Mathematical Physics

This talk will cover aspects of Feynman integral computations in configuration spaces, and some related mathematical problems, and the occurrence of motives and periods, focusing on joint work with Ozgur Ceyhan.

The first lecture will be devoted to the review of the classical theory of the Witten Laplacian, the second -- to the concepts of resurgent analysis. The third -- to applications of the resurgent analysis to the Witten Laplacian. Time permitting, we will touch upon some foundational questions of resurgent analysis.

These lectures will focus on the geometry of ambitwistor string theories. These are infinite tension analogues of conventional strings and provide the theory that leads to the remarkable formulae for tree amplitudes that have been developed by Cachazo, He and Yuan based on the scattering equations. Although the bosonic ambitwistor string action is expressed in space-time, it will be seen that its target is classically `ambitwistor space', the space of complexified null geodesics in the complexification of a space-time. The lectures will review Ambitwistor constructions from the 70's and 80's that extend the Penrose-Ward twistor constructions for self-dual Yang-Mills and gravitational fields in four dimensions to arbiitrary fields in general dimension. LeBrun showed that the conformal geometry of a space-time is encoded into the complex structure of ambitwistor space. The linearized version encodes linear fields on space-time into sheaf cohomology classes on ambitwistor space. In the case of momentum eigenstates, these give the `scattering equations' that underly the CHY formulae and the ambitwistor string can be used to compute amplitudes via these formulae. If there is time, the lectures will discuss how different matter theories can be obtained, different geometric realizations of ambitwistor space lead to different formulae, the relationship between the asymptotic symmetries of space-time and Weinberg's soft theorems concerning the behaviour of amplitudes when momenta become small, and/or extensions of the ideas to loop amplitudes.

Supersymmetric gauge theory computes a very special class of (generalized) polylogarithm functions known as scattering amplitudes that have remarkable mathematical properties. In particular, there is a rich connection between these amplitudes and the G(4,n) Grassmannian cluster algebra. To explain this connection I will review some basic facts about the Hopf algebra of polylogarithms and cluster Poisson varieties. I will then define cluster polylogarithm functions which roughly speaking are polylogarithm functions whose arguments are cluster X-coordinates of some cluster algebra A. I will describe an additional property of certain scattering amplitudes, that they are "local" in the algebra A, and describe the classification of cluster polylogarithm functions with this property. The computation of new amplitudes can be greatly aided by knowledge of the class of functions in terms of which they may be expressed, as I will illustrate via an example.

I will give an overview of the algebro-geometric approach to Feynman integral in perturbative quantum field theory and the occurrence of motives and periods in parametric Feynman integrals in momentum space, focusing on joint work with Paolo Aluffi.

Exact WKB analysis, developed by Voros et.al., is an effective method for the global study of differential equations (containing a large parameter) defined on a complex domain. In the first and second lecture I'll give an introduction to exact WKB analysis, and recall some basic facts about WKB solutions, Borel resummation, Stokes graphs etc. On the other hand, cluster algebras are a particular class of commutative subalgebras of the field of rational functions with distinguished generators. I'll explain about a hidden cluster algebraic structure in exact WKB analysis in the third lecture. This is a joint work with Tomoki Nakanishi (Nagoya).

In 2003 Witten introduced twistor string theory as a novel description of the scattering matrix of the maximally supersymmetric Yang-Mills theory in four dimensions. In these lectures I will give an introduction to the developments that have led to new formulations, also based on Riemann surfaces, of a large variety of theories, with and without supersymmetry, in arbitrary space-time dimensions.

Supersymmetric gauge theory computes a very special class of (generalized) polylogarithm functions known as scattering amplitudes that have remarkable mathematical properties. In particular, there is a rich connection between these amplitudes and the G(4,n) Grassmannian cluster algebra. To explain this connection I will review some basic facts about the Hopf algebra of polylogarithms and cluster Poisson varieties. I will then define cluster polylogarithm functions which roughly speaking are polylogarithm functions whose arguments are cluster X-coordinates of some cluster algebra A. I will describe an additional property of certain scattering amplitudes, that they are "local" in the algebra A, and describe the classification of cluster polylogarithm functions with this property. The computation of new amplitudes can be greatly aided by knowledge of the class of functions in terms of which they may be expressed, as I will illustrate via an example.

Buildings are higher dimensional analogues of trees. The goal of these lectures is to explain how the theory of harmonic maps to buildings affords a new perspective on certain aspects of the WKB analysis of differential equations that depend on a small parameter. We will also touch upon some motivation for developing this perspective, which derives from questions about compactifications of higher Teichmüller spaces, stability in Fukaya categories, and the work of Gaiotto, Moore and Neitzke on spectral networks and wall-crossing phenomena. These talks are based on joint work with Ludmil Katzarkov, Alexander Noll and Carlos Simpson. A central role in our discussion will be played by the notion of a versal pre-building associated with a given spectral cover of a Riemann surface. This notion generalizes to higher rank the leaf space of the foliation defined by a quadratic differential. We will see that spectral networks are closely related to the singular loci of versal buildings, and that distances in these buildings encode information about the asymptotic behavior at infinity of the Riemann-Hilbert correspondence.

Exact WKB analysis, developed by Voros et.al., is an effective method for the global study of differential equations (containing a large parameter) defined on a complex domain. In the first and second lecture I'll give an introduction to exact WKB analysis, and recall some basic facts about WKB solutions, Borel resummation, Stokes graphs etc. On the other hand, cluster algebras are a particular class of commutative subalgebras of the field of rational functions with distinguished generators. I'll explain about a hidden cluster algebraic structure in exact WKB analysis in the third lecture. This is a joint work with Tomoki Nakanishi (Nagoya).