Utrecht

S-duality is an intriguing symmetry of (twisted) N=4 supersymmetric Yang-Mills theory on a four-manifold. When the four-manifold underlies a complex projective surface, it leads to the Vafa-Witten invariants defined by Tanaka-Thomas in 2017. I will discuss some developments related to Azumaya algebras, universality, Seiberg-Witten invariants, wall-crossing for Nakajima quiver varieties, the structure of S-duality, and modular curves (including relations to the Rogers-Ramanujan continued fraction and Klein quartic).

In this talk I describe a holographic perspective to study field spaces that arise in string compactifications. The constructions are motivated by a general description of the asymptotic, near-boundary regions in complex structure moduli spaces of Calabi-Yau manifolds using Hodge theory. For real two-dimensional field spaces, I introduce an auxiliary bulk theory and describe aspects of an associated sl(2) boundary theory. The classical bulk reconstruction is provided by the sl(2)-orbit theorem, which is a famous and general result in Hodge theory. I then apply this correspondence to the flux landscape of Calabi-Yau fourfold compactifications and discuss how this allows us to prove that the number of self-dual flux vacua is finite. I will point out how the finiteness result for supersymmetric fluxes relates to the Hodge conjecture.

The main result of the talk is that two curves over a finite field are isomorphic, up to automorphisms of the ground field, if and only if there is an isomorphism of groups of Dirichlet characters such that the corresponding L-series are all equal. This can be shown by combining Uchida's proof of the anabelian theorem for global function fields with methods from (noncommutative) dynamical systems. I will also discuss how to turn this theorem into a theoretical algorithm that, given a listing of L-functions, determines an equation for the corresponding curve(s).

Introductory talk on topological quantum field theories (TQFTs) and the cobordism hypothesis, focusing on the conceptual issues involved.

The lecture will take place this Friday at 11am in Lecture Theatre A of the Department of Computer Science

The idea of three-tier conformal field theory (CFT) was first proposed by Greame Segal. It is an extension of the functorial approach to CFT, where one replaces the bordism category of Riemann surfaces by a suitable bordism 2-category, whose objects are points, whose morphism are 1-manifolds, and whose 2-morphisms are pieces of Riemann surface. The Baez-Dolan cobordism hypothesis is a meta-mathematical principle. It claims that functorial quantum field theory (i.e. quantum field theory expressed as a functor from some bordism category) becomes simper once "you go all the way down to points", i.e., once you replace the bordism category by a higher category. Three-tier CFT is an example of "going all the way down to points". We will apply the cobordism hypothesis to the case of three-tier CFT, and show how the modular invariance of the partition function can be derived as a consequence of the formalism, even if one only starts with genus-zero data.

There is a beautiful classification of full (rational) CFT due to

Fuchs, Runkel and Schweigert. The classification says roughly the

following. Fix a chiral algebra A (= vertex algebra). Then the set of

full CFT whose left and right chiral algebras agree with A is

classified by Frobenius algebras internal to Rep(A). A famous example

to which one can successfully apply this is the case when the chiral

algebra A is affine su(2): in that case, the Frobenius algebras in

Rep(A) are classified by A_n, D_n, E_6, E_7, E_8, and so are the

corresponding CFTs.

Recently, Kapustin and Saulina gave a conceptual interpretation of the

FRS classification in terms of 3-dimentional Chern-Simons theory with

defects. Those defects are also given by Frobenius algebras in Rep(A).

Inspired by the proposal of Kapustin and Saulina, we will (partially)

construct the three-tier CFT associated to a given Frobenius algebra.

This is an overview, mostly of work of others (Denef, Loeser, Merle, Heinloth-Bittner,..). In the first part of the talk we give a brief introduction to motivic integration emphasizing its application to vanishing cycles. In the second part we discuss a join construction and formulate the relevant Sebastiani-Thom theorem.

This is an overview, mostly of work of others (Denef, Loeser, Merle, Heinloth-Bittner,..). In the first part of the talk we give a brief introduction to motivic integration emphasizing its application to vanishing cycles. In the second part we discuss a join construction and formulate the relevant Sebastiani-Thom theorem.