String Theory Seminar  

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.

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

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.

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.

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

1. 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.
2. 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.
3. 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.

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

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

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.

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.​

We show that a certain class of supersymmetric (SUSY) observables in 3d $\mathcal{N}=2$ SUSY Chern-Simons (CS) matter theories has nontrivial resurgent structures with respect to coupling constants given by inverse CS levels, and that their exact results are expressed as appropriate resummations of weak coupling expansions given by transseries. With a real mass parameter varied, we encounter Stokes phenomena infinitely many times, where the perturbative series gets non-Borel-summable along positive real axis of the Borel plane. We also decompose integral representations of the exact results in terms of Lefschetz thimbles and study how they are related to the resurgent transseries. We further discuss connections between the non-perturbative effects appearing in the transseries and complexified SUSY solutions which formally satisfy SUSY conditions but are not on original path integral contour. We explicitly demonstrate the above for partition functions of rank-1 3d $\mathcal{N}=2$ CS matter theories on sphere. This talk is based on arXiv:1604.08653, 1710.05010, and an on-going collaboration with Toshiaki Fujimori, Syo Kamata, Tatsuhiro Misumi and Norisuke Sakai.​

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

I will discuss a string theoretic approach to integrable lattice models. This approach provides a unified perspective on various important notions in lattice models, and relates these notions to four-dimensional $N =1$ supersymmetric field theories and their surface operators. I will also explain Nekrasov-Shatashvili correspondence.

Persistent homology studies which homological features of a topological space persist over a long range of scales. I will discuss two applications of this formalism to the study of vacua in string theory. In the first application, I will discuss how to adapt such techniques to address the presence/absence of structure in a series of string compactifications (for example flux vacua in type IIB or heterotic vacua). In the second application, I will address the problem of studying vacua of certain two-dimensional Landau-Ginzburg models and see what information we can get about their algebraic structures.

We introduce a notion of attractors and attractive geometry in the context of static four-dimensional non-extremal black holes, and we establish constraints on the flow of scalars in these black hole backgrounds to ultimately derive area laws pertaining to horizons in these backgrounds that were hitherto known empirically.

We consider a specific type II string theory model with $N=2$ local supersymmetry, the so-called STU model. This is a model with exact duality symmetries. Using holomorphy and duality, we obtain exact results for this model that go beyond the perturbative formulation of topological string theory.

The singlet sectors of $O(N)$ vector field theories in three dimensions have been conjectured to be dual to theories of higher spins in $AdS_4$, in their large $N$ limit. We use the bilocal description of this invariant sector to obtain a constructive description of this correspondence.

In this talk, I will present some recent developments in computing the exact entropy of dyonic $1/4$-BPS black holes in four-dimensional $N=4$ supergravity theories originating from Type IIB string theory compactified on $K3 \times T_2$. The exact entropy is obtained in the Quantum Entropy Function formalism by means of supersymmetric localization techniques. The result can then be compared to the degeneracy of the brane/momentum system making up the black hole in the string theory picture. Such degeneracies are given by the Fourier coefficients of so-called mock Jacobi forms, a concept I will review. An exact formula for the coefficients can be obtained via a suitable generalization of the Hardy-Ramanujan-Rademacher circle method which takes into account the mock character of the counting functions. After presenting these results, I will outline some discrepancies (at sub-leading order in the charges) between the supergravity result for the exact entropy and the degeneracies of the brane/momentum system, and point to some aspects of the supergravity calculations which should be examined in more detail if one hopes to get a complete matching.

We explore the question of which shape a manifold is compelled to take when immersed in another one, provided it must be the extremum of some functional. We consider a family of functionals which depend quadratically on the extrinsic curvatures and on projections of the ambient curvatures. These functionals capture a number of physical setups ranging from holography to the study of membranes and elastica.

We study rotating attractor solutions from the point of view of a Riemann-Hilbert problem associated with the Breitenlohner-Maison linear system. We describe an explicit vectorial Riemann-Hilbert factorization method which we use to show that the near-horizon limit of these extremal solutions can be constructed by Riemann-Hilbert factorization of monodromy matrices with poles of second order.

This talk will give a lightning review on $K3$ theories, including some newer developments. In particular, we discuss a procedure recently devised in joint work with Anne Taormina in the context of Mathieu Moonshine. This procedure, which we call reflection, allows to transform certain superconformal field theories into super vertex operator algebras and their admissible modules, thus building a bridge between the two worlds.