arXiv subjects

Pavlo Gavrylenko

Publications and source records attributed to Pavlo Gavrylenko.

17 recordsLinked to original sources

arXiv2025

Modular transformations of tau functions and conformal blocks on the torus

The connection problem for isomonodromic tau functions on the one-punctured torus concerns the ratio between the tau function and its modular transform, associated to dual pants decompositions of the torus. In this paper, we study the modular transformations of the tau function and consequently derive the connection constant. Moreover, through the relation with two-dimensional Conformal Field Theory, we also obtain an exact closed formula for the $c=1$ Virasoro modular kernel, whose expression was previously unknown, and relate it to the $c\rightarrow\infty$ (semiclassical) modular kernel and $SL_2(\mathbb{C})$ complex Chern-Simons amplitudes. Finally, we prove that the connection constant and the two, $c=1$ and $c\to \infty$, modular kernels are generating functions of canonical transformations on the character variety of the one-punctured torus. Our results are also relevant for the $\mathcal{N}=2^*$ gauge theory.

math-ph↗

arXiv2025

On a 5D UV completion of Argyres-Douglas theories

We discuss a novel UV completion of a class of Argyres-Douglas (AD) theories in the $\Omega$-background by its embedding into the renormalisation group flow from five dimensional $\mathcal{N}=1$ superconformal field theories (SCFT) on $S^1$. This is obtained via analysing these theories in the light of ($q$-)Painlev\'e/gauge theory correspondence, which allows to compute the five dimensional BPS partition functions as an expansion in the Wilson loop vev with integer $q$-polynomials coefficients. These are derived formulating the gauge theory on a blown-up geometry and using a five-dimensional lift of (topological) operator/state correspondence. We discuss in detail the phase diagram of the four dimensional limits, pinpointing the special AD loci. Explicit computations are reported for $\tilde E_1$ SCFT and its limit to H$_0=(A_1,A_2)$ AD theory.

hep-th↗

arXiv2025

Blowing-up the edge: connection formulae and stability chart of the Lamé equation

We study periodic spectral problems through their connection with supersymmetric gauge theories and two-dimensional conformal field theory. To characterize the associated stability chart, we develop a novel and systematic approach for analyzing semi-classical Virasoro blocks near their poles. Via the AGT correspondence, these blocks correspond to SU(2) Nekrasov partition functions in the Nekrasov-Shatashvili limit, which we propose to resum using an appropriate limit of blow-up equations. We show that the analytic structure of the resulting resummed partition functions features branch cuts located precisely at the edges between bands and gaps in the spectrum of the associated quantum integrable system with periodic potential. We examine the Nekrasov partition functions of $\mathcal{N}=2$ SQCD with $N_f \le 4$ flavors and of the $\mathcal{N}=2^*$ theory, which are related to the Heun equation, its confluent forms, and the Lamé equation. In the latter case, we analyze the spectrum in detail and solve the associated connection problem. Finally, we compare our results with those obtained via isomonodromic deformation techniques and the computation of orbifold surface defect partition functions in the $\mathcal{N}=2^*$ gauge theory, finding perfect agreement.

hep-th↗

arXiv2025

2d Sigma Models on Non-compact Calabi-Yau and ${\mathcal N}=2$ Liouville Theory

We consider a class of two dimensional conformal ${\mathcal N}=2$ supersymmetric $U(1)$ gauge linear sigma models with $N$ fields of charges $+1$ and $N$ fields of charges $-1$, whose Higgs branches are non-compact toric Calabi-Yau manifolds of complex dimension $2N-1$. We show, starting from large-$N$ approximation, that the Coulomb branch of these models, which opens up at strong coupling, is described by ${\mathcal N}=2$ Liouville theory and then extrapolate it to exact equivalence demanding the central charge of the Liouville theory to be $\hat{c}=2N-1$. Next we concentrate on mostly physically attractive $N=2$ and $N \geq 3$ cases and find there a perfect agreement of the set of complex moduli on the Calabi-Yau side with the marginal deformations in ${\mathcal N}=2$ Liouville theory, supporting proposed exact equivalence.

hep-th↗

arXiv2025

Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general $q$-Painlevé III$_3$ tau functions

We reformulate the $q$-difference linear system corresponding to the $q$-Painlevé equation of type $A_7^{(1)'}$ as a Riemann-Hilbert problem on a circle. Then, we consider the Fredholm determinant built from the jump of this Riemann-Hilbert problem and prove that it satisfies bilinear relations equivalent to $P(A_7^{(1)'})$. We also find the minor expansion of this Fredholm determinant in explicit factorized form and prove that it coincides with the Fourier series in $q$-deformed conformal blocks, or partition functions of the pure $5d$ $\mathcal{N}=1$ $SU(2)$ gauge theory, including the cases with the Chern-Simons term. Finally, we solve the connection problem for these isomonodromic tau functions, finding in this way their global behavior.

math-ph↗

arXiv2024

Cluster Reductions, Mutations, and $q$-Painlevé Equations

We propose an extension of the Goncharov-Kenyon class of cluster integrable systems by their Hamiltonian reductions. This extension allows us to fill in the gap in cluster construction of the $q$-difference Painlevé equations, showing that all of them can be obtained as deautonomizations of the reduced Goncharov-Kenyon systems. Conjecturally, the isomorphisms of reduced Goncharov-Kenyon integrable systems are given by mutations in another, dual in some sense, cluster structure. These are the polynomial mutations of the spectral curve equations and polygon mutations of the corresponding decorated Newton polygons. In the Painlevé case the initial and dual cluster structures are isomorphic. It leads to self-duality between the spectral curve equation and the Painlevé Hamiltonian, and also extends the symmetry from affine to elliptic Weyl group.

nlin.SI↗

arXiv2024

Surface observables in gauge theories, modular Painlev\'e tau functions and non-perturbative topological strings

We study BPS surface observables of $\mathcal{N}=2$ four dimensional $SU(2)$ gauge theory in gravitational $\Omega$-background at perturbative and at Argyres-Douglas superconformal fixed points. This is done by formulating the equivariant gauge theory on the blow-up of $\mathbb{C}^2$ and considering the decoupling Nekrasov-Shatashvili limit. We show that in this limit the blow-up equations are solved by corresponding Painlev\'e $\mathcal{T}$-functions and exploit operator/state correspondence to compute their expansion in an integer basis, given in terms of the moduli of the quantum Seiberg-Witten curve. We study the modular properties of these solutions and show that they do directly lead to BCOV holomorphic anomaly equations for the corresponding topological string partition function. The resulting $\mathcal{T}$-functions are holomorphic and modular and as such they provide a natural non-perturbative completion of topological strings partition functions.

hep-th↗

arXiv2023

Connecting topological strings and spectral theory via non-autonomous Toda equations

We consider the Topological String/Spectral theory duality on toric Calabi-Yau threefolds obtained from the resolution of the cone over the $Y^{N,0}$ singularity. Assuming Kyiv formula, we demonstrate this duality in a special regime thanks to an underlying connection between spectral determinants of quantum mirror curves and the non-autonomous (q)-Toda system. We further exploit this link to connect small and large time expansions in Toda equations. In particular we provide an explicit expression for their tau functions at large time in terms of a strong coupling version of irregular $W_N$ conformal blocks at $c=N-1$. These are related to a special class of multi-cut matrix models which describe the strong coupling regime of four dimensional, $\mathcal{N}=2$ $SU(N)$ super Yang-Mills.

hep-th↗

arXiv2022

Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus

We compute the monodromy dependence of the isomonodromic tau function on a torus with $n$ Fuchsian singularities and $SL(N)$ residue matrices by using its explicit Fredholm determinant representation. We show that the exterior logarithmic derivative of the tau function defines a closed one-form on the space of monodromies and times, and identify it with the generating function of the monodromy symplectomorphism. As an illustrative example, we discuss the simplest case of the one-punctured torus in detail. Finally, we show that previous results obtained in the genus zero case can be recovered in a straightforward manner using the techniques presented here.

math-ph↗

arXiv2021

Quantum spectral problems and isomonodromic deformations

We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of $2\times 2$ linear systems (Riemann-Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé $\mathrm{III}_3$ equation. Then we extend the analysis to the 2-particle elliptic Calogero-Moser operator, which corresponds to a linear system on the torus. By using the Kiev formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions ($ε_1+ε_2=0$). Through blowup relations, we also find Nekrasov-Shatashvili type of quantizations ($ε_2=0$). In the case of the torus with one regular singularity we obtain certain results which are interesting by themselves. Namely, we derive blowup equations (filling some gaps in the literature) and we relate them to the bilinear form of the isomonodromic deformation equations. In addition, we extract the $ε_2\to 0$ limit of the blowup relations from the regularized action functional and CFT arguments.

math-ph↗

arXiv2021

Solution of tetrahedron equation and cluster algebras

We notice a remarkable connection between Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism we show how to construct integrable system with spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to general non-symmetric Newton polygons, and prove Lemma, which classifies conjugacy classes in double affine Weyl groups of $A$-type by Newton polygons.

nlin.SI↗

arXiv2021

Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions

We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in $GL(N,\mathbb{C})$ can be written in terms of a Fredholm determinant of Cauchy-Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a series labeled by charged partitions. As an example, we show that in the case of $SL(2,\mathbb{C})$ this combinatorial expression takes the form of a dual Nekrasov-Okounkov partition function, or equivalently of a free fermion conformal block on the torus. Based on these results, we also propose a definition of the tau function of the Riemann-Hilbert problem on a torus with generic jump on the A-cycle.

math-ph↗

arXiv2020

Irregular conformal blocks, Painlevé III and the blow-up equations

We study the relation of irregular conformal blocks with the Painlevé III$_3$ equation. The functional representation for the quasiclassical irregular block is shown to be consistent with the BPZ equations of conformal field theory and the Hamilton-Jacobi approach to Painlevé III$_3$. It leads immediately to a limiting case of the blow-up equations for dual Nekrasov partition function of 4d pure supersymmetric gauge theory, which can be even treated as a defining system of equations for both $c=1$ and $c\to\infty$ conformal blocks. We extend this analysis to the domain of strong-coupling regime where original definition of conformal blocks and Nekrasov functions is not known and apply the results to spectral problem of the Matheiu equations. Finally, we propose a construction of irregular conformal blocks in the strong coupling region by quantization of Painlevé III$_3$ equation, and obtain in this way a general expression, reproducing $c=1$ and quasiclassical $c\to\infty$ results as its particular cases. We have also found explicit integral representations for $c=1$ and $c=-2$ irregular blocks at infinity for some special points.

math-ph↗

arXiv2019

Crossing invariant correlation functions at $c=1$ from isomonodromic $τ$ functions

We present an approach that gives rigorous construction of a class of crossing invariant functions in $c=1$ CFTs from the weakly invariant distributions on the moduli space $\mathcal M_{0,4}^{SL(2,\mathbb{C})}$ of $SL(2,\mathbb{C})$ flat connections on the sphere with four punctures. By using this approach we show how to obtain correlation functions in the Ashkin-Teller and the Runkel-Watts theory. Among the possible crossing-invariant theories, we obtain also the analytic Liouville theory, whose consistence was assumed only on the basis of numerical tests.

math-ph↗

arXiv2019

Circular quiver gauge theories, isomonodromic deformations and $W_N$ fermions on the torus

We study the relation between class S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two dimensional torus with punctures. Turning on the self dual $\Omega$-background corresponds to a deautonomization of the Seiberg-Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $\tau$-function is proportional to the dual gauge theory partition function, the proportionality factor being a non trivial function of the solution of the deautonomized Seiberg-Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $W_N$ free fermion correlators on the torus.

hep-th↗

arXiv2019

$\mathcal{N}=2^*$ gauge theory, free fermions on the torus and Painlevé VI

In this paper we study the extension of Painlevé/gauge theory correspondence to circular quivers by focusing on the special case of $SU(2)$ $\mathcal{N}=2^*$ theory. We show that the Nekrasov-Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of $SL_2$ flat connections on the one-punctured torus. This is achieved by reformulating the Riemann-Hilbert problem associated to the latter in terms of chiral conformal blocks of a free-fermionic algebra. This viewpoint provides the exact solution of the renormalization group flow of the $SU(2)$ $\mathcal{N}=2^*$ theory on self-dual $Ω$-background and, in the Seiberg-Witten limit, an elegant relation between the IR and UV gauge couplings.

hep-th↗

arXiv2018

On solutions of the Fuji-Suzuki-Tsuda system

We derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlevé VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also initiate the study of algebraic braid group dynamics of semi-degenerate monodromy, and obtain as a byproduct a direct isomonodromic proof of the AGT-W relation for $c=N-1$.

math-ph↗