More on Supersymmetric and 2d Analogs of the SYK Model

TL;DR

This work extends SYK-type physics to supersymmetric and two-dimensional contexts, constructing a robust basis of (super)conformal eigenfunctions and applying shadow representations to derive exact order-\

Abstract

In this paper, we explore supersymmetric and 2d analogs of the SYK model. We begin by working out a basis of (super)conformal eigenfunctions appropriate for expanding a four-point function. We use this to clarify some details of the 1d supersymmetric SYK model. We then introduce new bosonic and supersymmetric analogs of SYK in two dimensions. These theories consist of $N$ fields interacting with random $q$-field interactions. Although models built entirely from bosons appear to be problematic, we find a supersymmetric model that flows to a large $N$ CFT with interaction strength of order one. We derive an integral formula for the four-point function at order $1/N$, and use it to compute the central charge, chaos exponent and some anomalous dimensions. We describe a problem that arises if one tries to find a 2d SYK-like CFT with a continuous global symmetry.

Figures (16)

• Figure 1: A kernel describing propagation of a two-particle system from $t_3,\theta_3$ and $t_4,\theta_4$ to $t_1,\theta_1$ and $t_2,\theta_2$. For the case of a $\widehat{q}$-fold interaction in superspace, points 3 and 4 are connected by $\widehat{q}-2$ propagators, as here for the case $\widehat{q}=7$.
• Figure 2: The lowest order contribution to a four-point function $\langle 1234\rangle$ comes from these "zero-rung" diagrams (a third diagram is missing because as usual we consider a four-point function with the disconnected contribution $\langle 12\rangle \langle 34\rangle$ subtracted out). The lines represent exact two-point functions.
• Figure 3: This diagram shows the arrangement of spurious poles that we meet in deforming the contour of integration in the direction of increasing ${\mathrm{Re}}\,h,{\widetilde{h}}$. The red poles (with $h$ or ${\widetilde{h}}$ negative) are poles of the conformal block $F_h(\chi)F_{\widetilde{h}}(\overline\chi)$, and the blue and green ones (with $h,{\widetilde{h}}$ positive) are poles of ${\mathcal{A}}(h,{\widetilde{h}})$. With the exception of the poles on the line $\ell=0$, which have to be treated more carefully, red and blue poles are related in pairs by $h\leftrightarrow 1-h$ or by ${\widetilde{h}}\leftrightarrow 1-{\widetilde{h}}$, but not both. A cancellation occurs between these pairs. Green poles on the lines $h=1/2$ and ${\widetilde{h}}=1/2$ do not contribute because the measure factor vanishes.
• Figure 4: Gluing in one more propagator connecting vertices 3 and 4 in fig. \ref{['kernel']}, we arrive at this diagram, with $q-1$ propagators connecting points 3 and 4. It is essentially the same diagram that one encounters in the Schwinger-Dyson equation that determines the two-point function of the model for large $N$.
• Figure 5: Propagation of $\phi\phi$ or $BB$ in a ladder diagram.