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Ethan Torres: Lectures

June 8, 2022
TITLE: Getting High on Gluing Orbifolds

ABSTRACT:
Quantum field theories (QFTs) engineered from M-theory on singular non-compact manifolds often enjoy a rich dictionary between physical data and geometric quantities. For instance, when real codimension-4 ADE orbifold singularities extend out to the asymptotic boundary, the local operators of the QFT form representations of a (so-called) flavor group whose Lie algebra is specified by the ADE-types. We present a novel geometric procedure to calculate the global structure of this group. Additionally, taking into account line operators of the QFT, this structure may generalize to a 2-group and we give a geometric picture for this as well. For part I of this talk series, we will discuss these ideas in the context of 5d superconformal field theories engineered from M-theory on quotients of C^3.

Jeffrey Streets: Lectures

January 14, 2022
TITLE: Singular sets of generalized Einstein metrics

ABSTRACT: Various considerations in geometry and physics lead to natural generalizations of Einstein metrics which are coupled to differential forms.  In this talk I will describe recent joint work with X. Fu and A. Naber on the singularity formation of sequences of such structures.  In particular we show that the limit spaces are regular outside of a set of codimension 4, and satisfy certain sharp integral estimates, leading to geometric applications.

Lecture Notes

Jihwan Oh: Lectures

January 12, 2022
TITLE: G2 instantons in twisted M-theory

ABSTRACT: I will discuss a string theory way to study G2 instanton moduli space and explain how to compute the instanton partition function for a certain G2 manifold. An important insight comes from the twisted M-theory on the G2 manifold. Building on the example, I will explain a possibility to extend the story to a large set of conjectural G2 manifolds and a possible connection to 4d N=1 SCFT via geometric engineering. This talk is based on a work with Michele Del Zotto and Yehao Zhou.

Slides of Lecture

Hiraku Nakajima: Lectures

January 10, 2022
TITLE: Symmetric bow varieties

ABSTRACT: In my joint work with Takayama, I showed that Coulomb branches of quiver gauge theories of affine type A, as defined in earlier joint work with Braverman and Finkelberg, are isomorphic to Cherkis’ bow varieties. For quiver gauge theories of affine type D, or more generally of classical affine types, we introduce symmetric bow varieties, which are fixed point loci of involution on bow varieties.

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Francis Kirwan: Lectures

January 10, 2022
TITLE: Hyperkahler implosion

ABSTRACT: The hyperkahler quotient construction, which allows us to construct new hyperkahler spaces from suitable group actions on hyperkahler manifolds, is an analogue of symplectic reduction (introduced by Marsden and Weinstein in the 1970s), and both are closely related to the quotient construction for complex reductive group actions in algebraic geometry provided by Mumford’s geometric invariant theory (GIT). Symplectic implosion was introduced twenty years ago by Guillemin, Jeffrey and Sjamaar; in some sense it abelianises symplectic reduction. Hyperkahler implosion is in turn an analogue of symplectic implosion; both are related to a generalised version of GIT providing quotients for non-reductive group actions in algebraic geometry.

Slide of Lecture

Cyril Closset: Lectures

January 11, 2022
TITLE: On SCFTs at canonical singularities

ABSTRACT: Canonical threefold singularities are used in string theory to ‘geometrically engineer’ superconformal field theories (SCFTs) in 4d and 5d. I will discuss various aspects of that intriguing relationship between physics and geometry, providing an overview of recent results, conjectures and open questions. 

Slides of Lecture

Mathew Bullimore: Lectures

January 11, 2022
TITLE: Moduli Stacks and Global Categorical Symmetry

ABSTRACT: I will discuss aspects of moduli spaces in supersymmetric gauge theories that take the form of conical symplectic singularities or resolutions thereof. They often admit Hamiltonian group actions arising from continuous global symmetries and feature prominently in geometric representation theory. I will argue that it is useful to promote moduli spaces to moduli stacks, which capture the presence of a topological sector such as a discrete gauge theory in the IR physics at a point on the underlying moduli space. I will explain how moduli stacks admit actions of higher-form and higher group global symmetries, generalizing the action of ordinary global symmetries on the underlying moduli space. I will present some examples of how such structures are mapped under three-dimensional mirror symmetry.

Slides of Lecture

Ragini Singhal: Lectures

January 14, 2022
TITLE: Deformations of G2-instantons on nearly G2 manifolds

ABSTRACT: We will talk about the deformation theory of instantons on manifolds with a nearly parallel G2-structure. We formulate the deformation theory in terms of spinors and Dirac operators and prove that the space of infinitesimal deformations of an instanton is isomorphic to the kernel of an elliptic operator. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to describe the deformation space of the characteristic connection on the four normal homogeneous nearly G2 manifolds. We also show that on three of these four spaces the deformations are genuine. 

Slides of Lecture

Nikita Nekrasov: Lectures

January 12, 2022
TITLE: Some progress in unification of enumerative and differential geometry and quantization(s)

ABSTRACT:
In the first part of the talk I will review the recent progress in our attempts to approach the hyperkahler geometry of the 3d Coulomb branches through the localization computations in 4d gauge theories (based on the joint work with S.Jeong and N.Lee). In the second  part I will talk on the global magnificence, i.e. an attempt to build an 8+1 dimensional gauge theory unifying K-theoretic Donaldson-Thomas theories of threefolds, a cohomological eleven dimensional supergravity and maybe more (based on the joint work with N.Piazzalunga). In the third part I will make some observations on the action of compact support cohomology on cohomology and its implications for 5d susy gauge theories realized by M-theory on toric Calabi-Yau threefolds (based on the joint work with N.Piazzalunga and M.Zabzine), and connections to my old formulas (proven in some cases by L.Gottsche, H.Nakajima and K.Yoshioka, and by E.Gasparim) for partition functions of 4d theory on toric surfaces (based on the joint work with M.del Zotto, N.Piazzalunga, and M.Zabzine). 

Slides of Lecture

Jan Manschot: Lectures

January 12, 2022
TITLE: Topological correlators of N=2* Yang-Mills theory

ABSTRACT: N=2* Yang-Mills theory is a mass deformation of N=4 Yang-Mills, which preserves N=2 supersymmetry. I will consider the topological twist of this theory with gauge group SU(2) on a smooth, compact four-manifold X. A consistent formulation requires coupling of the theory to a Spin-c structure, which is necessarily non-trivial if X is non-spin. I will discuss the contribution from the Coulomb branch to correlation functions in terms of the low energy effective field theory coupled to a Spin-c structure, and present how these are evaluated using mock modular forms. Upon varying the mass, the correlators can be shown to reproduce correlators of Donaldson-Witten theory as well as Vafa-Witten theory. Based on joint work with Greg Moore, arXiv:2104.06492.

Slides of Lecture