Supersymmetric Partition Functions and a String Theory in 4 Dimensions

Abstract

We propose a novel string theory propagating in a non-commutative deformation of the four dimensional space T* T^2 whose scattering states correspond to superconformal theories in 5 dimensions and the scattering amplitudes compute superconformal indices of the corresponding 5d theories. The superconformal theories are obtained by M-theory compactifications on singular CY 3-folds or equivalently from a web of 5-branes of type IIB strings. The cubic interaction of this string theory for primitive winding modes corresponds to the (refined) topological vertex. The oscillator modes of the string theory correspond to off-shell states and carry information about co-dimension 2 defects of the superconformal theory. Particular limits of a subset of scattering amplitudes of this string theory lead to the partition functions of A_n Gaiotto theories for all n, compactified on S^4, i.e., to amplitudes of all Toda theories.

Figures (13)

• Figure 1: An example of $(p,q)$ web with $N=6$ external lines and $g=3$ internal loops.
• Figure 2: A breathing mode corresponds to changing the web, without moving the external lines. There is one breathing mode per internal loop.
• Figure 3: An example of a $(p,q)$ web with $N=4$ external lines and $g=2$ internal loops. The convex outer polygon has 4 sides and its triangulation is dual to the web.
• Figure 4: An example of grouping 5-branes according to which 7-branes they end on. The dual grid contains white dots to depict the corresponding grouping. The vertices of the web correspond to faces of the grid. Vertex A corresponds to the bigger triangle. Vertex B the trapezoid and vertex C to the small triangle. The thicker edges of the web correspond to two 5-branes.
• Figure 5: The D3 branes (or Lagangian branes in the M theory setup), are suspended between a 5-brane at infinity (denoted by a dashed line) and the web. The D3 branes are perpendicular to the plane ending on the web at one of the two dots.