This paper examines knots in 3d Chern-Simons theory and their relation to 3d and 5d theories.
problem Understanding knots in 3d Chern-Simons theory and their correspondence with other 3d and 5d theories.
method Analyzing defects in 6d (2,0) theory compactified on a 3-manifold, computing partition functions.
result Identification and quantitative checks of defects in various 3d and 5d theories.
Proposes Teichmüller TQFT as a Chern-Simons theory with a new integration cycle.
problem Defining Teichmüller TQFT as a Chern-Simons theory with a novel integration cycle.
method Analytically continues Chern-Simons path-integral with an unusual integration cycle.
result Teichmüller TQFT is dual to complex SL(2,C) Chern-Simons theory at integer level k=1.
The paper explores the connection between 3d gravity and Chern-Simons theory using affine group connections.
problem Exploring the relationship between 3d gravity and Chern-Simons theory.
method A variational problem of Chern-Simons type on a principal fiber bundle with general affine group structure is studied. The connection is established through a generalized notion of extension and reduction of connections.
result Established a correspondence between 3d gravity and Chern-Simons theory using affine group connections.
New approach connects 3D Chern-Simons theory to spectral networks.
problem Understanding Chern-Simons invariants in 3D manifolds.
method Constructing equivalences between bundles and spectral networks.
result New formulas for Chern-Simons invariants of 3D manifolds.
We test the 3d-3d correspondence for theories that are labelled by Lens spaces. We find a full agreement between the index of the 3d N=2 "Lens space theory" T[L(p,1)] and the partition function of complex Chern-Simons theory on L(p,1). In particular, for p=1, we show how the familiar S3 partition func…
3D gauge theories link knot polynomials to vortex partition functions.
problem Connecting knot polynomials to gauge theory partition functions.
method Construct 3D N=2 abelian gauge theories on S2imesS1. result Colored Jones polynomials derived from vortex partition functions.
We use the 3d-3d correspondence together with the DGG construction of theories Tn[M] labelled by 3-manifolds M to define a non-perturbative state-integral model for SL(n,C) Chern-Simons theory at any level k, based on ideal triangulations. The resulting partition functions generalize a widely studied k=1 state-integ…
3D gravity reformulated in 6D Hitchin theory.
problem Formulating 3D gravity in a new 6D context.
method Reduction of 6D Hitchin theory to 3D gravity.
result 3D gravity emerges from 6D Hitchin theory.
We study S-dualities in analytically continued SL(2) Chern-Simons theory on a 3-manifold M. By realizing Chern-Simons theory via a compactification of a 6d five-brane theory on M, various objects and symmetries in Chern-Simons theory become related to objects and operations in dual 2d, 3d, and 4d theories. For example,…
The abstract discusses constructing 3d N=2 gauge theories using surgeries and M5-branes.
problem Constructing geometrically 3d N=2 gauge theories.
method Using surgeries and M5-branes wrapping on plumbing three-manifolds.
result Various dualities of 3d theories can be interpreted as Kirby moves and equivalent surgeries.
We consider topological field theories that compute the Reidemeister-Milnor-Turaev torsion in three dimensions. These are the psl(1|1) and the U(1|1) Chern-Simons theories, coupled to a background complex flat gauge field. We use the 3d mirror symmetry to derive the Meng-Taubes theorem, which relates the torsion and th…
Study gauged supergravity, M5-branes, and class R theories, constraining supergravity coefficients and calculating partition functions.
problem Constraints and calculations in higher-derivative supergravity and related theories.
method Holography, Chern-Simons theory, 3d-3d correspondence, wrapped M5-branes.
result Constrained coefficients in supergravity Lagrangian, calculated partition functions for class R theories.
Study of defects in 6d (2,0) theory compactified on 3-manifolds.
problem Understanding defects in 6d (2,0) theory compactified on 3-manifolds.
method State-integral models, cluster algebra techniques, domain wall theory, 5d SYM, holography.
result Established a dictionary for the 3d-3d correspondence with defects.
Study of 3-manifold homology using fivebrane compactifications.
problem Understanding homological invariants of 3-manifolds.
method Fivebrane compactifications and 3d/3d correspondence.
result Explicit calculations of 3-manifold homology invariants.
Complex Chern-Simons theory reveals peacock patterns in perturbative series.
problem Understanding the structure of partition functions in complex Chern-Simons theory.
method Analyzing the partition function as a holomorphic function and using resurgence theory.
result Perturbative series are resurgent, with trans-series involving non-perturbative variables.
New methods reveal symmetries in Chern-Simons theory.
problem Understanding symmetries in Chern-Simons theory.
method Introduced a special basis in the center of the universal enveloping algebra to present group factors in arbitrary representations.
result Computed Vassiliev invariants and proved the tug-the-hook symmetry of the colored HOMFLY polynomial.
The first part of this text is a gentle exposition of some basic constructions and results in the extended prequantum theory of Chern-Simons-type gauge field theories. We explain in some detail how the action functional of ordinary 3d Chern-Simons theory is naturally localized ("extended", "multi-tiered") to a map on t…
This paper quantizes geometry using SL(2,C) Chern-Simons theory and flat connections.
problem Quantizing four-dimensional quantum geometry.
method Using a correspondence between flat connections and four-dimensional simplices, the paper quantizes geometry via complex SL(2,C) Chern-Simons theory.
result The quantum geometrical states are represented by the 3d blocks of analytically continued Chern-Simons theory, and in the semiclassical limit, the three-dimensional Chern-Simons action becomes the discrete Einstein-Hilbert action of a 4-simplex.
New method uses resurgent analysis to determine growth rate of quantum field theory coefficients.
problem Determining the growth rate of quantum field theory coefficients.
method Resurgence analysis on the Stokes line, leading to transseries decomposition and continued across natural boundary.
result Essential exponent of growth has Cardy-like interpretation as effective central charge.
The abstract discusses resurgent functions in quantum knot invariants.
problem Understanding the asymptotic expansion of quantum knot invariants.
method Using resurgent functions and q-series to conjecture and compute knot invariants. result Explicit computations match conjectured values for specific knots.
Study of IR phases in 3D class R theories linked to non-hyperbolic 3-manifolds.
problem Understanding IR phases of 3D class R theories associated with non-hyperbolic 3-manifolds.
method Analysis of IR phenomena through `exceptional' Dehn fillings and gauging of flavor symmetries.
result 3D class R theories associated with certain atoroidal non-hyperbolic 3-manifolds exhibit supersymmetry enhancement at low energy.
We propose a new description of 3d N=2 theories which do not admit conventional Lagrangians. Given a quiver Q and a mutation sequence m on it, we define a 3d N=2 theory T[(Q,m)] in such a way that the Sb3 partition function of the theory coincides with the cluster partition f…
Study large N oscillations in 3D theories related to black hole physics.
problem Understanding large N sign oscillations in 3D theories via holography.
method Holographic computation of on-shell actions for Euclidean supergravity solutions, Wick rotation of magnetically charged AdS4 black holes.
result Proposed a non-trivial mathematical conjecture regarding phase factors of twisted Reidemeister-Ray-Singer torsion.
String theory connects lattice models, links, and geometric Langlands.
problem Connecting lattice models, links, and geometric Langlands.
method T-duality and worldvolume theories in string theory.
result Unified understanding of various mathematical concepts.
We study the Chern-Simons topological quantum field theory with an inhomogeneous gauge group, a non-semi-simple group obtained from a semi-simple one by taking its semi-direct product with its Lie algebra. We find that the standard knot observables (i.e. traces of holonomies along knots) essentially vanish, but yet, th…
We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell-Chern Simons field. For constant dilaton, the theory appears to obey a Birkhof…
We show how networks of Wilson lines realize quantum groups U_q(sl(m)), for arbitrary m, in 3d SU(N) Chern-Simons theory. Lifting this construction to foams of surface operators in 4d theory we find that rich structure of junctions is encoded in combinatorics of planar diagrams. For a particular choice of surface opera…
We identify a large class R of three-dimensional N=2 superconformal field theories. This class includes the effective theories T_M of M5-branes wrapped on 3-manifolds M, discussed in previous work by the authors, and more generally comprises theories that admit a UV description as abelian Chern-Simons-matter theories w…
Develops method to construct Lie algebra weight system kernel using Vogel algebra.
problem Detecting correlators and distinguishing knots in 3D Chern-Simons theory.
method Uses Vogel's Λ algebra and Jacobi diagrams.
result Explicitly provides Jacobi diagrams in the kernel of sl_N weight system.
3D BF theory on certain 3-manifolds evaluated via residues and large k limits.
problem Singular and ill-defined partition function of 3D BF theory.
method Direct evaluation of path integral for specific 3-manifolds, using residues and large k limits of Chern-Simons matrix integrals.
result 3 definitions of the integral offer insights into the sum/integral over all flat connections.
The trivial flat connection's Chern-Simons theory is resurgent, revealing its structure.
problem Understanding the resurgent structure of Chern-Simons theory at the trivial flat connection.
method Analyzing an extended square matrix of (x,q)-series to describe the resurgent structure and Stokes constants. result The resurgent structure and Stokes constants of the Chern-Simons series are completely described.
The paper explores BPS states in physics and their spectral sequences.
problem Understanding how BPS states vary with continuous parameters.
method Proposes a spectral sequence to describe the relationship between unperturbed and perturbed BPS spectra.
result Identifies a sliding property in colored HOMFLY homology and explains it using modular S-matrix.
Spin TFTs created by gauging line defects in 3D.
problem Creating spin TFTs from oriented TFTs with framed line defects.
method Constructing a spin TFT from an oriented TFT with framed line defects and a commutative Frobenius algebra.
result Spin TFTs extend earlier classifications and reproduce abelian spin Chern-Simons theories.
Summarizes quantum field theories with discrete symmetry, classifying representations and anomalies.
problem Classifying representations and anomalies in quantum field theories with discrete symmetry.
method Classification of representations and anomalies using the ring of profinite integers.
result Rich and complex classification of representations and anomalies.
We introduce and study the Wilson loops in a general 3D topological field theories (TFTs), and show that the expectation value of Wilson loops also gives knot invariants as in Chern-Simons theory. We study the TFTs within the Batalin-Vilkovisky (BV) and Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) framework, and the…
We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional 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, independ…
Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formula…
New invariant counts graph configurations in 3D manifolds.
problem Counting graph configurations in 3D manifolds.
method Using combings instead of parallelizations for a more flexible definition.
result Universal finite type invariant of three-manifolds.
Lecture notes on Lie groups and Chern-Simons theory for grad students.
problem Understanding Chern-Simons theory and its applications.
method Explains Lie groups and their relation to Chern-Simons theory.
result Motivated new fields in knot theory and topology.
Arithmetic Dijkgraaf-Witten theory constructs analogues in Chern-Simons TQFT.
problem Developing arithmetic analogues in Chern-Simons TQFT.
method Constructing arithmetic analogues of Chern-Simons 1-cocycle, prequantization bundle, and Chern-Simons functional.
result Decomposition and gluing formulas for arithmetic Chern-Simons invariants and arithmetic Dijkgraaf-Witten partition functions.
Chern-Simons theory on Seifert 3-manifolds evaluated.
problem Evaluating Chern-Simons theory on complex 3-manifolds.
method Abelianisation, background field method, Kawasaki Index theorem.
result Partition function evaluated on general Seifert 3-manifolds.
Teichmüller TQFT upgraded to level N, linked to quantum Chern-Simons Theory.
problem Constructing a Teichmüller TQFT at level N.
method Using results from [AK1] and [AK3], relating to quantum Chern-Simons Theory.
result Teichmüller TQFT at level N is connected to quantum Chern-Simons Theory.
Classifies extended Abelian Chern-Simons theories using quadratic modules.
problem Classifying extended Abelian Chern-Simons theories.
method Using quadratic modules to classify theories.
result Finite quadratic modules classify extended Abelian Chern-Simons theories.
New theory couples Chern-Simons to matter, topological.
problem Developing a new topological theory in 3D.
method Coupling Chern-Simons to matter, using transverse holomorphic foliation.
result The theory is equivalent to an N=2 supersymmetric Chern-Simons matter theory.
Categorifies Stokes coefficients in Chern-Simons theory models.
problem Stokes phenomenon in Chern-Simons theory around flat connections.
method Finite-dimensional model for analytically continued Chern-Simons theory, categorification of Stokes coefficients.
result Stokes coefficients can be promoted to graded vector spaces.
We show that Chern-Simons gauge theory with appropriate cutoffs is equivalent, term by term in perturbation theory, to a Fermionic theory with a nonlocal interaction term. When an additional cutoff is placed on the Fermi fields, this Fermionic theory gives rise to a convergent perturbation expansion. This leads us to c…
Chern-Simons and Reshetikhin-Turaev theories are shown equivalent for U(1) gauge group.
problem Equivalence between U(1) Chern-Simons and Reshetikhin-Turaev TQFTs. method Proof of natural isomorphism between theories for finite quadratic modules.
result Extended (2+1)-dimensional TQFTs are naturally isomorphic. Study evaluates Chern-Simons theory on Seifert fibred 3-manifolds.
problem Evaluate Chern-Simons theory on Seifert fibred 3-manifolds.
method Use Abelianisation method to evaluate the partition function and Wilson loops.
result Obtain closed formulae for expectation values of torus knots.