## Title and Abstract

### Bartlomiej Czech

**How to glue a spacetime from entanglement wedges**

If a holographic bulk spacetime is built out of quantum entanglement in the boundary theory, how do we understand the bulk connection? To inspect the entanglement structure of a boundary state, we dissect it into components and look at their quantum correlations. Each boundary component reconstructs a region of the bulk called entanglement wedge. The entanglement wedge (and its corresponding component subregion of the boundary) has an internal symmetry called modular flow, which has two properties that will be useful for our purposes. First, modular flow is a gauge symmetry because it relates to one another different ways of presenting the same physical system–the entanglement wedge. Second, modular flow is a generalization of choosing the phase of a pure quantum state in a Hilbert space. When we glue together two overlapping entanglement wedges to build a larger spacetime, we must specify how to map the observables in the first wedge (presented in some modular frame–in some gauge) to observables in the second wedge (also presented in some gauge). Thus, gluing together two component subregions of the boundary–as well as two entanglement wedges–requires a connection that relates their respective modular frames. This connection is analogous to specifying the phase of a quantum state that evolves under a time-dependent Hamiltonian, that is the Berry phase. I argue that the modular Berry connection is the boundary origin of the usual, geometric connection in the bulk. I will sketch some subtleties in the formal construction of the modular Berry connection, give examples and list key questions for the future.

### Yuji Sugimoto

**Open topological string on (periodic) chain geometry**

The mirror curves enable us to study B-model topological strings on non-compact toric Calabi–Yau threefolds. One of the methods to obtain the mirror curves is to calculate the partition function of the topological string with a single brane. In this talk, I will discuss two types of geometries; one is the chain of N P1’s which we call “N-chain geometry,” the other is the chain of N P1’s with a compactification which we call “periodic N-chain geometry.” I will explain how to calculate the partition functions of the open topological strings on these geometries, and obtain the mirror curves and their quantization, which is characterized by (elliptic) hypergeometric difference operator. After that, I will discuss a relation between the periodic chain and ∞-chain geometries, which implies a possible connection between 5d and 6d gauge theories in the large N limit.

### Hong-fei Shu

**TBA equations and resurgent Quantum Mechanics**

In this talk, we derive a system of Thermodynamic Bethe ansatz (TBA) equations governing the exact WKB periods in one-dimensional Quantum Mechanics with arbitrary polynomial potentials. These equations provide a generalization of the ODE/IM correspondence, and can be regarded as the solution of a Riemann-Hilbert problem in resurgent Quantum Mechanics formulated by Voros. We also show that our TBA equations, combined with exact quantization conditions, provide a powerful method to solve spectral problems in Quantum Mechanics. We illustrate our general analysis with a detailed study of cubic oscillators and quartic oscillators.

### Reona Arai

**Finite N Correction to the Superconformal Index of S-fold Theories**

Recently, Garcia-Etxebarria and Regalado constructed S-fold theories, which preserve twelve supercharges, in four-dimension. The remarkable features of the S-fold theories are that there is no marginal deformation and the non-perturbative effect is essential. That is why the S-fold theories have no Lagrangian description and we cannot use the localization technique to calculate the superconformal index in the finite N, where N is the number of background D3-branes. In this talk, we present a calculation method of the finite N correction to the index on the AdS side, by considering the fluctuation of D3-branes wrapping the non-trivial three cycle in AdS_5 × S^5/Z_k. The key idea is to consider the superconformal field theory on the wrapped D3-branes and we can compute the contribution of the wrapped D3-branes to the index. On the CFT side, these corrections can be interpreted as Pfaffian-like operators. We also see our formula gives correct results up to a certain order of the fugacity of the index for rank one and two cases by using the supersymmetry enhancement.