Blowup equations for refined topological strings

Abstract

Göttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the Nekrasov partition function of five dimensional N = 1 supersymmetric gauge theories compactified on a circle, which via geometric engineering correspond to the refined topological string theory on SU (N ) geometries. In this paper, we study the K-theoretic blowup equations for general local Calabi-Yau threefolds. We find that both vanishing and unity blowup equations exist for the partition function of refined topological string, and the crucial ingredients are the r fields introduced in our previous paper. These blowup equations are in fact the functional equations for the partition function and each of them results in infinite identities among the refined free energies. Evidences show that they can be used to determine the full refined BPS invariants of local Calabi-Yau threefolds. This serves an independent and sometimes more powerful way to compute the partition function other t han the refined topological vertex in the A-model and the refined holomorphic anomaly equations in the B-model. We study the modular properties of the blowup equations and provide a procedure to determine all the vanishing and unity r fields from the polynomial part of refined topological string at large radius point. We also find that certain form of blowup equations exist at generic loci of the moduli space. To Sheldon Katz on his 60th anniversary arXiv:1711.09884v2 [hep-th] 9 Nov 2018 7 Blowup equations at generic points of moduli space 70 7.1 Modular transformation 70 7.2 Conifold point 70 7.3 Orbifold point 73 -i -8 Outlook 75 References 79

Elliptic blowup equations for 6d SCFTs. Part II. Exceptional cases

“…In the 2d approach one needs for higher d to consider ever more complicated quiver gauge theories, while in the modular approach one has to deal with more and more complicated rings of weak Jacobi forms. For this reason we further develop in this paper the recursive approach based on the elliptic blowup equations [24] for the calculation of Z(t, 1 , 2 ) that is further based on a specialisation of the generalized blowup equation in [15] to the elliptic non-compact Calabi-Yau geometries. The main advantage of this approach is that it needs as input only 5 the classical topological data of X, i.e.…”

“…Let i=1 n i , the generalized blowup equations can be cast in the following form [15] (see also section 8 of [23]) n∈Z b c 4 (−1) |n| Z(t + 1 R n , 1 , 2 − 1 ) Z(t + 2 R n , 1 − 2 , 2 ) = 0, r ∈ S v , Λ( 1 , 2 , m, r)Z(t, 1 , 2 ) r ∈ S u .…”

“…Given the simple form of (1.2) and the method of proof in the gauge theory context [16,18], it seems reasonable to conjecture [15,23,24] that these equations, called the generalised blowup equations, should hold for the refined partition functions Z(t, 1 , 2 ) of all non-compact Calabi-Yau threefolds with a global U (1) R symmetry so that the refined invariants or equivalently the corresponding BPS index for the space time theory with an Ω background can be defined [11,12]. At the technical level the precise non-trivial claim is that S v ∪S u should be non-empty.…”

“…At the technical level the precise non-trivial claim is that S v ∪S u should be non-empty. In addition it was observed in [15] that the classical topological data of X mentioned above and the genus zero sector determine Z(t, 1 , 2 ) recursively, and many examples were already checked in great detail. In particular in [24] this approach was used to compute the refined BPS invariants of elliptic non-compact Calabi-Yau geometries associated to minimal 6d (1, 0) SCFTs with gauge group G = SU (3), SO (8).…”

“…This paper is organized as follows: in Section 2, we review the geometric construction of elliptic non-compact Calabi-Yau threefolds that engineer 6d (1, 0) minimal SCFTs, the basic properties of the generalized blowup equations in [15], and the de-affinisation procedure which was essentially already used in [24] to obtain elliptic blowup equations for G = SU (3) and SO (8). In Section 3, we discuss both unity and vanishing elliptic blowup equations in detail, and derive a universal recursion formula for the elliptic genera of all minimal SCFTs with G = SU (3), SO(8), F 4 , E 6,7,8 .…”