Thermodynamic limit of Nekrasov partition function for 5-brane web with O5-plane

Abstract

In this paper, we study 5d $$ \mathcal{N} $$ N = 1 Sp(N) gauge theory with Nf (≤ 2N + 3) flavors based on 5-brane web diagram with O5-plane. On the one hand, we discuss Seiberg-Witten curve based on the dual graph of the 5-brane web with O5-plane. On the other hand, we compute the Nekrasov partition function based on the topological vertex formalism with O5-plane. Rewriting it in terms of profile functions, we obtain the saddle point equation for the profile function after taking thermodynamic limit. By introducing the resolvent, we derive the Seiberg-Witten curve and its boundary conditions as well as its relation to the prepotential in terms of the cycle integrals. They coincide with those directly obtained from the dual graph of the 5-brane web with O5-plane. This agreement gives further evidence for mirror symmetry which relates Nekrasov partition function with Seiberg-Witten curve in the case with orientifold plane and shed light on the non-toric Calabi-Yau 3-folds including D-type singularities.

Five-dimensional gauge theories and the local B-model

“…We put this to the test in a wide array of cases. When the SW geometry arises from geometric engineering [13,51] or brane constructions in M-theory [41,54,58], we show that our constructions recover the microscopic gauge theory results with flying colours. We match the perturbative and instanton parts of the prepotential with the expressions arising from the K-theoretic version of the Nakajima-Yoshioka blow-up equations [71], as well as with their extrapolation to general gauge groups [53].…”

“…Although the two curves are closely related, and their periods are shown to be annihilated by the differential ideal generated by the second line of (2) for the same choice of structure constants C k ij , in this rank-1 case determining the full expression for the periods requires fixing a finite-dimensional ambiguity akin to imposing a quasi-homogeneity condition as in the first line of (2): our calculations show that this is done differently for the curves from [15] on one hand and of [41,58] on the other, ruling out the former and confirming the correctness of the latter with a direct prepotential calculation. We also match our B-model construction applied to the θ = 0 version of the brane web geometries with an O5-plane of [41,58] against the corresponding instanton calculation of the prepotentialagain finding perfect agreement.…”

“…We match the perturbative and instanton parts of the prepotential with the expressions arising from the K-theoretic version of the Nakajima-Yoshioka blow-up equations [71], as well as with their extrapolation to general gauge groups [53]. We also apply our proposal to the SW curves for classical non-simply-laced groups obtained from string theory constructions with orientifold planes [15,41,58]. As a byproduct of our construction we show, in type B n , that the curves proposed by [15] from an M-theory uplift of the Hanany-Witten construction with orientifold planes correctly reproduce the instanton prepotential for SO(2n + 1) gauge groups; and in type C n , we confirm a tension between the results of [15] for n = 1 and more recent proposals of Sp(1) curves with discrete θ-angle at θ = π in [41,58].…”

“…We also apply our proposal to the SW curves for classical non-simply-laced groups obtained from string theory constructions with orientifold planes [15,41,58]. As a byproduct of our construction we show, in type B n , that the curves proposed by [15] from an M-theory uplift of the Hanany-Witten construction with orientifold planes correctly reproduce the instanton prepotential for SO(2n + 1) gauge groups; and in type C n , we confirm a tension between the results of [15] for n = 1 and more recent proposals of Sp(1) curves with discrete θ-angle at θ = π in [41,58]. Although the two curves are closely related, and their periods are shown to be annihilated by the differential ideal generated by the second line of (2) for the same choice of structure constants C k ij , in this rank-1 case determining the full expression for the periods requires fixing a finite-dimensional ambiguity akin to imposing a quasi-homogeneity condition as in the first line of (2): our calculations show that this is done differently for the curves from [15] on one hand and of [41,58] on the other, ruling out the former and confirming the correctness of the latter with a direct prepotential calculation.…”

“…The difference between the two choices can be reabsorbed in a rescaling of the masses of the fundamental hypermultiplets when present, but it is physical for the pure gauge theory, leading to inequivalent prepotentials. Accordingly, two sets of curves {Q Sp(N );u,θ = 0} were put forward in [41,58] to account for this,…”