Large N duality beyond the genus expansion

We study non-perturbative aspects of the large N duality between Chern-Simons theory and topological strings, and we find a rich structure of large N phase transitions in the complex plane of the 't Hooft parameter. These transitions are due to large N instanton effects, and they can be regarded as a deformation of the Stokes phenomenon. Moreover, we show that, for generic values of the 't Hooft coupling, instanton effects are not exponentially suppressed at large N and they correct the genus expansion. This phenomenon was first discovered in the context of matrix models, and we interpret it as a generalization of the oscillatory asymptotics along anti-Stokes lines. In the string dual, the instanton effects can be interpreted as corrections to the saddle string geometry due to discretized neighboring geometries. As a mathematical application, we obtain the 1/N asymptotics of the partition function of Chern-Simons theory on L(2,1), and we test it numerically to high precision in order to exhibit the importance of instanton effects.Comment: 37 pages, 24 figures. v2: clarifications and references added, misprints corrected, to appear in JHE

(0,2) Trialities

Motivated by the connection between 4-manifolds and 2d N=(0,2) theories, we study the dynamics of a fairly large class of 2d N=(0,2) gauge theories. We see that physics of such theories is very rich, much as the physics of 4d N=1 theories. We discover a new type of duality that is very reminiscent of the 4d Seiberg duality. Surprisingly, the new 2d duality is an operation of order three: it is IR equivalence of three different theories and, as such, is actually a triality. We also consider quiver theories and study their triality webs. Given a quiver graph, we find that supersymmetry is dynamically broken unless the ranks of the gauge groups and flavor groups satisfy stringent inequalities. In fact, for most of the graphs these inequalities have no solutions. This supports the folklore theorem that generic 2d N=(0,2) theories break supersymmetry dynamically.Comment: 31 pages, 12 figure

(0,4)(0, 4) dualities

We study a class of two-dimensional ${\cal N}=(0, 4)$ quiver gauge theories that flow to superconformal field theories. We find dualities for the superconformal field theories similar to the 4d ${\cal N}=2$ theories of class ${\cal S}$, labelled by a Riemann surface ${\cal C}$. The dual descriptions arise from various pair-of-pants decompositions, that involves an analog of the $T_N$ theory. Especially, we find the superconformal index of such theories can be written in terms of a topological field theory on ${\cal C}$. We interpret this class of SCFTs as the ones coming from compactifying 6d ${\cal N}=(2, 0)$ theory on $\mathbb{CP}^1 \times {\cal C}$Comment: 41 pages, 12 figure

Vertex algebras and 4-manifold invariants

We propose a way of computing 4-manifold invariants, old and new, as chiral correlation functions in half-twisted 2d $\mathcal{N}=(0,2)$ theories that arise from compactification of fivebranes. Such formulation gives a new interpretation of some known statements about Seiberg-Witten invariants, such as the basic class condition, and gives a prediction for structural properties of the multi-monopole invariants and their non-abelian generalizations.Comment: 67 pages, 11 figure

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.Comment: 56 pages, 19 figures. v2: references adde

Spin-cobordisms, surgeries and fermionic modular bootstrap

We consider general fermionic quantum field theories with a global finite group symmetry $G$, focusing on the case of 2-dimensions and torus spacetime. The modular transformation properties of the family of partition functions with different backgrounds is determined by the 't Hooft anomaly of $G$ and fermion parity. For a general possibly non-abelian $G$ we provide a method to determine the modular transformations directly from the bulk 3d invertible topological quantum field theory (iTQFT) corresponding to the anomaly by inflow. We also describe a method of evaluating the character map from the real representation ring of $G$ to the group which classifies anomalies. Physically the value of the map is given by the anomaly of free fermions in a given representation. We assume classification of the anomalies/iTQFTs by spin-cobordisms. As a byproduct, for all abelian symmetry groups $G$, we provide explicit combinatorial expressions for corresponding spin-bordism invariants in terms of surgery representation of arbitrary closed spin 3-manifolds. We work out the case of $G=\mathbb{Z}_2$ in detail, and, as an application, we consider the constraints that 't Hooft anomaly puts on the spectrum of the infrared conformal field theory.Comment: 86 pages, 27 figure