3d-3d correspondence for mapping tori
- PDF / 1,241,570 Bytes
- 60 Pages / 595.276 x 841.89 pts (A4) Page_size
- 75 Downloads / 205 Views
Springer
Received: June 29, 2020 Accepted: August 25, 2020 Published: September 23, 2020
3d-3d correspondence for mapping tori
a
New High Energy Theory Center, Rutgers University, Piscataway, NJ 08854, U.S.A. b California Institute of Technology, Pasadena, CA 91125, U.S.A. c Max-Planck-Institut f¨ ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: 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 b 3 ) in the case of manifolds with b1 > 0. Although we use recent q-series invariants Z(M 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. Keywords: Conformal Field Models in String Theory, Supersymmetric Effective Theories, Topological Field Theories ArXiv ePrint: 1911.08456
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)152
JHEP09(2020)152
Sungbong Chun,a Sergei Gukov,b,c Sunghyuk Parkb and Nikita Sopenkob
Contents 1 Introduction and motivation
1 5 5 7 10 13 19
3 WRT invariants and q-series Zba (M3 ) 3.1 “Almost abelian” flat connections 3.2 Plumbings with loops 3.2.1 Genus-1 mapping tori 3.2.2 Example: a tadpole diagram 3.2.3 Example: double loop 3.3 0-surgery on knots 3.3.1 0-surgery on double twist knots Kn,m 3.3.2 0-surgery on torus knots Ts,t 3.3.3 Renormalon effects 3.3.4 0-surgery on the unknot: M3 = S 2 × S 1 3.4 Continuous versus discrete labels
22 24 28 30 31 32 33 33 37 38 40 42
4 Twisted indices and Hilbert spaces 4.1 Twisted Hilbert space on Fg 4.2 Twisted Hilbert space on D2 4.3 Twisted Hilbert space on D2 with impurity
43 44 46 49
5 Generalizations and future directions
51
1
Introduction and motivation
3d-3d correspondence, originally proposed in [1], relates (quantum) topology of 3-manifolds to physics of 3-dimensional supersymmetric gauge theories in various backgrounds. In particular, to an arbitrary 3-manifold M3 and a choice of ADE type group G it assigns a topological invariant T [M3 , G] valued in 3d N = 2 superconformal field theo
Data Loading...
