Eynard-Orantin Topological Recursion Research Articles

Topological recursion for Kadomtsev–Petviashvili tau functions of hypergeometric type

AbstractWe study the ‐point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their ‐deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so‐called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of ‐deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)‐type formulas discussed in the literature.

Masur–Veech volumes and intersection theory: The principal strata of quadratic differentials

We describe a conjectural formula via intersection numbers for the Masur–Veech volumes of strata of quadratic differentials with prescribed zero orders, and we prove the formula for the case when the zero orders are odd. For the principal strata of quadratic differentials with simple zeros, the formula reduces to computing the top Segre class of the quadratic Hodge bundle, which can be further simplified to certain linear Hodge integrals. An appendix proves that the intersection of this class with ψ-classes can be computed by Eynard–Orantin topological recursion. As applications, we analyze numerical properties of Masur–Veech volumes, area Siegel–Veech constants, and sums of Lyapunov exponents of the principal strata for fixed genus and varying number of zeros, which settles the corresponding conjectures due to Grivaux and Hubert, Fougeron, and elaborated in Andersen et al. We also describe conjectural formulas for area Siegel–Veech constants and sums of Lyapunov exponents for arbitrary affine invariant submanifolds, and verify them for the principal strata.

Genus Two Siegel Quasi-Modular Forms and Gromov–Witten Theory of Toric Calabi–Yau Threefolds

We first develop theories of differential rings of quasi-Siegel modular and quasi-Siegel Jacobi forms for genus two. Then we apply them to the Eynard-Orantin topological recursion of certain local Calabi-Yau threefolds equipped with branes, whose mirror curves are genus two hyperelliptic curves. By the proof of the Remodeling Conjecture, we prove that the corresponding open- and closed- Gromov-Witten potentials are essentially quasi-Siegel Jacobi and quasi-Siegel modular forms for genus two, respectively.

Nonperturbative effects and resurgence in Jackiw-Teitelboim gravity at finite cutoff

We investigate the nonperturbative structure of Jackiw-Teitelboim gravity at finite cutoff, as given by its proposed formulation in terms of a $T\bar{T}$-deformed Schwarzian quantum mechanics. Our starting point is a careful computation of the disk partition function to all orders in the perturbative expansion in the cutoff parameter. We show that the perturbative series is asymptotic and that it admits a precise completion exploiting the analytical properties of its Borel transform, as prescribed by resurgence theory. The final result is then naturally interpreted in terms of the nonperturbative branch of the $T\bar{T}$-deformed spectrum. The finite-cutoff trumpet partition function is computed by applying the same strategy. In the second part of the paper, we propose an extension of this formalism to arbitrary topologies, using the basic gluing rules of the undeformed case. The Weil-Petersson integrations can be safely performed due to the nonperturbative corrections and give results that are compatible with the flow equation associated with the $T\bar{T}$ deformation. We derive exact expressions for general topologies and show that these are captured by a suitable deformation of the Eynard-Orantin topological recursion. Finally, we study the "slope" and "ramp" regimes of the spectral form factor as functions of the cutoff parameter.

On topological recursion for Wilson loops in mathcal{N} = 4 SYM at strong coupling

We consider U(N) mathcal{N} = 4 super Yang-Mills theory and discuss how to extract the strong coupling limit of non-planar corrections to observables involving the frac{1}{2} -BPS Wilson loop. Our approach is based on a suitable saddle point treatment of the Eynard-Orantin topological recursion in the Gaussian matrix model. Working directly at strong coupling we avoid the usual procedure of first computing observables at finite planar coupling λ, order by order in 1/N, and then taking the λ ≫ 1 limit. In the proposed approach, matrix model multi-point resolvents take a simplified form and some structures of the genus expansion, hardly visible at low order, may be identified and rigorously proved. As a sample application, we consider the expectation value of multiple coincident circular supersymmetric Wilson loops as well as their correlator with single trace chiral operators. For these quantities we provide novel results about the structure of their genus expansion at large tension, generalising recent results in arXiv:2011.02885.

Combinatorics of Bousquet-Mélou–Schaeffer numbers in the light of topological recursion

In this paper we prove, in a purely combinatorial-algebraic way, a structural quasi-polynomiality property for the Bousquet-Mélou–Schaeffer numbers. Conjecturally, this property should follow from the Chekhov–Eynard–Orantin topological recursion for these numbers (or, to be more precise, the Bouchard–Eynard version of the topological recursion for higher order critical points), which we derive in this paper from the recent result of Alexandrov–Chapuy–Eynard–Harnad. To this end, the missing ingredient is a generalization to the case of higher order critical points on the underlying spectral curve of the existing correspondence between the topological recursion and Givental’s theory for cohomological field theories.

Asymptotic expansions of some Toeplitz determinants via the topological recursion

In this article, we study the large n asymptotic expansions of $$n\times n$$ Toeplitz determinants whose symbols are indicator functions of unions of arc intervals of the unit circle. In particular, we use a Hermitian matrix model reformulation of the problem to provide a rigorous derivation of the general form of the large n expansion when the symbol is an indicator function of either a single arc interval or several arc intervals with a discrete rotational symmetry. Moreover, we prove that the coefficients in the expansions can be reconstructed, up to some constants, from the Eynard–Orantin topological recursion applied to some explicit spectral curves. In addition, when the symbol is an indicator function of a single arc interval, we provide the corresponding normalizing constants using a Selberg integral and thus we obtain a generalization of Widom’s result.

Topological recursion in the Ramond sector

We investigate supereigenvalue models in the Ramond sector and their recursive structure. We prove that the free energy truncates at quadratic order in Grassmann coupling constants, and consider super loop equations of the models with the assumption that the 1/N expansion makes sense. Subject to this assumption, we obtain the associated genus-zero algebraic curve with two ramification points (one regular and the other irregular) and also the supersymmetric partner polynomial equation. Starting with these polynomial equations, we present a recursive formalism that computes all the correlation functions of these models. Somewhat surprisingly, correlation functions obtained from the new recursion formalism have no poles at the irregular ramification point due to a supersymmetric correction — the new recursion may lead us to a further development of supersymmetric generalizations of the Eynard-Orantin topological recursion.

Loop Equations for Gromov-Witten Invariant of P1

We show that non-stationary Gromov–Witten invariants of P1 can be extracted from open periods of the Eynard–Orantin topological recursion correlators ωg,n whose Laurent series expansion at ∞ compute the stationary invariants. To do so, we overcome the technical difficulties to global loop equations for the spectral x(z) = z + 1/z and y(z) = ln z from the local loop equations satisfied by the ωg,n, and check these global loop equations are equivalent to the Virasoro constraints that are known to govern the full Gromov–Witten theory of P1.

Topological recursion, topological quantum field theory and Gromov–Witten invariants of BG

The purpose of this paper is to give a decorated version of the Eynard–Orantin topological recursion using a 2D Topological Quantum Field Theory. We define a kernel for a 2D TQFT and use an algebraic reformulation of a topological recursion to define how to decorate a standard topological recursion by a 2D TQFT. The A-model side enumerative problem consists of counting cell graphs where in addition vertices are decorated by elements in a Frobenius algebra, and which are a decorated version of the generalized Catalan numbers. We show that the function that counts these decorated graphs, which is a decoration of the counting function of the generalized Catalan numbers by a Frobenius algebra, satisfies a topological recursion with respect to the edge-contraction axioms. The path we follow to pass from the A-model side to the remodeled B-model side is to use a discrete Laplace transform as a mirror symmetry map. We show that a decorated version by a 2D TQFT of the Eynard–Orantin differentials satisfies a decorated version of the Eynard–Orantin recursion formula. We illustrate these results using a toy model for the theory arising from the orbifold cohomology of the classifying space of a finite group. In this example, the graphs are orbifold cell graphs (graphs drawn on an orbifold punctured Riemann surface) defined out of the moduli space $\overline {\mathcal M}{g,n}(BG)$ of stable morphisms from twisted curves to the classifying space of a finite group $G$. In particular we show that the cotangent class intersection numbers on the moduli space $\overline {\mathcal{M}}{g,n}(BG)$ satisfy a decorated Eynard–Orantin topological recursion and we derive an orbifold DVV equation as a consequence of it. This proves from a different perspective the known result which states that the $\psi$-class intersection numbers on $\overline{\mathcal{M}}\_{g,n}(BG)$ satisfy the Virasoro constraint condition.

Supereigenvalue models and topological recursion

We show that the Eynard-Orantin topological recursion, in conjunction with simple auxiliary equations, can be used to calculate all correlation functions of supereigenvalue models.

Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems

We consider the Gopakumar–Ooguri–Vafa correspondence, relating U(N) Chern–Simons theory at large N to topological strings, in the context of spherical Seifert 3-manifolds. These are quotients S^Γ=Γ ∖ S^3 of the three-sphere by the free action of a finite isometry group. Guided by string theory dualities, we propose a large N dual description in terms of both A- and B-twisted topological strings on (in general non-toric) local Calabi–Yau threefolds. The target space of the B-model theory is obtained from the spectral curve of Toda-type integrable systems constructed on the double Bruhat cells of the simply-laced group identified by the ADE label of Γ. Its mirror A-model theory is realized as the local Gromov–Witten theory of suitable ALE fibrations on P^1, generalizing the results known for lens spaces.We propose an explicit construction of the family of target manifolds relevant for the correspondence, which we verify through a large N analysis of the matrix model that expresses the contribution of the trivial flat connection to the Chern–Simons partition function. Mathematically, our results put forward an identification between the 1/N expansion of the slN+1 LMO invariant of S^Γ and a suitably restricted Gromov–Witten/Donaldson–Thomas partition function on the A-model dual Calabi–Yau. This 1/N expansion, as well as that of suitable generating series of perturbative quantum invariants of fiber knots in S^Γ, is computed by the Eynard–Orantin topological recursion.

Quantizing Weierstrass

We study the connection between the Eynard-Orantin topological recursion and quantum curves for the family of genus one spectral curves given by the Weierstrass equation. We construct quantizations of the spectral curve that annihilate the perturbative and non-perturbative wave-functions. In particular, for the non-perturbative wave-function, we prove, up to order hbar^5, that the quantum curve satisfies the properties expected from matrix models. As a side result, we obtain an infinite sequence of identities relating A-cycle integrals of elliptic functions and quasi-modular forms.

WKB solutions of difference equations and reconstruction by the topological recursion

The purpose of this article is to analyze the connection between Eynard–Orantin topological recursion and formal WKB solutions of a -difference equation: with . In particular, we extend the notion of determinantal formulas and topological type property proposed for formal WKB solutions of -differential systems to this setting. We apply our results to a specific -difference system associated to the quantum curve of the Gromov–Witten invariants of for which we are able to prove that the correlation functions are reconstructed from the Eynard–Orantin differentials computed from the topological recursion applied to the spectral curve . Finally, identifying the large x expansion of the correlation functions, proves a recent conjecture made by Dubrovin and Yang regarding a new generating series for Gromov–Witten invariants of .

Chiodo formulas for the r-th roots and topological recursion

We analyze Chiodo’s formulas for the Chern classes related to the r-th roots of the suitably twisted integer powers of the canonical class on the moduli space of curves. The intersection numbers of these classes with psi -classes are reproduced via the Chekhov–Eynard–Orantin topological recursion. As an application, we prove that the Johnson-Pandharipande-Tseng formula for the orbifold Hurwitz numbers is equivalent to the topological recursion for the orbifold Hurwitz numbers. In particular, this gives a new proof of the topological recursion for the orbifold Hurwitz numbers.

Orbifold Hurwitz numbers and Eynard–Orantin invariants

We prove that a generalisation of simple Hurwitz numbers due to Johnson, Pandharipande and Tseng satisfy the topological recursion of Eynard and Orantin. This generalises the Bouchard-Marino conjecture and places Hurwitz-Hodge integrals, which arise in the Gromov--Witten theory of target curves with orbifold structure, in the context of the Eynard-Orantin topological recursion.

The Eynard–Orantin recursion for the total ancestor potential

It was proved recently that the correlation functions of a semisimple cohomological field theory satisfy the so-called local Eynard–Orantin topological recursion. We prove that in the settings of singularity theory, the local Eynard–Orantin recursion is equivalent to N copies of Virasoro constraints for the total ancestor potential. The latter follow easily from some known properties of the period integrals in singularity theory. Our approach generalizes easily to an arbitrary semisimple cohomological field theory, which yields a simple proof of the local Eynard–Orantin recursion for an arbitrary semisimple cohomological field theory.

A Generalized Topological Recursion for Arbitrary Ramification

The Eynard-Orantin topological recursion relies on the geometry of a Riemann surface S and two meromorphic functions x and y on S. To formulate the recursion, one must assume that x has only simple ramification points. In this paper we propose a generalized topological recursion that is valid for x with arbitrary ramification. We justify our proposal by studying degenerations of Riemann surfaces. We check in various examples that our generalized recursion is compatible with invariance of the free energies under the transformation (x,y) -> (y,x), where either x or y (or both) have higher order ramification, and that it satisfies some of the most important properties of the original recursion. Along the way, we show that invariance under (x,y) -> (y,x) is in fact more subtle than expected; we show that there exists a number of counter examples, already in the case of the original Eynard-Orantin recursion, that deserve further study.

The Remodeling Conjecture and the Faber–Pandharipande Formula

In this note, we prove that the free energies F g constructed from the Eynard–Orantin topological recursion applied to the curve mirror to \({\mathbb{C}^3}\) reproduce the Faber–Pandharipande formula for genus g Gromov–Witten invariants of \({\mathbb{C}^3}\) . This completes the proof of the remodeling conjecture for \({\mathbb{C}^3}\) .

The volume conjecture, perturbative knot invariants, and recursion relations for topological strings

We study the relation between perturbative knot invariants and the free energies defined by topological string theory on the character variety of the knot. Such a correspondence between SL ( 2 ; C ) Chern–Simons gauge theory and the topological open string theory was proposed earlier on the basis of the volume conjecture and AJ conjecture. In this paper we discuss this correspondence beyond the subleading order in the perturbative expansion on both sides. In the computation of the perturbative invariants for the hyperbolic 3-manifold, we adopt the state integral model for the hyperbolic knots, and the factorized AJ conjecture for the torus knots. On the other hand, we iteratively compute the free energies on the character variety using the Eynard–Orantin topological recursion relation. We discuss the correspondence for the figure eight knot complement and the once punctured torus bundle over S 1 with the monodromy L 2 R up to the fifth order. For the torus knots, we find trivial the recursion relations on both sides.