CMSA

Species Scale across String Moduli Spaces
2023-09-25 10:00 - 11:30 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar

Pre-talk Speaker: David Wu (Harvard Physics): 10:00-10:30 am
Speaker: Damian van de Heisteeg, CMSA

Title: Species Scale across String Moduli Spaces
Abstract: String theories feature a wide array of moduli spaces. We propose that the energy cutoff scale of these theories – the so-called species scale – can be determined through higher-curvature corrections. This species scale varies with the moduli; we use it both asymptotically to bound the diameter of the field space, as well as in the interior to determine a “desert point” where it is maximized.

Motivic decomposition of moduli space from brane dynamics
2023-10-02 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar

Pre-talk Speaker: Kai Xu (CMSA): 10:00-10:30 am

Speaker: Kai Xu (CMSA)
Title: Motivic decomposition of moduli space from brane dynamics
Abstract: Supersymmetric gauge theories encode deep structures in algebraic geometry, and geometric engineering gives a powerful way to understand the underlying structures by string/M theory. In this talk we will see how the dynamics of M5 branes tell us about the motivic and semiorthogonal decompositions of moduli of bundles on curves.

Moduli of boundary polarized Calabi-Yau pairs
2023-10-16 10:00 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Pre-talk Speaker: Rosie Shen (Harvard): 10:00-10:30 am
Pre-talk Title: Introduction to the singularities of the MMP

Speaker: Dori Bejleri (Harvard Math & CMSA)
Title: Moduli of boundary polarized Calabi-Yau pairs
Abstract: The theories of KSBA stability and K-stability furnish compact moduli spaces of general type pairs and Fano pairs respectively. However, much less is known about the moduli theory of Calabi-Yau pairs. In this talk I will present an approach to constructing a moduli space of Calabi-Yau pairs which should interpolate between KSBA and K-stable moduli via wall-crossing. I will explain how this approach can be used to construct projective moduli spaces of plane curve pairs. This is based on joint work with K. Ascher, H. Blum, K. DeVleming, G. Inchiostro, Y. Liu, X. Wang.

Gauged Linear Sigma Models and Cohomological Field Theories
2023-10-23 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar

Speaker: David Favero, University of Minnesota

Title: Gauged Linear Sigma Models and Cohomological Field Theories
Abstract: This talk is dedicated to the memory of my friend and collaborator Bumsig Kim and based on joint work with Ciocan-Fontanine-Guere-Kim-Shoemaker. Gauged Linear Sigma Models (GLSMs) serve as a means of interpolating between Kahler geometry and singularity theory. In enumerative geometry, they should specialize to both Gromov-Witten and Fan-Jarvis-Ruan-Witten theory. In joint work with Bumsig Kim (see arXiv:2006.12182), we constructed such enumerative invariants for GLSMs. Furthermore, we proved that these invariants form a Cohomological Field Theory. In this lecture, I will describe GLSMs and Cohomological Field Theories, review the history of their development in enumerative geometry, and discuss the construction of these general invariants. Briefly, the invariants are obtained by forming the analogue of a virtual fundamental class which lives in the twisted Hodge complex over a certain “moduli space of maps to the GLSM”. This virtual fundamental class roughly comes as the Atiyah class of a “virtual matrix factorization” associated to the GLSM data.

A p-adic Laplacian on the Tate curve
2023-11-27 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: An Huang (Brandeis)
Pre-talk Speaker: TBA: 10:00-10:30 am

Title: A p-adic Laplacian on the Tate curve
Abstract: We shall first explain the relation between a family of deformations of genus zero p-adic string worldsheet action and Tate’s thesis. We then propose a genus one p-adic string worldsheet action. The key is the definition of a p-adic Laplacian operator on the Tate curve. We show that the genus one p-adic Green’s function exists, is unique under some obvious constraints, is locally constant off diagonal, and has a reflection symmetry. It can also be numerically computed exactly off the diagonal, thanks to some simplifications due to the p-adic setup. Numerics suggest that at least in some special cases, the asymptotic behavior of the Green’s function near the diagonal is a direct p-adic counterpart of the familiar Archimedean case, although the p-adic Laplacian is not a local operator. Joint work in progress with Rebecca Rohrlich.

M-theory on nodal Calabi-Yau 3-folds and torsion refined GV-invariants
2023-12-11 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Thorsten Schimannek (Utrecht University)

Title: M-theory on nodal Calabi-Yau 3-folds and torsion refined GV-invariants
Abstract: The physics of M-theory and Type IIA strings on a projective nodal CY 3-folds is determined by the geometry of a small resolution, even if the latter is not Kähler. We will demonstrate this explicitly in the context of a family of Calabi-Yau double covers of P^3. Using conifold transitions, we prove that the exceptional curves in any small resolution are torsion while M-theory develops a discrete gauge symmetry.This leads to a torsion refinement of the ordinary Gopakumar-Vafa invariants, that is associated to the singular Calabi-Yau and captures the enumerative geometry of the non-Kähler resolutions. We further argue that twisted circle compactifications of the 5d theory are dual to IIA compactifications on the nodal CY 3-fold with a flat but topologically non-trivial B-field. As a result, the torsion refined invariants are encoded in the topological string partition functions with different choices for the global topology of a flat B-field.
The talk is based on 2108.09311, 2212.08655 (with S. Katz, A. Klemm, and E. Sharpe) and 2307.00047 (with S. Katz).

Algebraic billiards and dynamical degrees
2024-02-01 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Max Weinreich (Harvard)
Title: Algebraic billiards and dynamical degrees
Abstract: Billiards is one of the most-studied dynamical systems, modeling the behavior of a point particle bouncing around some space. If the space is a plane region bounded by an algebraic curve, then we may use techniques from algebraic geometry to study its billiards map. We explain how to view billiards as a complex algebraic correspondence, and we prove upper and lower bounds on the dynamical degree, the growth rate of the degrees of the iterates, in terms of the degree of the boundary curve. These degree growth rates are studied in mathematical physics, broadly speaking, as a way to identify integrable (exactly solvable) physical models. In our setting, this theory gives us an upper bound on the entropy, or chaos, of billiards in curves.

On (semi)stable reduction and KSBA moduli in positive characteristic
2024-02-08 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Iacopo Brivio (Harvard CMSA)
Title: On (semi)stable reduction and KSBA moduli in positive characteristic
Abstract: The moduli space M_g of genus g stable curves is perhaps the most studied of all algebraic varieties. Its higher-dimensional generalization is the moduli functor M_{n,v} of n-dimension stable varieties of volume v. It was proven only recently, and thanks to the joint effort of many over many years, that such functors are represented by projective algebraic spaces when working over the complex numbers. In this talk I will give some examples showing that the same moduli functors in positive characteristic are not even proper and, more in general, that the MMP fails to be functorial even in very nice families. In the second part I am going to explore some possible generalizations of the notion of stable variety that could be used as a replacement in positive characteristic.

Geometric origins of values of the Riemann Zeta functions at positive integers
2024-02-22 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Yan Zhou, Northeastern
Title: Geometric origins of values of the Riemann Zeta functions at positive integers
Abstract: Given a Fano manifold, Iritani proposed that the asymptotic behavior of solutions to the quantum differential equation of the Fano should be given by the so-called ‘Gamma class’ in its cohomology ring. Later, Abouzaid-Ganatra-Iritani-Sheridan reformulated the ‘Gamma conjecture’ for Calabi-Yau manifolds via the tropical SYZ mirror symmetry and proposed that values of the Riemann Zeta function at positive integers have geometric origins in the tropical periods and singularities of the SYZ geometry. In this talk, we will first review the content of the Gamma conjecture. Then, we will discuss the first step of generalizing AGIS’ approach to Gamma conjecture for the Gross-Siebert mirror families of a Fano manifold in dimension 2 cases, based on joint work with Bohan Fang and Junxiao Wang.

Classifying curves on Fano varieties
2024-02-29 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Brian Lehmann (Boston College)
Title: Classifying curves on Fano varieties
Abstract: How can we understand the set of curves on a Fano variety? One perspective is provided by Geometric Manin’s Conjecture, a collection of conjectures with roots in arithmetic and topology. While I will mention some recent progress, the main focus will be developing a conceptual framework for thinking about our question.

Geometric construction of toric NCRs
2024-03-07 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Jesse Huang, University of Alberta
Title: Geometric construction of toric NCRs
Abstract: The Rouquier dimension of a toric variety is recently shown to be achieved by the Frobenius pushforward of O via coherent-constructible correspondence. From the perspective of noncommutative geometry, this result leads to a geometric construction of toric NCR of the invariant ring of the Cox ring with respect to a multi-grading which also gives the information about its global dimension. From the perspective of mirror symmetry, the same construction provides a universal “wall skeleton” capturing VGIT wall-crossings, which contains a window for each chamber as a full subcategory. From the perspective of commutative algebra, the same construction indicates the existence of virtual resolutions of the multigraded diagonal bimodule, which agrees with a recent result of Hanlon-Hicks-Larzarev constructing one such resolution explicitly. In this talk, I will survey these perspectives. The talk is based on joint works with P. Zhou, joint works with D. Favero, and work in progress with D. Favero.

The KSBA moduli space of log Calabi-Yau surfaces
2024-03-21 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Pierrick Bousseau, University of Georgia
Title: The KSBA moduli space of log Calabi-Yau surfaces
Abstract: The KSBA moduli space, introduced by Kollár–Shepherd-Barron, and Alexeev, is a natural generalization of “the moduli space of stable curves” to higher dimensions. It parametrizes stable pairs (X,B), where X is a projective algebraic variety satisfying certain conditions and B is a divisor such that K_X+B is ample. This moduli space is described concretely only in a handful of situations: for instance, if X is a toric variety and B=D+epsilon C, where D is the toric boundary divisor and C is an ample divisor, it is shown by Alexeev that the KSBA moduli space is a toric variety. Generally, for a log Calabi-Yau variety (X,D) consisting of a projective variety X and an anticanonical divisor D, with B=D+epsilon C where C is an ample divisor, it was conjectured by Hacking–Keel–Yu that the KSBA moduli space is still toric (up to passing to a finite cover). In joint work with Alexeev and Argüz, we prove this conjecture for all log Calabi-Yau surfaces. This uses tools from the minimal model program, log smooth deformation theory and mirror symmetry.

Derived categories of genus one curves and torsors over abelian varieties
2024-04-04 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Jonathan Rosenberg, University of Maryland

Title: Derived categories of genus one curves and torsors over abelian varieties

Abstract: Studying orientifold string theories on elliptic curves or abelian
varieties motivates studying the derived category of coherent sheaves on
a genus one curve or a torsor over an abelian variety over the reals
(as opposed to the complex numbers).

In joint work with Nirnajan Ramachandran (to appear in MRL), we show that
a genus one curve over a perfect field determines a class in the relative
Brauer group of the Jacobian elliptic curve, and that there is a natural
Mukai-type derived equivalence between the original genus one curve
and the Jacobian twisted by the Brauer class. The proof extends to
torsors over abelian varieties (of any dimension).

Mirror symmetry for fibrations and degenerations of K3 surfaces
2024-04-11 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Alan Thompson (Loughborough University)
Title: Mirror symmetry for fibrations and degenerations of K3 surfaces
Abstract: In 2016, Doran, Harder, and I conjectured a mirror symmetric relationship between Tyurin degenerations and splittings of codimension 1 fibrations on Calabi-Yau manifolds. In this talk I will discuss recent work to make this conjecture rigorous in the case of K3 surfaces. I will give a precise definition of what it means for a Tyurin degeneration of K3’s to be mirror to a splitting of an elliptically fibred K3, and show that this definition enjoys the following compatibilities with existing mirror symmetric theories: 1) The general fibre of the Tyurin degeneration is mirror to the elliptically fibred K3, in the sense of Dolgachev-Nikulin. 2) Components of the Tyurin degeneration and pieces of the splitting satisfy a homological version of the (quasi-) Fano-LG correspondence. 3) Components of the Tyurin degeneration which are weak del Pezzo are mirror to pieces of the splitting that arise as restrictions of the corresponding lattice polarised LG models to discs. This is joint work with Luca Giovenzana.

Geometric local systems on very general curves
2024-04-18 10:15 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Aaron Landesman, MIT
Title: Geometric local systems on very general curves
Abstract: What is the smallest genus h of a non-isotrivial curve over the generic genus g curve? In joint work with Daniel Litt, we show h is more than $sqrt{g}$ by proving amore general result about variations of Hodge structure on sufficiently general curves. As a consequence, we show that local systems on a sufficiently general curve of geometric origin are not Zariski dense in the character variety parameterizing such local systems. This gives counterexamples to conjectures of Esnault-Kerz and Budur-Wang.

The logarithmic double ramification locus
2024-04-25 10:30 - 11:30 | Virtual
Algebraic Geometry in String Theory Seminar
Speaker: Alessandro Chiodo, IMJ-Paris Rive Gauche (Jussieu)
Title: The logarithmic double ramification locus
Abstract: Given a family of smooth curves C -> S with a line bundle L on C, it is natural to study the locus of points x in S where L_x is trivial on C_x. When the family is stable, the definition can be extended, not directly on the base scheme S, but more naturally on a (logarithmic) blow-up S’ of S. The problem is in many ways analogue to the problem of defining a Néron model on the moduli space of stable curves (instead of a DVR). Over the past years, David Holmes and his collaborators pioneered a new approach on a logarithmic modification of the entire moduli space of curves. In this talk, we determine this logarithmic double ramification cycle and several variants and alternative presentations of it (work in collaboration with David Holmes).

Computing periods of hypersurfaces and elliptic surfaces via effective homology
2024-05-09 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Eric Pichon-Pharabod, Universite Paris-Saclay
Title: Computing periods of hypersurfaces and elliptic surfaces via effective homology
Abstract: The period matrix of a smooth complex projective variety X encodes the isomorphism between its singular homology and its algebraic De Rham cohomology. Numerical approximations with sufficient precision of the entries of this matrix, called periods, allow to recover some algebraic invariants of the variety, such as the Néron-Severi group in the case of surfaces. In this talk, we will present a method relying on the computation of an effective description of the homology for obtaining such numerical approximations of the periods of hypersurfaces and elliptic surfaces.

Mirror symmetry and log del Pezzo surfaces
2024-05-16 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Franco Rota, University of Glasgow
Title: Mirror symmetry and log del Pezzo surfaces
Abstract: The homological mirror symmetry conjecture predicts a duality, expressed in terms of categorical equivalences, between the complex geometry of a variety X (the B side) and the symplectic geometry of its mirror object Y (the A side). Motivated by this, we study a series of singular surfaces (called log del Pezzo). I will describe the category arising in the B side, using the McKay correspondence and explicit birational geometry. I will discuss early results on the A side, using the language of pseudolattices to focus on the special case of a smooth degree 2 del Pezzo surface. This is joint work with Giulia Gugiatti.
