Any four dimensional N=2 superconformal field theory (SCFT) contains a subsector of local operator which is isomorphic to a two dimensional chiral algebra. If the 4d theory possesses N= 4 superconformal symmetry, the corresponding chiral algebra is an extension of the (small) N=4 super-Virasoro algebra. In this talk I will present some results on the classification of N=4 chiral algebras and discuss the conditions they should satisfy in order to correspond to a 4d theory.

# Past String Theory Seminar

We discuss mathematical studies on the Vafa-Witten theory, one of topological twists of N=4 super Yang-Mills theory in four dimensions, from the viewpoints of both differential and algebraic geometry. After mentioning backgrounds and motivation, we describe some issues to construct mathematical theory of this Vafa-Witten one, and explain possible ways to sort them out by analytic and algebro-geometric methods, the latter is joint work with Richard Thomas.

I will talk about a generalisation of the Seiberg-Witten equations introduced by Taubes and Pidstrygach, in dimension 3 and 4 respectively, where the spinor representation is replaced by a hyperKahler manifold admitting certain symmetries. I will discuss the 4-dimensional equations and their relation with the almost-Kahler geometry of the underlying 4-manifold. In particular, I will show that the equations can be interpreted in terms of a PDE for an almost-complex structure on 4-manifold. This generalises a result of Donaldson.

I will talk about three-dimensional N=2 supersymmetric gauge theories on a class of Seifert manifold. More precisely, I will compute the supersymmetric partition functions and correlation functions of BPS loop operators on M_{g,p}, which is defined by a circle bundle of degree p over a genus g Riemann surface. I will also talk about four-dimensional uplift of this construction, which computes the generalized index of N=1 gauge theories defined on elliptic fiberation over genus g Riemann surface. We will find that the partition function or the index can be written as a sum over "Bethe vacua” of two-dimensional A-twisted theory obtained by a circle compactification. With this framework, I will show how the partition functions on manifolds with different topologies are related to each other. We will also find that these observables are very useful to study the action of Seiberg-like dualities on co-dimension two BPS operators.

Compactifications of 6D Superconformal Field Theories (SCFTs) on four-manidolfds lead to novel interacting 2D SCFTs. I will describe the various Lagrangian and non-Lagrangian sectors of the resulting 2D theories, as well as their interactions. In general this construction can be embedded in compactifications of the physical superstring, providing a general template for realizing 2D conformal field theories coupled to worldsheet gravity, i.e. a UV completion for non-critical string theories.

Nekrasov, Rosly and Shatashvili observed that the generating function of a certain space of SL(2) opers has a physical interpretation as the effective twisted superpotential for a four-dimensional N=2 quantum field theory. In this talk we describe the ingredients needed to generalise this observation to higher rank. Important ingredients are spectral networks generated by Strebel differentials and the abelianization method. As an example we find the twisted superpotential for the E6 Minahan-Nemeschansky theory.

A K3 surface is called attractive if and only if its Picard number is 20: The maximal possible. Attractive K3 surfaces possess complex multiplication. This property endows attractive K3 surfaces with rich and well understood arithmetic. For example, the associated Galois representation turns out to be a product of well known two dimensional representations and the Hasse-Weil L-function turns out to be a product of well known L-functions. On the other hand, attractive K3 surfaces show up as solutions of the attractor equations in type IIB string theory compactified on the product of a K3 surface with an elliptic curve. As such, these surfaces dictate the near horizon geometry of a charged black hole in this theory. We will try to see which arithmetic properties of the attractive K3 surfaces lend a stringy interpretation and use them to shed light on physical properties of the charged black hole.

Transport properties of liquids and gases in the regime of weak coupling (or effective weak coupling) are determined by the solutions of relevant kinetic equations for particles or quasiparticles, with transport coefficients being proportional to the minimal eigenvalue of the linearized kinetic operator. At strong coupling, the same physical quantities can sometimes be determined from dual gravity, where quasinormal spectra enter as the eigenvalues of the linearized Einstein's equations. We discuss the problem of interpolating between the two regimes using results from higher derivative gravity.

Understanding the structure of hadrons in terms of their fundamental constituents requires an understanding of QCD at large distances, a vastly complex and unsolved dynamical problem. I will discuss in this talk a new approach to hadron structure based on superconformal quantum mechanics in the light-front and its holographic embedding in a higher dimensional gravity theory. This approach captures essential aspects of the confinement dynamics which are not apparent from the QCD Lagrangian, such as the emergence of a mass scale and confinement, the occurrence of a zero mode: the pion, universal Regge trajectories for mesons and baryons and precise connections between the light meson and nucleon spectra. This effective semiclassical approach to relativistic bound-state equations in QCD can be extended to heavy-light hadrons where heavy quark masses break the conformal invariance but the underlying dynamical supersymmetry holds.