arXiv subjects

Dimitri Zvonkine

Publications and source records attributed to Dimitri Zvonkine.

At least 19 recordsLinked to original sources

arXiv2025

The Chern character of the Laughlin vector bundle in the Fractional Quantum Hall Effect

We begin by explaining how a physical problem of studying the quantum Hall effect on a closed surface $C$ leads, via Laughlin's approach, to a mathematical question of describing the rank and the first Chern class of a particular vector bundle on the Picard group ${\rm Pic}^g(C)$. Then we formulate and solve the problem mathematically, proving several important conjectures made by physicists, in particular the Wen-Niu topological degeneracy conjecture and the Wen-Zee shift formula. Let $C$ be a closed Riemann surface of genus~$g$ and $S^NC$ its $N$th symmetric power. The product $C \times {\rm Pic}^d(C)$ carries a universal line bundle. On the product $C^N \times {\rm Pic}^d(C)$ we consider the product of $N$ pull-backs of this universal line bundle and twist it by a power of the diagonal on $C^N$. The resulting line bundle descends onto $S^NC \times {\rm Pic}^d(C)$. Its push-forward (as a sheaf) to ${\rm Pic}^d(C)$ is a vector bundle that we call Laughlin's vector bundle. We determine all the Chern characters of the Laughlin vector bundle via a Grothendieck-Riemann-Roch calculation.

math.AG↗

arXiv2023

Gromov-Witten Theory of Complete Intersections via Nodal Invariants

We provide an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov-Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten invariants, we introduce the new notion of nodal relative Gromov-Witten invariants. We then prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov-Witten theory are stated in complete generality and are of independent interest.

math.AG↗

arXiv2021

Moduli spaces of residueless meromorphic differentials and the KP hierarchy

We prove that the cohomology classes of the moduli spaces of residueless meromorphic differentials, i.e., the closures, in the moduli space of stable curves, of the loci of smooth curves whose marked points are the zeros and poles of prescribed orders of a meromorphic differential with vanishing residues, form a partial cohomological field theory (CohFT) of infinite rank. To this partial CohFT we apply the double ramification hierarchy construction to produce a Hamiltonian system of evolutionary PDEs. We prove that its reduction to the case of differentials with exactly two zeros and any number of poles coincides with the KP hierarchy up to a change of variables.

math.AG↗

arXiv2021

Geometric test for topological states of matter

We generalize the flux insertion argument due to Laughlin, Niu-Thouless-Tao-Wu, and Avron-Seiler-Zograf to the case of fractional quantum Hall states on a higher-genus surface. We propose this setting as a test to characterise the robustness, or topologicity, of the quantum state of matter and apply our test to the Laughlin states. Laughlin states form a vector bundle, the Laughlin bundle, over the Jacobian - the space of Aharonov-Bohm fluxes through the holes of the surface. The rank of the Laughlin bundle is the degeneracy of Laughlin states or, in presence of quasiholes, the dimension of the corresponding full many-body Hilbert space; its slope, which is the first Chern class divided by the rank, is the Hall conductance. We compute the rank and all the Chern classes of Laughlin bundles for any genus and any number of quasiholes, settling, in particular, the Wen-Niu conjecture. Then we show that Laughlin bundles with non-localized quasiholes are not projectively flat and that the Hall current is precisely quantized only for the states with localized quasiholes. Hence our test distinguishes these states from the full many-body Hilbert space.

cond-mat.str-el↗

arXiv2019

Double Hurwitz numbers and multisingularity loci in genus 0

In the Hurwitz space of rational functions on CP^1 with poles of given orders, we study the loci of multisingularities, that is, the loci of functions with a given ramification profile over 0. We prove a recursion relation on the Poincare dual cohomology classes of these loci and deduce a differential equation on Hurwitz numbers.

math.AG↗

arXiv2018

Hurwitz numbers for real polynomials

We consider the problem of defining and computing real analogs of polynomial Hurwitz numbers, in other words, the problem of counting properly normalized real polynomials with fixed ramification profiles over real branch points. We show that, provided the polynomials are counted with an appropriate sign, their number does not depend on the order of the branch points on the real line. We study generating series for the invariants thus obtained, determine necessary and sufficient conditions for the vanishing and nonvanishing of these generating series, and obtain a logarithmic asymptotic for the invariants as the degree of the polynomials tends to infinity.

math.AG↗

arXiv2016

Chiodo formulas for the r-th roots and topological recursion

We analyze Chiodo's formulas for the Chern classes related to the r-th roots of the suitably twisted integer powers of the canonical class on the moduli space of curves. The intersection numbers of these classes with psi-classes are reproduced via the Chekhov-Eynard-Orantin topological recursion. As an application, we prove that the Johnson-Pandharipande-Tseng formula for the orbifold Hurwitz numbers is equivalent to the topological recursion for the orbifold Hurwitz numbers. In particular, this gives a new proof of the topological recursion for the orbifold Hurwitz numbers.

math-ph↗

arXiv2015

Universal cohomological expressions for singularities in families of genus 0 stable maps

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 above properties.

math.AG↗

arXiv2015

Relations on Mbar_{g,n} via 3-spin structures

Witten's class on the moduli space of 3-spin curves defines a (non-semisimple) cohomological field theory. After a canonical modification, we construct an associated semisimple CohFT with a non-trivial vanishing property obtained from the homogeneity of Witten's class. Using the classification of semisimple CohFTs by Givental-Teleman, we derive two main results. The first is an explicit formula in the tautological ring of Mbar_{g,n} for Witten's class. The second, using the vanishing property, is the construction of relations in the tautological ring of Mbar_{g,n}. Pixton has previously conjectured a system of tautological relations on Mbar_{g,n} (which extends the established Faber-Zagier relations on M_g). Our 3-spin construction exactly yields Pixton's conjectured relations. As the classification of CohFTs is a topological result depending upon the Madsen-Weiss theorem (Mumford's conjecture), our construction proves relations in cohomology. The study of Witten's class and the associated tautological relations for r-spin curves via a parallel strategy will be taken up in a following paper.

math.AG↗

arXiv2009

A group action on Losev-Manin cohomological field theories

We discuss an analog of the Givental group action for the space of solutions of the commutativity equation. There are equivalent formulations in terms of cohomology classes on the Losev-Manin compactifications of genus 0 moduli spaces; in terms of linear algebra in the space of Laurent series; in terms of differential operators acting on Gromov-Witten potentials; and in terms of multi-component KP tau-functions. The last approach is equivalent to the Losev-Polyubin classification that was obtained via dressing transformations technique.

math.AG↗

arXiv2009

Cycle factorizations and one-faced graph embeddings

Consider factorizations into transpositions of an n-cycle in the symmetric group S_n. To every such factorization we assign a monomial in variables w_{ij} that retains the transpositions used, but forgets their order. Summing over all possible factorizations of n-cycles we obtain a polynomial that happens to admit a closed expression. From this expression we deduce a formula for the number of 1-faced embeddings of a given graph.

math.CO↗

arXiv2009

Tautological relations and the r-spin Witten conjecture

In a series of two preprints, Y.-P. Lee studied relations satisfied by all formal Gromov-Witten potentials, as defined by A. Givental. He called them "universal relations" and studied their connection with tautological relations in the cohomology ring of moduli spaces of stable curves. Building on Y.-P. Lee's work, we give a simple proof of the fact that every tautological relation gives rise to a universal relation (which was also proved by Y.-P. Lee modulo certain results announced by C. Teleman). In particular, this implies that in any semi-simple Gromov-Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the formal and the geometric Gromov-Witten potentials coincide. As the most important application, we show that our results suffice to deduce the statement of a 1991 Witten conjecture on r-spin structures from the results obtained by Givental for the corresponding formal Gromov-Witten potential. The conjecture in question states that certain intersection numbers on the moduli space of r-spin structures can be arranged into a power series that satisfies the r-KdV (or r-th higher Gelfand-Dikii) hierarchy of partial differential equations.

math.AG↗

arXiv2007

Twisted Gromov-Witten r-spin potential and Givental's quantization

The universal curve p:C->\Mbar over the moduli space \Mbar of stable r-spin maps to a target Kähler manifold X carries a universal spinor bundle L->C. Therefore the moduli space \Mbar itself carries a natural K-theory class Rp_*L. We introduce a twisted r-spin Gromov-Witten potential of X enriched with Chern characters of Rp_*L. We show that the twisted potential can be reconstructed from the ordinary r-spin Gromov-Witten potential of X via an operator that assumes a particularly simple form in Givental's quantization formalism.

math.AG↗

arXiv2007

Changes of variables in ELSV-type formulas

In [5] I.P. Goulden, D.M. Jackson, and R. Vakil formulated a conjecture relating certain Hurwitz numbers (enumerating ramified coverings of the sphere) to the intersection theory on a conjectural Picard variety. We are going to use their formula to study the intersection theory on this variety (if it is ever to be constructed) by methods close to those of M. Kazarian and S. Lando in [7]. In particular, we prove a Witten-Kontsevich-type theorem relating the intersection theory and integrable hierarchies. We also extend the results of [7] to include the Hodge integrals over the moduli spaces, involving one lambda-class.

math.AG↗

arXiv2006

Intersection numbers with Witten's top Chern class

Witten's top Chern class is a particular cohomology class on the moduli space of Riemann surfaces endowed with r-spin structures. It plays a key role in Witten's conjecture relating to the intersection theory on these moduli spaces. Our first goal is to compute the integral of Witten's class over the so-called double ramification cycles in genus 1. We obtain a simple closed formula for these integrals. This allows us, using the methods of [15], to find an algorithm for computing the intersection numbers of the Witten class with powers of the ψ-classes (or tautological classes) over any moduli space of r-spin structures, in short, all numbers involved in Witten's conjecture.

math.AG↗

arXiv2005

Enumeration of ramified coverings of the sphere and 2-dimensional gravity

Let A be the algebra generated by the power series \sum n^{n-1} q^n/n! and \sum n^n q^n /n! . We prove that many natural generating functions lie in this algebra: those appearing in graph enumeration problems, in the intersection theory of moduli spaces M_{g,n} and in the enumeration of ramified coverings of the sphere. We argue that ramified coverings of the sphere with a large number of sheets provide a model of 2-dimensional gravity. Our results allow us to compute the asymptotic of the number of coverings as the number of sheets goes to infinity. The leading terms of such asymptotics are the values of certain observables in 2-dimensional gravity. We prove that they coincide with the values provided by other models. In particular, we recover a solution of the Painleve I equation and the string solution of the KdV hierarchy.

math.AG↗

arXiv2005

Enumeration of almost polynomial rational functions with given critical values

Enumerating ramified coverings of the sphere with fixed ramification types is a well-known problem first considered by A. Hurwitz. Up to now, explicit solutions have been obtained only for some families of ramified coverings, for instant, those realized by polynomials in one complex variable. In this paper we obtain an explicit answer for a large new family of coverings, namely, the coverings realized by simple almost polynomials, defined below. Unlike most other results in the field, our formula is obtained by elementary methods.

math.CO↗

arXiv2004

An algebra of power series arising in the intersection theory of moduli spaces of curves and in the enumeration of ramified coverings of the sphere

A bracket is a function that assigns a number to each monomial in variables τ_0, τ_1, ... We show that any bracket satisfying the string and the dilaton relations gives rise to a power series lying in the algebra A generated by the series \sum n^{n-1} q^n/n! and \sum n^n q^n /n! . As a consequence, various series from A appear in the intersection theory of moduli spaces of curves. A connection between the counting of ramified coverings of the sphere and the intersection theory on moduli spaces allows us to prove that some natural generating functions enumerating the ramified coverings lie, yet again, in A. As an application, one can find the asymptotic of the number of such coverings as the number of sheets tends to infinity. We believe that the leading terms of the asymptotics like that correspond to observables in 2-dimensional gravity.

math.AG↗