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Todor Milanov

Publications and source records attributed to Todor Milanov.

At least 19 recordsLinked to original sources
arXiv2026

Elliptic orbifold lines and integrable hierarchies

We prove that the Gromov--Witten invariants of the elliptic orbifold lines $\mathbf{P}^1_{3,3,3}$, $\mathbf{P}^1_{2,4,4}$, and $\mathbf{P}^1_{2,3,6}$ satisfy a certain system of Hirota Quadratic (or Bilinear) Equations. Our result is the analogue of the so-called Toda conjecture in the Gromov-Witten theory of $\mathbf{P}^1$ or more precisely its non-extended version. A new feature in our constructions is a certain bilinear operator whose principal symbol can be expressed in terms of elliptic theta functions.

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arXiv2024

Dubrovin conjecture and the second structure connection

We give a reformulation of the Dubrovin conjecture about the semisimplicity of quantum cohomology in terms of the so-called second structure connection of quantum cohomology. The key ingredient in our work is the notion of a twisted reflection vector which allows us to give an elegant description of the monodromy data of the quantum connection in terms of the monodromy data of its Laplace transform.

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arXiv2024

Confluence in quantum K-theory of weak Fano manifolds and q-oscillatory integrals for toric manifolds

For a smooth projective variety whose anti-canonical bundle is nef, we prove confluence of the small $K$-theoretic $J$-function, i.e., after rescaling appropriately the Novikov variables, the small $K$-theoretic $J$-function has a limit when $q\to 1$, which coincides with the small cohomological $J$-function. Furthermore, in the case of a Fano toric manifold $X$ of Picard rank 2, we prove the $K$-theoretic version of an identity due to Iritani that compares the $I$-function of the toric manifold and the oscillatory integral of the toric mirror. In particular, our confluence result yields a new proof of Iritani's identity in the case of a toric manifold of Picard rank 2.

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arXiv2024

Reflection Vectors and Quantum Cohomology of Blowups

Let $X$ be a smooth projective variety with a semisimple quantum cohomology. It is known that the blowup $\operatorname{Bl}_{\rm pt}(X)$ of $X$ at one point also has semisimple quantum cohomology. In particular, the monodromy group of the quantum cohomology of $\operatorname{Bl}_{\rm pt}(X)$ is a reflectiongroup. We found explicit formulas for certain generators of the monodromy group of the quantum cohomology of $\operatorname{Bl}_{\rm pt}(X)$ depending only on the geometry of the exceptional divisor.

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arXiv2024

K-theoretic Heisenberg algebras and permutation-equivariant Gromov--Witten theory

We found an interesting application of the K-theoretic Heisenberg algebras of Weiqiang Wang to the foundations of permutation equivariant K-theoretic Gromov--Witten theory. We also found an explicit formula for the genus 0 correlators in the permutation equivariant Gromov--Witten theory of the point. In the non-equivariant limit our formula reduces to a well known formula due to Y.P. Lee.

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arXiv2021

Hirota Quadratic Equations for the Gromov--Witten Invariants of $\mathbb{P}_{n-2,2,2}^1$

Fano orbifold lines are classified by the Dynkin diagrams of type $A,D,$ and $E$. It is known that the corresponding total descendant potential is a tau-function of an appropriate Kac--Wakimoto hierarchy. It is also known that in the A-case the Kac--Wakimoto hierarchies admit an extension and that the total descendant potential is a tau-function of an extended Kac--Wakimoto hierarchy. The goal of this paper is to prove that in the D-case the total descendent potential is also a tau-function of an extended Kac--Wakimoto hierarchy.

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arXiv2020

Integral Structure for Simple Singularities

We compute the image of the Milnor lattice of an ADE singularity under a period map. We also prove that the Milnor latticecan be identified with an appropriate relative $K$-group defined through the Berglund-Hübsch dual of the corresponding singularity.

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arXiv2019

The extended D-Toda hierarchy

In a companion paper to this one, we proved that the Gromov--Witten theory of a Fano orbifold line of type $D$ is governed by a system of Hirota Bilinear Equations. The goal of this paper is to prove that every solution to the Hirota Bilinear Equations determines a solution to a new integrable hierarchy of Lax equations. We suggest the name extended D-Toda hierarchy for this new system of Lax equations, because it should be viewed as the analogue of Carlet's extended bi-graded Toda hierarchy, which is known to govern the Gromov--Witten theory of Fano orbifold lines of type $A$

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arXiv2019

The Period map for quantum cohomology of $\mathbb{P}^2$

We invert the period map defined by the second structure connection of quantum cohomology of $\mathbb{P}^2$. For small quantum cohomology the inverse is given explicitly in terms of the Eisenstein series $E_4$ and $E_6$, while for big quantum cohomology the inverse is determined perturbatively as a Taylor series expansion whose coefficients are quasi-modular forms.

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arXiv2018

The 2-component BKP Grassmanian and simple singularities of type D

It was proved in 2010 that the principal Kac--Wakimoto hierarchy of type $D$ is a reduction of the 2-component BKP hierarchy. On the other hand, it is known that the total descendant potential of a singularity of type $D$ is a tau-function of the principal Kac--Wakimoto hierarchy. We find explicitly the point in the Grassmanian of the 2-component BKP hierarchy (in the sense of Shiota) that corresponds to the total descendant potential. We also prove that the space of tau-functions of Gaussian type is parametrized by the base of the miniversal unfolding of the simple singularity of type $D$.

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arXiv2017

Primitive forms and Frobenius structures on the Hurwitz spaces

The main goal of this paper is to introduce the notion of a primitive form for a generic family of Hurwitz covers of $\mathbb{P}^1$ with a fixed ramification profile over infinity. We prove that primitive forms are in one-to-one correspondence with semi-simple Frobenius structures on the base of the family. Furthermore, we introduce the notion of a polynomial primitive form and show that the corresponding class of Frobenius manifolds contains the Hurwitz Frobenius manifolds of Dubrovin. Finally, we apply our theory to investigate the relation between the Eynard--Orantin recursion and Frobenius manifolds.

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arXiv2016

Gromov-Witten Theory of Quotient of Fermat Calabi-Yau varieties

We construct a global B-model for weighted homogeneous polynomials based on K. Saito's theory of primitive forms. Our main motivation is to give a rigorous statement of the so called global mirror symmetry conjecture relating Gromov-Witten invariants and Fan--Jarvis--Ruan--Witten invariants. Furthermore, our construction allows us to generalize the notion of a quasi-modular form and holomorphic anomaly equations. Finally, we prove the global mirror symmetry conjecture for the Fermat polynomials.

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arXiv2016

$\mathcal{W}$-algebra constraints and topological recursion for $A_N$-singularity

We derive a Bouchard--Eynard type topological recursion for the total descendant potential of $A_N$-singularity. Our argument relies on a certain twisted representation of a Heisenberg Vertex Operator Algebra (VOA) constructed via the periods of $A_N$-singularity. In particular, our approach allows us to prove that the topological recursion for the total descendant potential is equivalent to a certain generating set of $\mathcal{W}$-algebra constraints.

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arXiv2015

The phase factors in singularity theory

The paper \cite{BM} proposed a construction of a twisted representation of the lattice vertex algebra corresponding to the Milnor lattice of a simple singularity. The main difficulty in extending the above construction to an arbitrary isolated singularity is in the so called {\em phase factors} -- the scalar functions produced by composing two vertex operators. They are certain family of multivalued analytic functions on the space of miniversal deformations. The first result in this paper is an explicit formula for the unperturbed phase factors in terms of the classical monodromy operator and the polylogorithm functions. Our second result is that with respect to the deformation parameters the phase factors are analytic functions on the monodromy covering space.

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arXiv2015

The Eynard--Orantin recursion for simple singularities

According to \cite{BOSS} and \cite{M1}, the ancestor correlators of any semi-simple cohomological field theory satisfy {\em local} Eynard--Orantin recursion. In this paper, we prove that for simple singularities, the local recursion can be extended to a global one. The spectral curve of the global recursion is an interesting family of Riemann surfaces defined by the invariant polynomials of the corresponding Weyl group. We also prove that for genus 0 and 1, the free energies introduced in \cite{EO} coincide up to some constant factors with respectively the genus 0 and 1 primary potentials of the simple singularity.

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