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Based on the observed behavior of the superconformal index in three-dimensional 풩=2 theories, we propose a quantity that can be considered as an analogue of the “effective central charge.” We discuss the general properties of this quantity and ways of computing it in a variety of different theories, including simple Lagrangian theories as well as more interesting strongly coupled examples that come from 3d-3d correspondence. [ABSTRACT FROM AUTHOR]

We construct special Lagrangian 3-spheres in non-Kähler compact threefolds equipped with the Fu–Li–Yau geometry. These non-Kähler geometries emerge from topological transitions of compact Calabi-Yau threefolds. From this point of view, a conifold transition exchanges holomorphic 2-cycles for special Lagrangian 3-cycles. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1, 0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0, 1) supersymmetry and, possibly, a residual flavor symmetry. The equivariant topological Witten genus of this 2d theory then produces a new invariant of the 4-manifold equipped with a principle bundle, valued in the ring of equivariant weakly holomorphic (topological) modular forms. We describe basic properties of this map and present a few simple examples. As a byproduct, we obtain some new results on 't Hooft anomalies of 6d (1, 0) theories and a better understanding of the relation between 2d (0, 1) theories and TMF spectra. [ABSTRACT FROM AUTHOR]

We study reductions of 6d theories on a d-dimensional manifold Md, focusing on the interplay between symmetries, anomalies, and dynamics of the resulting (6 −d)-dimensional theory T[Md]. We refine and generalize the notion of "polarization" to polarization on Md, which serves to fix the spectrum of local and extended operators in T[Md]. Another important feature of theories T[Md] is that they often possess higher-group symmetries, such as 2-group and 3-group symmetries. We study the origin of such symmetries as well as physical implications including symmetry breaking and symmetry enhancement in the renormalization group flow. To better probe the IR physics, we also investigate the 't Hooft anomaly of 5d Chern-Simons matter theories. The present paper focuses on developing the general framework as well as the special case of d = 0 and 1, while an upcoming paper will discuss the case of d = 2, 3 and 4. [ABSTRACT FROM AUTHOR]

We construct a new class of three-dimensional topological quantum field theories (3d TQFTs) by considering generalized Argyres–Douglas theories on S1 × M3 with a non-trivial holonomy of a discrete global symmetry along the S1. For the minimal choice of the holonomy, the resulting 3d TQFTs are non-unitary and semisimple, thus distinguishing themselves from theories of Chern–Simons and Rozansky–Witten types respectively. Changing the holonomy performs a Galois transformation on the TQFT, which can sometimes give rise to more familiar unitary theories such as the and Chern–Simons theories. Our construction is based on an intriguing relation between topologically twisted partition functions, wild Hitchin characters, and chiral algebras which, when combined together, relate Coulomb branch and Higgs branch data of the same 4d theory. We test our proposal by applying localization techniques to the conjectural UV Lagrangian descriptions of the (A1, A2), (A1, A3) and (A1, D3) theories. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 3d N = 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 invariants 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We provide a physical definition of new homological invariants ℋ a (M 3) of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on M 3 times a 2-disk, D 2 , whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d 𝒩 = 2 theory T [ M 3 ] : D 2 × S 1 half-index, S 2 × S 1 superconformal index, and S 2 × S 1 topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern–Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of M 3 . The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds. [ABSTRACT FROM AUTHOR]

We take a peek at a general program that associates vertex (or chiral) algebras to smooth 4-manifolds in such a way that operations on algebras mirror gluing operations on 4-manifolds and, furthermore, equivalent constructions of 4-manifolds give rise to equivalences (dualities) of the corresponding algebras. [ABSTRACT FROM AUTHOR]

In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class S theory T [Σ, G] on L(k, 1) × S¹, the other is the LG "equivariant Verlinde formula", or equivalently partition function of LGC complex Chern-Simons theory on Σ × S¹. We first derive this equivalence using the M-theory geometry and show that the gauge groups appearing on the two sides are naturally G and its Langlands dual LG. When G is not simply-connected, we provide a recipe of computing the index of T [Σ, G] as summation over the indices of T [Σ, Ğ] with non-trivial background 't Hooft fluxes, where Ğ is the universal cover of G. Then we check explicitly this relation between the Coulomb index and the equivariant Verlinde formula for G = SU(2) or SO(3). In the end, as an application of this newly found relation, we consider the more general case where G is SU(N) or PSU(N) and showthat equivariantVerlinde algebra can be derived using field theory via (generalized) Argyres-Seiberg duality. We also attach a Mathematica notebook that can be used to compute the SU(3) equivariant Verlinde coefficients. [ABSTRACT FROM AUTHOR]

We study complex Chern-Simons theory on a Seifert manifold M by embedding it into string theory. We show that complex Chern-Simons theory on M is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the dimensional reduction of this theory to 2d gives the low energy dynamics of vortices in four-dimensional gauge theory, the fact apparently overlooked in the vortex literature. We also generalize the relations between (1) the Verlinde algebra, (2) quantum cohomology of the Grassmannian, (3) Chern-Simons theory on $${\Sigma\times S^1}$$ and (4) index of a spin Dirac operator on the moduli space of flat connections to a new set of relations between (1) the 'equivariant Verlinde algebra' for a complex group, (2) the equivariant quantum K-theory of the vortex moduli space, (3) complex Chern-Simons theory on $${\Sigma \times S^1}$$ and (4) the equivariant index of a spin Dirac operator on the moduli space of Higgs bundles. [ABSTRACT FROM AUTHOR]

Interpreting RG flows as dynamical systems in the space of couplings we produce a variety of constraints, global (topological) as well as local. These constraints, in turn, rule out some of the proposed RG flows and also predict new phases and fixed points, surprisingly, even in familiar theories such as O ( N ) model, QED 3 , or QCD 4 . [ABSTRACT FROM AUTHOR]

The goal of this paper is twofold. First, we find a natural home for the double affine Hecke algebras (DAHA) in the physics of BPS states. Second, we introduce new invariants of torus knots and links called hyperpolynomials that address the 'problem of negative coefficients' often encountered in DAHA-based approaches to homological invariants of torus knots and links. Furthermore, from the physics of BPS states and the spectra of singularities associated with Landau-Ginzburg potentials, we also describe a rich structure of differentials that act on homological knot invariants for exceptional groups and uniquely determine the latter for torus knots. [ABSTRACT FROM AUTHOR]

Abstract: We introduce and compute a 2-parameter family deformation of the A-polynomial that encodes the color dependence of the superpolynomial and that, in suitable limits, reduces to various deformations of the A-polynomial studied in the literature. These special limits include the t-deformation which leads to the “refined A-polynomial” introduced in the previous work of the authors and the Q-deformation which leads, by the conjecture of Aganagic and Vafa, to the augmentation polynomial of knot contact homology. We also introduce and compute the quantum version of the super-A-polynomial, an operator that encodes recursion relations for -colored HOMFLY homology. Much like its predecessor, the super-A-polynomial admits a simple physical interpretation as the defining equation for the space of SUSY vacua (= critical points of the twisted superpotential) in a circle compactification of the effective 3d theory associated to a knot or, more generally, to a 3-manifold M. Equivalently, the algebraic curve defined by the zero locus of the super-A-polynomial can be thought of as the space of open string moduli in a brane system associated with M. As an inherent outcome of this work, we provide new interesting formulas for colored superpolynomials for the trefoil and the figure-eight knot. [Copyright &y& Elsevier]

To every 3-manifold M one can associate a two-dimensional $${\mathcal{N}=(2, 2)}$$ supersymmetric field theory by compactifying five-dimensional $${\mathcal{N}=2}$$ super-Yang-Mills theory on M. This system naturally appears in the study of half-BPS surface operators in four-dimensional $${\mathcal{N}=2}$$ gauge theories on one hand, and in the geometric approach to knot homologies, on the other. We study the relation between vortex counting in such two-dimensional $${\mathcal{N}=(2, 2)}$$ supersymmetric field theories and the refined BPS invariants of the dual geometries. In certain cases, this counting can also be mapped to the computation of degenerate conformal blocks in two-dimensional CFT's. Degenerate limits of vertex operators in CFT receive a simple interpretation via geometric transitions in BPS counting. [ABSTRACT FROM AUTHOR]

When combined with mirror symmetry, the A-model approach to quantization leads to a fairly simple and tractable problem. The most interesting part of the problem then becomes finding the mirror of the coisotropic brane. We illustrate how it can be addressed in a number of interesting examples related to representation theory and gauge theory, in which mirror geometry is naturally associated with the Langlands dual group. Hyperholomorphic sheaves and ( B, B, B) branes play an important role in the B-model approach to quantization. [ABSTRACT FROM AUTHOR]

We study refined and motivic wall-crossing formulas in $${{\mathcal N}=2}$$ supersymmetric gauge theories with SU(2) gauge group and N < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that 'refined = motivic.' [ABSTRACT FROM AUTHOR]

We establish a direct map between refined topological vertex and sl( N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the Hopf link, whose components are colored by arbitrary representations of sl( N). At present, the mathematical formulation of such homological invariants is available only for the fundamental representation (the Khovanov-Rozansky theory) and the relation with the refined topological vertex should be useful for categorizing quantum group invariants associated with other representations ( R1, R2). Our result is a first direct verification of a series of conjectures which identifies link homologies with the Hilbert space of BPS states in the presence of branes, where the physical interpretation of gradings is in terms of charges of the branes ending on Lagrangian branes. [ABSTRACT FROM AUTHOR]

It is well known that in string compactifications on toric Calabi–Yau manifolds one can introduce refined BPS invariants that carry information not only about the charge of the BPS state but also about the spin content. In this paper we study how these invariants behave under wall crossing. In particular, by applying a refined wall crossing formula, we obtain the refined BPS degeneracies for the conifold in different chambers. The result can be interpreted in terms of a new statistical model that counts “refined” pyramid partitions; the model provides a combinatorial realization of wall crossing and clarifies the relation between refined pyramid partitions and the refined topological vertex. We also compare the wall crossing behavior of the refined BPS invariants with that of the motivic Donaldson–Thomas invariants introduced by Kontsevich–Soibelman. In particular, we argue that, in the context of BPS state counting, the three adjectives in the title of this paper are essentially synonymous. [ABSTRACT FROM AUTHOR]

It has been proposed that the asymptotic behavior of the colored Jones polynomial is equal to the perturbative expansion of the Chern–Simons gauge theory with complex gauge group $${SL(2, \mathbb{C})}$$ on the hyperbolic knot complement. In this note we make the first step toward verifying this relation beyond the semi-classical approximation. This requires a careful understanding of some delicate issues, such as normalization of the colored Jones polynomial and the choice of polarization in Chern–Simons theory. Addressing these issues allows us to go beyond the volume conjecture and to verify some predictions for the behavior of the subleading terms in the asymptotic expansion of the colored Jones polynomial. [ABSTRACT FROM AUTHOR]

We conjecture a relation between the sl( N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl( N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the sl( N) knot homology groups for all values of N. We verify that our predictions pass some non-trivial checks [ABSTRACT FROM AUTHOR]

We study three-dimensional Chern-Simons theory with complex gauge groupSL(2,C), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between theSL(2,C) partition function and the colored Jones polynomial. [ABSTRACT FROM AUTHOR]

We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed study of RCFT’s corresponding to the T2 target and identify the Cardy branes with geometric branes. The T2’s leading to RCFT’s admit ‘‘complex multiplication’’ which characterizes Cardy branes as specific D0-branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary Calabi-Yau n-folds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for Calabi-Yau n-folds for n>2. RCFT’s on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of Calabi-Yau n-folds in connection with freezing geometric moduli. [ABSTRACT FROM AUTHOR]

M theory compactifications on G2 holonomy manifolds, whilst supersymmetric, require singularities in order to obtain non-Abelian gauge groups, chiral fermions and other properties necessary for a realistic model of particle physics. We review recent progress in understanding the physics of such singularities. Our main aim is to describe the techniques which have been used to develop our understanding of M theory physics near these singularities. In parallel, we also describe similar sorts of singularities in Spin(7) holonomy manifolds which correspond to the properties of three dimensional field theories. As an application, we review how various aspects of strongly coupled gauge theories, such as confinement, mass gap and non-perturbative phase transitions may be given a simple explanation in M theory. [Copyright &y& Elsevier]

Discusses the fibration of G[sub2] manifolds for M-theory admitting string theory Calabi-Yau duals are fibered by coassociative submanifolds. Example of string duality; Arguments of mirror symmetry as T-duality; Dualities between M-theory on manifold X with G[sub2] holonomy and string theory on Calabi-Yau manifolds.

Looking for string vacua with fixed moduli, we study compactifications of type IIA string theory on Calabi–Yau fourfolds in the presence of generic Ramond–Ramond fields. We explicitly derive the (super)potential induced by Ramond–Ramond fluxes performing a Kaluza–Klein reduction of the ten-dimensional effective action. This can be conveniently achieved in a formulation of the massive type IIA supergravity where all Ramond–Ramond fields appear in a democratic way. The result agrees with the general formula for the superpotential written in terms of calibrations. We further notice that for generic Ramond–Ramond fluxes all geometric moduli are stabilized and one finds non-supersymmetric vacua at positive values of the scalar potential. [Copyright &y& Elsevier]

We study M-theory on two classes of manifolds of Spin(7) holonomy that are developing an isolated conical singularity. We construct explicitly a new class of Spin(7) manifolds and analyse in detail the topology of the corresponding classical spacetimes. We discover also an intricate interplay between various anomalies in M-theory, string theory, and gauge theory within these models, and in particular find a connection between half-integral G-fluxes in M-theory and Chern–Simons terms of the N=1, D=3 effective theory. [Copyright &y& Elsevier]

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 3-manifolds in terms of affine Grassmannians and a generalization of Akutsu-Deguchi-Ohtsuki knot invariants. [ABSTRACT FROM AUTHOR]

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 F K = Z ^ (M K) , where M K is the complement of a knot K in the 3-sphere, and explore the relationship between an a-deformed ( a = q N ) version of F K and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of F K in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for a-deformed F K for (2 , 2 p + 1) torus knots and an order-by-order construction for other cases. They both 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 F K transforms under natural geometric operations on K indicates relations to quantum modularity in a new setting. [ABSTRACT FROM AUTHOR]

By studying the properties of q-series Z ^ -invariants, we develop a dictionary between 3-manifolds and vertex algebras. In particular, we generalize previously known entries in this dictionary to Lie groups of higher rank, to 3-manifolds with toral boundaries, and to BPS partition functions with line operators. This provides a new physical realization of logarithmic vertex algebras in the framework of the 3d-3d correspondence and opens new avenues for their future study. For example, we illustrate how invoking a knot-quiver correspondence for Z ^ -invariants leads to many infinite families of new fermionic formulae for VOA characters. [ABSTRACT FROM AUTHOR]

We generalize the F K invariant, i.e. Z ˆ for the complement of a knot K in the 3-sphere, the knots-quivers correspondence, and A -polynomials of knots, and find several interconnections between them. We associate an F K invariant to any branch of the A -polynomial of K and we work out explicit expressions for several simple knots. We show that these F K invariants can be written in the form of a quiver generating series, in analogy with the knots-quivers correspondence. We discuss various methods to obtain such quiver representations, among others using R -matrices. We generalize the quantum a -deformed A -polynomial to an ideal that contains the recursion relation in the group rank, i.e. in the parameter a , and describe its classical limit in terms of the Coulomb branch of a 3d-5d theory. We also provide t -deformed versions. Furthermore, we study how the quiver formulation for closed 3-manifolds obtained by surgery leads to the superpotential of 3d N = 2 theory T [ M 3 ] and to the data of the associated modular tensor category MTC [ M 3 ]. [ABSTRACT FROM AUTHOR]

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional $${\mathcal{N} = 2}$$ gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that $${S^{3}_{b}}$$ partition functions of two mirror 3d $${\mathcal{N} = 2}$$ gauge theories are equal. Three-dimensional $${\mathcal{N} = 2}$$ field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional $${\mathcal{N} = 2}$$ SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing. [ABSTRACT FROM AUTHOR]

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d N = 2 theories where such structures a priori are not manifest. These modular structures include: mock modular forms, SL(2,ℤ) Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs. [ABSTRACT FROM AUTHOR]