59 research outputs found
3d-3d correspondence revisited
No full textIn fivebrane compactifications on 3-manifolds, we point out the importance of all flat connections in the proper definition of the effective 3d N=2
theory. The Lagrangians of some theories with the desired properties can be constructed with the help of homological knot invariants that categorify colored Jones polynomials. Higgsing the full 3d theories constructed this way recovers theories found previously by Dimofte-Gaiotto-Gukov. We also consider the cutting and gluing of 3-manifolds along smooth boundaries and the role played by all flat connections in this operation
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3d-3d Correspondence Revisited
No full textIn fivebrane compactifications on 3-manifolds, we point out the importance of all
flat connections in the proper definition of the effective 3d N=2 theory. The Lagrangians
of some theories with the desired properties can be constructed with the help of
homological knot invariants that categorify colored Jones polynomials. Higgsing the full 3d
theories constructed this way recovers theories found previously by Dimofte-Gaiotto-Gukov.
We also consider the cutting and gluing of 3-manifolds along smooth boundaries and the role
played by all flat connections in this operation
Perturbative and nonperturbative aspects of complex Chern–Simons theory
No full textWe present an elementary review of some aspects of Chern-Simons theory with complex gauge group SL(N,C). We discuss some of the challenges in defining the theory as a full-fledged TQFT, as well as some successes inspired by the 3d-3d correspondence. The 3d-3d correspondence relates partition functions (and other aspects) of complex Chern-Simons theory on a 3-manifold M to supersymmetric partition functions (and other observables) in an associated 3d theory T[M]. Many of these observables may be computed by supersymmetric localization. We present several prominent applications to 3-manifold topology and number theory in light of the 3d-3d correspondence
3d Superconformal Theories from Three-Manifolds
No full textThis is the 10th article in the collection of reviews "Exact results on N=2 supersymmetric gauge theories", ed. J.Teschner. It reviews correspondences between three-dimensional gauge theories and complex Chern-Simons theory on suitable three-manifolds. Such correspondences are in many cases deeply related to the 4d/2d correspondence discussed in this collection. The partition functions of the three-dimensional gauge theories are calculable by means of localisation (as discussed in article no. 9 by K. Hosomichi), and the results are related to the quantum theory obtained by quantisation of Hitchin's moduli spaces
Complex Chern-Simons theory at level k via the 3d-3d correspondence
No full textWe 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-integral as well as the 3d index, which is k=0. The Chern-Simons partition functions correspond to partition functions of Tn[M] on squashed lens spaces L(k,1). At any k, they admit a holomorphic-antiholomorphic factorization, corresponding to the decomposition of L(k,1) into two solid tori, and the associated holomorphic block decomposition of the partition functions of T_n[M]. A generalization to L(k,p) is also presented. Convergence of the state integrals, for any k, requires triangulations to admit a positive angle structure; we propose that this is also necessary for the DGG gauge theory T_n[M] to flow to a desired IR SCFT
Quantum modularity and complex Chern-Simons theory
No full textThe Quantum Modularity Conjecture of Zagier predicts the existence of a formal power series with arithmetically interesting coefficients that appears in the asymptotics of the Kashaev invariant at each root of unity. Our goal is to construct a power series from a Neumann-Zagier datum (i.e., an ideal triangulation of the knot complement and a geometric solution to the gluing equations) and a complex root of unity ζ. We prove that the coefficients of our series lie in the trace field of the knot, adjoined a complex root of unity. We conjecture that our series are those that appear in the Quantum Modularity Conjecture and confirm that they match the numerical asymptotics of the Kashaev invariant (at various roots of unity) computed by Zagier and the first author. Our construction is motivated by the analysis of singular limits in Chern-Simons theory with gauge group SL(2,C) at fixed level k, where ζk=1
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Recent Developments in Representation Theory and Mathematical Physics
No full textThis mini-workshop was devoted to foster the interactions between mathematicians and mathematical physicists who are working on questions related to representation theory. This includes for example the representation theory of supergroups, vertex operator algebras and quantum groups. Another focus was on link and manifold invariants and TQFTs
QFT's for non-semisimple TQFT's
No full textThirty years ago, work of Witten and Reshetikhin-Turaev activated the study of quantum invariants of links and three-manifolds. A cornerstone of subsequent developments, leading up to our current knot-homology conference, was a three-pronged approach involving 1) quantum field theory (Chern-Simons); 2) rational VOA's (WZW); and 3) semisimple representation theory of quantum groups. The second and third perspectives have since been extended, to logarithmic VOA's and related non-semisimple quantum-group categories. I will propose a family of 3d quantum field theories that similarly extend the first perspective to a non-semisimple (and more so, derived) regime. The 3d QFT's combine Chern-Simons theory with a topologically twisted supersymmetric theory. They support boundary VOA's whose module categories are dual to modules for Feigin-Tipunin algebras and (correspondingly) to modules for small quantum groups at even roots of unity. The QFT is also compatible with deformations by flat connections, related to the Frobenius center of quantum groups at roots of unity.
This is joint work with T. Creutzig, N. Garner, and N. Geer. I will mention potential connections to related recent work of Gukov-Hsin-Nakajima-Park-Pei-Sopenko and promising routes to categorification, from a physics perspective.Non UBCUnreviewedAuthor affiliation: University of EdinburghResearche
The Coulomb Branch of 3d N = 4 Gauge Theories
No full textThe moduli space of a 3d N = 4 gauge theory contains at least two branches, typically referred to as Higgs and Coulomb. Both are hyperkahler manifolds with some special properties, but while the Higgs branch has a straightforward classical construction, the Coulomb branch is affected by quantum corrections and has long remained mysterious. I will discuss a physically motivated construction of both the ring of holomorphic functions on the Coulomb branch and its hyperkahler structure. I also hope to touch upon boundary conditions in 3d N = 4 gauge theories, and their images on the Higgs and Coulomb branches, which tie 3d N = 4 gauge theory to geometric representation theory in mathematics. This project was initially motivated by constructions of knot homology using the 6d (2,0) theory; 3d N = 4
3 theories also arise naturally from compactification of the 6d (2,0) theory on a surface and an additional circle. (Joint work with M. Bullimore, D. Gaiotto, and J. Hilburn.)Non UBCUnreviewedAuthor affiliation: Institute for Advanced StudyPostdoctora
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