$\partial \psi / \partial t$ = $- \frac{1}{2} G \partial^2 \psi /\partial x^2$

The functional form of this equation is: $\psi(x_f,t_i) = \int dx_i e^{\frac{(x_f-x_i)^2}{G(t_f)-G(t_i)}} \psi(x_i,t_i)$

Now one can promote x to a field variable x(p). The above equation can be written recast in path integral language

$\psi(x_f, T) = \int dx _{x(0)=x_i,x(T)=x_f}\, \int \,\mathcal{D}X(t) \, e^{\frac{1}{2} \int_0^T \, dt \frac{\dot{x}^2}{G}} \psi(x_i,0)$

Finally one makes a field redefinition $x=yf$ and chooses f such that

$(zd/{dz})^2e^{-ln f} = (z^2p^2+m^2)e^{-ln f}$

where $z = e^t$ . Then the action in the path integral expression becomes:

$\displaystyle \int \frac{dz}{z} [z^2 (dy/dz)^2 + y^2(z^2p^2 + m^2)]$

which is the action for a free scalar field in AdS_1! This can be generalized to higher dimensional AdS. So bulk action can be directly derived from RG flow, without invoking string theory.

This was for a Gaussian fixed point. For non-trivial fixed point, Bala showed that the bulk action is more complicated and involves interactions. He also showed how anamolous dimension can be incorporated in this approach.

Prafulla Oak told us about RG flow for generalized Sine-Gordon model. This is based on his recent paper with Bala Sathiapalan.

The motivation to study this is twofold. First, to understand the connection between the ERG equations described in Bala’s talk and what is usually known as holographic RG. Secondly, Sine Gordon models are closely connected to string propagation in tachyonic condensate backgrounds. In their paper Prafulla and Bala considered a generalized Sine Gordon model

$S=1/{4\pi} \int \, d^2 x \,(\partial_\mu \vec{X} ).(\partial^\mu \vec{X}) +m^2\vec{X}.\vec{X} + F/(a(0)^2)cos (\vec{b_1}.\vec{X} ) +G/(a(0)^2)cos (\vec{b_2}.\vec{X}) + H/(a(0)^2)cos (\vec{b_3}.\vec{X})$

Here a(0) is a UV cutoff.

Prafulla took us through their calculation of the leading and subleading terms in the beta function for F for this generalized Sine Gordon model using techniques of Exact Renormalization Group.

Then he computed the same beta function from the bulk dual, which is scalar fields in the bulk with cubic coupling. The contribution from the interaction term shows up in the subleading terms in the beta function. Prafulla showed us the AdS computation using both position and momentum space techniques and matched the results with those obtained from ERG.

Partha Mukhopadhyay spoke about small string quantization in curved space. In this talk Partha summarized the research program he has been carrying out for the past several years.

Partha started by outlining how obtaining a field theory of particle excitations from string theory in the limit of string length going to zero is an open problem in curved spacetimes. In both and curved flat spacetimes, usual field theory can be described both through a spacetime and a worldline formulation. Both give the same results. But how would one get back field theory from string theory in the $\alpha' \to 0$ limit? To be able to do this first one needs to construct a string field theory.

As Partha explained, as long as the worldline string theory is an exactly solvable CFT, one can construct the corresponding String Field Theory as the CFT correlators become couplings in the SFT. This can be done in flat and exactly solvable backgrounds and the $\alpha' \to 0$ limit can also be obtained. But in a general background solving the CFT and constructing the SFT are open issues and therefore the $\alpha' \to 0$ limit cannot be taken. Nor does the background field method, which is usually used to study string theory in a curved background, help. One does not get a particle by taking the $\alpha' \to 0$ limit in this case.

To tackle this problem, Partha has been developing a new framework. He started explaining it by noting that a string (or any extended object/ bound state) would look like a point from far away, which he calls the centre of mass (CM). All the possible configurations of the CM (all the points it can occupy) in a general spacetime M will be a copy of M itself. It will also be a subspace of the full configuration space of the string. The full configuration space of a closed string is the space of maps between $S^1$ and the spacetime M it is embedded in( $LM = C^{\infty}(S,M)$ ). This space is known as loop space.

Then the evolution of a string in M is equivalent to the evolution of a point in loop space. In other words worldsheet string theory is equivalent to quantum mechanics in loop space. A formal quantization can be carried out in loop space. Now one can try once again to take the $\alpha' \to 0$ limit. The strategy is to arrange things such that the potential term in the loop space quantum mechanical Hamiltonian is supported only on zero sized loops, i.e on the configuration space of particles M. Deviations away from the particle limit can in principle be obtained by taking a tubular expansion of this theory in loop space LM in the neighbourhood of M. To be able to do this Partha has also developed the method of making covariant tubular expansions. The goal is to find the effective theory in the neighbourhood of M order by order in $\alpha'$ expansion.

Partha also described other related ideas about getting the semiclassical limit such as the cut off loop space (where one divides the string into points and takes their combined configuration space) and conformal string bits.

Chennai String Meet day 2

Published by Nirmalya Kajuri

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