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Author Archives: Victoria Hain
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.
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).
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.
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.
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.
Shih-Kai Chiu: Lectures
- 5/16/2024: Special Lagrangians in K3-fibered Calabi-Yau 3-folds
- 1/11/2022: Rigidity of Calabi-Yau metrics with maximal volume growth
May 16, 2024
TITLE: Special Lagrangians in K3-fibered Calabi-Yau 3-folds
ABSTRACT:
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.
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
Fabian Lehmann: Lectures
- 9/08/2022: An embedding problem for closed 3-forms on 5-manifolds
- 9/13/2021: Non-compact Spin(7)-manifolds
September 8, 2022
TITLE: An embedding problem for closed 3-forms on 5-manifolds
ABSTRACT: The real part of a holomorphic volume form restricts to a closed 3-form on a 5-dimensional submanifold of a complex 3-fold with a Calabi-Yau structure. Vice versa, in analogy with the embedding problem for abstract CR-structures, this leads to the question which closed 3-forms on a given 5-manifold can be realised by an embedding into a Calabi-Yau 3-fold. We describe the structure induced on the 5-manifold by a closed 3-form and introduce a convexity notion. The main result is that in the “strongly pseudoconvex” case the embedding problem can be solved perturbatively if a finite dimensional vector space of obstructions vanishes. The proof uses the theory of sub-elliptic operators and the Nash-Moser inverse function theorem. The main example is the standard embedding of S^5 in C^3, for which the obstruction space vanishes. This is joint work with Simon Donaldson.
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.
Max Hübner: Lectures
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- 3/15/2023: Living on the Edge: Interfaces of SCFTs with G2-Orbifolds
- 9/13/2022: Generalized Symmetries and Compact Models
- 9/9/2022: Higher Symmetries via Cutting and Gluing of Orbifolds
- 6/8/2022: Getting High on Gluing Orbifolds (Part II)
- 9/13/2021: Higgs Bundles for G2 Manifolds and Brane/Particle Probes
March 15, 2023
TITLE: Living on the Edge: Interfaces of SCFTs with G2-Orbifolds
ABSTRACT: Consider the asymptotically conical G2 metric on the bundle of anti-self-dual two-forms over a four-manifold base B. When B is smooth such metrics exhibit large isometry groups and taking quotients by discrete subgroups of these produces a network of codimension-4 and -6 singularities. When B has orbifold singularities there are similar networks prior to any isometry quotient. M-theory maps such spaces to 4D minimally supersymmetric theories. In the conical limit these describe superconformal interfaces between 5D SCFTs. We describe how geometry parametrizes some properties of these strongly coupled interfaces between strongly coupled theories.
September 13, 2022
TITLE: Generalized Symmetries and Compact Models
ABSTRACT: When coupling quantum field theories (QFTs) to each other and gravity their symmetries are believed to be gauged or broken. We consider this process for supersymmetric QFTs engineered in M-theory by local geometries of special holonomy and characterize the breaking and gauging of their higher symmetries. Here, the coupling of theories is geometrized by the embedding of local models into one compact model where topological data of the embedding determines the fate of n-form and n-group symmetries. With local and global K3 surfaces we take our starting point in 7d. We analyze the global structure of the resulting supergravity gauge group, generalizing and simplifying methods centered on Mordell-Weil groups for elliptic K3s, and give results for torus orbifolds. Next, we consider Calabi-Yau threefolds geometrizing couplings between localized 5d supersymmetric conformal sectors and determine the fate of their generalized symmetries.
September 9, 2022
TITLE: Higher Symmetries via Cutting and Gluing of Orbifolds
ABSTRACT: We study the higher symmetry structures of 4d, 5d, 7d quantum field theories (QFTs) which are geometrically engineered in M-theory by non-compact geometries with non-compact ADE loci. We characterize flavor and gauge Wilson lines of such QFTs via orbifold homology groups of the asymptotic boundary geometry and argue that the corresponding 0-form, 1-form and 2-group symmetries are mapped onto the Mayer-Vietoris sequence with respect to a covering derived from the asymptotic orbifold locus and its complement. Further, applying related cutting and gluing constructions to various compact geometries with ADE loci we discuss the global structure of the engineered supergravity gauge group.
June 8, 2022
TITLE: Getting High on Gluing Orbifolds (Part II)
ABSTRACT: In this second talk we extend our discussion from global quotients to more general classes of non-compact orbifolds. In particular we consider M-theory on elliptically fibered Calabi-Yau threefolds with non-compact discriminant loci and G2-spaces constructed as D6 brane uplifts. These setups engineer 5d conformal matter theories, 5d gauge theories and QCD-like theories in 4d and in all these cases we determine the global structure of flavor symmetries and possible 2-groups from geometry. With these results we turn to compact geometries constructed via gluing from local geometries. The gluing couples the QFTs associated with individual local patches to each other and gravity, and as a consequence flavor symmetries are broken or gauged. We characterize both processes geometrically.
September 13, 2021
TITLE: Higgs Bundles for G2 Manifolds and Brane/Particle Probes
ABSTRACT:
We consider M-theory on a local, ALE-fibered G2 manifold. At low energies the effective physics is described by a partially twisted 7d SYM theory. The BPS equations describe a Higgs bundle associated to the ALE fibration of the G2 manifold. We describe how M2-branes probing the G2 manifold descend to particles probing the Higgs bundle. Such probe particles attach a Morse-Witten complex to the geometry. This complex is generated by the singular subloci of the G2 manifold. Supersymmetric three-spheres are in correspondence with flow trees and give rise to boundary maps and cup products.
Jingxiang Wu: Lectures
- 3/15/2023: 3-manifolds and modular tensor categories
- 9/13/2021: On the Vafa-Witten theory on closed four-manifolds
March 15, 2023
TITLE: 3-manifolds and modular tensor categories
ABSTRACT: I will report on some progress in understanding a novel correspondence between modular tensor categories (MTC) and 3-manifolds. In particular, it has been conjectured that to any Seifert manifold M3 with three singular fibres, one can canonically associate a MTC, which is expected to describe the infrared of T[M3], a 3d QFT obtained from wrapping M5 branes over special Lagrangians inside CY3. We examine the conjecture by studying the generalised symmetries and the SymTFT from compactifying topological sectors of M theory using the framework of differential cohomology. We generalise the conjecture to more general Seifert manifolds with arbitrary number of singular fibres. I will also comment on some interesting connections to G2 manifolds. The talk is based on our work in progress with Federico Bonetti and Sakura Schafer-Nameki.
September 13, 2021
TITLE: Kondo line defect and affine oper/Gaudin correspondence
ABSTRACT: It is well-known that the spectral data of the Gaudin model associated to a finite semisimple Lie algebra is encoded by the differential data of certain flat connections associated to the Langlands dual Lie algebra on the projective line with regular singularities, known as oper/Gaudin correspondence. Recently, some progress has been made in understanding the correspondence associated with affine Lie algebras.
I will present a physical perspective from Kondo line defects, physically describing a local impurity chirally coupled to the bulk conformal field theory. The Kondo line defects exhibit interesting integrability properties and wall-crossing behaviors, which are encoded by the generalized monodromy data of affine opers. In the trigonometric setting, this reproduces the known ODE/IM correspondence in the physics literature. I will explain how the recently proposed 4d Chern Simons theory provides a new perspective which suggests the possibility of a physicists’ proof. Along the way, I will also present new examples of ODE/IM correspondences. The talk is based on [2003.06694][2010.07325][2106.07792] in collaboration with D. Gaiotto, J Lee, B. Vicedo.
