# 2002 seminars

## 01/03/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Jan Manschot, School of Mathematics, Trinity College, Dublin
Vafa-Witten theory and quivers

Supersymmetric D-branes supported on the complex two-dimensional base $S$ of the local Calabi-Yau threefold $K_S$ are described by semi-stable coherent sheaves on $S$. Under suitable conditions, the BPS indices counting these objects (known as generalized Donaldson-Thomas invariants) coincide with the Vafa-Witten invariants of $S$ (which encode the Betti numbers of the moduli space of semi-stable sheaves). For surfaces which admit a strong collection of exceptional sheaves, we develop a general method for computing these invariants by exploiting the isomorphism between the derived category of coherent sheaves and the derived category of representations of a suitable quiver with potential $(Q,W)$ constructed from the exceptional collection. We spell out the dictionary between the Chern class $\gamma$ and polarization $J$ on $S$ vs. the dimension vector $\vec N$ and stability parameters $\vec\zeta$ on the quiver side. For all examples that we consider, which include all del Pezzo and Hirzebruch surfaces, we find that the BPS indices $\Omega_\star(\gamma)$ at the attractor point (or self-stability condition) vanish, except for dimension vectors corresponding to simple representations and pure D0-branes. This opens up the possibility to compute the BPS indices in any chamber using either the flow tree or the Coulomb branch formula. In all cases we find precise agreement with independent computations of Vafa-Witten invariants based on wall-crossing and blow-up formulae. This agreement suggests that i) generating functions of DT invariants for a large class of quivers coming from strong exceptional collections are mock modular functions of higher depth and ii) non-trivial single-centered black holes and scaling solutions do not exist quantum mechanically in such local Calabi-Yau geometries.

## 15/03/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

David Berman, Queen Mary University of London
Double Field Theory and Geometric Quantisation

We examine various properties of double field theory and the doubled string sigma model in the context of geometric quantisation. In particular we look at T-duality as the symplectic transformation related to an alternative choice of polarisation in the construction of the quantum bundle for the string. Following this perspective we adopt a variety of techniques from geometric quantisation to study the doubled space. One application is the construction of the double coherent state that provides the shortest distance in any duality frame and a stringy deformed Fourier transform.

## 29/03/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Dmitry Melnikov, ITEP Moscow
Entanglement and Complexity in Topological Quantum Field Theories

One of the attractive ideas of building a quantum computer is based on the topological properties of matter. In such a realization, the Topological Quantum Field Theories (TQFT) become the main language to describe the functioning of the quantum computer. In my talk I will discuss some basic elements of the topological quantum computing. I will start from a description of TQFTs as instances of quantum mechanics in terms of category theory. Then I will review the notion of quantum entanglement in this context. As a further preparation to quantum computations I will discuss the question of complexity of quantum algorithms and quantum states. I will introduce a complexity measure for a simple class of the "torus knot states" and review some alternative recent measures and approaches from the literature.

## 12/04/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Murad Alim, University of Hamburg
Intrinsic non-perturbative topological strings

We study difference equations which are obtained from the asymptotic expansion of topological string theory on the deformed and the resolved conifold geometries as well as for topological string theory on arbitrary families of Calabi-Yau manifolds near generic singularities at finite distance in the moduli space. Analytic solutions in the topological string coupling to these equations are found. The solutions are given by known special functions and can be used to extract the strong coupling expansion as well as the non-perturbative content. The strong coupling expansions show the characteristics of D-brane and NS5-brane contributions, this is illustrated for the quintic Calabi-Yau threefold. For the resolved conifold, an expression involving both the Gopakumar-Vafa resummation as well as the refined topological string in the Nekrasov-Shatashvili limit is obtained and compared to expected results in the literature. Furthermore, a precise relation between the non-perturbative partition function of topological strings and the generating function of non-commutative Donaldson-Thomas invariants is given. Moreover, the expansion of the topological string on the resolved conifold near its singular small volume locus is studied. Exact expressions for the leading singular term as well as the regular terms in this expansion are provided and proved. The constant term of this expansion turns out to be the known Gromov-Witten constant map contribution.

## 26/04/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Michal P. Heller, Albert Einstein Institute
Spacetime as a quantum circuit

We propose that finite cutoff regions of holographic spacetimes represent quantum circuits that map between boundary states at different times and Wilsonian cutoffs, and that the complexity of those quantum circuits is given by the gravitational action. The optimal circuit minimizes the gravitational action. This is a generalization of both the "complexity equals volume" conjecture to unoptimized circuits, and path integral optimization to finite cutoffs. Using tools from holographic $T\bar{T}$, we find that surfaces of constant scalar curvature play a special role in optimizing quantum circuits. We also find an interesting connection of our proposal to kinematic space, and discuss possible circuit representations and gate counting interpretations of the gravitational action. Based on arXiv:2101.01185.

## 10/05/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Tarek Anous, University of Amsterdam
An invitation to the principal series

Scalar unitary representations of the isometry group of d-dimensional de Sitter space $SO(1,d)$ are labeled by their conformal weights $\Delta$. A salient feature of de Sitter space is that scalar fields with sufficiently large mass compared to the de Sitter scale $1/l$ have complex conformal weights, and physical modes of these fields fall into the unitary continuous principal series representation of $SO(1,d)$. Our goal is to study these representations in $d=2$, where the relevant group is $SL(2,R)$. We show that the generators of the isometry group of $dS_2$ acting on a massive scalar field reproduce the quantum mechanical model introduced by de Alfaro, Fubini and Furlan (DFF) in the early/late time limit. Motivated by the ambient $dS_2$ construction, we review in detail how the DFF model must be altered in order to accommodate the principal series representation. We point out a difficulty in writing down a classical Lagrangian for this model, whereas the canonical Hamiltonian formulation avoids any problem. We speculate on the meaning of the various de Sitter invariant vacua from the point of view of this toy model and discuss some potential generalizations.

## 24/05/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Daniel Waldram, Imperial College London
G-algebroids, consistent truncations and Poisson-Lie U-duality

“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.

## 07/06/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Syo Kamata, National Centre for Nuclear Research, Warsaw
Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$

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:

1. On exact-WKB analysis, resurgent structure, and quantization conditions, N. Sueishi, S. K., T. Misumi, and M. Ünsal, arXiv:2008.00379.
2. 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

### Video

WKB_S1_Lisbon-1.pdf

## 21/06/2021, Monday, 11:30–12:30 Europe/Lisbon — Online

Sven Krippendorf, Ludwig-Maximilians University Munich
Searching for Integrability with Artificial Intelligence

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.

Unusual time.

## 05/07/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Ingmar Saberi, Ludwig-Maximilians University Munich
Networks of perfect tensors via symplectic geometry over finite fields

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.

## 19/07/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

ChunJun Cao, University of Maryland
Building Bulk Geometry from the Tensor Radon Transform

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.

## 04/10/2021, Monday, 16:15–17:15 Europe/Lisbon — Online

Ben Heidenreich, University of Massachusetts, Amherst
The Weak Gravity Conjecture and BPS Particles

Motivated by the Weak Gravity Conjecture, we uncover an intricate interplay between black holes, BPS particle counting, and Calabi-Yau geometry in five dimensions. In particular, we point out that extremal BPS black holes exist only in certain directions in the charge lattice, and we argue that these directions fill out a cone that is dual to the cone of effective divisors of the Calabi-Yau threefold. The tower and sublattice versions of the Weak Gravity Conjecture require an infinite tower of BPS particles in these directions, and therefore imply purely geometric conjectures requiring the existence of infinite towers of holomorphic curves in every direction within the dual of the cone of effective divisors. We verify these geometric conjectures in a number of examples by computing Gopakumar-Vafa invariants.

Unusual hour!

## 18/10/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Sergey Mozgovoy, Trinity College Dublin
Attractor invariants, brane tilings and crystals

Given a CY3-fold $X$, we define its refined DT invariants $\Omega_Z(d)$ by counting objects in the derived category $D^b(X)$, semistable with respect to a stability condition $Z$ and having Chern character $d$. Attractor invariants $\Omega_*(d)$ correspond to a special stability condition that depends on the Chern character $d$. They usually have a particularly simple form. If known, attractor invariants can be used to determine DT invariants for all other stability conditions using wall-crossing formulas or flow tree formulas. A wide class of non-compact toric CY3-folds is encoded by combinatorial data called brane tilings or by associated quivers with potentials. In this setting the derived category of $X$ can be substituted by the derived category of a quiver with potential and the counting problems can be reduced to representation theoretic problems and then solved under suitable conditions. I will survey known results about DT invariants and some new conjectures about attractor invariants in this setting. I will also explain how these formulas in the unrefined limit correspond to the counting of molten crystals associated with brane tilings.

## 08/11/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Joerg Teschner, University of Hamburg
Mathematical structures of non-perturbative topological string theory: from GW to DT invariants

We study the Borel summation of the Gromov-Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson-Thomas invariants of the resolved conifold, having a direct relation to the Riemann-Hilbert problem formulated by T. Bridgeland. There exist distinguished integration contours for which the Borel summation reproduces previous proposals for the non-perturbative topological string partition functions of the resolved conifold. These partition functions are shown to have another asymptotic expansion at strong topological string coupling. We demonstrate that the Stokes phenomena of the strong-coupling expansion encode the DT invariants of the resolved conifold in a second way. Mathematically, one finds a relation to Riemann-Hilbert problems associated to DT invariants which is different from the one found at weak coupling. The Stokes phenomena of the strong-coupling expansion turn out to be closely related to the wall-crossing phenomena in the spectrum of BPS states on the resolved conifold studied in the context of supergravity by D. Jafferis and G. Moore.

## 29/11/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Shira Chapman, Ben Gurion University of the Negev
Holographic Complexity and de Sitter Space

We compute the length of spacelike geodesics anchored at opposite sides of certain double-sided flow geometries in two dimensions. These geometries are asymptotically anti-de Sitter but they admit either a de Sitter or a black hole event horizon in the interior. While in the geometries with black hole horizons, the geodesic length always exhibit linear growth at late times, in the flow geometries with de Sitter horizons, geodesics with finite length only exist for short times of the order of the inverse temperature and they do not exhibit linear growth. We comment on the implications of these results towards understanding the holographic proposal for quantum complexity and the holographic nature of the de Sitter horizon.

## 06/12/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Nathan Benjamin, Princeton University
Harmonic analysis of $2d$ CFT partition functions

We apply the theory of harmonic analysis on the fundamental domain of $SL(2,\mathbb{Z})$ to partition functions of two-dimensional conformal field theories. We decompose the partition function of $c$ free bosons on a Narain lattice into eigenfunctions of the Laplacian of worldsheet moduli space $H/SL(2,\mathbb{Z})$, and of target space moduli space $O(c, c; \mathbb{Z})\backslash O(c, c; \mathbb{R})/O(c) × O(c)$. This decomposition manifests certain properties of Narain theories and ensemble averages thereof. We extend the application of spectral theory to partition functions of general two-dimensional conformal field theories, and explore its meaning in connection to $AdS_3$ gravity. An implication of harmonic analysis is that the local operator spectrum is fully determined by a certain subset of degeneracies.

## 13/12/2021, Monday, 15:00–16:00 Europe/Lisbon — Online

Sourav Roychowdhury, Chennai Mathematical Institute
Non-Abelian T-dual of Klebanov-Tseytlin background and its Penrose limits

In this talk I will discuss Klebanov-Tseytlin background and its non-Abelian T-dual geometry. In particular I will show that the T-dual background admits pp-wave geometry in the neighbourhood of appropriate null geodesic. I will make comments on possible dual gauge theory for our pp-wave background.