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## Feynman transform and cohomological field theories

hal.archives-ouvertes.fr (CNRS)
,
2018.
No. 01804639.

Barannikov S.

The construction from [B06], see also [B10], of cohomology classes of compactified moduli spaces of Riemann surfaces, starting from a derivation of associative whose square is nonzero, is generalized to the case of A-infinity algebras. It is shown that the constructed cohomology classes define Cohomological Field Theory.

Buryak A., Mathematical Research Letters 2016 Vol. 23 No. 3 P. 675-683

In a previous paper we proved that after a simple transformation the generating series of the linear Hodge integrals on the moduli space of stable curves satisfies the hierarchy of the Intermediate Long Wave equation. In this paper we present a much shorter proof of this fact. Our new proof is based on an explicit ...

Added: September 28, 2020

Galkin S., Rybakov S., / Cornell University. Series math "arxiv.org". 2019. No. 1910.14379.

For a family of K3 surfaces we implement a variation of a general construction of towers of algebraic curves over finite fields given in a previous paper. As a result we get a good tower over k=𝔽_{p^2}, that is optimal if p=3. ...

Added: November 6, 2019

Galkin S., / Cornell University. Series math "arxiv.org". 2014. No. 1404.7388.

Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates. As a consequence we obtain a positive answer ...

Added: May 4, 2014

Gusein-Zade S., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2020 Vol. 16 No. 051 P. 1-15

P. Berglund, T. Hübsch, and M. Henningson proposed a method to construct mirror symmetric Calabi–Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group
of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal ...

Added: October 27, 2020

Coates T., Galkin S., Kasprzyk A. et al., Experimental Mathematics 2020 Vol. 29 No. 2 P. 183-221

We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles. ...

Added: September 1, 2018

Buryak A., Communications in Mathematical Physics 2015 Vol. 336 No. 3 P. 1085-1107

It this paper we present a new construction of a hamiltonian hierarchy associated to a cohomological field theory. We conjecture that in the semisimple case our hierarchy is related to the Dubrovin-Zhang hierarchy by a Miura transformation and check it in several examples. ...

Added: September 29, 2020

Barannikov S., Arnold Mathematical Journal 2019 Vol. 5 No. 1 P. 97-104

The EA-matrix integrals, introduced in Barannikov (Comptes Rendus Math 348:359–362, 2006), are studied in the case of graded associative algebras with odd or even scalar product. I prove that the EA-matrix integrals for associative algebras with scalar product are integrals of equivariantly closed differential forms with respect to the Lie algebra glN(A)glN(A). ...

Added: June 4, 2019

Khoroshkin A., Willwacher T., / Cornell University. Серия "Working papers by Cornell University". 2019. № 1905.04499.

We give a description of the operad formed by the real locus of the moduli space of stable genus zero curves with marked points $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ in terms of a homotopy quotient of an operad of associative algebras. We use this model to find different Hopf models of the algebraic operad of Chains and ...

Added: October 30, 2019

Buryak A., Rossi P., Letters in Mathematical Physics 2016 Vol. 106 No. 3 P. 289-317

In this paper we define a quantization of the Double Ramification Hierarchies using intersection numbers of the double ramification cycle, the full Chern class of the Hodge bundle and psi-classes with a given cohomological field theory. We provide effective recursion formulae which determine the full quantum hierarchy starting from just one Hamiltonian, the one associated with ...

Added: September 28, 2020

Buryak A., Dubrovin B., Guere J. et al., Communications in Mathematical Physics 2018 Vol. 363 No. 1 P. 191-260

In this paper we continue the study of the double ramification hierarchy introduced by the first author. After showing that the DR hierarchy satisfies tau-symmetry we define its partition function as the (logarithm of the) tau-function of the string solution and show that it satisfies various properties (string, dilaton and divisor equations plus some important degree ...

Added: September 27, 2020

Sawada T., Li Y., Pizlo Z., Symmetry 2011 Vol. 3 No. 2 P. 365-388

Added: September 23, 2014

Galkin S., Iritani H., / Cornell University. 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., / Cornell University. Series math "arxiv.org". 2012. No. 1212.1722.

We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas. ...

Added: September 14, 2013

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

Kazaryan M., Zvonkine D., Lando S., International Mathematics Research Notices 2018 No. 22 P. 6817-6843

We consider families of curve-to-curve maps that have no singularities except those of genus 0 stable maps and that satisfy a versality condition at each singularity. We provide a universal expression for the cohomology class Poincaré dual to the locus of any given singularity. Our expressions hold for any family of curve-to-curve maps satisfying the ...

Added: July 10, 2017

Cheltsov I., Przyjalkowski V., / Cornell University. Series arXiv "math". 2018.

We verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds. ...

Added: December 3, 2018

Ebeling W., Gusein-Zade S., International Mathematics Research Notices 2021 Vol. 2021 No. 16 P. 12305-12329

A.Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito duality between Burnside rings to a case of non-abelian groups and prove a "non-abelian" generalization of the statement about the equivariant Saito duality ...

Added: August 26, 2021

Akhtar M., Coates T., Galkin S. et al., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2012 Vol. 8 No. 094 P. 1-707

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with ...

Added: September 14, 2013

Ilten N. O., Lewis J., Victor Przyjalkowski, Journal of Algebra 2013 Vol. 374 P. 104-121

We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of ...

Added: July 2, 2013

Coates T., Galkin S., Kasprzyk A. et al., / Cornell University. Series math "arxiv.org". 2014. No. 1406.4891.

We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles. ...

Added: June 20, 2014

Buryak A., Moscow Mathematical Journal 2017 Vol. 17 No. 1 P. 1-13

In this paper, using the formula for the integrals of the psi-classes over the double ramification cycles found by S. Shadrin, L. Spitz, D. Zvonkine and the author, we derive a new explicit formula for the n-point function of the intersection numbers on the moduli space of curves. ...

Added: September 27, 2020

Buryak A., Shadrin S., Spitz L. et al., American Journal of Mathematics 2015 Vol. 137 No. 3 P. 699-737

DR-cycles are certain cycles on the moduli space of curves. Intuitively, they parametrize curves that allow a map to the complex projective line with some specified ramification profile over two points. They are known to be tautological classes, but in general there is no known expression in terms of standard tautological classes. In this paper, ...

Added: September 30, 2020

Kalashnikov E. G., / arXiv. 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

Buryak A., Moscow Mathematical Journal 2020 Vol. 20 No. 3 P. 475-493

By a famous result of K. Saito, the parameter space of the miniversal deformation of the $A_{r-1}$-singularity carries a Frobenius manifold structure. The Landau-Ginzburg mirror symmetry says that, in the flat coordinates, the potential of this Frobenius manifold is equal to the generating series of certain integrals over the moduli space of $r$-spin curves. In ...

Added: May 22, 2020