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Sergey Shadrin

Publications and source records attributed to Sergey Shadrin.

At least 19 recordsLinked to original sources
arXiv2026

Functional relations for higher-order free cumulants

We establish the functional relations between generating series of higher-order free cumulants and moments in higher-order free probability, solving an open problem posed fifteen years ago by Collins, Mingo, Śniady and Speicher. We propose an extension of free probability theory, which governs the all-order topological expansion in unitarily invariant matrix ensembles, with a corresponding notion of free cumulants and give as well their relation to moments via functional relations. Our approach is based on the study of a master transformation involving double monotone Hurwitz numbers via semi-infinite wedge techniques, building on the recent advances of the last-named author with Bychkov, Dunin-Barkowski and Kazarian. We illustrate our formulas by computing the first few decaying terms of the correlation functions of an ensemble of spiked GUE matrices, going beyond the law of large numbers and the central limit theorem.

math.OA↗
arXiv2026

Augmentation curve for torus knots

We prove a version of topological recursion for the correlation differentials of colored HOMFLY-PT polynomials of torus knots on the corresponding augmentation curves, following the proposal of Gu, Jockers, Klemm, and Soroush. To this end, we explicitly construct the spectral curve from the so-called auxiliary curve of Brini, Eynard, and Mariño and prove that it coincides with the augmentation curve for every torus knot with coprime $(r,s)$. Finally, our version of topological recursion proves the higher-genus closed-string prediction of the Aganagic--Vafa conjecture for all torus knot augmentation curves.

math-ph↗
arXiv2026

Topological recursion for spin enumeration of hypermaps

We establish topological recursion for the spin enumeration of hypermaps, or, in the dual language, spin dessins d'enfants. This problem has remarkable features distinguishing it from other applications of topological recursion to enumerative combinatorics. First, the spectral curve is elliptic rather than rational, as in most cases studied before. Second, it contains degenerate ramification points, and one has to apply a version of degenerate topological recursion. Finally, the meromorphic correlator differentials have "moving" poles on the antidiagonals between pairs of distinct arguments, and one has to adjust the loop equations to this kind of degeneracy as well. The computations involve concepts of different kinds, such as BKP integrability, Gaussian integrals, the x-y swap, and symplectic duality.

math-ph↗
arXiv2026

KP integrability of non-perturbative differentials

We prove the KP integrability of non-perturbative topological recursion, which can be considered as a formal $\hbar$-deformation of the Krichever construction of algebro-geometric solutions of the KP hierarchy. This property goes back to a 2011 conjecture of Borot and Eynard.

math-ph↗
arXiv2026

Theta classes: generalized topological recursion, integrability and $\mathcal{W}$-constraints

We study the intersection theory of the $Θ^{r,s}$-classes, where $r \geq 2$ and $1 \le s \le r-1$, which are cohomological field theories obtained as the top degrees of Chiodo classes. We show that the recently introduced generalized topological recursion on the $(r,s)$ spectral curves computes the descendant integrals of the $Θ^{r,s}$-classes. As a consequence, we deduce that the descendant potential of the $Θ^{r,s}$-classes is a tau function of the $r$-KdV hierarchy, generalizing the Brézin--Gross--Witten tau function (the special case $r=2$, $s=1$). We also explicitly compute the $\mathcal{W}$-constraints satisfied by the descendant potential, obtained as differential representations of the $\mathcal{W}(\mathfrak{gl}_r)$-algebra at self-dual level. This work extends previously known results on the Witten $r$-spin class, the $r$-spin $Θ$-classes (the case $s=r-1$), and the Norbury $Θ$-classes (the special case $r=2$, $s=1$).

math.AG↗
arXiv2026

Lifts of partial cohomological field theories and examples of bi-Hamiltonian structures in the non-semisimple case

We define the lift of a partial cohomological field theory with respect to a Frobenius algebra and the corresponding lift of local polyvector fields, extending the lift procedure proposed by Della Vedova, Lorenzoni, and Savoldi. This allows us to systematically produce examples of non-semisimple homogeneous partial cohomological field theories whose integrable systems possess a second Hamiltonian structure, thus confirming the conjecture of Buryak et al. on an explicit formula for the second Poisson bracket in new non-semisimple cases. Moreover we present some relations of these lift constructions with the Morimoto theory of the lift of geometric structures to the Weil bundle of infinitely near points associated to a local algebra.

math-ph↗
arXiv2026

A new spin on polynomial relations among kappa classes

We prove a recent conjecture of the fourth named author with P. Norbury that states a system of universal polynomial relations among the kappa classes on the moduli spaces of algebraic curves. The proof involves localization and materialization analysis of the spin Gromov-Witten theory of the projective line and is dictated by $\mathbb{Z}_2$-equivariant topological recursion.

math.AG↗
arXiv2026

Beyond descendants: integrable observables for cohomological field theories

We introduce the concept of integrable observables and propose them as alternatives to the standard Witten's psi classes (a.k.a. descendants in $2D$ quantum gravity) to be coupled with cohomological field theories and their generalisations. The main property of integrable observables is that they retain the integrability properties. We present three examples of integrable observables. The first two recover the Dubrovin-Zhang and double ramification hierarchies, while revealing new structural features in this framework. The third, a new example, builds on recently established properties of the so-called $\mathbbΠ$-class, extending them and placing this class naturally within the theory of integrable systems. Notably, our integrable observables framework yields a proof that the new $\mathbbΠ$-hierarchies are Miura equivalent both to the Dubrovin-Zhang hierarchies and to the double ramification hierarchies. A new very short proof of Witten's conjecture is also provided.

math.AG↗
arXiv2026

Any topological recursion on a rational spectral curve is KP integrable

We prove that for any initial data on a genus zero spectral curve the corresponding correlation differentials of topological recursion are KP integrable. As an application we prove KP integrability of partition functions associated via ELSV-type formulas to the $r$-th roots of the twisted powers of the log canonical bundles.

math-ph↗
arXiv2026

A new family of weighted double Hurwitz numbers and a new ELSV-type formula with $Ω$-classes

We analyze a new family of weighted double Hurwitz numbers that was introduced as a notable example in the context of the $x-y$ duality for logarithmic topological recursion. We use this family to systematically demonstrate, refine and develop techniques that play a crucial role in the interaction of hypergeometric (Orlov--Scherbin) KP tau functions and intersection theory of moduli spaces of curves. In particular, we discuss the subtleties related to the derivation of the ELSV-type formulas in this context and derive a new, explicit ELSV-type formula in terms of the so-called $Ω$-classes.

math.AG↗
arXiv2025

The quantum integrable hierarchy for the Gromov-Witten theory of elliptic curves

We construct the quantum double ramification hierarchy associated with the Gromov-Witten theory of elliptic curves. We use results of Oberdieck and Pixton on the intersection numbers of the double ramification cycle, the Gromov-Witten classes of the elliptic curve and the Hodge class $λ_{g-1}$ together with vanishing results for $λ_{g-2}$ to produce a closed, modular expression for the resulting integrable hierarchy. It is the first explicit nontrivial example of a quantum integrable hierarchy from a cohomological field theory containing fermionic fields, which correspond to the odd classes in the cohomology of the elliptic curve.

math.AG↗
arXiv2025

Operads of moduli spaces of points in $\mathbb{C}^d$ revisited

We study the operad structure on the homology of moduli spaces of pointed rooted trees of $d$-dimensional projective spaces, introduced by Chen, Gibney and Krashen a couple of decades ago. We describe this operad by generators and relations, show that it is homotopy Koszul, exhibit a Givental-type action on representations of that operad, and prove that this operad represents the homotopy quotient of the operad of chains of $S^1$-framed little $2d$-disks by its natural circle action. Our approach also sheds new light on the $d=1$ case, revealing a new combinatorial way to write the original Givental formulas.

math.AT↗
arXiv2025

Taking limits in topological recursion

When does topological recursion applied to a family of spectral curves commute with taking limits? This problem is subtle, especially when the ramification structure of the spectral curve changes at the limit point. We provide sufficient (straightforward-to-use) conditions for checking when the commutation with limits holds, thereby closing a gap in the literature where this compatibility has been used several times without justification. This takes the form of a stronger result of analyticity of the topological recursion along suitable families. To tackle this question, we formalise the notion of global topological recursion and provide sufficient conditions for its equivalence with local topological recursion. The global version facilitates the study of analyticity and limits. For nondegenerate algebraic curves, we reformulate these conditions purely in terms of the structure of its underlying singularities. Finally, we apply this to study deformations of $ (r,s) $-spectral curves, spectral curves for weighted Hurwitz numbers, and provide several other examples and non-examples (where the commutation with limits fails).

math.AG↗
arXiv2025

On the strong DR/DZ equivalence conjecture

We establish the Miura equivalence of two integrable systems associated to a semi-simple cohomological field theory: the double ramification hierarchy of Buryak and the Dubrovin-Zhang hierarchy. This equivalence was conjectured by Buryak and further refined by Buryak, Dubrovin, Guéré, and Rossi.

math.AG↗
arXiv2025

Structure of Dubrovin-Zhang free energy functions and universal identities

We prove a structural theorem relating the higher genera free energy functions of the Dubrovin-Zhang hierarchies to the Witten-Kontsevich free energy function of the Korteweg-de Vries hierarchy. As an important application, for any given genus $g\geq 1$, we construct a set of universal identities valid for the free energy functions of any Dubrovin-Zhang hierarchy. In particular, we present some techniques that can be used to derive universal identities without relying on the geometry of the moduli space of stable curves of higher genus.

math-ph↗
arXiv2025

KP integrability through the $x-y$ swap relation

We discuss a universal relation that we call the $x-y$ swap relation, which plays a prominent role in the theory of topological recursion, Hurwitz theory, and free probability theory. We describe in a very precise and detailed way the interaction of the $x-y$ swap relation and KP integrability. As an application, we prove a recent conjecture that relates some particular instances of topological recursion to the Mironov-Morozov-Semenoff matrix integrals.

math-ph↗