Five-dimensional gauge theories and the local B-model. (English) Zbl 1495.14063
Five-dimensional supersymmetric gauge theories play an important role in both low-energy descriptions of 5d SCFTs and non-trivial quantum field theories which can be engineered by M-theory compactification on a Calabi-Yau 3-fold. In this paper, the authors propose a powerful method to compute prepotential for five-dimensional \(N = 1\) gauge theories on a circle with simple gauge groups. They put forward a systematic construction of Picard-Fuchs operators(i.e. operator generalizations of Picard-Fuchs equations) which annihilating the period integrals of the Seiberg-Witten curve. The method using Picard-Fuchs equations was known for toric Calabi-Yau 3-folds corresponding to the gauge theories with SU gauge groups. This paper aims to extend this idea to all the general simple gauge groups. For the simply-laced gauge groups, a beautiful formulation to construct Picard-Fuchs operators from the corresponding Frobenius manifold was established by generalizing the toric cases. As for non-simply laced gauge groups, the generalization is quite non-trivial because the analogous interpretation in terms of Frobenius manifolds is not clear. Moreover, the authors successfully generalized the computational method even for the non-simply-laced cases by focusing on the algebraic structure associated with the Seiberg-Witten curve. The method developed in this paper can be applied to many other known cases of the Seiberg-Witten curves, whose prepotentials were not computed before. For example, it had been believed for a long while that Seiberg-Witten curves for 5d gauge theories could be obtained from the relativistic Toda systems for the corresponding gauge groups. However, the Seiberg-Witten curves for non-simply laced gauge groups which can be predicted from integrable systems as spectral curves turn out to disagree with the correct prepotential computed from Nakajima-Yoshioka’s K-theoretic blow-up formula.
In conclusion, the paper under review presents a new and significant technique exploring the relationship between Seiberg-Witten prepotential obtained by blow-up formula and prepotential of spectral curve from some certain integrable system. They match well for the cases of ADE and do not match for the cases of BCFG. It is worth mentioning that it shed light on finding the proper integrable systems giving the correct Seiberg-Witten curves for 5d gauge theories with non-simply laced gauge groups.
In conclusion, the paper under review presents a new and significant technique exploring the relationship between Seiberg-Witten prepotential obtained by blow-up formula and prepotential of spectral curve from some certain integrable system. They match well for the cases of ADE and do not match for the cases of BCFG. It is worth mentioning that it shed light on finding the proper integrable systems giving the correct Seiberg-Witten curves for 5d gauge theories with non-simply laced gauge groups.
Reviewer: Xiaobin Li (Chengdu)
MSC:
| 14J80 | Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
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