L3

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 present a new approach to study the RG flow in 3d N=4 gauge theories, based on an analysis of the Coulomb branch of vacua. The Coulomb branch is described as a complex algebraic variety and important information about the strongly coupled fixed points of the theory can be extracted from the study of its singularities. I will use this framework to study the fixed points of U(N) and Sp(N) gauge theories with fundamental matter, revealing some surprising scenarios at low amount of matter.

Lein Applied Diagnostics has a novel optical measurement technique that is used to measure various parameters in the body for medical applications.

Two particular areas of interest are non-invasive glucose measurement for diabetes care and the diagnosis of diabetes. Both measurements are based on the eye and involve collecting complex data sets and modelling their links to the desired parameter.

If we take non-invasive glucose measurement as an example, we have two data sets – that from the eye and the gold standard blood glucose reading. The goal is to take the eye data and create a model that enables the calculation of the glucose level from just that eye data (and a calibration parameter for the individual). The eye data consists of measurements of apparent corneal thickness, anterior chamber depth, optical axis orientation; all things that are altered by the change in refractive index caused by a change in glucose level. So, they all correlate with changes in glucose as required but there are also noise factors as these parameters also change with alignment to the meter etc. The goal is to get to a model that gives us the information we need but also uses the additional parameter data to discount the noise features and thereby improve the accuracy.

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. 
 

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.

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.