The Gravitational Cavalry Surrounds the Conformal Mountain

Today’s morning session began with the first of the four lectures on Supersymmetric Gauge Theories and Localization by Victor Ivan Giraldo-Rivera (ICTS). Before starting his main lectures, Victor continued the review of \mathcal{N}=1 supersymmetry in 4-dimensional flat space, which was initiated by Madhusudhan Raman (IMSc) in the evening talk of yesterday. He briefly reviewed chiral superfields and wrote down the Lagrangian for non-linear sigma model. Then he demonstrated a non-renormalization theorem in theories of chiral superfields, which says that the superpotential does not receive perturbative quantum corrections.

Then he moved on to discuss Supersymmetry on Curved Backgrounds, in particular, focusing on \mathcal{N}=1 supersymmetry on 4-dimensional curved manifolds. To get a supersymmetric theory on a curved manifold \mathcal{M}, one couples the supersymmetric theory to off-shell supergravity and takes a rigid limit i.e., M_{Pl}\rightarrow\infty while keeping the metric to be some fixed background. In this limit, gravity becomes non-dynamical and one obtains a supersymmetric theory on a fixed curved manifold (i.e., classical background) \mathcal{M}.

Victor introduced the Ferrara-Zumino stress-tensor multiplet, which is given by a real superfield \mathcal{J}^{FZ}_{\mu}=(j_{\mu},x,S_{\mu\alpha},T_{\mu\nu}), such that \bar{D}^{\dot{\alpha}}\mathcal{J}^{FZ}_{\alpha\dot{\alpha}}=D_{\alpha}X, \bar{D}^{\dot{\alpha}}X=0, where X=(x,\sigma^{\mu}_{\alpha\dot{\alpha}}\bar{S}^{\dot{\alpha}}_{\mu},T^{\mu}_{\mu}+i\partial_{\mu}j^{\mu}), a chiral superfield, is the trace submultiplet of the FZ-multiplet. When X=0, the FZ-multiplet reduces to the superconformal multiplet. He then introduced the old minimal supergravity multiplet \mathcal{H}_{\mu}=(g_{\mu},\psi_{\mu\alpha},b_{\mu},M,\tilde{M}), which couples to the FZ multiplet. b_{\mu}, M, \tilde{M} are auxiliary fields and characterize the classical background manifold, as will be seen later. For theories with U(1)_{R}-symmetry, there is a R_{\mu} multiplet which couples to the new minimal supergravity multiplet.

He then wrote down the Lagrangian for chiral superfields coupled to supergravity (chap. 23 of Wess and Bagger), which is invariant under the supergravity transformations \delta e^{a}_{\mu}, \delta \psi^{\alpha}_{\mu}, \delta \tilde{\psi}_{\mu\dot{\alpha}}. Here one does not integrate out auxiliary fields but only imposes supersymmetry i.e., \delta \psi_{\mu\alpha}=0 and \delta \tilde{\psi}_{\mu\dot{\alpha}}=0. These are called generalized Killing spinor equations and are solved for the background supergravity fields and the spinors \zeta, the parameters of supersymmetry transformations; one way to see their origin is to take the M_{pl} \to \infty limit of the full supergravity action, and demand that we work around a saddle-point which is a normal manifold, in which case these are the equations that tell you that this saddle-point is invariant under supersymmetry transformations. He then wrote down the Lagrangian, which is obtained by taking the rigid limit as \mathcal{L}_{B}/e +\mathcal{L}^{0}_{F}/e +\delta\mathcal{L}_{F}/e. Here \mathcal{L}_{B}/e is the bosonic part of the Lagrangian, \mathcal{L}^{0}_{F}/e is the fermionic part with \partial_{\mu} replaced by \nabla_{\mu} (with respect to background manifold \mathcal{M}) and \delta\mathcal{L}_{F}/e comes from replacing \partial_{\mu} by \nabla_{\mu} in \mathcal{L}^{0}_{F}/e. Unlike flat space supersymmetry, this Lagrangian for supersymmetric theory of chiral superfields on fixed curved background manifold is invariant under deformed Kahler transformations viz., the Kahler transformation of K together with a transformation of the superpotential. This allows one to transform away the superpotential and so it does not have a special significance here unlike in flat space theories.

After that crash course on how all supersymmetric theories live on manifolds that are saddle-points of the supergravity action, and lunch (one must never forget about lunch), we returned to see Pranjal Nayak (TIFR) jump through flaming hoops while juggling knives — a feat better known as working out the SYK four-point function by diagonalising the conformal Casimir and finding the correct set of eigenfunctions.

First, he reminded us where he’d left us yesterday: the four-point function was a sum of ladder diagrams, and so the main thing to do was work out the eigenfunction decomposition of each rung. Since the rung function commuted with the dilatation, the strategy was to diagonalise the dilatation, i.e. solve the conformal Casimir equation,

C f(\chi) = h(h-1) f(\chi).

This was in the complexity class MP,  or Mathematica-solvable problems. But not all solutions of this were permissible: one had to impose various conditions to make sure that the solutions belonged to a reasonable Hilbert space. They were that \psi'_h (2) = 0, that the singularity at \chi = 1 was not singular enough to spoil the normalisability of the function, and that the conformal Casimir was Hermitian on this set. That gave an allowed spectrum

h = \frac{1}{2} + i s,\quad s \in R, and

h = 2 n,\quad n \in Z^+.

Using this and the exact form of the eigenfunctions and eigenvalues of the rung, he wrote down a beautiful expression for the four-point function,

F_4 (x) = \alpha_0 \int_C \frac{ds}{2\pi i} \frac{h - \frac{1}{2}}{\pi \tan \frac{\pi h}{2}} \frac{K(h)}{1-K(h)} \psi_h (\chi),

where the contour C goes from \frac{1}{2} - i \infty to \frac{1}{2} + i \infty and circles all the even integers beginning from 2 in a counter-clockwise manner. Apart from the poles at the even integers because of the tan function, the integrand has poles whenever K(h) = 1.

The most important of these poles is the pole at h = 2, which makes that point a double pole. For now, he just ignored that pole. Apart from that, all the K(h) = 1 poles were in the upper-half plane, as he showed simply from the fact that all the residues of the eigenvalue were positive, in this marvellously simple diagram


Then, he deformed the contour to go around these poles and get an infinite-sum representation of the four-point function. Taking an “OPE limit” (the state-operator map hasn’t been made precise in 1d CFT, but we must assume it exists, because why not), he interpreted these poles that contributed as the exchange of operators of the form \psi \partial^{2n+1} \psi.

Finally he was able to use this to calculate the late-time behaviour of the out-of-time-order four-point function that is used as a diagnostic of chaos to be t e^{\frac{2\pi}{\beta} t} (another problem which is in complexity class MP). However, this was a faster growth than was allowed by the theorem of Maldacena, Shenker and Stanford, where they said that the most chaotic growth is e^{\frac{2\pi}{\beta} t}. He explained that it was because of that double pole contribution at h = 2 that we’d dropped; the coefficient of this too-fast growth would turn out to be infinitely suppressed compared to the coefficient of the bound-saturating growth that would come from that contribution; thus, in classic saas-bahu serial fashion, he ended with a cliffhanger.

We returned for the second talk that worked around a gravitational saddle-point, the evening talk on bulk reconstruction by Nirmalya Kajuri (IIT-M). He reviewed recent developments on how to reconstruct the bulk information from the CFT side. This is a new perspective and an approach of the AdS/CFT correspondence. He nicely explained how this is done in three equivalent but different ways.

A first approach is very straightforward in some sense; giving a boundary condition to the bulk fields (which is a relation between the CFT operator and a boundary value of the bulk field) and solving the equation of motion for the bulk field. The bulk field is constructed by the smearing function and the boundary data. After he explained how this method works for the free scalar theory, he also developed the discussion to the interacting case.

The second approach uses the micro-causality and the CFT properties. In this picture, a bulk field is basically related to the higher dimension operators as well as the CFT operator discussed in a previous method. The precise relation is fixed by the requirement of the micro-causality.

The third approach is more “symmetry-based” argument in some sense. This was first proposed by Ooguri-Nakayama. In this approach, the bulk field is constructed by the isometries of the bulk spacetime and is related to the Ishibashi state of the CFT.

Finally, Nirmalya gave some comments on the future directions to be more clarified. A first problem is the back-reaction. In this talk the geometry is fixed. The simple and important generalization would be to take into account the back reaction to the geometry. A next problem is to study the corrections to these analysis. In addition to the usual perturbative corrections like cubic couplings, we also have the 1/N corrections. This should be also considered. Third one is to consider the more general CFT states and reconstruct the corresponding bulk. The final, very non-trivial problem is the bulk reconstruction on the BH geometries. In this case, naively speaking, we cannot reconstruct the bulk at some causal patch from the CFT side. This is a nontrivial and interesting question to be studied.


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