We propose an equivalence of the partition functions of two different 3d gauge theories. On one side of the correspondence we consider the partition function of 3d SL(2, R) Chern-Simons theory on a 3-manifold, obtained as a punctured Riemann surface times an interval. On the other side we have a partition function of a 3d N = 2 superconformal field theory on S 3 , which is realized as a duality domain wall in a 4d gauge theory on S 4 . We sketch the proof of this conjecture using connections with quantum Liouville theory and quantum Teichmüller theory, and study in detail the example of the once-punctured torus. Motivated by these results we advocate a direct Chern-Simons interpretation of the ingredients of (a generalization of) the Alday-Gaiotto-Tachikawa relation. We also comment on M5-brane realizations as well as on possible generalizations of our proposals.

We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2d discrete lattice, and moreover show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4d to 3d. More concretely, we propose an equality between (1) the 4d superconformal index of a 4d N = 1 superconformal quiver gauge theory described by a bipartite graph on T 2 and (2) the partition function of a classical integrable spin chain on T 2 . The 2d spin system is lifted to a hyperbolic 3-manifold after the dimensional reduction and the Higgsing of the 4d gauge theory.Introduction.-The concept of spacetime has been of crucial importance in our understanding of Nature. However, in the theory of quantum gravity, it is widely believed that even the notion of classical spacetime is of secondary nature, and emerges from a more fundamental structure. One proposal for such a structure is the spin network [1], a spin system defined on a discrete lattice.In a different line of development, more recently there have been important developments in supersymmetric gauge theories suggesting that the spacetime geometry could be traded for another "internal" geometry. This has been discussed for a class of supersymmetric gauge theories compactified on a compact curved manifold, which is thought of as the Euclidean version of the spacetime for the theory. The idea is simple; we begin with a D-dimensional field theory and compactify the theory on a class of d 1 -dimensional manifolds C. The resulting d 2 -dimensional theory is defined on a fixed d 2 -dimensional compact manifold S, where d 1 + d 2 = D. We could instead first compactify on S, and then we have a d 1 -dimensional theory on C. Thus we have a correspondence between the d 2 -dimensional field theory on S and the d 1 -dimensional field theory on C.While the idea itself is rather general, in practice it is a rather difficult problem to make a precise identification between the observables of the two theories, since a quantity on one side could take a rather different form on the other. A successful example of such a quantitative identification is the relation between the S 4 partition function of 4d N = 2 superconformal field theories (SCFT) arising from a compactification of 6d (2, 0) theory on a Riemann surface C [2] and a correlation function of 2d Liouville theory on C [3].The goal of this Letter is to unify these two apparently unrelated ideas in supersymmetric gauge theories and gravity. This gives new perspectives on the emergence of classical geometry, and surprisingly the process has a counterpart in the supersymmetric gauge theory.We analyze the 4d superconformal index for quiver gauge theories dual to toric Calabi-Yau 3-folds, and find that the 4d index is equivalent to the partition function of an integrable spin system in 2d. We then discuss dimensional reduction from the 4d index to the 3d partition

We provide quantitative evidence for our previous conjecture which states an equivalence of the partition function of a 3d N ¼ 2 gauge theory on a duality wall and that of the SLð2; RÞ Chern-Simons theory on a mapping torus, for a class of examples associated with once-punctured torus. In particular, we demonstrate that a limit of the 3d N ¼ 2 partition function reproduces the hyperbolic volume and the Chern-Simons invariant of the mapping torus. This is shown by analyzing the classical limit of the trace of an element of the mapping class group in the Hilbert space of the quantum Teichmüller theory. We also show that the subleading correction to the partition function reproduces the Reidemeister torsion.

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 S 3 b partition function of the theory coincides with the cluster partition function defined from the pair (Q, m).Our formalism includes the case where 3d N = 2 theories arise from the compactification of the 6d (2, 0) A N −1 theory on a large class of 3-manifolds M , including complements of arbitrary links in S 3 . In this case the quiver is defined from a 2d ideal triangulation, the mutation sequence represents an element of the mapping class group, and the 3-manifold is equipped with a canonical ideal triangulation. Our partition function then coincides with that of the holomorphic part of the SL(N ) Chern-Simons partition function on M .1 Note that in correspondence (1.1) the same data appears in rather different guises on the two sides. For example, the geometry of M for the right hand side determines the choice of the theory T [M ] itself on the left. Similarly, the deformation of the geometry, the parameter b, on the left hand side is translated into a parameter of the Lagrangian on the right.2 The first evidence for this conjecture [1] came from a chain of arguments involving quantum Liouville and Teichmüller theories. The semiclassical (t → ∞) expansion of the right-hand side of Eq. (1.1) reproduce hyperbolic volumes and Reidemeister torsions of 3-manifolds [13,14]. See [15][16][17][18][19] for further developments in the 3d/3d correspondence. SummaryThe results of this paper are summarized as follows ( Figure 2):(1) We introduce a new combinatorial object, a mutation network (section 2.3), which encodes the combinatorial data of a quiver Q and a mutation sequence m.3 Suppose that we have a 3-manifold M , and a link L inside. The complement of a link is defined as a complement of a thickened link. More formally, the link complement is a complement of the tubular neighborhood N (L) of M , i.e, M \N (L). By construction the boundary of the link complement is a disjoint union of 2d tori. 4 However, it is worthing emphasizing that their braid (branched locus) is not our braid; see further comments in section 4.3.5 All our theories are contained in the theories of "class R" in [16]. For comparison one might be tempted to call the theories T [M ] to be of "class M" (M for manifold) and theories T [(Q, m)] to be of "class C" (C for cluster algebras). 6 The theory in addition depends on the choice of the boundary condition, as will be explained in section 3.2. 2/42

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