Gauge Theories Labelled by Three-Manifolds

Abstract

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional N=2 gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that S^3_b partition functions of two mirror 3d N=2 gauge theories are equal. Three-dimensional N=2 field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional N=2 SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.

Figures (15)

• Figure 1: $(a)$ A cobordism $M$ between $\mathcal{C}$ and $\mathcal{C}'$ gives rise to a domain wall $(b)$ between 4d $\mathcal{N}=2$ theories $T(\mathcal{C})$ and $T(\mathcal{C}')$.
• Figure 2: $(a)$ A 3-manifold $M$ stretched along a 'neck' ${\mathbb R} \times \mathcal{C}$ becomes a 4d $\mathcal{N}=2$ superconformal theory $(b)$ on ${\mathbb R}^3 \times I$ coupled to 3-dimensional theories $T(M^+)$ and $T(M^-)$ at the boundary. The 4d $\mathcal{N}=2$ gauge theory in the bulk is determines by the cross-section $\mathcal{C}$ of the 3-manifold $M$.
• Figure 3: Natural polarizations for a tetrahedron, with the thickened pairs of opposite edges corresponding to the "position" coordinate.
• Figure 4: Triangulations by Euclidean vertex triangles of (a) an annular cusp attached to a geodesic boundary, and (b) a torus cusp.
• Figure 5: Forming a bipyramid from three tetrahedra.