Speakers

Gokce Basar

University of Maryland

Resurgence, Uniform WKB and Complex Instantons

The theory of resurgence connects perturbative and non-perturbative physics. Focusing on certain one-dimensional quantum mechanical systems with degenerate harmonic minima, I will explain how the resurgent trans-series expansions for the low lying energy eigenvalues follow from the exact quantization condition via the uniform WKB approach. In the opposite spectral region (with high lying eigenvalues), in contrast to the divergent asymptotic expansions expressed as trans-series, the relevant expansions are convergent. However, due to the poles in the expansion coefficients, they contain non-perturbative contributions which can be identified with complex instantons. I will demonstrate that in each spectral region there are striking relation between perturbative and non-perturbative expansions even though the nature of these expansions are very different. Notably, the quantum mechanical examples that I will discuss encode the vacua of certain supersymmetric gauge theories in their spectra.

Videos

Lecture 1

Freddy Cachazo

Perimeter Institute

Scattering Amplitudes and Riemann Surfaces

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.

Olivia Dumitrescu

Leibniz Universität Hannover

Topological Recursion for Higgs Bundles and Cohomological Field Theory

I will give a brief overview of Topological Recursion and present the general setting and our contribution to this field via geometry and topology techniques. In particular, I will discuss the topological recursion applied to the family of spectral curves of Hitchin moduli spaces of Higgs bundles over a smooth base curve C. We study meromorphic Higgs fields of rank two and we realize their spectral curves as divisors in the compactifed cotangent bundle. Topological recursion gives a way to quantize the spectral curve of a Higgs bundle.

I will present as typical examples of our theory some well-known constructions as the recursion of Witten–Kontsevich intersection numbers and the recursion of Catalan numbers, that count the number of cellular graphs on a Riemann Surfaces. In particular, we present a model for the twisted version of the Topological Recursion via Cohomological Field Theory for Mg,n(BG). We prove that edge contraction axioms of cellular graphs leads to a TQFT.

Videos

Lecture 1

Alexander Getmanenko

Universidad de los Andes

Resurgent Analysis and its Applications to the Witten Laplacian

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.

Kohei Iwaki

Nagoya University

Introduction to Exact WKB Analysis

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.

Exact WKB Analysis and Cluster Algebras

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).

Matilde Marcolli

California Institute of Technology

Colloquium: Motives in Quantum Field Theory

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.

Feynman Integrals and Motives in Configuration Spaces

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 Özgür Ceyhan.

A Motivic Approach to Potts Models

In this talk I will discuss how techniques similar to those adopted to study the algebro-geometric aspects of Feynman integrals in momentum space can be applied to Potts models and generalizations with magnetic field, and investigate various complexity questions. This talk is based on joint work with Paolo Aluffi and with my students Jessica Su and Shival Dasu.

Lionel Mason

University of Oxford

Ambitwistor-strings and Amplitudes

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.

Pranav Pandit

Universität Wien

Buildings, WKB Analysis, and Spectral networks

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.

Videos

Lecture 1 | Lecture 2

Marcus Spradlin

Brown University

Cluster Algebras and Scattering 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.

Videos

Lecture 1 | Lecture 2

Karen Yeats

Simon Fraser University

Some Combinatorial Comments on Amplitudes

I will begin with the perspective that the perturbative expansion is an augmented generating function and then discuss some of the results which follow from this perspective.

Videos

Lecture 1 | Lecture 2

Participants

Name Affiliation
Gökçe Başar University of Maryland
Francis Bischoff University of Toronto
Anton Borissov University of Waterloo
Dylan Butson Perimeter Institute
Freddy Cachazo Perimeter Institute
Sean Carrell University of Waterloo
Linqing Chen Perimeter Institute
Raymond Cheng University of Waterloo
Clair Dai University of Waterloo
Olivia Dumitrescu Leibniz Universität Hannover
Mohamed El Alami University of Waterloo
Travis Ens University of Toronto
Ali Fathi University of Western Ontario
Antonia Frassino Perimeter Institute
Laurent Freidel Perimeter Institute
Alberto García-Raboso University of Toronto
Alexander Getmanenko Universidad de los Andes
Steven Gindi University of Waterloo
Henrique Gomes Perimeter Institute
Humberto Gómez Perimeter Institute
Marco Gualtieri University of Toronto
Krystal Guo Simon Fraser University
Shengda Hu Wilfrid Laurier University
Kohei Iwaki Nagoya University
Lisa Jeffrey University of Toronto
Masoud Khalkhali University of Western Ontario
Maximilian Klambauer University of Toronto
Carson Li University of Guelph
Yan Rong Li University of Waterloo
Matilde Marcolli California Institute of Technology
Lionel Mason University of Oxford
Mykola Matviichuk University of Toronto
Matt McTaggart Royal Military College of Canada
Sonali Mohapatra Perimeter Institute
Ruxandra Moraru University of Waterloo
Victor Mouquin University of Toronto
Robert Myers Perimeter Institute
Timothy Nguyen Michigan State University
Nikita Nikolaev University of Toronto
Comron Nouri Pennsylvania State University
Pranav Pandit Universität Wien
Gideon Providence University of Toronto
Trevor Rempel Perimeter Institute
Geoffrey Scott University of Toronto
Sam Selmani McGill University
Reza Seyyedali University of Waterloo
Barak Shoshany Perimeter Institute
Marcus Spradlin Brown University
Prashant Subbarao University of Pennsylvania
David Svoboda Perimeter Institute
Brett Teeple University of Toronto
Baris Ugurcan University of Western Ontario
David Wagner University of Waterloo
McKenzie Wang McMaster University
Karen Yeats Simon Fraser University
Ahmed Jihad Zerouali University of Toronto
Nosiphiwo Zwane Perimeter Institute

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GAP2015 Poster

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