Emergent 3-Manifolds from Four Dimensional Superconformal Indices

Abstract

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

Superconformal indices, Sasaki-Einstein manifolds, and cyclic homologies

“…The superconformal index of quivers for toric Calabi-Yau cones were also studied in [48,49] from a rather different perspective. It will be of interest to see what connections, if any, there are with our work.…”

Lens Generalisation of τ-Functions for the Elliptic Discrete Painlevé Equation

“…In another direction, the lens elliptic gamma function, which is a central function for this paper, first appeared in the study of four-dimensional N = 1 supersymmetric gauge theories on a circle times the lens space S 3 /Z r [7]. This connection suggests that there exists an interpretation of the results of this paper in terms of supersymmetric gauge theories and associated integrable lattice models [8,[11][12][13][14][15][16][17][18][19][20].…”

“…In terms of the regular elliptic gamma function (22), the functions (15), and (16), are simply γ σ (z, z; σ, τ) = Γ 1 (z + σz; rσ, σ + τ), γ τ (z, z; σ, τ) = Γ 1 (z + τ(r − z); rτ, σ + τ).…”

Integrability as duality: The Gauge/YBE correspondence

“…The previous statement that SSR implies YBE is now an exercise in graphical calculus: we need to show the equality of the zig-zag paths in 21 The idea that the R-charge can be identified with an angle also appeared in the context of the 3d-3d correspondence [57,58,59,60,61,62]. The relation between Gauge/YBE and 3d-3d deserves further exploration, see [11,12,18] for preliminary discussion. 22 The R-charge in itself is a real parameter.…”

“…This approach is called the Gauge/YBE correspondence, since this correspondence claims rather direct relation between the YBE and the partition functions of gauge theories. This approach was initiated in 2012-2013 in my papers (partly in collaboration with Terashima) [11,12,13]. Our work is based on several previous important ideas in the literature, most notably the integrable models (known as the "master solution") constructed by Bazhanov and Sergeev [14,15].…”