arXiv subjects

Marcos Marino

Publications and source records attributed to Marcos Marino.

At least 19 recordsLinked to original sources

arXiv2026

Resurgence of Chern-Simons theory at the trivial flat connection

Some years ago, it was conjectured by the first author that the Chern-Simons perturbation theory of a 3-manifold at the trivial flat connection is a resurgent power series. We describe completely the resurgent structure of the above series (including the location of the singularities and their Stokes constants) in the case of a hyperbolic knot complement in terms of an extended square matrix of $(x,q)$-series whose rows are indexed by the boundary parabolic $\text{SL}_2(\mathbb{C})$-flat connections, including the trivial one. We use our extended matrix to describe the Stokes constants of the above series, to define explicitly their Borel transform and to identify it with state-integrals. Along the way, we use our matrix to give an analytic extension of the Kashaev invariant and of the colored Jones polynomial and to complete the matrix valued holomorphic quantum modular forms as well as to give an exact version of the refined quantum modularity conjecture of Zagier and the first author. Finally, our matrix provides an extension of the 3D-index in a sector of the trivial flat connection. We illustrate our definitions, theorems, numerical calculations and conjectures with the two simplest hyperbolic knots.

math.GT↗

arXiv2026

Thou shalt not tunnel: Complex instantons and tunneling suppression in deformed quantum mechanics

The quantization of the Seiberg-Witten curve of ${\cal N}=2$ super Yang-Mills theory leads to a deformation of one-dimensional quantum mechanics with unconventional behavior. Most notably, quantum tunneling is suppressed at special points in parameter space. In this paper we examine these deformed models in the case of double-well and cubic potentials, and we find that they have a rich phase structure. In what we call the strong coupling phase, the theory behaves like conventional quantum mechanics, instantons are real, and tunneling is not suppressed. In the weak coupling phase, the instantons responsible for tunneling become complex, and tunneling suppression takes place at the so-called Toda lattice points. At the critical point between the two phases, which corresponds to a monopole point in super Yang-Mills theory, the non-perturbative amplitudes display an anomalous scaling as a function of $\hbar$. This phase structure reflects the physics of the underlying super Yang-Mills theory and can be regarded as a physical manifestation of wall-crossing behavior of the BPS spectrum, which we determine in our problem by using resurgent techniques.

hep-th↗

arXiv2025

Les Houches lectures on non-perturbative topological strings

In these lecture notes for the Les Houches School on Quantum Geometry I give an introductory overview of non-perturbative aspects of topological string theory. After a short summary of the perturbative aspects, I first consider the non-perturbative sectors of the theory as unveiled by the theory of resurgence. I give a self-contained derivation of recent results on non-perturbative amplitudes, and I explain the conjecture relating the resurgent structure of the topological string to BPS invariants. In the second part of the lectures I introduce the topological string/spectral theory (TS/ST) correspondence, which provides a non-perturbative definition of topological string theory on toric Calabi-Yau manifolds in terms of the spectral theory of quantum mirror curves

hep-th↗

arXiv2025

On the Structure of Trans-Series in Quantum Field Theory

Many observables in quantum field theory can be expressed in terms of trans-series, in which one adds to the perturbative series a typically infinite sum of exponentially small corrections, due to instantons or to renormalons. Even after Borel resummation of the series in the coupling constant, one has to sum this infinite series of small exponential corrections. It has been argued that this leads to a new divergence, which is sometimes called the divergence of the OPE. We show that, in some interesting examples in quantum field theory, the series of small exponential corrections is convergent, order by order in the coupling constant. In particular, we give numerical evidence for this convergence property in the case of the free energy of integrable asymptotically free theories, which has been intensively studied recently in the framework of resurgence. Our results indicate that, in these examples, the Borel resummed trans-series leads to a well defined function, and there are no further divergences.

hep-th↗

arXiv2025

Anatomy of the simplest renormalon

Perhaps the simplest IR renormalon occurs in the ground state energy of a superrenormalizable model, the scalar $O(N)$ theory in two dimensions with a quartic potential and negative squared mass. We show that this renormalon, found previously in perturbation theory at next-to-leading order (NLO) in the $1/N$ expansion, gives indeed the correct asymptotic expansion of the exact large $N$ solution of the model, and we determine explicitly the complete trans-series of non-perturbative corrections to the perturbative result. We also use this framework to study the $O(N)$-invariant two-point function of the scalar field. As expected, it is IR finite in perturbation theory, but it is afflicted as well with an IR renormalon singularity and is not Borel summable. The pole mass is purely non-perturbative and its trans-series can be also fully determined at NLO in $1/N$

hep-th↗

arXiv2025

Trans-series from condensates

The Shifman-Vainshtein-Zakharov (SVZ) sum rules provide a method to obtain trans-series expansions in many quantum field theories, in which exponentially small corrections are calculated by combining the operator product expansion with the assumption of vacuum condensates. In some solvable models, exact expressions for trans-series can be obtained from non-perturbative results, and this makes it possible to test the SVZ method by comparing its predictions to these exact trans-series. In this paper we perform such a precision test in the example of the fermion self-energy in the Gross-Neveu model. Its exact trans-series expansion can be extracted from the large $N$ solution, at the first non-trivial order in $1/N$. It is given by an infinite series of exponentially small corrections involving factorially divergent power series in the 't Hooft parameter. We show that the first two corrections are associated to two-quark and four-quark condensates, and we reproduce the corresponding power series exactly, and at all loops, by using the SVZ method. In addition, the numerical values of the condensates can be extracted from the exact result, up to order $1/N$.

hep-th↗

arXiv2024

Large N instantons from topological strings

The $1/N$ expansion of matrix models is asymptotic, and it requires non-perturbative corrections due to large $N$ instantons. Explicit expressions for large $N$ instanton amplitudes are known in the case of Hermitian matrix models with one cut, but not in the multi-cut case. We show that the recent exact results on topological string instanton amplitudes provide the non-perturbative contributions of large $N$ instantons in generic multi-cut, Hermitian matrix models. We present a detailed test in the case of the cubic matrix model by considering the asymptotics of its $1/N$ expansion, which we obtain at relatively high genus for a generic two-cut background. These results can be extended to certain non-conventional matrix models which admit a topological string theory description. As an application, we determine the large $N$ instanton corrections for the free energy of ABJM theory on the three-sphere, which correspond to D-brane instanton corrections in superstring theory. We also illustrate the applications of topological string instantons in a more mathematical setting by considering orbifold Gromov-Witten invariants. By focusing on the example of $\mathbb{C}^3/\mathbb{Z}_3$, we show that they grow doubly-factorially with the genus and we obtain and test explicit asymptotic formulae for them.

hep-th↗

arXiv2024

Resurgent Structure of the Topological String and the First Painlevé Equation

We present an explicit formula for the Stokes automorphism acting on the topological string partition function. When written in terms of the dual partition function, our formula implies that flat coordinates in topological string theory transform as quantum periods, and according to the Delabaere-Dillinger-Pham formula. We first show how the formula follows from the non-linear Stokes phenomenon of the Painlevé I equation, together with the connection between its $τ$-function and topological strings on elliptic curves. Then, we show that this formula is also a consequence of a recent conjecture on the resurgent structure of the topological string, based on the holomorphic anomaly equations, and it is in fact valid for arbitrary Calabi-Yau threefolds.

hep-th↗

arXiv2024

Large N instantons, BPS states, and the replica limit

We study the relation between large N instantons and conventional instantons, focusing on matrix models and topological strings. We show that the resurgent properties of the perturbative series at fixed but arbitrary N, including the replica limit N = 0, can be obtained from large N instantons. In the case of topological strings, it has been conjectured that the resurgent structure encoded by large N instantons is closely related to the spectrum of BPS states. We give direct evidence for this connection in the case of Seiberg-Witten theory and other topological string models, and we show in detail how the resurgent properties at fixed N follow from the large N theory, and therefore can be used to obtain information on BPS invariants.

hep-th↗

arXiv2023

Exact multi-instantons in topological string theory

Topological string theory has multi-instanton sectors which lead to non-perturbative effects in the string coupling constant and control the large order behavior of the perturbative genus expansion. As proposed by Couso, Edelstein, Schiappa and Vonk, these sectors can be described by a trans-series extension of the BCOV holomorphic anomaly equations. In this paper we find exact, closed form solutions for these multi-instanton trans-series in the case of local Calabi-Yau manifolds. The resulting multi-instanton amplitudes turn out to be very similar to the eigenvalue tunneling amplitudes of matrix models. Their form suggests that the flat coordinates of the Calabi-Yau manifold are naturally quantized in units of the string coupling constant, as postulated in large $N$ dualities. Based on our results we obtain a general picture for the resurgent structure of the topological string in the local case, which we illustrate with explicit calculations in the case of local $\mathbb{P}^2$.

hep-th↗

arXiv2023

Instantons, renormalons and the theta angle in integrable sigma models

Some sigma models which admit a theta angle are integrable at both $\vartheta=0$ and $\vartheta=π$. This includes the well-known $O(3)$ sigma model and two families of coset sigma models studied by Fendley. We consider the ground state energy of these models in the presence of a magnetic field, which can be computed with the Bethe ansatz. We obtain explicit results for its non-perturbative corrections and we study the effect of the theta angle on them. We show that imaginary, exponentially small corrections due to renormalons remain unchanged, while instanton corrections change sign, as expected. We find in addition corrections due to renormalons which also change sign as we turn on the theta angle. Based on these results we present an explicit non-perturbative formula for the topological susceptibility of the $O(3)$ sigma model in the presence of a magnetic field, in the weak coupling limit.

hep-th↗

arXiv2023

Non-perturbative topological string theory on compact Calabi-Yau 3-folds

We obtain analytic and numerical results for the non-perturbative amplitudes of topological string theory on arbitrary, compact Calabi-Yau manifolds. Our approach is based on the theory of resurgence and extends previous special results to the more general case. In particular, we obtain explicit trans-series solutions of the holomorphic anomaly equations. Our results predict the all orders, large genus asymptotics of the topological string free energies, which we test in detail against high genus perturbative series obtained recently in the compact case. We also provide additional evidence that the Stokes constants appearing in the resurgent structure are closely related to integer BPS invariants.

hep-th↗

arXiv2023

On the resurgent structure of quantum periods

Quantum periods appear in many contexts, from quantum mechanics to local mirror symmetry. They can be described in terms of topological string free energies and Wilson loops, in the so-called Nekrasov-Shatashvili limit. We consider the trans-series extension of the holomorphic anomaly equations satisfied by these quantities, and we obtain exact multi-instanton solutions for these trans-series. Building on this result, we propose a unified perspective on the resurgent structure of quantum periods. We show for example that the Delabaere-Pham formula, which was originally obtained in quantum mechanical examples, is a generic feature of quantum periods. We illustrate our general results with explicit calculations for the double-well in quantum mechanics, and for the quantum mirror curve of local $\mathbb{P}^2$.

hep-th↗

arXiv2023

New renormalons from analytic trans-series

We study the free energy of integrable, asymptotically free field theories in two dimensions coupled to a conserved charge. We develop methods to obtain analytic expressions for its trans-series expansion, directly from the Bethe ansatz equations, and we use this result to determine the structure of its Borel singularities. We find a new class of infrared renormalons which does not fit the traditional expectations of renormalon physics proposed long ago by 't Hooft and Parisi. We check the existence of these new singularities with detailed calculations based on the resurgent analysis of the perturbative expansion. Our results show that the structure of renormalons in asymptotically free theories is more subtle than previously thought, and that large $N$ estimates of their location might be misleading.

hep-th↗

arXiv2022

New Instantons for Matrix Models

The complete, nonperturbative content of random matrix models is described by resurgent-transseries -- general solutions to their corresponding string-equations. These transseries include exponentially-suppressed multi-instanton amplitudes obtained by eigenvalue tunneling, but they also contain exponentially-enhanced and mixed instanton-like sectors with no known matrix model interpretation. This work shows how these sectors can be also described by eigenvalue tunneling in matrix models -- but on the non-physical sheet of the spectral curve describing their large-N limit. This picture further explains the full resurgence of random matrices via analysis of all possible eigenvalue integration-contours. How to calculate these "anti" eigenvalue-tunneling amplitudes is explained in detail and in various examples, such as the cubic and quartic matrix models, and their double-scaling limit to Painleve I. This further provides direct matrix-model derivations of their resurgent Stokes data, which were recently obtained by different techniques.

hep-th↗

arXiv2022

Peacock patterns and resurgence in complex Chern-Simons theory

The partition function of complex Chern-Simons theory on a 3-manifold with torus boundary reduces to a finite dimensional state-integral which is a holomorphic function of a complexified Planck's constant $τ$ in the complex cut plane and an entire function of a complex parameter $u$. This gives rise to a vector of factorially divergent perturbative formal power series whose Stokes rays form a peacock-like pattern in the complex plane. We conjecture that these perturbative series are resurgent, their trans-series involve two non-perturbative variables, their Stokes automorphism satisfies a unique factorization property and that it is given explicitly in terms of a fundamental matrix solution to a (dual) linear $q$-difference equation. We further conjecture that entries of the Stokes automorphism matrix are the 3D-indices of Dimofte-Gaiotto-Gukov. We provide proofs of our statements regarding the $q$-difference equations and their properties of their fundamental solutions and illustrate our conjectures regarding the Stokes matrices with numerical calculations for the two simplest hyperbolic $4_1$ and $5_2$ knots.

math.GT↗

arXiv2022

Peacock patterns and new integer invariants in topological string theory

Topological string theory near the conifold point of a Calabi-Yau threefold gives rise to factorially divergent power series which encode the all-genus enumerative information. These series lead to infinite towers of singularities in their Borel plane (also known as "peacock patterns"), and we conjecture that the corresponding Stokes constants are integer invariants of the Calabi-Yau threefold. We calculate these Stokes constants in some toric examples, confirming our conjecture and providing in some cases explicit generating functions for the new integer invariants, in the form of q-series. Our calculations in the toric case rely on the TS/ST correspondence, which promotes the asymptotic series near the conifold point to spectral traces of operators, and makes it easier to identify the Stokes data. The resulting mathematical structure turns out to be very similar to the one of complex Chern-Simons theory. In particular, spectral traces correspond to state integral invariants and factorize in holomorphic/anti-holomorphic blocks.

hep-th↗

arXiv2021

Resurgence and renormalons in the one-dimensional Hubbard model

We use resurgent analysis to study non-perturbative aspects of the one-dimensional, multicomponent Hubbard model with an attractive interaction and arbitrary filling. In the two-component case, we show that the leading Borel singularity of the perturbative series for the ground-state energy is determined by the energy gap, as expected for superconducting systems. This singularity turns out to be of the renormalon type, and we identify a class of diagrams leading to the correct factorial growth. As a consequence of our analysis, we propose an explicit expression for the energy gap at weak coupling in the multi-component Hubbard model, at next-to-leading order in the coupling constant. In the two-component, half-filled case, we use the Bethe ansatz solution to determine the full trans-series for the ground state energy, and the exact form of its Stokes discontinuity.

hep-th↗