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## Geometry and Analysis of Moduli Spaces, Imperial College, 6-10 January 2020

Arrival Date: **Sunday January 5, 2020**. Departure Date: **Saturday January 11, 2020**.

**All talks will be at Huxley Bldg 140. [ click for map ] **

Registration and coffee/tea are in the Huxley 5th floor common room.

Scientific Organizers: Dominic Joyce joyce@maths.ox.ac.uk, Aleksander Doan doan@math.columbia.edu.

The workshop will have two main themes:

(A) The Donaldson-Segal programme for defining enumerative invariants of compact G2-manifolds by counting G2 instantons, with correction terms from associative 3-folds. Analysis of G2-instanton moduli spaces, singularities of G2-instantons. Analysis of Seiberg-Witten type equations on 3-manifolds used to define correction terms in Donaldson-Segal programme. Related gauge theory problems, including singularities of Hermitian-Yang Mills connections.

(B) The Pantev-Toën-Vaquié-Vezzosi theory of shifted symplectic derived algebraic geometry, giving geometric structures on Calabi-Yau moduli spaces, and its applications to generalizations of Donaldson-Thomas theory of Calabi-Yau 3- and 4-folds. One particular aim is to present the theory in a way accessible to String Theorists, to encourage communication between mathematicians and physicists on this subject, and to facilitate interpretation of the implications of the theory in String Theory terms. Algebro-geometric enumerative invariants (and Floer theories, etc), particularly of Calabi-Yau manifolds, related to the PTVV theory: Donaldson-Thomas, Gromov-Witten, and Vafa-Witten invariants, Donaldson-Thomas type invariants of Calabi-Yau 4-folds, the Gopakumar-Vafa conjecture.

Attendance at the workshop is by invitation only (except for London locals). Non-invitees wishing to attend may contact the organizers, but are likely to be disappointed, as we are already inviting as many as the lecture theatre will comfortably hold, so we can only include others if we have refusals.

### Speakers:

- Aleksander Doan (Columbia)
- Simon Donaldson (Stony Brook and Imperial College London)
- Penka Georgieva (Paris)
- Andriy Haydys (Freiburg)
- Eleny Ionel (Stanford)
- Dominic Joyce (Oxford)
- Yang Li (Institute for Advanced Study)
- Davesh Maulik (MIT)
- Tony Pantev (University of Pensylvania)
- Tristan Riviere (Zurich)
- Pavel Safronov (Zurich)
- Vivek Shende (U. C. Berkeley)
- Song Sun (U. C. Berkeley)
- Richard Thomas (Imperial College London)

## Albrecht Klemm: Lectures

### September 11, 2019

TITLE: CY 3-folds over finite fields, Black hole attractors, and D-brane masses

ABSTRACT: The integer coefficients of the numerator of the Hasse-Weil Zeta function for one parameter Calabi-Yau 3-folds are expected to be Hecke eigenvalues of Siegel modular forms. For rigid CY 3-folds as well as at conifold — and rank two attractor points this numerator contains factors of lower degree, which can be shown to be the Hecke eigenvalues of weight two or four of modular cusp forms of . We show that the Hecke L-function at integer arguments or more generally the periods of these modular forms give the -brane masses as well as the value and the curvature of the Weil-Peterssen metric at the points. The coefficients of the connection matrix from the integer symplectic basis to a Frobenius basis at the conifold are given by the quasi periods of these modular forms.

## Cumrun Vafa: Lectures

### September 11, 2019

TITLE: G2 Structure and Physical Interpretation of Taubes Construction of SW Invariants

ABSTRACT: In this talk I review a joint recent work with Sergio Cecotti, where we use the structure to shed light on Taubes reformulation of invariant for symplectic 4-manifolds in terms of Gromov invariants.

## Andrei Moroianu: Lectures

### September 10, 2019

TITLE: Toric nearly Kähler 6-manifolds

ABSTRACT: Nearly Kähler manifolds are a particular class of almost Hermitian manifolds in the Gray-Hervella classification, but for several reasons they are mostly relevant in dimension 6, where (normalized, strict) nearly Kähler 6-manifolds can be characterized by the fact that their metric cone has holonomy contained in . In this talk, based on a joint work with P.-A. Nagy, I will show that strict nearly Kähler 6-manifolds admitting effective actions by automorphisms are characterized in the neigbourhood of each point by a function on satisfying a certain Monge–Ampère-type equation.

## Chris Hull: Lectures

### September 10, 2019

TITLE: Special Holonomy Metrics, Degenerate Limits and Intersecting Branes

ABSTRACT: The starting point for this talk is the degenerate limit of K3 constructed by Hein, Sun, Viaclovsky and Zhang, with a long neck that can be thought of as a 3-dimensional nilmanifold fibred over a line, with gravitational instanton insertions. Special holonomy generalisations of this neck region that are given by higher dimensional nilmanifolds fibred over a line will be discussed. They are dual to intersecting brane solutions, and this relation leads to more general solutions. The possibility of these arising as part of a degenerate limit of a compact special holonomy space will be considered, and applications to string theory will be explored.

## Brian Willett: Lectures

### September 9, 2019

TITLE: Global aspects of the 3d-3d correspondence

ABSTRACT: We discuss various aspects of the 3d supersymmetric quantum field theories obtained by compactifying a stack of M5 branes on a three-manifold, arising in the so-called “3d-3d correspondence.” In particular, we emphasize the role of higher-form symmetries of the M5 brane theory, which provide a refinement of the 3d-3d correspondence related to the global structure of the 3d gauge theories. We focus in particular on the case where the three-manifold is a Seifert manifold, where we write 3d Lagrangians for these theories. The vacuum equations take the form of Bethe-ansatz type equations, which we solve in several examples. This leads to new tests of the 3d-3d correspondence, in particular, of the action of the higher symmetry on the set of vacua. We comment on implications for the 3d-3d correspondence, arising from compactification on an associative cycle in a manifold.

## Nelvis Fornasin: Lectures

### September 9, 2019

TITLE: The eta invariant under cone-edge degeneration

ABSTRACT: I will discuss an analytic approach to the computation of the \bar\nu invariant introduced by Crowley, Goette and Nordström.

The \bar\nu invariant is made up of \eta invariants: following work by Sher, I will talk about the behaviour of these invariants under cone-edge degeneration of the underlying manifold. The results can be applied to the computation of the \bar\nu invariant of Joyce’s G_2 manifolds… with some restrictions.

## Ronan Conlon: Lectures

### September 9, 2019

TITLE: Classification results for expanding and shrinking gradient Kähler-Ricci solitons

ABSTRACT: A complete Kahler metric on a Kahler manifold is a “gradient Kahler-Ricci soliton” if there exists a smooth real-valued function with holomorphic such that for a real number. I will present some classification results for such manifolds. This is joint work with Alix Deruelle (Université Paris-Sud) and Song Sun (UC Berkeley).

## Amihay Hanany: Lectures

### September 9, 2019

TITLE: Hasse diagrams for symplectic singularities

ABSTRACT: Symplectic singularities appear in physics as the so called Higgs branches and Coulomb branches of supersymmetric gauge theories. They naturally form a structure of symplectic leaves with a partial order that can be depicted by a Hasse diagram.

The physical interpretation of this geometric structure is the so called “partial Higgs mechanism” where, on each leaf, the so-called “unbroken gauge group” is fixed. Simple examples of such symplectic singularities are closures of nilpotent orbits, and each leaf is associated with a smaller nilpotent orbit.

In this talk, we will explain these points, present the physical intuition, and suggest some results for Hasse diagrams of various Higgs branches that are of interest in some physical systems that arise in string theory.

## Shing-Tung Yau: Lectures

### September 8, 2019

TITLE: String duality and G_{2} manifolds

ABSTRACT: In this talk, I will begin by reviewing the mirror construction of G_{2} manifolds motivated by string duality (Gukov-Zaslow-Yau) and then discuss the physical interpretation of K stability of 3-dimensional Ricci-flat conical metrics (equivalently, Sasaki-Einstein metric and Kahler-Einstein metrics on Fano orbifolds) and how such metrics can be used to construct non-compact G_{2}-manifolds.