We consider the geometric quantisation of Chern--Simons theory for closed genus-one surfaces and semisimple complex groups. First we introduce the natural complexified analogue of the Hitchin connection in Kähler quantisation, with polarisations coming from the nonabelian Hodge hyper-Kähler geometry of the moduli spaces of flat connections, thereby complementing the real-polarised approach of Witten. Then we consider the connection of Witten, and we identify it with the complexified Hitchin connection using a version of the Bargmann transform on polarised sections over the moduli spaces.

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 a distinguished entry of the Stokes automorphism matrix is the 3D-index 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 41 and 52 knots.

We consider meromorphic transforms given by meromorphic kernels and study their asymptotic expansions under a certain rescaling. Under decay assumptions we establish the full asymptotic expansion in the rescaling parameter of these transforms and provide global estimates for error terms. We show that the resulting asymptotic series is Borel resummable and we provide formulae for the resulting resurgent function, which allows us to give formulae for the Stokes coefficients. A number of classical functions are obtained by applying such meromorphic transforms to elementary functions, such as Faddeev's quantum dilogarithm, and our general theory applies to these cases.

We determine analytically the energy gap at weak coupling in the attractive multi-component Gaudin--Yang model, an integrable model which describes interacting fermions in one dimension with κ components. We use three different methods. The first one is based on a direct analysis of the Bethe ansatz equations. The second method uses the theory of resurgence and the large order behavior of the perturbative series for the ground state energy. The third method is based on a renormalization group analysis. The three methods lead to the same answer, providing in this way a non-trivial test of the ideas of resurgence and renormalons as applied to non-relativistic many-body systems.

We study the combinatorial Teichmüller space and construct on it global coordinates, analogous to the Fenchel-Nielsen coordinates on the ordinary Teichmüller space. We prove that these coordinates form an atlas with piecewise linear transition functions, and constitute global Darboux coordinates for the Kontsevich symplectic structure on top-dimensional cells. We then set up the geometric recursion in the sense of Andersen-Borot-Orantin adapted to the combinatorial setting, which naturally produces mapping class group invariant functions on the combinatorial Teichmüller spaces. We establish a combinatorial analogue of the Mirzakhani-McShane identity fitting this framework. As applications, we obtain geometric proofs of Witten conjecture/Kontsevich theorem (Virasoro constraints for ψ-classes intersections) and of Norbury's topological recursion for the lattice point count in the combinatorial moduli spaces. These proofs arise now as part of a unified theory and proceed in perfect parallel to Mirzakhani's proof of topological recursion for the Weil-Petersson volumes. We move on to the study of the spine construction and the associated rescaling flow on the Teichmüller space. We strengthen former results of Mondello and Do on the convergence of this flow. In particular, we prove convergence of hyperbolic Fenchel-Nielsen coordinates to the combinatorial ones with some uniformity. This allows us to effectively carry natural constructions on the Teichmüller space to their analogues in the combinatorial spaces. For instance, we obtain the piecewise linear structure on the combinatorial Teichmüller space as the limit of the smooth structure on the Teichmüller space. To conclude, we provide further applications to the enumerative geometry of multicurves, Masur-Veech volumes and measured foliations in the combinatorial setting.

We give elements towards the classification of quantum Airy structures based on the W(glr)-algebras at self-dual level based on twisted modules of the Heisenberg VOA of glr for twists by arbitrary elements of the Weyl group Sr. In particular, we construct a large class of such quantum Airy structures. We show that the system of linear ODEs forming the quantum Airy structure and determining uniquely its partition function is equivalent to a topological recursion à la Chekhov--Eynard--Orantin on singular spectral curves. In particular, our work extends the definition of the Bouchard--Eynard topological recursion (valid for smooth curves) to a large class of singular curves, and indicates impossibilities to extend naively the definition to other types of singularities. We also discuss relations to intersection theory on moduli spaces of curves and give precise conjectures for application in open r-spin intersection theory.

Many BPS partition functions depend on a choice of additional structure: fluxes, Spin or Spin-c structures, etc. In a context where the BPS generating series depends on a choice of Spin-c structure we show how different limits with respect to the expansion variable q and different ways of summing over Spin-c structures produce different invariants of homology cobordisms out of the BPS q-series.

The quantum dilogarithm function of Faddeev is a special function that plays a key role as the building block of quantum invariants of knots and 3-manifolds, of quantum Teichmüller theory and of complex Chern-Simons theory. Motivated by conjectures on resurgence and recent interest in wall-crossing phenomena, we prove that the Borel summation of a formal power series solution of a linear difference equation produces Faddeev's quantum dilogarithm. Along the way, we give an explicit formula for the meromorphic function in Borel plane, locate its poles and residues, and describe the Stokes phenomenon of its Laplace transforms along the Stokes rays.

The asymptotic expansion of quantum knot invariants in complex Chern-Simons theory gives rise to factorially divergent formal power series. We conjecture that these series are resurgent functions whose Stokes automorphism is given by a pair of matrices of q-series with integer coefficients, which are determined explicitly by the fundamental solutions of a pair of linear q-difference equations. We further conjecture that for a hyperbolic knot, a distinguished entry of those matrices equals to the Dimofte-Gaiotto-Gukov 3D-index, and thus is given by a counting of BPS states. We illustrate our conjectures explicitly by matching theoretically and numerically computed integers for the cases of the 4(1) and the 5 (2) knots.

Using derivatives of primary fields (null or not) with respect to the conformal dimension, we build infinite families of non-trivial logarithmic representations of the conformal algebra at generic central charge, with Jordan blocks of dimension 2 or 3. Each representation comes with one free parameter, which takes fixed values under assumptions on the existence of degenerate fields. This parameter can be viewed as a simpler, normalization-independent redefinition of the logarithmic coupling. We compute the corresponding non-chiral conformal blocks, and show that they appear in limits of Liouville theory four-point functions. As an application, we describe the logarithmic structures of the critical two-dimensional O(n) and Q-state Potts models at generic central charge. The validity of our description is demonstrated by semi-analytically bootstrapping four-point connectivities in the Q-state Potts model to arbitrary precision. Moreover, we provide numerical evidence for the Delfino--Viti conjecture for the three-point connectivity. Our results hold for generic values of Q in the complex plane and beyond.

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.

Reducing a 6d fivebrane theory on a 3-manifold Y gives a q-series 3-manifold invariant Z^(Y). We analyse the large-N behaviour of FK = Z^(MK), where MK is the complement of a knot K in the 3-sphere, and explore the relationship between an a-deformed (a = qN) version of FK and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of FK in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for a-deformed FK for (2,2p + 1)-torus knots. They suggest a further t-deformation based on superpolynomials, which can be used to obtain a t-deformation of ADO polynomials, expected to be related to categorification. Moreover, studying how FK transforms under natural geometric operations on K indicates relations to quantum modularity in a new setting.

We introduce the notion of analytic stability data on the Lie algebra of vector fields on a torus. We prove that the subspace of analytic stability data is open and closed in the topological space of all stability data. We formulate a general conjecture which explains how analytic stability data give rise to resurgent series. This conjecture is checked in several examples.

By studying Rozansky-Witten theory with non-compact target spaces we find new connections with knot invariants whose physical interpretation was not known. This opens up several new avenues, which include a new formulation of q-series invariants of 3manifolds in terms of affine Grassmannians and a generalization of Akutsu-Deguchi-Ohtsuki knot invariants.

Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of CP1 with prescribed ramification profile over ∞, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers. In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first involves Hodge classes on moduli spaces of stable maps to classifying spaces; the second involves Chiodo classes on moduli spaces of spin curves; and the third involves tautological classes on moduli spaces of stable curves. We proceed to discuss the merits of these formulas against a list of desired properties enunciated by Goulden, Jackson and Vakil. Our formulas lead to non-trivial relations between tautological intersection numbers on moduli spaces of stable curves and hints at further structure underlying Chiodo classes. The paper concludes with generalisations of our results to the context of spin Hurwitz numbers.

We introduce a framework in noncommutative geometry consisting of a ∗-algebra A, a bimodule Ω1 endowed with a derivation A→Ω1 and with a Hermitian structure Ω1⊗Ω¯1→A (a "noncommutative Kähler form"), and a cyclic 1-cochain A→C whose coboundary is determined by the previous structures. These data give moment map equations on the space of connections on an arbitrary finitely-generated projective A-module. As particular cases, we obtain a large class of equations in algebra (King's equations for representations of quivers, including ADHM equations), in classical gauge theory (Hermitian Yang-Mills equations, Hitchin equations, Bogomolny and Nahm equations, etc.), as well as in noncommutative gauge theory by Connes, Douglas and Schwarz. We also discuss Nekrasov's beautiful proposal for re-interpreting noncommutative instantons on Cn≃R2n as infinite-dimensional solutions of King's equation


where H is a Hilbert space completion of a finitely-generated C[T1,…,Tn]-module (e.g. an ideal of finite codimension).

Double Hurwitz numbers enumerate branched covers of CP1 with prescribed ramification over two points and simple ramification elsewhere. In contrast to the single case, their underlying geometry is not well understood. In previous work by the second- and third-named authors, the double Hurwitz numbers were conjectured to satisfy a polynomiality structure and to be governed by the topological recursion, analogous to existing results concerning single Hurwitz numbers. In this paper, we resolve these conjectures by a careful analysis of the semi-infinite wedge representation for double Hurwitz numbers, by pushing further methods previously used for other Hurwitz problems. We deduce a preliminary version of an ELSV-like formula for double Hurwitz numbers, by deforming the Johnson-Pandharipande-Tseng formula for orbifold Hurwitz numbers and using properties of the topological recursion under variation of spectral curves. In the course of this analysis, we unveil certain vanishing properties of the Chiodo classes.

According to standard lore, perturbative series of super-renormalizable theories have only instanton singularities. In this paper we show that two-dimensional scalar theories with a spontaneously broken O(N) symmetry at the classical level, which are super-renormalizable, have an IR renormalon singularity at large N. Since perturbative expansions in these theories are made around the “false vacuum” in which the global symmetry is broken, this singularity can be regarded as a manifestation of the non-perturbative absence of Goldstone bosons. We conjecture that the Borel singularity in the ground state energy of the Lieb–Liniger model is a non-relativistic manifestation of this phenomenon. We also provide en passant a detailed perturbative calculation of the Lieb–Liniger energy up to two-loops, and we check that it agrees with the prediction of the Bethe ansatz.

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 compute 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-Hubert, Fougeron, and elaborated in [the7]. 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.

One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d N = 2 SCFT T[M3] — or, rather, a “collection of SCFTs” as we refer to it in the paper — for all types of 3-manifolds that include, for example, a 3-torus, Brieskorn spheres, and hyperbolic surgeries on knots. The goal of this paper is to overcome this challenge by a more systematic study of 3d-3d correspondence that, first of all, does not rely heavily on any geometric structure on M3 and, secondly, is not limited to a particular supersymmetric partition function of T[M3]. In particular, we propose to describe such “collection of SCFTs” in terms of 3dN = 2 gauge theories with “non-linear matter” fields valued in complex group manifolds. As a result, we are able to recover familiar 3-manifold invariants, such as Turaev torsion and WRT invariants, from twisted indices and half-indices of T[M3], and propose new tools to compute more recent q-series invariantsb Z(M3) in the case of manifolds with b1 > 0. Although we use genus-1 mapping tori as our “case study,” many results and techniques readily apply to more general 3-manifolds, as we illustrate throughout the paper.

Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of spectral curves with a global involution satisfy a system of partial differential equations, whose equations can be seen as quantizations of the original spectral curves. The families of spectral curves can be parametrized with the so-called times, defined as periods on second type cycles. These equations can be used to prove that the WKB solution of many isomonodromic systems coincides with the topological recursion wave function, and thus show that the topological recursion wave function is annihilated by a quantum curve. This recovers many known quantum curves for genus zero spectral curves and generalizes to hyperelliptic curves. In the particular case of a degenerate elliptic curve, apart from giving the quantum curve, we prove that the wave function satisfies the first Painlevé isomonodromic system and equation just from loop equations, making use of our system of PDEs. In general, we are able to recover the Gelfand--Dikii isomonodromic systems just from topological recursion.

We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of spectral Darboux coordinates on the moduli space of meromorphic sl2-connections on P1 and argue that the topological recursion produces a 2g-parameter family of associated tau functions, where 2g is the dimension of the moduli space considered. We apply this procedure to the 6 Painlevé equations which correspond to g=1 and consider a g=2 example.

We study dynamics of two-dimensional N = (0,1) supersymmetric gauge theories. In particular, we propose that there is an infrared triality between certain triples of theories with orthogonal and symplectic gauge groups. The proposal is supported by matching of anomalies and elliptic genera. This triality can be viewed as a (0,1) counterpart of the (0,2) triality proposed earlier by two of the authors and A. Gadde. We also describe the relation between global anomalies in gauge theoretic and sigma-model descriptions, filling in a gap in the present literature.

While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in G2 holonomy spaces and to Spin(7) metrics on 8-manifolds with T2 fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories T[M4] on the other.

In integrable field theories in two dimensions, the Bethe ansatz can be used to compute exactly the ground state energy in the presence of an external field coupled to a conserved charge. We generalize previous results by Volin and we extract analytic results for the perturbative expansion of this observable, up to very high order, in various asymptotically free theories: the non-linear sigma model and its supersymmetric extension, the Gross–Neveu model, and the principal chiral field. We study the large order behavior of these perturbative series and we give strong evidence that, as expected, it is controlled by renormalons. Our analysis is sensitive to the next-to-leading correction to the asymptotics, which involves the first two coefficients of the beta function. We also show that, in the supersymmetric non-linear sigma model, there is no contribution from the first IR renormalon, in agreement with general arguments.