Online Seminar
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11032020
Albrecht Klemm: Analytic Structure of all Loop Banana Amplitudes
Abstract: Using the Gelfand-Kapranov-Zelevinsk\uı system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana amplitudes with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel Γˆ-class evaluation in the ambient spaces of the mirror, while the imaginary part of the amplitude in this regime is determined by the Γˆ-class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius κ-constants, which determine the behaviour of the amplitudes, when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogenous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the amplitude as well as other master integrals with raised powers of the propagators in very short time to very high numerical precision for all values of the physical parameters. Using a recent p-adic analysis of the periods we determine the value of the maximal cut equal mass four-loop amplitude at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.
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10272020
Tseng, Hsian-Hua: Relative Gromov-Witten theory without log geometry
Abstract: We describe a new Gromov-Witten theory of a space relative to a simple normal-crossing divisor constructed using multi-root stacks
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10212020
Hiraku Nakajima:Quiver gauge theories with symmetrizers
Abstract: We generalize the mathematical definition of Coulomb branches of 3-dimensional N=4 SUSY quiver gauge theories in arXiv:1503.03676, arXiv:1601.03686, arXiv:1604.03625 to the cases with symmetrizers. We obtain generalized affine Grassmannian slices of type BCFG as examples of the construction, and their deformation quantizations via truncated shifted Yangians. Finally, we study modules over these quantizations and relate them to the lower triangular part of the quantized enveloping algebra of type ADE. (Based on the joint work with Alex Weekes, arXiv:1907.06522.)
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10092020
Richard Thomas: Square root Euler classes and counting sheaves on Calabi-Yau 4-folds
Abstract: I will explain a nice characteristic class of SO(2n,C) bundles in both Chow cohomology and K-theory, and how to localise it to the zeros of an isotropic section. This builds on work of Edidin-Graham, Polishchuk-Vaintrob, Anderson and many others.This can be used to construct an algebraic virtual cycle (and virtual structure sheaf) on moduli spaces of stable sheaves on Calabi-Yau 4-folds.It recovers the real derived differential geometry virtual cycle of Borisov-Joyce but has nicer properties, like a torus localisation formula. Joint work with Jeongseok Oh (KIAS).Seminar slides:
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07282020
Kentaro Hori: D-brane central charges
Abstract: I will describe what are known as D-brane central charges in various contexts including string compactifications, tt^* geometry, hemisphere partition function, Gromov-Witten theory, and old matrix models. I will then discuss their relationships.
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07212020
Shuai Guo:The Landau-Ginzburg/Calabi-Yau correspondence for the quintic threefold
Abstract: In this talk, we will first introduce the physical and mathematical versions of the Landau-Ginzburg/Calabi-Yau correspondence conjecture for the Calabi-Yau threefolds. Then we will explain our approach to prove this conjecture for the most simple Calabi-Yau threefold - the quintic threefold. This is a work in progress joint with Felix Janda and Yongbin Ruan.
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07142020
Huai-Liang Chang: BCOV Feynman structure in A side
Abstract: Gromov Witten invariants Fg encodes the numbers of genus g curves in Calabi Yau threefolds. They play a role in enumerative geometry and are not easy to be determined. In 1993 Bershadsky, Cecotti, Ooguri, Vafa solved holomorphic anomaly equations for Fg's in B side and their solution exhibited a hidden ``Feynman structure” governing all Fg’s at once. The method was via path integral while its counterpart in mathematic has been missing for decades. In 2018, a large N topological string theory is developed in math and provides the wanted ``Feynman structure”. New features are (i) dynamically quantizing the Kaehler moduli parameterin Witten’s GLSM, and (ii) enhancing the Calabi Yau target to a large N bulk with Calabi Yau boundary. Both are achieved within a Landau Ginzburg type construction (P fields and cosections in math terms). This theory, called “N-Mixed Spin P field” in math, is thus responsible for BCOV theory in A side. In it the A model theory of the large N bulk encodes the B model propagators of the boundary’s mirror. This is a new phenomenon intertwining bulk-boundary correspondence, mirror symmetry, and Gauge-String(B-A) duality.In this talk we will see genuine ideas behind these features, along a new angle to realize NMSP from people familiar with Givental theory.
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07072020
Abstract: The Crepant Transformation Conjecture, proposed by Ruan, asserts certain equivalence between Gromov-Witten theory of two manifolds/orbifolds which are related by a crepant transformation. In general, the higher genus Crepant Transformation Conjecture is quite hard to study. Even the formulation of the Crepant Transformation Conjecture in the higher genus case is subtle. In this talk, I will explain the proof of the all genus Crepant Transformation Conjecture for general toric Calabi-Yau 3-orbifolds. We will consider the higher genus Gromov-Witten theory of toric Calabi-Yau 3-orbifolds in both open-string sector and closed-string sector. This talkis based on an ongoing project joint with Bohan Fang, Chiu-Chu Melissa Liu, and Song Yu.
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06302020
Mauricio Romo:Hemisphere partition function, Landau-Ginzburg orbifolds and FJRW invariants
Abstract: We consider LG orbifolds and the central charges of their B-branes (equivariant matrix factorizations). We will focus on the hemisphere partition function on the gauged linear sigma model (GLSM) extension of certain LG orbifolds (even though our proposal does not require a GLSM embedding), and how this provides information about their Gamma class, I/J-function and some predictions about FJRW invariants.
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06232020
Hans Jockers: Wilson Loop Algebras and Quantum K-Theory
Abstract: In this talk we review certain aspects of Wilson line operators in 3d N=2 supersymmetric gauge theories with a Higgs branch that is geometrically described by complex Grassmannians. We discuss the relationship between Wilson loop algebras of the gauge theory and the quantum K-theoretic ring of Schubert structure sheaves of the complex Grassmannians.
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06162020
Du Pei:On Quantization of Coulomb Branches
Abstract: The study of Coulomb branches of supersymmetric quantum field theories have in recent years led to many interesting insights into geometry and representation theory. In this talk, I will discuss the A-models with Coulomb branch targets, and show how to use them to better understand the representation theory of the double affine Hecke algebra.
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06092020
Hiroshi Iritani:Quantum cohomology and birational transformation
Abstract: A famous conjecture of Yongbin Ruan says that quantum cohomology of birational varieties becomes isomorphic after analytic continuation when the birational transformation preserves the canonical class (the so-called crepant transformation). When the transformation is not crepant, the quantum cohomology becomes non-isomorphic, but it is conjectured that one side is a direct summand of the other. In this talk, I will explain a conjecture that a semiorthogonal decomposition of topological K-groups (or derived categories) should induce a relationship between quantum cohomology. The relationship between quantum cohomology can be described in terms of solutions to a Riemann-Hilbert problem.Seminar note:https://www.math.kyoto-u.ac.jp/~iritani/talk_QC_birat.pdf
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06022020
Qile Chen: The logarithmic gauged linear sigma model
Abstract: We introduce the notion of log R-maps generalizing stable maps with p-fields, and develop a proper moduli stack of stable log R-maps in the case of a hybrid gauged linear sigma model. This moduli stack carries two virtual fundamental classes -- the canonical virtual cycle and the reduced virtual cycle. The main results are two comparison theorems: (1) We identify the reduced virtual cycle with the Kiem-Li cosection localized virtual cycle which was shown to recover Gromov-Witten theory of certain critical locus. (2) We relate the reduced virtual cycle to the canonical virtual cycle which can have larger symmetry in many interesting examples. This is part of a project aiming at the foundation of a new technique for computing higher genus Gromov--Witten invariants of complete intersections. The talk consists of joint work with Felix Janda, Yongbin Ruan, and Adrien Sauvaget.
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05262020
Bumsig Kim: Virtual Factorizations
Abstract: In this talk, based on joint work with David Favero, we explain how to define an A-side cohomological field theory for a given gauged linear sigma model (GLSM). For this, we follow the approach by Polishchuk -- Vaintrob, i.e., we construct a virtual factorization on a suitable smooth stack U containing the moduli space LGQ of Landau-Ginzburg stable (quasi)maps. The latter moduli space LGQ was introduced by Fan, Jarvis and Ruan. The state space taken is the hypercohomology of Hodge complex twisted by the superpotential of GLSM in the inertia stack of the associated DM stack of GLSM. We will use the notion of Atiyah class of global matrix factorizations from a joint work with A. Polishchuk and a simplified construction of the ambient stack U from a previous joint work with I. Ciocan-Fontanine, D. Favero, J. Guere, M. Shoemaker.