Younghan Bae (Michigan)
Surfaces on Calabi-Yau fourfolds
Curve counting theory on Calabi-Yau and Fano threefolds has been a central topic in enumerative geometry. For Calabi-Yau fourfolds, DT4 virtual cycles can be defined on the Hilbert scheme of two-dimensional subschemes using the Borisov-Joyce/Oh-Thomas theory. These classes, however, become trivial when the Hodge locus of the surface class has positive codimension. We reduce the theory and prove that the resulting invariants remain deformation invariant along the Hodge locus. In pursuit of understanding the structure of invariants counting surfaces on Calabi-Yau fourfolds, we turn our attention to the moduli space of stable two-dimensional sheaves. For surfaces with mild singularities, we propose a conjecture that the pushforward of the (reduced) virtual cycle to the Chow variety has modular properties. This is joint work with M. Kool and H. Park.
Andreas Braun (Durham) [Slides - PDF]
G2 Mirror Symmetry
Strings on G2 manifolds exhibit a phenomenon similar to Calabi-Yau mirror symmetry in which strings propagating on topologically different spaces may give rise to equivalent physics. This talk will give a broad overview of the basic ideas, results, and conjectures. There are proposals for how to construct mirrors in most classes of examples of G2 manifolds, and we'll explain how these are motivated and what evidence we have to support them.
François Greer (Michigan State)
Mock modularity of special cycles
Let X be a non-compact Shimura variety associated to the group O(2,n) or U(1,n). Using the theta correspondence, Kudla and Millson showed that certain generating series of special divisors are modular forms with values in the cohomology of X. The analogous generating series of special cycles in codimension g are Siegel forms of genus g. For the purposes of intersection theory, it is natural to pass to a smooth toroidal compactification of X and then take the Zariski closures of the special cycles. I will explain recent work showing that the generating series of these cycle closures are quasi-modular or mixed mock modular. In codimension >1, we find an interesting duality between the Lefschetz filtration and a certain filtration on quasi-Siegel forms.
Damian van de Heisteeg (Harvard) [Slides - PDF]
The Structure of the Flux Landscape
Identifying flux vacua in string theory with stabilized complex structure moduli presents a significant challenge, necessitating the minimization of a scalar potential complicated by infinitely many exponential corrections. In order to obtain exact results we connect three central topics: transcendentality or algebraicity of coupling functions, emergent symmetries, and the distribution of vacua. We demonstrate these ideas on an explicit example where we determine the landscape of exact flux vacua with a vanishing superpotential. We examine the implications of the tadpole bound, which intriguingly confines flux vacua to real values of the moduli, providing a potential avenue for addressing the strong CP problem.
James Hotchkiss (Columbia)
The period-index problem for fields, varieties, and spaces
The period-index problem is an elementary question about central simple algebras over a field, whose origins lie in the calculation of the Brauer groups of number fields from the early part of the twentieth century. The problem is widely open. I will give an introduction to the subject, and explain its connection to a variety of topics in algebraic geometry and topology (derived categories, DT invariants, twisted K-theory). Finally, I will explain how to construct a topological solution to the problem, leaving only the question of algebraizability.
Hans Jockers (Johannes Gutenberg University, Mainz)
Rational Conformal Field Theories and Hodge Structure with Complex Multiplication
It is known that two-dimensional toroidal conformal field theories are rational, if the toroidal target space $T^2$ yields an elliptic curves with complex multiplication. In this talk I review the property of complex multiplication of elliptic curves from a Hodge theoretic perspective. This formulation admits a generalization for two-dimensional sigma models with Calabi-Yau target spaces. Based on examples, I present some evidence that two-dimensional rational conformal field theories associated to Calabi-Yau target spaces give rise to Hodge structures with complex multiplication.
Dominic Joyce (Oxford) and Markus Upmeier (Aberdeen) [Slides Joyce - PDF] [Slides Upmeier - PDF]
Orientations on moduli spaces of coherent sheaves on Calabi-Yau 4-folds. I and II
Let X be a compact Calabi-Yau 4-fold. To define DT4 invariants of X, one needs an orientation on moduli spaces of semistable coherent sheaves on X, or (better) an orientation on the moduli stack M of all perfect complexes on X, in the sense of Borisov-Joyce 2017. Cao-Gross-Joyce 2020 claimed to prove that M is orientable for any Calabi-Yau 4-fold X. Unfortunately, we have found a mistake in their proof, and the theorem itself may be false, though we do not have a counterexample. We will explain how to fix the mistake in Cao-Gross-Joyce under an extra hypothesis on the cohomology H^3(X,Z) (for example, H^3(X,Z)=0 is sufficient). We also explain a choice of extra data (a "flag structure” on H^4(X,Z)) which determines a canonical orientation on M. This is part of a larger project studying orientability and orientations of moduli spaces of connections in gauge theory, and of moduli spaces of special submanifolds. We define "bordism categories” Bord_n(BG) with objects pairs (X,P) of a compact spin n-manifold X and a principal G-bundle P —> X. Then orientations of gauge theory moduli spaces of connections on P can be encoded in a functor from Bord_n(BG) to Z_2-torsors, and an orientation of the gauge theory moduli space B_P corresponds to a trivialization of this functor on a subcategory [X,P] of Bord_n(BG). It turns out that Bord_n(BG) is a "Picard groupoid”, and can be understood in terms of the spin bordism groups Omega_m^{Spin}(BG) for m=n,n+1. So we reduce orientability questions to (difficult) calculations involving spin bordism groups of classifying spaces in Algebraic Topology.
Hee-Cheol Kim (POSTECH) [Slides - PDF]
Finiteness of 6d (1,0) Supergravity Landscape
I will talk about a bottom-up proof of the finiteness of the number of massless fields in any 6d supergravity theories preserving 8 supercharges. From this, sharp bounds on the number of tensor multiplets and the rank of gauge algebras will be derived. I will also show that these bounds are saturated by specific theories realized through F-theory compactifications on elliptic Calabi-Yau threefolds.
Albrecht Klemm (University of Bonn)
The 1,2,3,4,... of applying the period geometry of CY n-folds to Feynman integrals
We explain how geometric and arithmetic properties of Calabi-Yau period geometry serve to make physical predictions in perturbative quantum field theory and general relativity.
Bruno Klingler (Humboldt University)
Recent progress on Hodge loci
Given a quasi projective family S of complex algebraic varieties, its Hodge locus is the locus of points of S where the corresponding fiber admits exceptional Hodge classes (conjecturally: exceptional algebraic cycles). In this talk I will survey the many recent advances in our understanding of such loci, both geometrically and arithmetically, discussing in particular families of Calabi-Yau manifolds.
Pyry Kuusela (Johannes Gutenberg University, Mainz) [Slides - PDF]
Studying black holes, flux vacua, and M-theory via arithmetic geometry
Calabi-Yau compactifications can be used to describe a variety of physical setups including asymptotically flat black holes and flux vacua. Existence of many interesting solutions of this type and their properties are intricately related to the structure of the cohomology of the compactification manifold. In this talk, I discuss how arithmetic geometry and number theory can be used to find these solutions and compute many of their relevant physical quantities, concentrating mainly on the relation between modularity of Calabi-Yau manifolds and supersymmetric flux vacua. I introduce the zeta function of a manifold, discuss briefly how it is computed and how it can be used to find these vacua. In special cases, the zeta function is related to (elliptic) modular forms, and conjectures in number theory relate the modular forms to physical quantities, such as the vacuum expectation value of the axiodilaton field.
Henry Liu (IPMU)
The 4-fold Pandharipande-Thomas vertex
I will explain a conjectural explicit description of the Oh-Thomas virtual class for the K-theoretic Pandharipande-Thomas vertex on C^4. Interestingly, torus-fixed loci in this setting carry both an Oh-Thomas virtual class as well as a Behrend-Fantechi virtual class, and the conjecture is that they are equal, up to some explicit signs. This generalizes previous conjectures by Nekrasov-Piazzalunga and Cao-Kool-Monavari, and satisfies all low-degree checks of the K-theoretic DT4/PT4 vertex correspondence.
Jeongseok Oh (Seoul National University)
A pullback between sheaves on log Calabi-Yau 4-folds
Given a section of a bundle with quadratic form, we construct a specialisation map between K-groups of matrix factorisations of the quadratic function from the space to the normal cone of the zero locus. When the section is isotropic, the quadratic function becomes zero and the construction recovers usual specialisation map in Fulton-MacPherson's intersection theory. When a log Calabi-Yau 4-fold (X,D) is given, we apply the construction to define a pullback from sheaves on D to ones on X after assuming the space of sheaves on D is a critical locus globally. For a Calabi-Yau 4-fold X, we artificially define "the space of sheaves on D" to be a point so that its structure sheaf pulls back to the virtual structure sheaf of the space of sheaves on X. This is a joint work in progress with Dongwook Choa and Richard Thomas.
Nicolò Piazzalunga (Rutgers)
Quasimaps of surfaces
Let S be a compact Kahler surface, and consider two line bundles on it, such that their product is isomorphic to the canonical bundle, and the total space X is a non-compact Calabi-Yau fourfold. We propose a correspondence between Pandharipande-Thomas PT_1 theory on X, as introduced by Bae-Kool-Park, and a supersymmetric gauge theory on S, whose matter content we determine. For S toric, we can use equivariant localization and match the corresponding generating functions. This determines all signs of PT_1 theory on X from first principles. Based on upcoming work with E. Diaconescu.
Boris Pioline (CNRS) [Slides - PDF]
Counting BPS states with scattering diagrams
Generalized Donaldson-Thomas invariants are the mathematical incarnation of the BPS index counting supersymmetric bound states in type IIA strings compactified on a Calabi-Yau threefold $X$. They depend sensitively on the choice of K\"ahler moduli, or more generally Bridgeland stability conditions. When $X$ is the resolution of a toric CY3 singularity, BPS states are described by semi-stable representations of a certain quiver with potential near the singular locus, and by semi-stable coherent sheaves at infinite volume. Scattering diagrams provide a powerful way of encoding (generalized) DT invariants at an arbitrary point in the space of stability conditions, interpolating between these two regimes. I will describe the scattering diagram for the simplest local del Pezzo surfaces, namely the local projective space $K_{P^2}$ and local Hirzebruch surface $K_{P^1\times P^1]$, thereby completely characterizing the BPS spectrum in these examples. Hopefully, similar ideas should also prove useful for local CY4 singularities, or Fano 3-folds. Based on [arXiv:2210.10712, 2412.07680] in collaboration with Pierrick Bousseau, Pierre Descombes, Bruno Le Floch, and Rishi Raj.
Elli Pomoni (Hamburg) [Slides - PDF]
Hidden Symmetries of 4D N=2 Gauge Theories
In this talk we will study the global symmetries of orbifolds of N = 4 Super-Yang-Mills theory and their marginal deformations. The process of orbifolding to obtain an N = 2 theory would appear to break the SU(4) R-symmetry down to SU(2)×U(1). We show that the broken generators can be recovered by moving beyond the Lie algebraic setting to that of a Lie algebroid. This remains true when marginally deforming away from the orbifold point by allowing the different gauge couplings to vary independently. The information about the marginal deformation is captured by a Drinfeld-type twist of this SU(4) Lie algebroid. The twist is read off from the F- and D- terms, and thus directly from the Lagrangian. We will show that the planar Lagrangian of the theory is invariant under this twisted version of the SU(4) algebroid and discuss implications of this hidden symmetry for the spectrum of the N = 2 theories.
Jørgen V. Rennemo (Oslo)
K-theoretic sheaf counting on C^4
Oh and Thomas have defined a K-theoretic sheaf counting invariant for moduli spaces of sheaves on a Calabi-Yau 4-fold. One of the simplest examples of such a moduli space is the Hilbert scheme of n points on C^4. The topic of this talk is the proof of a formula for a generating function of K-theoretic invariants on these Hilbert schemes, verifying a conjecture of Nekrasov (as well as its generalisation to Quot schemes on C^4, conjectured by Nekrasov and Piazzalunga). This is joint work in progress with Martijn Kool.
Yukinobu Toda (IPMU)
Dolbeault Geometric Langlands conjecture via quasi-BPS categories
In this talk, I will introduce the notion of `limit category' for cotangents of smooth stacks, which is expected to give a categorical degeneration of the category of D-modules on them. I show that the limit category for the moduli stack of Higgs bundles admits a semiorthogonal decomposition into products of quasi-BPS categories, which are categorifications of BPS invariants of some non-compact Calabi-Yau 3-folds. I propose the formulation of Dolbeault Geometric Langlands conjecture using the limit category, which is regarded as a classical limit of Geometric Langlands correspondence. I also show that the limit category admits Hecke operators. This is a joint work in progress with Tudor Padurariu.
Richard P. Thomas (Imperial College) [Slides Thomas - PDF]
A Hopf index for isotropic sections of bundles with quadratic forms
The Hopf index theorem counts numbers of solutions of equations via a winding number. I’ll describe a number of analogues for equations which form isotropic sections of bundles with quadratic form. There are applications to “cosection localised virtual cycles” and to “DT4 virtual cycles”. This is joint work with Martijn Kool, Jeongseok Oh and Jørgen Rennemo.
Timo Weigand (Hamburg) [Slides - PDF]
Asymptotics in quantum gravity beyond geometry
The asymptotic degeneration of a quantum gravity theory near infinite distance boundaries of its moduli space is believed to be among the universal properties which every consistent such theory must exhibit.Specifically, the Distance Conjecture and a refined version, the Emergent String Conjecture, predict the emergence of light towers of states towards the boundaries of moduli space in every consistent quantum gravity. In compactifications of string or M-theory on Calabi-Yau varieties, these predictions have been verified explicitly by tracing them back to non-trivial properties of the underlying Calabi-Yau moduli spaces. This leads to the question under which conditions said asymptotic properties of a quantum gravity theory can be argued for without having to assume a geometric origin via string or M-theory from the beginning. In this talk we analyse this question in the context of general five-dimensional supergravity theories with eight supercharges, focusing on infinite distance limits in the vector multiplet moduli space. Our main assumption is the existence of certain string-like solutions in the effective field theory known as supergravity strings, as suggested by the Completeness Hypothesis of quantum gravity. With this, we argue that the infinite distance limits are precisely as predicted by the Emergent String Conjecture.The key ingredient is that consistency conditions imposed by the existence of supergravity strings lead to structures completely analogous to the asymptotic behaviour of classical Kahler moduli spaces of Calabi-Yau threefolds, reproducing key geometric insights without having to use geometry to begin with. As a result, we provide evidence that every consistent five-dimensional N=1 supergravity with a non-compact vector multiplet moduli space is either a compactification from six to five dimensions or a critical string theory.