Geometric engineering of Seiberg–Witten theories with massive hypermultiplets

Abstract

We analyze the geometric engineering of the N = 2 SU (2) gauge theories with 1 ≤ N f ≤ 3 massive hypermultiplets in the vector representation. The set of partial differential equations satisfied by the periods of the Seiberg-Witten differential is obtained from the Picard-Fuchs equations of the local B-model. The differential equations and its solutions are consistent with the massless case. We show that the Yukawa coupling of the local A-model gives rise to the correct instanton expansion in the gauge theory, and propose the pattern of the distribution of the world-sheet instanton number from it. As a side result, we obtain the asymptotic form of the instanton amplitude in the gauge theories with massless hypermultiplets.

Asymptotic form of Gopakumar–Vafa invariants from instanton counting

“…We computed the Gopakumar-Vafa invariants of A 2 -fibration over P 1 , for g = 0 and d B ≤ 2, d 1 , d 2 ≤ 21 by the local B-model calculation [21]. The results agree with the results from the partition function of Iqbal and Kashani-Poor (11) and the results in [21] (section 6.4) for m = −1 and m = 2. In this section, we list the relevant data.…”

“…The asymptotic form of the Gopakumar-Vafa invariants was first studied in the quintic case [9]. Other cases studied so far are: the canonical bundle of P 2 and other one modulus local mirror systems with Picard-Fuchs equations given by Meijer's equation [10]; the canonical bundle of Hirzebruch surface F 0 , F 1 , F 2 [1]; the canonical bundle of F 2 blown up at 1,2,3-points [11]. The last two cases are the results of the geometric engineering of SU(2) gauge theory.…”

“…, d n ≫ d B (g + 1). When this condition is satisfied, The asymptotic form for the A 1 case at genus zero was studied in [1,11]. In this section we study the case n = 2.…”