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Marcos Marino

Publications and source records attributed to Marcos Marino.

At least 55 records · Page 3Linked to original sources
arXiv2017

Exact eigenfunctions and the open topological string

Mirror curves to toric Calabi-Yau threefolds can be quantized and lead to trace class operators on the real line. The eigenvalues of these operators are encoded in the BPS invariants of the underlying threefold, but much less is known about their eigenfunctions. In this paper we first develop methods in spectral theory to compute these eigenfunctions. We also provide a matrix integral representation which allows to study them in a 't Hooft limit, where they are described by standard topological open string amplitudes. Based on these results, we propose a conjecture for the exact eigenfunctions which involves both the WKB wavefunction and the standard topological string wavefunction. This conjecture can be made completely explicit in the maximally supersymmetric, or self-dual case, which we work out in detail for local P1xP1. In this case, our conjectural eigenfunctions turn out to be closely related to Baker-Akhiezer functions on the mirror curve, and they are in full agreement with first-principle calculations in spectral theory.

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arXiv2017

Operators and higher genus mirror curves

We perform further tests of the correspondence between spectral theory and topological strings, focusing on mirror curves of genus greater than one with nontrivial mass parameters. In particular, we analyze the geometry relevant to the SU(3) relativistic Toda lattice, and the resolved C^3/Z_6 orbifold. Furthermore, we give evidence that the correspondence holds for arbitrary values of the mass parameters, where the quantization problem leads to resonant states. We also explore the relation between this correspondence and cluster integrable systems.

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arXiv2016

Resurgence in complex Chern-Simons theory

We study resurgence properties of partition function of SU(2) Chern-Simons theory (WRT invariant) on closed three-manifolds. We check explicitly that in various examples Borel transforms of asymptotic expansions posses expected analytic properties. In examples that we study we observe that contribution of irreducible flat connections to the path integral can be recovered from asymptotic expansions around abelian flat connections. We also discuss connection to Floer instanton moduli spaces, disk instantons in 2d sigma models, and length spectra of "complex geodesics" on the A-polynomial curve.

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arXiv2016

Localization at large N in Chern-Simons-matter theories

We review some exact results for the matrix models appearing in the localization of Chern-Simons-matter theories, focusing on the structure of non-perturbative effects and onthe M-theory expansion of ABJM theory. We also summarize some of the results obtained for other Chern-Simons-matter theories, as well as recent applications to topological strings.

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arXiv2016

Exact quantization conditions for cluster integrable systems

We propose exact quantization conditions for the quantum integrable systems of Goncharov and Kenyon, based on the enumerative geometry of the corresponding toric Calabi-Yau manifolds. Our conjecture builds upon recent results on the quantization of mirror curves, and generalizes a previous proposal for the quantization of the relativistic Toda lattice. We present explicit tests of our conjecture for the integrable systems associated to the resolved C^3/Z_5 and C^3/Z_6 orbifolds.

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arXiv2016

Quantization conditions and functional equations in ABJ(M) theories

The partition function of ABJ(M) theories on the three-sphere can be regarded as the canonical partition function of an ideal Fermi gas with a non-trivial Hamiltonian. We propose an exact expression for the spectral determinant of this Hamiltonian, which generalizes recent results obtained in the maximally supersymmetric case. As a consequence, we find an exact WKB quantization condition determining the spectrum which is in agreement with numerical results. In addition, we investigate the factorization properties and functional equations for our conjectured spectral determinants. These functional equations relate the spectral determinants of ABJ theories with consecutive ranks of gauge groups but the same Chern-Simons coupling.

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arXiv2016

Matrix models from operators and topological strings, 2

The quantization of mirror curves to toric Calabi--Yau threefolds leads to trace class operators, and it has been conjectured that the spectral properties of these operators provide a non-perturbative realization of topological string theory on these backgrounds. In this paper, we find an explicit form for the integral kernel of the trace class operator in the case of local P1xP1, in terms of Faddeev's quantum dilogarithm. The matrix model associated to this integral kernel is an O(2) model, which generalizes the ABJ(M) matrix model. We find its exact planar limit, and we provide detailed evidence that its 1/N expansion captures the all genus topological string free energy on local P1xP1.

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arXiv2015

Spectral Theory and Mirror Curves of Higher Genus

Recently, a correspondence has been proposed between spectral theory and topological strings on toric Calabi-Yau manifolds. In this paper we develop in detail this correspondence for mirror curves of higher genus, which display many new features as compared to the genus one case studied so far. Given a curve of genus g, our quantization scheme leads to g different trace class operators. Their spectral properties are encoded in a generalized spectral determinant, which is an entire function on the Calabi-Yau moduli space. We conjecture an exact expression for this spectral determinant in terms of the standard and refined topological string amplitudes. This conjecture provides a non-perturbative definition of the topological string on these geometries, in which the genus expansion emerges in a suitable 't Hooft limit of the spectral traces of the operators. In contrast to what happens in quantum integrable systems, our quantization scheme leads to a single quantization condition, which is elegantly encoded by the vanishing of a quantum-deformed theta function on the mirror curve. We illustrate our general theory by analyzing in detail the resolved C3/Z5 orbifold, which is the simplest toric Calabi-Yau manifold with a genus two mirror curve. By applying our conjecture to this example, we find new quantization conditions for quantum mechanical operators, in terms of genus two theta functions, as well as new number-theoretic properties for the periods of this Calabi-Yau.

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arXiv2015

Topological Strings from Quantum Mechanics

We propose a general correspondence which associates a non-perturbative quantum-mechanical operator to a toric Calabi-Yau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an M-theoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the Nekrasov-Shatashvili limit of the refined topological string, but there are non-perturbative corrections determined by the conventional topological string. We analyze in detail the cases of local P2, local P1xP1 and local F1. In all these cases, the predictions for the spectrum agree with the existing numerical results. We also show explicitly that our conjectured spectral determinant leads to the correct spectral traces of the corresponding operators. Physically, our results provide a non-perturbative formulation of topological strings on toric Calabi-Yau manifolds, in which the genus expansion emerges as a 't Hooft limit of the spectral traces. Since the spectral determinant is an entire function on moduli space, it leads to a background independent formulation of the theory. Mathematically, our results lead to precise, surprising conjectures relating the spectral theory of functional difference operators to enumerative geometry

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arXiv2015

Operators from mirror curves and the quantum dilogarithm

Mirror manifolds to toric Calabi-Yau threefolds are encoded in algebraic curves. The quantization of these curves leads naturally to quantum-mechanical operators on the real line. We show that, for a large number of local del Pezzo Calabi-Yau threefolds, these operators are of trace class. In some simple geometries, like local P2, we calculate the integral kernel of the corresponding operators in terms of Faddeev's quantum dilogarithm. Their spectral traces are expressed in terms of multi-dimensional integrals, similar to the state-integrals appearing in three-manifold topology, and we show that they can be evaluated explicitly in some cases. Our results provide further verifications of a recent conjecture which gives an explicit expression for the Fredholm determinant of these operators, in terms of enumerative invariants of the underlying Calabi-Yau threefolds.

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arXiv2015

Exact solutions to quantum spectral curves by topological string theory

We generalize the conjectured connection between quantum spectral problems and topological strings to many local almost del Pezzo surfaces with arbitrary mass parameters. The conjecture uses perturbative information of the topological string in the unrefined and the Nekrasov-Shatashvili limit to solve non-perturbatively the quantum spectral problem. We consider the quantum spectral curves for the local almost del Pezzo surfaces of $\mathbb{F}_2$, $\mathbb{F}_1$, the blowup of $\mathbb{P}^2$ in two points and a mass deformation of the $E_8$ del Pezzo corresponding to different deformations of the three-term operators $\mathsf{O}_{1,1}$, $\mathsf{O}_{1,2}$ and $\mathsf{O}_{2,3}$. To check the conjecture, we compare the predictions for the spectrum of these operators with numerical results for the eigenvalues. We also compute the first few fermionic spectral traces from the conjectural spectral determinant, and we compare them to analytic and numerical results in spectral theory. In all these comparisons, we find that the conjecture is fully validated with high numerical precision. For local $\mathbb{F}_2$ we expand the spectral determinant around the orbifold point and find intriguing relations for Jacobi theta functions. We also give an explicit map between the geometries of $\mathbb{F}_0$ and $\mathbb{F}_2$ as well as a systematic way to derive the operators $\mathsf{O}_{m,n}$ from toric geometries.

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arXiv2015

Exact results in N=8 Chern-Simons-matter theories and quantum geometry

We show that, in ABJ(M) theories with N=8 supersymmetry, the non-perturbative sector of the partition function on the three-sphere simplifies drastically. Due to this simplification, we are able to write closed form expressions for the grand potential of these theories, which determines the full large N asymptotics. Moreover, we find explicit formulae for the generating functionals of their partition functions, for all values of the rank N of the gauge group: they involve Jacobi theta functions on the spectral curve associated to the planar limit. Exact quantization conditions for the spectral problem of the Fermi gas are then obtained from the vanishing of the theta function. We also show that the partition function, as a function of N, can be extended in a natural way to an entire function on the full complex plane, and we explore some possible consequences of this fact for the quantum geometry of M-theory and for putative de Sitter extensions.

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arXiv2015

Matrix models from operators and topological strings

We propose a new family of matrix models whose 1/N expansion captures the all-genus topological string on toric Calabi-Yau threefolds. These matrix models are constructed from the trace class operators appearing in the quantization of the corresponding mirror curves. The fact that they provide a non-perturbative realization of the (standard) topological string follows from a recent conjecture connecting the spectral properties of these operators, to the enumerative invariants of the underlying Calabi-Yau threefolds. We study in detail the resulting matrix models for some simple geometries, like local P^2 and local F_2, and we verify that their weak 't Hooft coupling expansion reproduces the topological string free energies near the conifold singularity. These matrix models are formally similar to those appearing in the Fermi-gas formulation of Chern-Simons-matter theories, and their 1/N expansion receives non-perturbative corrections determined by the Nekrasov-Shatashvili limit of the refined topological string.

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arXiv2015

Resumming the string perturbation series

We use the AdS/CFT correspondence to study the resummation of a perturbative genus expansion appearing in the type II superstring dual of ABJM theory. Although the series is Borel summable, its Borel resummation does not agree with the exact non-perturbative answer due to the presence of complex instantons. The same type of behavior appears in the WKB quantization of the quartic oscillator in Quantum Mechanics, which we analyze in detail as a toy model for the string perturbation series. We conclude that, in these examples, Borel summability is not enough for extracting non-perturbative information, due to non-perturbative effects associated to complex instantons. We also analyze the resummation of the genus expansion for topological string theory on local $\mathbb P^1 \times \mathbb P^1$, which is closely related to ABJM theory. In this case, the non-perturbative answer involves membrane instantons computed by the refined topological string, which are crucial to produce a well-defined result. We give evidence that the Borel resummation of the perturbative series requires such a non-perturbative sector.

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arXiv2015

The topological open string wavefunction

We show that, in local Calabi-Yau manifolds, the topological open string partition function transforms as a wavefunction under modular transformations. Our derivation is based on the topological recursion for matrix models, and it generalizes in a natural way the known result for the closed topological string sector. As an application, we derive results for vevs of 1/2 BPS Wilson loops in ABJM theory at all genera in a strong coupling expansion, for various representations.

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arXiv2014

M-theoretic matrix models

Some matrix models admit, on top of the usual 't Hooft expansion, an M-theory-like expansion, i.e. an expansion at large N but where the rest of the parameters are fixed, instead of scaling with N. These models, which we call M-theoretic matrix models, appear in the localization of Chern-Simons-matter theories, and also in two-dimensional statistical physics. Generically, their partition function receives non-perturbative corrections which are not captured by the 't Hooft expansion. In this paper, we discuss general aspects of these type of matrix integrals and we analyze in detail two different examples. The first one is the matrix model computing the partition function of N=4 supersymmetric Yang-Mills theory in three dimensions with one adjoint hypermultiplet and N_f fundamentals, which has a conjectured M-theory dual, and which we call the N_f matrix model. The second one, which we call the polymer matrix model, computes form factors of the 2d Ising model and is related to the physics of 2d polymers. In both cases we determine their exact planar limit. In the N_f matrix model, the planar free energy reproduces the expected behavior of the M-theory dual. We also study their M-theory expansion by using Fermi gas techniques, and we find non-perturbative corrections to the 't Hooft expansion.

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arXiv2014

The uses of the refined matrix model recursion

We study matrix models in the beta ensemble by building on the refined recursion relation proposed by Chekhov and Eynard. We present explicit results for the first beta-deformed corrections in the one-cut and the two-cut cases, as well as two applications to supersymmetric gauge theories: the calculation of superpotentials in N=1 gauge theories, and the calculation of vevs of surface operators in superconformal N=2 theories and their Liouville duals. Finally, we study the beta deformation of the Chern-Simons matrix model. Our results indicate that this model does not provide an appropriate description of the Omega-deformed topological string on the resolved conifold, and therefore that the beta-deformation might provide a different generalization of topological string theory in toric Calabi-Yau backgrounds.

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