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.
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 b Z(Y ). We analyse the large-N behaviour of FK = b 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.
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).
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
-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.
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.