?
Degenerations, transitions and quantum cohomology
P. 1269–1273.
Galkin S.
Given a singular variety I discuss the relations between quantum cohomology of its resolution and smoothing. In particular, I explain how toric degenerations help with computing Gromov–Witten invariants, and the role of this story in “Fanosearch” programme. The challenge is to formulate enumerative symplectic geometry of complex 3-folds in a way suitable for extracting invariants under blowups, contractions, and transitions.
Publication based on the results of:
Kalashnikov E. G., / Series arXiv "arXiv". 2020.
We introduce a superpotential for partial flag varieties of type A. This is a map W:Y∘→C, where Y∘ is the complement of an anticanonical divisor on a product of Grassmannians. The map W is expressed in terms of Plücker coordinates of the Grassmannian factors. This construction generalizes the Marsh--Rietsch Plücker coordinate mirror for Grassmannians. We show that in a distinguished cluster ...
Added: November 26, 2020
Losev A. S., Mnev P., Youmans D., Communications in Mathematical Physics 2020 Vol. 376 P. 993–1052
We study two-dimensional non-abelian BF theory in Lorenz gauge and prove that it is a topological conformal field theory. This opens the possibility to compute topological string amplitudes (Gromov–Witten invariants). We found that the theory is exactly solvable in the sense that all correlators are given by finite-dimensional convergent integrals. Surprisingly, this theory turns out to be ...
Added: November 11, 2020
Gusein-Zade S., Manuscripta Mathematica 2018 Vol. 155 No. 3-4 P. 335–353
For a germ of a quasihomogeneous function with an isolated critical point at the origin invariant with respect to an appropriate action of a finite abelian group, H. Fan, T. Jarvis, and Y. Ruan defined the so-called quantum cohomology group. It is considered as the main object of the quantum singularity theory (FJRW-theory). We define ...
Added: October 27, 2020
Gorbounov V., Корфф К., Строппель К., Успехи математических наук 2020 Т. 75 № 5(455) С. 3–58
We survey a recent development which connects quantum integrable models with Schubert calculus for quiver varieties: there is a purely geometric construction of solutions to the Yang-Baxter equation and their associated Yang-Baxter algebras which play a central role in quantum integrable systems and exactly solvable lattice models in statistical physics. We will provide a simple ...
Added: September 9, 2020
Barannikov S., / Series arXiv "math". 2017.
The equivariantly closed matrix integrals introduced in [B06], see also [B10], are studied in the case of the graded associative algebras with odd or even scalar product. ...
Added: October 25, 2018
Barannikov S., Letters in Mathematical Physics 2019 Vol. 109 No. 3 P. 699–724
https://arxiv.org/abs/1803.11549
I describe a combinatorial construction of the cohomology classes in compactified moduli spaces of curves ZˆI∈H∗(barM_g,n) starting from the following data: an odd derivation I, whose square is non-zero in general, I2≠0, acting on a ℤ/2ℤ-graded associative algebra with odd scalar product. The constructed cocycles were first described in the theorem 2 in the author's paper "Noncommmutative Batalin-Vilkovisky geometry and ...
Added: October 5, 2018
Galkin S., / Series math "arxiv.org". 2018. No. 1809.02737.
Given a singular variety I discuss the relations between quantum cohomology of its resolution and smoothing. In particular, I explain how toric degenerations helps with computing Gromov--Witten invariants, and the role of this story in Fanosearch programme. The challenge is to formulate enumerative symplectic geometry of complex 3-folds in a way suitable for extracting invariants ...
Added: September 25, 2018
Galkin S., Iritani H., , in: Primitive Forms and Related Subjects — Kavli IPMU 2014.: Tokyo: Mathematical Society of Japan, 2019. P. 55–115.
The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class.
Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class associated to Euler's Gamma-function.
We illustrate in the case of ...
Added: September 1, 2018
Anno R., Bezrukavnikov R., Mirkovic I., Moscow Mathematical Journal 2015 Vol. 15 No. 2 P. 187–203
The paper provides new examples of an explicit submani-fold in Bridgeland stabilities space of a local Calabi-Yau. More precisely, let X be the standard resolution of a transversal slice to an adjoint nilpotent orbit of a simple Lie algebra over C. An action of the affine braid group on the derived category Db(Coh(X)) and a ...
Added: September 4, 2015
Galkin S., Iritani H., / Series math "arxiv.org". 2015. No. 1508.00719.
The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class G_F associated to Euler's Γ-function. We illustrate in ...
Added: August 5, 2015
Coates T., Corti A., Galkin S. et al., Geometry and Topology 2016 Vol. 20 No. 1 P. 103–256
The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3-dimensional Fano manifolds. In particular we show that 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by ...
Added: November 18, 2014
Galkin S., Golyshev V., Iritani H., Duke Mathematical Journal 2016 Vol. 165 No. 11 P. 2005–2077
We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the ...
Added: November 18, 2014
Galkin S., Mellit A., Smirnov M., International Mathematics Research Notices 2015 Vol. 2015 No. 18 P. 8847–8859
We show that the big quantum cohomology of the symplectic isotropic Grassmanian IG(2,6) is generically semisimple, whereas its small quantum cohomology is known to be non-semisimple. This gives yet another case where Dubrovin's conjecture holds and stresses the need to consider the big quantum cohomology in its formulation. ...
Added: October 20, 2014
Galkin S., Mellit A., Smirnov M., / Series math "arxiv.org". 2014. No. 1405.3857.
We show that the big quantum cohomology of the symplectic isotropic Grassmanian IG(2,6) is generically semisimple, whereas its small quantum cohomology is known to be non-semisimple. This gives yet another case where Dubrovin's conjecture holds and stresses the need to consider the big quantum cohomology in its formulation. ...
Added: May 16, 2014