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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. 

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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.

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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. 

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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). 

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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.

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Martijn Kool: Lectures

January 12, 2022
TITLE: Counting surfaces on Calabi-Yau fourfolds

ABSTRACT: On the sheaf side, there are two well-known ways to count curves on threefolds: via 1-dimensional subschemes (DT theory) and via stable pairs (PT theory). We show that for surfaces on fourfolds there are three theories: 2-dimensional subschemes and two types of stable pairs. Using Oh-Thomas/Borisov-Joyce, this allows us to define DT, PT0, PT1 invariants of Calabi-Yau fourfolds. We reduce the theory and prove that the resulting invariants are deformation invariant over the Hodge locus. This can be used to show the variational Hodge conjecture in some examples. We conjecture a DT-PT0 correspondence, which we check in non-compact examples using toric geometry and in compact examples using virtual pull-back. We conjecture a PT0-PT1 correspondence on Weierstrass elliptic fourfolds, which we prove for certain vertical classes. Joint work with Y. Bae and H. Park.

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Andrew Dancer: Lectures

January 10, 2022
TITLE: Hyperkahler implosions and symplectic duality

ABSTRACT: We present candidates for magnetic quivers for the universal hyperkahler implosion for special unitary groups, that is, quivers whose associated Coulomb branch gives the implosion space. We also discuss the orthosymplectic case.

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Shih-Kai Chiu: Lectures

January 11, 2022
TITLE: Rigidity of Calabi-Yau metrics with maximal volume growth

ABSTRACT: Calabi-Yau manifolds with maximal volume growth can be seen as 1)  higher dimensional generalizations of ALE gravitational instantons and 2) generalizations of asymptotically conical (AC) Calabi-Yau manifolds allowing singular tangent cones at infinity. We first show that on a Calabi-Yau manifold with maximal volume growth, any subquadratic harmonic function must be pluriharmonic. This can be seen as the linearization of the rigidity of the complex Monge-Ampere equation. Next we show that under the additional condition that the metric is ddbar-exact, the metric must be rigid under perturbation by a Kahler potential with subquadratic growth. Part of the talk is joint work in progress with Székelyhidi.

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Benjamin Aslan: Lectures

January 13, 2022
TITLE: Transverse J-holomorpic curves in nearly Kähler CP^3

ABSTRACT: Nearly Kähler manifolds in dimension six are certain almost Hermitian manifolds which admit a real Killing spinor, meaning they are Einstein manifolds. The most famous example is the round six-sphere but CP^3 also carries a nearly Kähler structure. J-holomorphic curves in nearly Kähler manifolds are closely related to associatives in G_2 geometry. In this talk, we will introduce the class of transverse J-holomorphic curves in CP^3, discuss their relationship to Toda lattice equations and construct a moment-type map for torus actions on CP^3 to study U(1)-invariant examples

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Fabian Lehmann: Lectures

September 13, 2021
TITLE: Non-compact Spin(7)-manifolds

ABSTRACT:
In the non-compact setting, symmetry reduction methods can be used to simplify the condition for Spin(7)-holonomy, which in general is given by a large, non-linear, first order PDE system, to a system of ODEs. I will talk about a particular example with symmetry group SU(3). I will outline a rigorous proof for the existence of two families of complete Spin(7)-metrics, where all members are either asymptotically locally conical (ALC), or asymptotically conical (AC).

These families were conjectured to exist earlier and fit into the landscape of other known families of non-compact G2 and Spin(7) holonomy spaces. Time permitting, I will also discuss the deformation theory of AC Spin(7)-manifolds. The talk is based on arXiv:2012.11758 and arXiv:2101.10310.

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