# Chaos in Supersymmetry-Constrained Melons

Victor Ivan Giraldo-Rivera (ICTS) began the ninth day’s morning session with his second lecture on SUSY and localization. He wrote the generalized Killing spinor (GKS) equations as $\mathcal{D}_{\mu}\zeta^{\alpha}=\delta \psi_{\mu\alpha}=0$, $\mathcal{D}_{\mu}\tilde{\zeta}_{\dot{\alpha}}=\delta \tilde{\psi}_{\mu\dot{\alpha}}=0$ and introduced an integrability condition $[\mathcal{D}_{\mu},\mathcal{D}_{\mu}]\zeta^{\alpha}=0$. Note that in Minkowski signature $\tilde{\psi}=(\psi)^*$ and in Euclidean signature $\psi$ and $\tilde{\psi}$ are independent. Combining $\zeta$ and $\tilde{\zeta}$ as a $4$-component spinor $\epsilon$, the integrability condition gave $[\nabla_{\mu},\nabla_{\nu}]\epsilon_L=(\dots)\epsilon_L+(\dots)\gamma^{\rho}\gamma_5\epsilon_L+(\dots)\gamma_{\alpha}\epsilon_L+(\dots)\gamma_{\rho\alpha}\epsilon_L$. Since the $\gamma$ matrices and their products form the basis for $4\times 4$ complex matrices, the coefficients of the above equation give $Mb=0$, $\tilde{M}b=0$, $\nabla_{\mu}b_{\nu}=0$, $\partial_{\mu}M=0=\partial_{\mu}\tilde{M}$, $\mathcal{R}_{\mu\nu}=-\frac{2}{9}(b_{\mu}b_{\nu}-g_{\mu\nu}b^2)+\frac{1}{3}M\tilde{M}g_{\mu\nu}$. The last expression shows how the auxiliary fields of the old minimal supergravity multiplet characterize the background manifold. Then Victor chose one solution for the above equations $b=0$, $M=constant$, $\tilde{M}=constant$ and gave two examples; (i) in Minkowski signature, for $M=\tilde{M}=-\frac{3}{r}$, one gets $AdS_4$ and (ii) in Euclidean signature, for $M=\tilde{M}=-\frac{3i}{r}$, one gets $S^4$ as the background.

These backgrounds, for $b=0$, $M$, $\tilde{M}$ constants are conformally flat manifolds and require stress tensor to be traceless. From the coupling of the FZ-multiplet to old minimal supergravity multiplet, one gets $\dots +T^{\mu\nu}g_{\mu\nu}+M\bar{X}+\tilde{M}X$, where traceless of stress-tensor requires $X=0=\bar{X}$ as was mentioned in the first lecture.

Then Victor moved on to solve the GKS equations, not fully but tried to get a some solutions and defined the supercharge by these solutions, $Q=(\zeta,\tilde{\zeta})$. When $(\zeta,\tilde{\zeta})\neq 0$, he wrote a few tensors on $\mathcal{M}$, $J_{\mu\nu}$, $\tilde{J}_{\mu\nu}$, $K_{\mu}$, $\bar{K}_{\mu}$, etc. in terms of the spinors $\zeta$ and $\tilde{\zeta}$. Here $K_{\mu}$ is a Killing vector and generates generalized translations on $\mathcal{M}$ i.e., $\{Q,\tilde{Q}\}^{\mathcal{M}}\sim \mathcal{L}_{K}$.

Then he considered the solution $Q=(\zeta,0)$, where $J^{\mu\nu}$ is an almost complex structure. He then mentioned that when $J^{\mu\nu}$ satisfies an integrability condition, $\mathcal{M}$ is a complex manifold locally. He then argued that any even-dimensional real manifold is a complex manifold.

Coming back to the solution $(\zeta,\tilde{\zeta})\neq 0$, he wrote the metric on $\mathcal{M}$ in terms of above tensors, $K_{\mu}$, etc.

$g_{\mu\nu}=\frac{1}{2|\zeta|^2|\tilde{\zeta}|^2}(K_{\mu}\bar{K}_{\nu}+K_{\nu}\bar{K}_{\mu}+X_{\mu}\bar{X}_{\nu}+X_{\nu}\bar{X}_{\mu})$

Then he discussed two possibilities; (i) $[K_{\mu},K_{\nu}]=0$, where the metric written in complex coordinates describes a Torus fibration over a $2$-dimensional complex surface and (ii) $[K_{\mu},K_{\nu}]\neq 0$, where the metric describes an $S^3$ fibration over a line. These examples demonstrate how the metric of the background manifold is constrained by the supersymmetry placed on it.

After lunch, in his last lecture, Pranjal Nayak (TIFR) completed the analysis of dealing with the soft modes contributing to the four point function. These were modes (eigen functions to $\tilde{K}$) with eigen value $1$, thus making $\frac{1}{1-\tilde{K}}$ blow up at the conformal fixed point (large J limit).

He first shows that at the conformal fixed point these eigenfunctions are indeed generated by infinitesimal diffeomorphisms of $G_c(\tau_1,\tau_2)$: $K_c\star\delta_\epsilon G_c=\delta_\epsilon G_c$ where $\tau=\tau+\epsilon(\tau)$. Also since $\frac{1}{1-K_c}$ is singular for these modes we would also need $1/(\beta J)$ corrections to (the eigen-values of)$K$ away from $K_c$. This is first done by writing $K$ (and $\tilde{K}$) using the exact in $J$ solution for the 2-pt function $G(\tau_1,\tau_2)$ in the large $q$ limit. Now Maldacena $\&$ Stanford show that the form of $\frac{1}{\beta J}$ correction to $\frac{\delta G}{G}$ at finite $q$ is basically the same as that in the large $q$ limit ($1/q$ expansion). They fix the $q$ dependent co-efficient by doing hardcore numerics. Having thus found the change in $\delta K$ away from $K_c$ in $\frac{1}{\beta J}$ for any $q$, they finally show that the 4-pt function saturates the chaos bound with $(\beta J)e^{\lambda_L t}$ where $\lambda_L=\frac{2\pi}{\beta}\left( 1+\frac{k'(2)q \alpha_G}{k'_R(-1)\beta {\mathcal{J}}}+\dots \right)$.

We broke off for a much needed tea break and for also taking a group photo for the conference.

So the 4pt function gets contribution from the heavy modes of the form $t e^{\lambda_L t}$ and a leading contribution from the soft modes as $(\beta J )e^{\lambda_L t}$ in $\frac{1}{\beta J}$ expansion. The authors also find the effective action which governs the soft modes to be that of a Schawrzian: $S=\frac{\alpha_S N}{{\mathcal{J}}}\int_0^\beta d\tau \,\,\frac{1}{2}\left((\epsilon''(\tau))^2-\left(\frac{2\pi}{\beta}\right)^2 (\epsilon'(\tau))^2\right)$, where $\epsilon(\tau)$ parametrizes infinitesimal diffeos from the conformal fixed point. So basically its the zero modes of the conformal fixed point, governed by the Schwarzian effective action closed to the conformal fixed point, which are responsible for saturating the chaos bound in the out of time ordered correlator.

Pranjal then discussed the possibility of having a bulk dual to such a model. The bulk model must necessarily have similar soft modes with a similar Schwarzain effective action. If looking at $AdS_2$, of which there are till now 2 candidates: Jackiw-Teitelboim theory proposed by Maldacena and Stanford; Polyakov action in $AdS_2$ proposed by Pranjal, Gautam Mandal $\&$ Spenta Wadia. Pranjal mentions that the key ingredient needed to yield a Schawrzian effective action at the boundary is the necessity to have a boundary term of the from $\int_{\partial AdS_2}\sqrt{-\gamma}K\phi$ where $\phi$ is an additional parameter which needs to be held fixed during the variation of the bulk theory.

Junggi Yoon (ICTS) gave a broad review of the vast and varied activity that has happened, and is currently happening, to understand the SYK and tensor-like models in the evening. The high-level overview focused mostly on the melonic aspects of the models.

The talk began by mentioning the quick proliferation of indices as one moves from vector models to tensor models and then from tensor models with a lower rank to tensor models with higher ranks. This makes it very difficult to calculate correlation functions for such models, and in fact there is no known low-energy effective action for any tensor models. One necessarily has to resort to some simplified version of the theory one is interested in. The crucial realization here is that as one takes the large $N$ limit for, e.g., $O(N)$ or $SU(N)$ models, only the “melonic” diagrams contribute to the maximal chaotic behavior.

With this in mind, one can study many kinds of fermion lattices, depending on whether the links are colored, the sites labelled, gauge groups distinguished, and so on and so forth. One of the better known models is the Gurau-Witten models, which get the maximal chaotic behavior from the $q$ simplex for $q$ species of fermions in the Lagrangian. Another popular model is the Klebanov-Tarnopolsky $O(N)$ model, where one distinguishes the various gauge group links between lattice sites.

The speaker then mentioned the close relation between the two models, and proceeded to counting the orders of the various melonic contributions to two, three and four point functions. Here, one has to make some fine-tuning choices to ensure that the “Cooper pairing” diagrams are the dominant contributors instead of the “pillow” diagrams (which do not give the desired maximal chaotic behavior). Doing the diagrammatics, one also sees that one needs to distinguish broken and unbroken diagrams which are just disconnected and connected diagrams. The speaker also showed that summing over these diagrams does indeed give the much coveted chaotic behavior.

Interspersed among the many colorful melons, there was also a high-level overview of the literature, divided into the old papers and papers from the last 18 odd months. We learnt about what models are being explored and the motivations for most of them. The talk finally concluded after about 145 minutes(!) with a summary of the speaker’s own very interesting work on tensor models.