Searcharxiv⌕ Search

arXiv subjects

Fabrizio Del Monte

Publications and source records attributed to Fabrizio Del Monte.

15 recordsLinked to original sources
arXiv2025

Non-toric 5d SCFTs from Reid's Pagoda

We construct new families of non-toric 5d SCFTs via abelian orbifolds of the Reid Pagoda, including a surprising infinite family of rank-1 theories, that evade all known classifications. Using the McKay correspondence, we derive their BPS quivers and superpotentials. The hallmark of these theories is a novel sector we dub Pagoda matter, whose vacuum expectation values obstruct the Kaehler moduli. This mechanism freezes the gauge coupling to infinite value, precluding a weak-coupling limit and rendering the theories intrinsically strongly coupled. Finally, we interpret these results as 5d SCFTs deformed by non-constant flavor backgrounds.

hep-th↗
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

Joyce structures and poles of Painlevé equations

Joyce structures are a class of geometric structures that first arose in relation to Donaldson-Thomas theory. There is a special class of examples, called class $S[A_1]$, whose underlying manifold parameterises Riemann surfaces of some fixed genus equipped with a meromorphic quadratic differential with poles of fixed orders. We study two Joyce structures of this type using the isomonodromic systems associated to the Painlevé II and III$_3$ equations. We give explicit formulae for the Plebański functions of these Joyce structures, and compute several associated objects, including their tau functions, which we explicitly relate to the corresponding Painlevé tau functions. We show that the behaviour of the Joyce structure near the zero-section can be studied analytically through poles of Painlevé equations. The systematic treatment gives a blueprint for the study of more general Joyce structures associated to meromorphic quadratic differentials on the Riemann sphere.

math-ph↗
arXiv2024

Invariant Stability Conditions on Certain Calabi-Yau Threefolds

We apply results on inducing stability conditions to local Calabi-Yau threefolds and obtain applications to Donaldson-Thomas (DT) theory. A basic example is the total space of the canonical bundle of $Z=\mathbb{P}^1\times \mathbb{P}^1$. We use a result of Dell to construct stability conditions on the derived category of $X$ for which all stable objects can be explicitly described. We relate them to stability conditions on the resolved conifold $Y=\mathscr{O}_{\mathbb{P}^1}(-1)^{\oplus 2}$ in two ways: geometrically via the McKay correspondence, and algebraically via a quotienting operation on quivers with potential. These stability conditions were first discussed in the physics literature by Closset and del Zotto, and were constructed mathematically by Xiong by a different method. We obtain a complete description of the corresponding DT invariants, from which we can conclude that they define analytic wall-crossing structures in the sense of Kontsevich and Soibelman. In the last section we discuss several other examples of a similar flavour.

math.AG↗
arXiv2024

Monodromies of Second Order $q$-difference Equations from the WKB Approximation

This paper studies the space of monodromy data of second order $q$-difference equations through the framework of WKB analysis. We compute the connection matrices associated to the Stokes phenomenon of WKB wavefunctions and develop a general framework to parameterize monodromies of $q$-difference equations. Computations of monodromies are illustrated with explicit examples, including a $q$-Mathieu equation and its degenerations. In all examples we show that the monodromy around the origin of $\mathbb{C}^*$ admits an expansion in terms of Voros symbols, or exponentiated quantum periods, with integer coefficients. Physically these monodromies correspond to expectation values of Wilson line operators in five dimensional quantum field theories with minimal supersymmetry. In the case of the $q$-Mathieu equation, we show that the trace of the monodromy can be identified with the Hamiltonian of a corresponding $q$-Painlevé equation.

math-ph↗
arXiv2023

BPS Spectra and Algebraic Solutions of Discrete Integrable Systems

This paper extends the correspondence between discrete Cluster Integrable Systems and BPS spectra of five-dimensional $\mathcal{N}=1$ QFTs on $\mathbb{R}^4\times S^1$ by proving that algebraic solutions of the integrable systems are exact solutions for the system of TBA equations arising from the BPS spectral problem. This statement is exemplified in the case of M-theory compactifications on local del Pezzo Calabi-Yau threefolds, corresponding to q-Painlevé equations and $SU(2)$ gauge theories with matter. A degeneration scheme is introduced, allowing to obtain closed-form expression for the BPS spectrum also in systems without algebraic solutions. By studying the example of local del Pezzo 3, it is shown that when the region in moduli space associated to an algebraic solution is a ``wall of marginal stability'', the BPS spectrum contains states of arbitrarily high spin, and corresponds to a 5d uplift of a four-dimensional nonlagrangian theory.

hep-th↗
arXiv2023

Generating function of monodromy symplectomorphism for $2\times 2$ Fuchsian systems and its WKB expansion

We study the WKB expansion of $2\times 2$ system of linear differential equations with four fuchsian singularities. The main focus is on the generating function of the monodromy symplectomorphism which, according to a recent paper is closely related to the Jimbo-Miwa tau-function. We compute the first three terms of the WKB expansion of the generating function and establish the link to the Bergman tau-function.

math-ph↗
arXiv2022

The threefold way to quantum periods: WKB, TBA equations and q-Painlevé

We show that TBA equations defined by the BPS spectrum of $5d$ $\mathcal{N}=1$ $SU(2)$ Yang-Mills on $S^1\times \mathbb{R}^4$ encode the q-Painlevé III$_3$ equation. We find a fine-tuned stratum in the physical moduli space of the theory where solutions to TBA equations can be obtained exactly, and verify that they agree with the algebraic solutions to q-Painlevé. Switching from the physical moduli space to that of stability conditions, we identify a one-parameter deformation of the fine-tuned stratum, where the general solution of the q-Painlevé equation in terms of dual instanton partition functions continues to provide explicit TBA solutions. Motivated by these observations, we propose a further extensions of the range of validity of this correspondence, under a suitable identification of moduli. As further checks of our proposal, we study the behavior of exact WKB quantum periods for the quantum curve of local $\mathbb{P}^1\times\mathbb{P}^1$.

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

Quiver symmetries and wall-crossing invariance

We study the BPS particle spectrum of five-dimensional superconformal field theories (SCFTs) on $\mathbb{R}^4\times S^1$ with one-dimensional Coulomb branch, by means of their associated BPS quivers. By viewing these theories as arising from the geometric engineering within M-theory, the quivers are naturally associated to the corresponding local Calabi-Yau threefold. We show that the symmetries of the quiver, descending from the symmetries of the Calabi-Yau geometry, together with the affine root lattice structure of the flavor charges, provide equations for the Kontsevich-Soibelman wall-crossing invariant. We solve these equations iteratively: the pattern arising from the solution is naturally extended to an exact conjectural expression, that we provide for the local Hirzebruch $\mathbb{F}_0$, and local del Pezzo $dP_3$ and $dP_5$ geometries. Remarkably, the BPS spectrum consists of two copies of suitable $4d$ $\mathcal{N}=2$ spectra, augmented by Kaluza-Klein towers.

hep-th↗
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

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↗
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 $Ω$-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 $τ$-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↗