We study D-brane instantons in systems of D3-branes at toric CY 3-fold singularities. The instanton effect can be described as a backreaction modifying the geometry of the mirror configuration, in which the breaking of U (1) symmetries by the instanton translates into the recombination of gauge D-branes, which also directly generates the instanton-induced charged field theory operator. In this paper we describe the D-brane instanton backreaction in terms of a combinatorial operation in the bipartite dimer diagram of the original theory. Interestingly, the resulting theory is a general Bipartite Field Theory (BFT), defined by a bipartite graph tiling a general (possibly higher-genus) Riemann surface. This provides the first string theory realization of such general BFTs. We study the general properties of the resulting theories, including the construction of the higher-dimensional toric diagrams and the interplay between backreaction and Seiberg duality. In cases where the non-perturbative effects relate to complex deformations, we show that the procedure reproduces and explains earlier existing combinatorial recipes. The combinatorial operation and its properties generalize to an operation on the class of general BFTs, even including boundaries, relating BFTs defined on Riemann surfaces of different genus. arXiv:1805.00011v2 [hep-th] 6 Sep 2018 7 The Inverse Problem 43 8 Conclusions 46 -1 -1 Closely related theories were introduced in [22] and studied further in [23].-3 -depiction of the handle attachment surgery which simplifies the discussion of the genus increase. In §4 we describe the computation of the new toric data corresponding to the backreacted BFTs. The change in perfect matchings between the original and final theories is discussed in §4.1, yielding the construction of the new toric diagram in §4.2. A direct construction based on relating new toric coordinates with the bridges identifying formerly different nodes of the original theory is provided in §4.3. These concepts are illustrated in a detailed example in §4.4. In §5 we describe instances of the interplay of instanton backreaction and Seiberg duality: in §5.1, when they are applied to the same dimer face, and in §5.2 when applied to neighboring faces. In §6 we consider the generalization to backreaction of multi-instantons, focusing on cases corresponding to complex deformations of the original CY 3-fold, and recover earlier results in the literature. In §7 we show the non-uniqueness of the inverse problem of reconstructing initial theories for a given final one. We present our concluding remarks in §8.
Review of D-brane Instanton Backreaction on D-Branes at SingularitiesIn this section we review some background material on dimer diagrams as tools to describe systems of D3-branes at toric singularities. Subsequently we review the backreaction of instantons in the mirror of these systems, to lay the ground for their novel discussion in the dimer diagram, and the connection with general BFTs.