Using the tensor Radon transform and related numerical methods, we study how bulk geometries can be explicitly reconstructed from boundary entanglement entropies in the specific case of AdS$_3$/CFT$_2$. We find that, given the boundary entanglement entropies of a 2d CFT, this framework provides a quantitative measure that detects whether the bulk dual is geometric in the perturbative (near AdS) limit. In the case where a well-defined bulk geometry exists, we explicitly reconstruct the unique bulk metric tensor once a gauge choice is made. We then examine the emergent bulk geometries for static and dynamical scenarios in holography and in many-body systems. Apart from the physics results, our work demonstrates that numerical methods are feasible and effective in the study of bulk reconstruction in AdS/CFT.

There has been much excitement in the recent literature about the idea that tensor network models, which construct states whose entanglement properties mimic those of CFT vacua, correspond geometrically to a bulk space one dimension higher. Such discrete models of holography tend to miss at least two interesting features of the continuum AdS/CFT correspondence: there, an essential ingredient is the group of isometries, which acts via conformal transformations of the boundary. Furthermore, the local data in tensor networks (a choice of a finite-rank tensor, which is taken to be "perfect" in one interesting class of models) is not obviously related to any dynamics that resemble field theory. I'll give a rather biased outsider's perspective on approaches to both of these issues, taking inspiration in each case from algebraic geometry.

Artificial Intelligence (AI) can be used for finding structures in data from scratch, i.e. without knowing about them beforehand. One might wonder whether AI can be used to decide whether a dynamical system is integrable or not? In this talk, I report on how we can set up machines to successfully search for the Lax pair and Lax connection in some classical examples. We shall also see how symbolic regression can be used to render the output interpretable for a mathematical physicist.

I briefly give an overview on the key Machine Learning frameworks involved in this analysis (neural networks, auto-differentiation, representation learning). This talk is mainly based on 2103.07475, and further related work can be found in 2104.14444, 2003.13679.

We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with $N $ minima on $S^{1}$. We describe the Stokes graphs of a general potential problem as a network of Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of $N$-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with 't Hooft anomaly for even $N$ and global inconsistency for odd $N$. By using Delabaere Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence.

This talk is based on:

On exact-WKB analysis, resurgent structure, and quantization conditions, N. Sueishi, S. K., T. Misumi, and M. Ünsal, arXiv:2008.00379.

Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$, N. Sueishi, S.K., T. Misumi, and M. Ünsal, arXiv.2103.06586

“G-algebroids” are natural extension of Lie and Courant algebroids that give a unified picture of the symmetries that underlie generalised and exceptional geometry as well as new “non-exact” versions. We analyse their structure in the exceptional case, and translate the problem of finding maximally supersymmetric consistent truncations to a relatively simple algebraic condition. We then show how Poisson-Lie U-duality is encoded in this framework and prove, in particular, that it is compatible with the supergravity equations of motion.