Algebraic Geometry in String Theory

During the 2021–22 academic year, the CMSA will be hosting a seminar on Algebraic Geometry in String Theory, organized by Yun Shi and Tsung-Ju Lee. This seminar will take place on Tuesdays at 9:30am – 10:30am (Boston time). The meetings will take place virtually on Zoom. To learn how to attend, please fill out this form, or contact the organizers Yun (yshi@cmsa.fas.harvard.edu) and Tsung-Ju (tjlee@cmsa.fas.harvard.edu).

The schedule below will be updated as talks are confirmed.

DateSpeakerTitle/Abstract
9/7/2021Fei Xie, University of EdinburghTitle: Derived categories of nodal quintic del Pezzo threefolds

Abstract: Conifold transitions are important algebraic geometric constructions that have been of special interests in mirror symmetry, transforming Calabi-Yau 3-folds between A- and B-models. In this talk, I will discuss the change of the quintic del Pezzo 3-fold (Fano 3-fold of index 2 and degree 5) under the conifold transition at the level of the bounded derived category of coherent sheaves. The nodal quintic del Pezzo 3-fold X has at most 3 nodes. I will construct a semiorthogonal decomposition for D^b(X) and in the case of 1-nodal X, detail the change of derived categories from its smoothing to its small resolution.
9/14/2021Will Donovan, Tsinghua UniversityTitle: Simplices in the Calabi–Yau web

Abstract: Calabi–Yau manifolds of a given dimension are connected by an intricate web of birational maps. This web has deep consequences for the derived categories of coherent sheaves on such manifolds, and for the associated string theories. In particular, for 4-folds and beyond, I will highlight certain simplices appearing in the web, and identify corresponding derived category structures.
9/21/2021Xujia Chen, Harvard UniversityTitle: What do bounding chains look like, and why are they related to linking numbers?

Abstract: Gromov-Witten invariants count pseudo-holomorphic curves on a symplectic manifold passing through some fixed points and submanifolds. Similarly, open Gromov-Witten invariants are supposed to count disks with boundary on a Lagrangian, but in most cases such counts are not independent of some choices as we would wish. Motivated by Fukaya’11, J. Solomon and S. Tukachinsky constructed open Gromov-Witten invariants in their 2016 papers from an algebraic perspective of $A_{\infty}$-algebras of differential forms, utilizing the idea of bounding chains in Fukaya-Oh-Ohta-Ono’06. On the other hand, Welschinger defined open invariants on sixfolds in 2012 that count multi-disks weighted by the linking numbers between their boundaries. We present a geometric translation of Solomon-Tukachinsky’s construction. From this geometric perspective, their invariants readily reduce to Welschinger’s.
9/28/2021Alan Thompson, Loughborough UniversityTitle: The Mirror Clemens-Schmid Sequence

Abstract: I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a “mirror P=W” conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting. This is joint work with Charles F. Doran.
10/7/2021

*Special Time: 10:00PM – 11:00PM (ET)*
Mark Shoemaker, Colorado State UniversityTitle: A mirror theorem for GLSMs

Abstract: A gauged linear sigma model (GLSM) consists roughly of a complex vector space V, a group G acting on V, a character \theta of G, and a G-invariant function w on V.  This data defines a GIT quotient Y = [V //_\theta G] and a function on that quotient.  GLSMs arise naturally in a number of contexts, for instance as the mirrors to Fano manifolds and as examples of noncommutative crepant resolutions. GLSMs provide a broad setting in which it is possible to define an enumerative curve counting theory, simultaneously generalizing FJRW theory and the Gromov-Witten theory of hypersurfaces. Despite a significant effort to rigorously define the enumerative invariants of a GLSM, very few computations of these invariants have been carried out.  In this talk I will describe a new method for computing generating functions of GLSM invariants.  I will explain how these generating functions arise as derivatives of generating functions of Gromov-Witten invariants of Y.
10/12/2021Qingyuan Jang, University of EdinburghTitle: Derived projectivizations of two-term complexes

Abstract: For a given two-term complex of vector bundles on a derived scheme (or stack), there are three natural ways to define its “derived projectivizations”: (i) as the derived base-change of the classical projectivization of Grothendieck; (ii) as the derived moduli parametrizing one-dimensional locally free quotients; (iii) as the GIT quotient of the total space by $\mathbb{G}_m$-action. In this talk, we first show that these three definitions are equivalent. Second, we prove a structural theorem about the derived categories of derived projectivizations and study the corresponding mutation theory. Third, we apply these results to various moduli situations, including the moduli of certain stable pairs on curves and the Hecke correspondences of one-point modification of moduli of stable sheaves on surfaces. If time allowed, we could also discuss the generalizations of these results to the derived Quot schemes of locally free quotients.
10/19/2021Andrea T. Ricolfi, Università di BolognaTitle: D-critical structure(s) on Quot schemes of points of Calabi-Yau 3-folds

Abstract: D-critical schemes and Artin stacks were introduced by Joyce in 2015, and play a central role in Donaldson-Thomas theory. They typically occur as truncations of (-1)-shifted symplectic derived schemes, but the problem of constructing the d-critical structure on a “DT moduli space” without passing through derived geometry is wide open. We discuss this problem, and new results in this direction, when the moduli space is the Hilbert (or Quot) scheme of points on a Calabi-Yau 3-fold. Joint work with Michail Savvas.
10/26/2021Xiaowen HuTitle: On singular Hilbert schemes of points

Abstract: It is well known that the Hilbert schemes of points on smooth surfaces are smooth. In higher dimensions the Hilbert schemes of points are in general singular. In this talk we will present some examples and conjectures on the local structures of the Hilbert scheme of points on $\mathbb{P}^3$. As an application we study a conjecture of Wang-Zhou on the Euler characteristics of the tautological sheaves on Hilbert schemes of points.
11/2/2021Hossein Movasati, IMPATitle: Gauss-Manin connection in disguise: Quasi Jacobi forms of index zero

Abstract: We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields. This is a joint work with J. Cao and R. Villaflor. (arXiv:2109.00587)
11/9/2021Michail Savvas, UT AustinTitle: Cosection localization for virtual fundamental classes of d-manifolds and Donaldson-Thomas invariants of Calabi-Yau fourfolds

Abstract: Localization by cosection, first introduced by Kiem-Li in 2010, is one of the fundamental techniques used to study invariants in complex enumerative geometry. Donaldson-Thomas (DT) invariants counting sheaves on Calabi-Yau fourfolds were first defined by Borisov-Joyce in 2015 by combining derived algebraic and differential geometry. 
In this talk, we develop the theory of cosection localization for derived manifolds in the context of derived differential geometry of Joyce. As a consequence, we also obtain cosection localization results for (-2)-shifted symplectic derived schemes. This provides a cosection localization formalism for the Borisov-Joyce DT invariant. As an immediate application, the stable pair invariants of hyperkähler fourfolds, constructed by Maulik-Cao-Toda, vanish, as expected.
11/16/2021Pierrick Bousseau, ETH ZürichTitleGromov-Witten theory of complete intersections

Abstract: I will describe an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten invariants, we introduce the new notion of nodal relative Gromov-Witten invariants. This is joint work with Hülya Argüz, Rahul Pandharipande, and Dimitri Zvonkine (arxiv:2109.13323).
11/23/2021Dori Bejleri, HarvardTitle: Wall crossing for moduli of stable log varieties

Abstract: Stable log varieties or stable pairs (X, D) are the higher dimensional generalization of pointed stable curves. They form proper moduli spaces which compactify the moduli space of normal crossings, or more generally klt, pairs. These stable pairs compactifications depend on a choice of parameters, namely the coefficients of the boundary divisor D. In this talk, after introducing the theory of stable log varieties, I will explain the wall-crossing behavior that governs how these compactifications change as one varies the coefficients. I will also discuss some examples and applications. This is joint work with Kenny Ascher, Giovanni Inchiostro, and Zsolt Patakfalvi.
11/30/2021Charles Doran, University of AlbertaTitle: K_2 and Quantum Curves

Abstract: The talk will be based on my paper of the same name with Matt Kerr and Soumya Sinha Babu as arXiv:2110.08482.
12/07/2021Xingyang Yu, NYU Title: 2d N=(0,1) gauge theories, Spin(7) orientifolds and triality

Abstract: I will introduce a new brane engineering for 2d minimally supersymmetric, i.e. N=(0,1), gauge theories. Starting with 2d N=(0,2) gauge theories on D1-branes probing Calabi-Yau 4-folds, a brand new orientifold configuration named ’Spin(7) orientifold’ is constructed and the resultant 2d N=(0,1) theories on D1-branes are derived. Using this method, one can build an infinite family of 2d N=(0,1) gauge theories explicitly. Furthermore, the N=(0,1) triality, proposed by Gukov, Pei and Putrov, enjoys a geometric interpretation as the non-uniqueness of the map between gauge theories and Spin(7) orientifolds. The (0,1) triality can then be regarded as inherited from the N=(0,2) triality of gauge theories associated with Calabi-Yau 4-folds. Furthermore, there are theories with N=(0,1) sector coupled to (0,2) sector, where both sectors respectively enjoy (0,1) and (0,2) trialities.


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