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09/12/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Paolo Benincasa, *Niels Bohr Institute*

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Understanding $\operatorname{AdS}_2$: From Calogero-like models and SLE to $4d$ black hole microstate entropy

Extremal black holes show the presence of an $\operatorname{AdS}_2$ factor as an universal feature. This fact provides a strong motivation for getting a deeper understanding of $\operatorname{AdS}_2$ spacetimes. In this talk, I will argue how $\operatorname{AdS}_2$ systems have a natural description in terms of $1d$ Calogero-type models and, in turn to SLE curves, which describe the geodesic motion of particles in $\operatorname{AdS}_2$. This treatment allows to compute the dimension of the phase space of these geodesics, linking it to the leading Bekenstein-Hawking black hole entropy and the black hole degeneracy.

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18/11/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Francesca Ferrari, *SISSA Trieste*

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A look into $3d$ modularity

Since the 1980s, the study of invariants of 3-dimensional manifolds has benefited from the connections between topology, physics and number theory. Motivated by the recent discovery of a new homological invariant (corresponding to the half-index of certain $3d$ $N=2$ theories), in this talk I describe the role of quantum modular forms, mock and false theta functions in the study of $3$-manifold invariants. The talk is based on 1809.10148 and work in progress with Cheng, Chun, Feigin, Gukov, and Harrison.

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22/10/2019, 16:00 — 17:00 — Room P3.10, Mathematics Building

Michele Cirafici, *University of Trieste*

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Supersymmetric line operators and their spectral problem

I will discuss BPS invariants associated with quantum line operators in certain supersymmetric quantum field theories. Such operators can be specified via geometric engineering in the UV by assigning a path on a certain curve. In the IR they are described by representation theory data. I will discuss the associated BPS spectral problem and the relevant indices.

Note: unusual date

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27/09/2019, 16:00 — 17:00 — Room P3.10, Mathematics Building

Debashis Ghoshal, *Jawaharlal Nehru University*

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Designing matrix models for zeta functions

The apparently random pattern of the non-trivial zeroes of the Riemann zeta function (all on the critical line, according to the Riemann hypothesis) has led to the suggestion that they may be related to the spectrum of an operator. It has also been known for some time that the statistical properties of the eigenvalue distribution of an ensemble of random matrices resemble those of the zeroes of the zeta function. With the objective to identify a suitable operator, we start by assuming the Riemann hypothesis and construct a unitary matrix model (UMM) for the zeta function. Our approach, however, could be termed *piecemeal*, in the sense that, we consider each factor (in the Euler product representation) of the zeta function to get a UMM for each prime, and then assemble these to get a matrix model for the full zeta function. This way we can write the partition function as a trace of an operator. Similar construction works for a family of related zeta functions.

Note: unusual date

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24/06/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Stefano Andriolo, *Hong Kong University of Science and Technology*

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The Weak Gravity Conjecture

We discuss various versions of the weak gravity conjecture.

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20/05/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Vishnu Jejjala, *University of the Witwatersrand*

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Experiments with Machine Learning in Geometry & Physics

Identifying patterns in data enables us to formulate questions that can lead to exact results. Since many of the patterns are subtle, machine learning has emerged as a useful tool in discovering these relationships. We show that topological features of Calabi–Yau geometries are machine learnable. We indicate the broad applicability of our methods to existing large data sets by finding relations between knot invariants, in particular, the hyperbolic volume of the knot complement and the Jones polynomial.

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07/05/2019, 10:00 — 11:00 — Room P5.18, Mathematics Building

Nils Carqueville, *University of Vienna*

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TQFTS, Orbifolds and Topological Quantum Computation

I will review basic notions and results in topological quantum field theory and discuss its orbifolds, with the aim to apply them in the context of topological quantum computation.

Unusual day and hour and room.

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06/05/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Ceyda Simsek, *University of Groningen*

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Spacetime geometry of non-relativistic string theory

Non-relativistic string theory is described by a sigma model that maps a two dimensional string worldsheet to a non-relativistic spacetime geometry. We discuss recent developments in understanding the spacetime geometry of non-relativistic string theory trying to provide several new insights. We show that the non-relativistic string action admits a surprisingly large number of symmetries. We introduce a non-relativistic limit to obtain the non-relativistic string action which also provides us the non-relativistic T-duality transformation rules and spacetime equations of motion.

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01/04/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Davide Masoero, *Faculdade de Ciências, Universidade de Lisboa*

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Meromorphic opers and the Bethe Ansatz

The Bethe Ansatz equations were initially conceived as a method to solve some particular Quantum Integrable Models (IM), but are nowadays a central tool of investigation in a variety of physical and mathematical theories such as string theory, supersymmetric gauge theories, and Donaldson-Thomas invariants. Surprisingly, it has been observed, in several examples, that the solutions of the same Bethe Ansatz equations are provided by the monodromy data of some ordinary differential operators with an irregular singularity (ODE/IM correspondence).

In this talk I will present the results of my investigation on the ODE/IM correspondence in quantum $g$-KdV models, where $g$ is an untwisted affine Kac-Moody algebra. I will construct solutions of the corresponding Bethe Ansatz equations, as the (irregular) monodromy data of a meromorphic $L(g)$-oper, where $L(g)$ denotes the Langlands dual algebra of $g$.

The talk is based on:

- D Masoero, A Raimondo, D Valeri, Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case. Comm. Math. Phys. (2016)
- D Masoero, A Raimondo, D Valeri, Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections II: The nonsimply-laced case. Comm. Math. Phys. (2017)
- D Masoero, A Raimondo, Opers corresponding to Higher States of the $g$-Quantum KdV model. arXiv 2018.

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26/11/2018, 15:00 — 16:00 — Room P3.10, Mathematics Building

Davide Polini, *Instituto Superior Técnico*

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Counting formulae for extremal black holes in an STU-model

We present microstate counting formulae for BPS black holes in an $N=2$ STU-model.

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12/11/2018, 15:00 — 16:00 — Room P3.10, Mathematics Building

Alexandre Belin, *University of Amsterdam*

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Siegel Modular Forms in AdS/CFT

I will discuss the application of Siegel modular forms for extracting the degeneracy of states of symmetric orbifold CFTs. These modular forms are closely related to the generating function for the elliptic genera of such CFTs and I will present an efficient technic for extracting their Fourier coefficients. I will then discuss to what extent symmetric orbifold CFTs can admit nice gravity duals and thus make an interesting connection between number theory and quantum gravity.

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05/11/2018, 15:00 — 16:00 — Room P3.10, Mathematics Building

Zhihao Duan, *École Normale Supérieure Paris*

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Instantons in the Hofstadter butterfly: resurgence and quantum mirror curves

Recently an interesting connection between topological string theory and lattice models in condensed matter physics was discussed by several authors. In this talk, we will focus on the Harper-Hofstadter Hamiltonian. For special values of the magnetic flux, its energy spectrum can be exactly solved and its graph has a beautiful shape known as Hofstadter's butterfly. We are interested in the non-perturbative information inside the spectrum. First we consider the weak magnetic field limit and write down a trans-series ansatz for the energies. We then discuss fluctuations around instanton sectors as well as resurgence relations. For the second half of the talk, our goal is to present another powerful way to compute those fluctuations using the topological string formalism, after reviewing all the necessary background. The talk will be based on arXiv: 1806.11092.

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03/10/2018, 14:30 — 15:30 — Room P3.10, Mathematics Building

Abhiram Kidambi, *Technical University of Vienna*

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BPS algebras and Moonshine

We give a brief introduction to BPS algebras and Moonshine in this informal seminar.

Unusual hour and day.

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01/10/2018, 15:00 — 16:00 — Room P3.10, Mathematics Building

Abhiram Kidambi, *Tecnical University of Vienna*

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$\Gamma_0(N)$, quantum black holes and wall crossing

The degeneracies of supersymmetric dyonic black holes are known to be encoded in the Fourier coefficients of certain modular objects. For the case of $N = 4$, $d=4$ theory which I shall discuss, the spectrum of quarter BPS dyons is prone to wall crossing phenomena. The number theory machinery behind wall crossing in $4d$ $N = 4$ theories was described systematically in a comprehensive paper by Atish Dabholkar, Sameer Murthy and Don Zagier. There have also been supergravity localisation calculations thereafter which confirm some of the results that were shown by DMZ.

In this talk, I shall provide some of the number theoretic background for BPS state counting and review some of the key results known so far from both the microscopic and macroscopic side. I shall comment on black hole metamorphosis studied by Sen (and collaborators) and Nampuri et.al from a number theoretic framework. The remainder of the talk will be devoted to the generalisation of the number theory machinery of DMZ to congruence subgroups of $\operatorname{SL}(2,\mathbb{Z})$ i.e. for orbifolded CHL black holes and the supergravity approach for the CHL case.

This talk summarises some of the ongoing work with Sameer Murthy, Valentin Reys, Abhishek Chowdhury and Timm Wrase.

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09/07/2018, 15:00 — 16:00 — Room P3.10, Mathematics Building

Vladislav Kupriyanov, *Ludwig-Maximilians-Universität München*

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$L_{\infty}$ bootstrap approach to non-commutative gauge theories

Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying $L_{\infty}$ algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory. In this talk I will give a brief introduction to $L_{\infty}$ algebras and discuss in more details the $L_{\infty}$ bootstrap program: the existence of the solution, uniqueness and particular examples. The talk is mainly based on: arXiv:1803.00732 and 1806.10314.

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04/06/2018, 15:00 — 16:00 — Room P3.10, Mathematics Building

Frank Ferrari, *Université Libre de Bruxelles*

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On Melonic Matrix Models and SYK-like Black Holes

I will illustrate three aspects of the new large $D$ limit of matrix models and their applications to black hole physics:

*Graph theory aspect*: I will review the basic properties of the new large $D$ limit of matrix models and provide a simple graph-theoretic argument for its existence, independent of standard tensor model techniques, using the concepts of Tait graphs and Petrie duals.*Phase diagrams*: I will outline the interesting phenomena found in the phase diagrams of simple fermionic matrix quantum mechanics/tensor/SYK models at strong coupling, including first and second order phase transitions and quantum critical points. Some of these phase transitions can be argued to provide a quantum mechanical description of the phenomenon of gravitational collapse.*Probe analysis*: I will briefly describe how the matrix point of view allows to naturally define models of D-particles probing an SYK-like black hole and discuss the qualitative properties of this class of models, emphasizing the difference between models based on fermionic and on bosonic strings. This approach provides an interesting strategy to study the emerging geometry of melonic/SYK black holes. In particular, it will be explained how a sharply defined notion of horizon emerges naturally.

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28/05/2018, 15:00 — 16:00 — Room P3.10, Mathematics Building

Salvatore Baldino, *Instituto Superior Técnico*

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Introduction to resurgence (III)

This is the third in a series of talks introducing the subject of resurgence in quantum mechanics, field theory and string theory.

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30/04/2018, 16:00 — 17:00 — Room P3.10, Mathematics Building

Maximilian Schwick, *Instituto Superior Técnico*

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Introduction to resurgence (II)

This is the second in a series of talks introducing the subject of resurgence in quantum mechanics, field theory and string theory.

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24/04/2018, 11:00 — 12:00 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa

Panagiotis Betzios, *University of Crete*

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Matrix Quantum Mechanics and the $S^1/\mathbb{Z}_2$ orbifold

We revisit $c=1$ non-critical string theory and its formulation via Matrix Quantum Mechanics (MQM). In particular we study the theory on an $S^1/\mathbb{Z}_2$ orbifold of Euclidean time and try to compute its partition function in the grand canonical ensemble that allows one to study the double scaling limit of the matrix model and connect the result to string theory (Liouville theory). The result is expressed as the Fredholm Pfaffian of a Kernel which we describe in several bases. En route we encounter interesting mathematics related to Jacobi elliptic functions and the Hilbert transform. We are able to extract the contribution of the twisted states at the orbifold fixed points using a formula by Dyson for the determinant of the sine kernel. Finally, we will make some comments regarding the possibility of using this model as a toy model of a two dimensional big-bang big-crunch universe.

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23/04/2018, 15:00 — 16:00 — Room P3.10, Mathematics Building

Olga Papadoulaki, *University of Southampton*

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FZZT branes and non-singlets of Matrix Quantum Mechanics

We will discuss the non-singlet sectors of the matrix model associated with two dimensional non-critical string theory. These sectors of the matrix model contain rich physics and are expected to describe non-trivial states such as black holes. I will present how one can turn on the non-singlets by adding $N_f \times N$ fundamental and anti-fundamental fields in the gauge matrix quantum mechanics model as well as a Chern-Simons term. Then, I will show how one can rewrite our model as a spin-Calogero model in an external magnetic field. By introducing chiral variables we can define spin-currents that in the large $N$ limit satisfy an $SU(2N_f )_k$ Kac-Moody algebra. Moreover, we can write down the canonical partition function and study different limits of the parameters and possible phase transitions. In the grand canonical ensemble the partition function is a $\tau$ - function obeying discrete soliton equations. Also, in a certain limit we recover the matrix model of Kazakov-Kostov-Kutasov conjectured to describe the two dimensional black hole. Finally, I will discuss several implications that our model has for the understanding of the thermodynamics and the physics of such string theory states.