Aspects of defects in 3d-3d correspondence

Published for SISSA by Springer Received: July 13, 2016 Accepted: September 26, 2016 Published: October 12, 2016 Aspects of defects in 3d-3d correspondence a Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Chiba 277-8583, Japan b Department of Physics and Research Institute of Basic Science, Kyung Hee University, Seoul 02447, Korea c School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea d School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, U.S.A. E-mail: , , , Abstract: In this paper we study supersymmetric co-dimension 2 and 4 defects in the compactification of the 6d (2, 0) theory of type AN −1 on a 3-manifold M . The so-called 3d-3d correspondence is a relation between complexified Chern-Simons theory (with gauge group SL(N, C)) on M and a 3d N = 2 theory TN [M ]. We study this correspondence in the presence of supersymmetric defects, which are knots/links inside the 3-manifold. Our study employs a number of different methods: state-integral models for complex ChernSimons theory, cluster algebra techniques, domain wall theory T [SU(N )], 5d N = 2 SYM, and also supergravity analysis through holography. These methods are complementary and we find agreement between them. In some cases the results lead to highly non-trivial predictions on the partition function. Our discussion includes a general expression for the cluster partition function, which can be used to compute in the presence of maximal and certain class of non-maximal punctures when N > 2. We also highlight the non-Abelian description of the 3d N = 2 TN [M ] theory with defect included, when such a description is available. This paper is a companion to our shorter paper [1], which summarizes our main results. Keywords: Chern-Simons Theories, M-Theory, Wilson, ’t Hooft and Polyakov loops ArXiv ePrint: 1510.05011 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP10(2016)062 JHEP10(2016)062 Dongmin Gang,a Nakwoo Kim,b,c Mauricio Romoa,d and Masahito Yamazakia,d Contents 1 1 2 4 2 3d-3d correspondence with defects 2.1 Co-dimension 2 defects 2.2 Co-dimension 4 defects 5 5 11 3 From state integral model 3.1 Generalities on state-integral models 3.2 Co-dimension 2 defects 3.3 Co-dimension 4 defects 12 12 19 20 4 From cluster partition function 4.1 General formula 4.2 Applications to 3-manifolds 4.3 Relation with state-integral models 4.4 Inclusion of co-dimsion 4 defects 4.5 Examples 4.5.1 Co-dimension 2 defects: ρ = maximal 4.5.2 Co-dimension 2 defects: ρ = non-maxiaml 4.5.3 Co-dimension 4 defects 25 25 28 32 36 38 39 44 47 5 From domain wall theory T [SU(N )] 5.1 Necessity of non-Abelian description for TN [M ] 5.2 Co-dimension 2 defects: ρ = simple from T [SU(N )] theory 5.3 Co-dimension 4 defects 49 49 50 54 6 From 5d N = 2 SYM 6.1 Co-dimension 4 defects as Wilson lines 6.2 Co-dimension 2 defect: Higgsing and refinement 56 57 59 7 From large N holography 7.1 Supergravity background 7.2 ‘Simple’ co-dimension 2 defects 7.2.1 Single probe M5 7.2.2 Large N of TN [(Σ1,1 × S 1 )ϕ , simple] 7.3 Co-dimension 4 defects 7.3.1 Fundamental representation as M2-brane 7.3.2 Antisymmetric representation as M5-brane 7.4 Chern-Simons perturbation 62 62 64 64 65 66 66 67 70 8 Discussion and outlook 72 –i– JHEP10(2016)062 1 Introduction and outline 1.1 M5-branes on 3-manifolds 1.2 Supersymmetric defects 1.3 Computational methods 74 B Quantum dilogarithm function 76 C Derivation of cluster partition function C.1 Detailed derivation C.2 Inclusion of Wilson lines 76 76 82 D Proof of (4.7) 84 E Direct computation of Tr(ϕ̂) on Hk=0 N =3 (Σ1,1 , simple) 85 F Index for T [SU(3)] 89 G Derivation of eq. (7.22) 91 H Verification of eq. (7.43) in Chern-Simons theory 93 1 Introduction and outline We have learned over the past few years that compactification of M5-branes on various manifolds generates a class of lower-dimensional supersymmetric field theories labeled by the geometrical data. This has led to fruitful interplay between the physics of supersymmetric gauge theories (and in particular their non-perturbative dualities) and the geometry of the compactification manifolds (see e.g. [2] and references therein). When we choose to compactify on a 3-manifold M , we have the correspondence between complex ChernSimons (CS) theory on M and 3d N = 2 theory T [M ]. This has been worked out in a number of papers [3–9], and the appearance of complex Chern-Simons theory has recently been derived in [10, 11] (see also [12]). In this paper we include supersymmetric defects to this story, inherited from co-dimension 2 and co-dimension 4 defects in the 6d (2, 0) theory. In the rest of this introduction we provide more detailed outline of this paper. 1.1 M5-branes on 3-manifolds Let us consider N > 1 M5-branes, whose low energy world-volume theory is the 6d AN −1 (2,0) theory. We wrap the M5-branes on a closed 3-manifold M̂ : 1,2,3 3,4,5 z}|{ z}|{ N M5s on R1,2 × M̂ . (1.1) Since M̂ is a curved manifold, we perform a partial topological twisting along M̂ , and turn on an R-symmetry flux mixing the SO(3) connection on M̂ with an SO(3) current inside SO(5) R-symmetry of 6d (2, 0) theory. The resulting theory has four supercharges with the –1– JHEP10(2016)062 A Conditions on boundary holonomies remaining SO(2) R-symmetry. Thus such a compactification generates a 3d N = 2 theory, which we denote by TN [M̂ ]. The 3d–3d correspondence relates1 3d N = 2 theory TN [M̂ ] ⇐⇒ SL(N ) CS theory on M̂ . (1.2) We will comment on more precise versions of this relation momentarily. 1.2 Supersymmetric defects Co-dimension 2 defects. The brane configuration is R1,2 M̂ z}|{ z}|{ N M5: 012345 Defect M5: 0 1 2 3 78 (1.3) For the 6d AN −1 (2, 0) theory, the co-dimension 2 defect is labelled by an embedding ρ : SU(2) → SU(N ) or equivalently a partition [n1 , . . . , ns ] of N . Let denote by K the trajectory of the defect inside M̂ . Since the defect fills the whole R1,2 , the effect of this defect is to replace the 3d N = 2 theory TN [M̂ ] by a new theory,4 which we denote by TN [M̂ \K, ρ]. Geometrically, this is to replace a closed 3-manifold M̂ by a knot/link complement, which we denote by M := M̂ \K . (1.4) In the SL(N ) CS theory, the defect will be a loop defect along the knot K. We propose that the loop defect of type ρ can be identified with monodromy defect associated to 1 This has generalizations to other gauge groups G, as is clear from the derivation of [10, 11]. The same comment applies to our discussion in section 6.1. 2 In this paper, co-dimensions always refer to co-dimensions inside the 6d theory. In 3d–3d correspondence, we have two ‘3d’ directions, and we also consider compactification of 6d theory to 5d N = 2 SYM. In each of these cases the co-dimensions in these (3d or 5d) spaces will be different from those in 6d. 3 There are many discussions of supersymmetric defects i (...truncated)