Abstract

A class of 3d $\mathcal{N}=2$ supersymmetric gauge theories are constructed and shown to encode the simplicial geometries in 4-dimensions. The gauge theories are defined by applying the Dimofte-Gaiotto-Gukov construction in 3d/3d correspondence to certain graph complement 3-manifolds. Given a gauge theory in this class, the massive supersymmetric vacua of the theory contain the classical geometries on a 4d simplicial complex. The corresponding 4d simplicial geometries are locally constant curvature (either dS or AdS), in the sense that they are made by gluing geometrical 4-simplices of the same constant curvature. When the simplicial complex is sufficiently refined, the simplicial geometries can approximate all possible smooth geometries on 4-manifold. At the quantum level, we propose that a class of holomorphic blocks defined in arXiv:1211.1986 from the 3d $\mathcal{N}=2$ gauge theories are wave functions of quantum 4d simplicial geometries. In the semiclassical limit, the asymptotic behavior of holomorphic block reproduces the classical action of 4d Einstein-Hilbert gravity in the simplicial context.

Highlights

  • We propose that a class of holomorphic blocks defined in [2] from the 3d N = 2 gauge theories are wave functions of quantum 4d simplicial geometries

  • The asymptotic behavior of holomorphic block reproduces the classical action of 4d Einstein-Hilbert gravity in the simplicial context

  • We propose that a class of holomorphic blocks from the 3d N = 2 gauge theories are wave functions of quantum 4d simplicial geometries

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Summary

Holomorphic block of 3-dimensional supersymmetric gauge theories

The holomorphic block has been firstly proposed in [2, 18] as a BPS index for 3d supersymmetric gauge theory (it has been generalized to 4d gauge theory [19]). TS3\Γ5 has the gauge group U(1) and the flavor symmetry group U(1)15 It is straight-forward to obtain the perturbative expression of holomorphic block integral. The leading order contribution to the holomorphic block is given by a contour integral of the Liouville 1-form in terms of the right symplectic coordinates: BSα3\Γ5 (x, q) = exp (xI ,yI(α)). The changes of variables results in the two sets of algebraic equations AI (x, y) = 0 defining 2 copies LSUSY(TS3\Γ5). The leading order contribution to the holomorphic block is given by a contour integral: BM α 3 (x, q) = exp (x,y(α)). This result will be important in deriving the relation with 4-dimensional simplicial gravity

Supersymmetric vacua of TM3 and 4-dimensional simplicial geometry
Complex Fenchel-Nielsen coordinate and geometrical quantities
Holomorphic block and 4-dimensional quantum geometry
P in front
B Review of holomorphic block
C Exponents in the product of theta functions inside block integral
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