arXiv subjects

Giulio Bonelli

Publications and source records attributed to Giulio Bonelli.

At least 19 recordsLinked to original sources

arXiv2025

Bilinear tau forms of quantum Painlev\'e equations and $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations in SUSY gauge theories

We derive bilinear tau forms of the canonically quantized Painlev\'e equations, thereby relating them to those previously obtained from the $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations for the $\mathcal{N}=2$ supersymmetric gauge theory partition functions on a general $\Omega$-background. We fully fix the refined Painlev\'e/gauge theory dictionary by formulating the proper equations for the quantum nonautonomous Painlev\'e Hamiltonians. We also describe the quantum Painlev\'e symmetries at the level of tau functions and, as a byproduct of this analysis, obtain several $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations including those in nontrivial holonomy sectors of the 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 Lam\'e 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 Lam\'e 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

One loop corrections to the thermodynamics of near-extremal Kerr-(A)dS black holes from Heun equation

We compute one-loop corrections to the euclidean gravitational path integral of near-extremal (anti-)de Sitter-Kerr black hole in terms of the connection coefficients of the Heun equation describing the black hole linear perturbations in the Teukolsky formalism. We show that different near-extremal limits lead to distinct physical properties of the gravitational configuration, as they get described by distinct limiting differential equations. As a result, the light modes emerging in the limit determine different scaling properties in the temperature of the one-loop determinants. We show that the cold case displays distinctive universal log(T) corrections to the entropy of the system, including the ultracold regime. On the contrary, these do not appear in the limit in which the event horizon superimposes onto the cosmological one. In the Schwarzschild-de Sitter case, a further check is performed by comparison with the Denef-Hartnoll-Sachdev formula.

hep-th↗

arXiv2024

One-loop corrections to near extremal Kerr thermodynamics from semiclassical Virasoro blocks

We propose a method to perform an exact calculation of one-loop quantum corrections to black hole entropy in terms of Virasoro semiclassical blocks. We analyse in detail four-dimensional Kerr black hole and show that in the near-extremal limit a branch of long-lived modes arises. We prove that the contribution of these modes accounts for a $(s-1/2)\log T_{\text{Hawking}}$ correction to the entropy for massless particles of spin $s=1,2$. We show that in the full calculation performed in the exact Kerr background the leading contribution actually is sourced by the near-horizon region only, and as such has a universal validity for any asymptotic behavior at infinity.

hep-th↗

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↗

arXiv2024

One loop effective actions in Kerr-(A)dS Black Holes

We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ODEs with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasi-normal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulae for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss the exact properties of the thermal version of the BH effective actions by providing a proof of the DHS formula and explicitly computing it for new relevant cases. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulae can be given for higher spin matter and radiation fields in more general gravitational backgrounds.

hep-th↗

arXiv2023

Black Hole Perturbation Theory Meets CFT$_2$: Kerr Compton Amplitudes from Nekrasov-Shatashvili Functions

We present a novel study of Kerr Compton amplitudes in a partial wave basis in terms of the Nekrasov-Shatashvili (NS) function of the confluent Heun equation (CHE). Remarkably, NS-functions enjoy analytic properties and symmetries that are naturally inherited by the Compton amplitudes. Based on this, we characterize the analytic dependence of the Compton phase-shift in the Kerr spin parameter and provide a direct comparison to the standard post-Minkowskian (PM) perturbative approach within General Relativity (GR). We also analyze the universal large frequency behavior of the relevant characteristic exponent of the CHE -- also known as the renormalized angular momentum -- and find agreement with numerical computations. Moreover, we discuss the analytic continuation in the harmonics quantum number $\ell$ of the partial wave, and show that the limit to the physical integer values commutes with the PM expansion of the observables. Finally, we obtain the contributions to the tree level, point-particle, gravitational Compton amplitude in a covariant basis through $O(a_{\text{BH}}^8)$, without the need to take the super-extremal limit for Kerr spin.

hep-th↗

arXiv2023

Black hole perturbation theory and multiple polylogarithms

We study black hole linear perturbation theory in a four-dimensional Schwarzschild (anti) de Sitter background. When dealing with a positive cosmological constant, the corresponding spectral problem is solved systematically via the Nekrasov-Shatashvili functions or, equivalently, classical Virasoro conformal blocks. However, this approach can be more complicated to implement for certain perturbations if the cosmological constant is negative. For these cases, we propose an alternative method to set up perturbation theory for both small and large black holes in an analytical manner. Our analysis reveals a new underlying recursive structure that involves multiple polylogarithms. We focus on gravitational, electromagnetic, and conformally coupled scalar perturbations subject to Dirichlet and Robin boundary conditions. The low-lying modes of the scalar sector of gravitational perturbations and its hydrodynamic limit are studied in detail.

hep-th↗

arXiv2022

On the convergence of Nekrasov functions

In this note we present some results on the convergence of Nekrasov partition functions as power series in the instanton counting parameter. We focus on $U(N)$ ${\mathcal N}=2$ gauge theories in four dimensions with matter in the adjoint and in the fundamental representations of the gauge group respectively and find rigorous lower bounds for the convergence radius in the two cases: if the theory is {\it conformal}, then the series has at least a {\it finite} radius of convergence, while if it is {\it asymptotically free} it has {\it infinite} radius of convergence. Via AGT correspondence, this implies that the related irregular conformal blocks of $W_N$ algebrae admit a power expansion in the modulus converging in the whole plane. By specifying to the $SU(2)$ case, we apply our results to analyse the convergence properties of the corresponding Painlev\'e $\tau$-functions.

hep-th↗

arXiv2022

M2-branes and $\mathfrak{q}$-Painlevé equations

In this paper we investigate a novel connection between the effective theory of M2-branes on $(\mathbb{C}^2/\mathbb{Z}_2\times \mathbb{C}^2/\mathbb{Z}_2)/\mathbb{Z}_k$ and the $\mathfrak{q}$-deformed Painlevé equations, by proposing that the grand canonical partition function of the corresponding four-nodes circular quiver $\mathcal{N}=4$ Chern-Simons matter theory solves the $\mathfrak{q}$-Painlevé VI equation. We analyse how this describes the moduli space of the topological string on local $\text{dP}_5$ and, via geometric engineering, five dimensional $N_f=4$ $\text{SU}(2)$ $\mathcal{N}=1$ gauge theory on a circle. The results we find extend the known relation between ABJM theory, $\mathfrak{q}$-Painlevé $\text{III}_3$, and topological strings on local ${\mathbb P}^1\times{\mathbb P}^1$. From the mathematical viewpoint the quiver Chern-Simons theory provides a conjectural Fredholm determinant realisation of the $\mathfrak{q}$-Painlevé VI $τ$-function. We provide evidence for this proposal by analytic and numerical checks and discuss in detail the successive decoupling limits down to $N_f=0$, corresponding to $\mathfrak{q}$-Painlevé$\,\,$III${}_3$.

hep-th↗

arXiv2022

Irregular Liouville correlators and connection formulae for Heun functions

We perform a detailed study of a class of irregular correlators in Liouville Conformal Field Theory, of the related Virasoro conformal blocks with irregular singularities and of their connection formulae. Upon considering their semi-classical limit, we provide explicit expressions of the connection matrices for the Heun function and a class of its confluences. Their calculation is reduced to concrete combinatorial formulae from conformal block expansions.

hep-th↗

arXiv2022

$tt^*$ Toda equations for surface defects in ${\mathcal N}=2$ SYM and instanton counting for classical Lie groups

The partition function of $\mathcal{N}=2$ super Yang-Mills theories with arbitrary simple gauge group coupled to a self-dual $Ω$-background is shown to be fully determined by studying the renormalization group equations relevant to the surface operators generating its one-form symmetries. The corresponding system of equations results in a ${\it non-autonomous}$ Toda chain on the root system of the Langlands dual, the evolution parameter being the RG scale. A systematic algorithm computing the full multi-instanton corrections is derived in terms of recursion relations whose gauge theoretical solution is obtained just by fixing the perturbative part of the IR prepotential as its asymptotic boundary condition for the RGE. We analyse the explicit solutions of the $τ$-system for all the classical groups at the diverse levels, extend our analysis to affine twisted Lie algebras and provide conjectural bilinear relations for the $τ$-functions of linear quiver gauge theory.

hep-th↗

arXiv2021

Exact solution of Kerr black hole perturbations via CFT$_2$ and instanton counting. Greybody factor, Quasinormal modes and Love numbers

We give explicit expressions for the finite frequency greybody factor, quasinormal modes and Love numbers of Kerr black holes by computing the exact connection coefficients of the radial and angular parts of the Teukolsky equation. This is obtained by solving the connection problem of the confluent Heun equation in terms of the explicit expression of irregular Virasoro conformal blocks as sums over partitions via the AGT correspondence. In the relevant approximation limits our results are in agreement with existing literature. The method we use can be extended to solve the linearized Einstein equation in other interesting gravitational backgrounds.

hep-th↗

arXiv2021

Instantons to the people: the power of one-form symmetries

We show that the non-perturbative dynamics of $\mathcal{N}=2$ super Yang-Mills theories in a self-dual $Ω$-background and with an arbitrary simple gauge group is fully determined by studying renormalization group equations of vevs of surface operators generating one-form symmetries. The corresponding system of equations is a {\it non-autonomous} Toda chain, the time being the RG scale. We obtain new recurrence relations which provide a systematic algorithm computing multi-instanton corrections from the tree-level one-loop prepotential as the asymptotic boundary condition of the RGE. We exemplify by computing the $E_6$ and $G_2$ cases up to two-instantons.

hep-th↗

arXiv2020

Gauge theories on compact toric manifolds

We compute the ${\cal N}=2$ supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on $\mathbb{C}^2$. The evaluation of these residues is greatly simplified by using an "abstruse duality" that relates the residues at the poles of the one-loop and instanton parts of the $\mathbb{C}^2$ partition function. As particular cases, our formulae compute the $SU(2)$ and $SU(3)$ {\it equivariant} Donaldson invariants of $\mathbb{P}^2$ and $\mathbb{F}_n$ and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the $SU(2)$ case. Finally, we show that the $U(1)$ self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a $\mathcal{N}=2$ analog of the $\mathcal{N}=4$ holomorphic anomaly equations.

hep-th↗

arXiv2020

ADHM in 8d, coloured solid partitions and Donaldson-Thomas invariants on orbifolds

We study the moduli space of $SU(4)$ invariant BPS conditions in supersymmetric gauge theory on non-commutative ${\mathbb C}^4$ by means of an ADHM-like quiver construction and we classify the invariant solutions under the natural toric action in terms of solid partitions. In the orbifold case ${\mathbb C}^4/G$, $G$ being a finite subgroup of $SU(4)$, the classification is given in terms of coloured solid partitions. The statistical weight for their counting is defined through the associated equivariant cohomological gauge theory. We explicitly compute its partition function on ${\mathbb C}^4$ and ${\mathbb C}^2\times\left({\mathbb C}^2/{\mathbb Z}_2\right)$ which conjecturally provides the corresponding orbifold Donaldson-Thomas invariants.

hep-th↗

arXiv2020

BPS quivers of five-dimensional SCFTs, Topological Strings and q-Painlevé equations

We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi-Yau geometries describing five dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $τ$-functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five dimensional $SU(2)$ pure Super Yang-Mills and $N_f = 2$ on a circle.

hep-th↗