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Register a new user5d Higgs Branch Localization, Seiberg-Witten Equations and Contact Geometry
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Authors
Yiwen Pan
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In this paper we apply the idea of Higgs branch localization to 5d supersymmetric theories of vector multiplet and hypermultiplets, obtained as the rigid limit of $mathcal{N} = 1$ supergravity with all auxiliary fields. On supersymmetric K-contact/Sasakian background, the Higgs branch BPS equations can be interpreted as 5d generalizations of the Seiberg-Witten equations. We discuss the properties and local behavior of the solutions near closed Reeb orbits. For $U(1)$ gauge theories, we show the suppression of the deformed Coulomb branch, and the partition function is dominated by 5d Seiberg-Witten solutions at large $zeta$-limit. For squashed $S^5$ and $Y^{pq}$ manifolds, we show the matching between poles in the perturbative Coulomb branch matrix model, and the bound on local winding numbers of the BPS solutions.
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We show how to map Grothendiecks dessins denfants to algebraic curves as Seiberg-Witten curves, then use the mirror map and the AGT map to obtain the corresponding 4d $mathcal{N}=2$ supersymmetric instanton partition functions and 2d Virasoro conformal blocks. We explicitly demonstrate the 6 trivalent dessins with 4 punctures on the sphere. We find that the parametrizations obtained from a dessin should be related by certain duality for gauge theories. Then we will discuss that some dessins could correspond to conformal blocks satisfying certain rules in different minimal models.
We give a pedagogical review of the computation of Gromov-Witten invariants via localization in 2D gauged linear sigma models. We explain the relationship between the two-sphere partition function of the theory and the Kahler potential on the conformal manifold. We show how the Kahler potential can be assembled from classical, perturbative, and non-perturbative contributions, and explain how the non-perturbative contributions are related to the Gromov-Witten invariants of the corresponding Calabi-Yau manifold. We then explain how localization enables efficient calculation of the two-sphere partition function and, ultimately, the Gromov-Witten invariants themselves.
Coulomb branch chiral rings of $mathcal N=2$ SCFTs are conjectured to be freely generated. While no counter-example is known, no direct evidence for the conjecture is known either. We initiate a systematic study of SCFTs with Coulomb branch chiral rings satisfying non-trivial relations, restricting our analysis to rank 1. The main result of our study is that (rank-1) SCFTs with non-freely generated CB chiral rings when deformed by relevant deformations, always flow to theories with non-freely generated CB rings. This implies that if they exist, they must thus form a distinct subset under RG flows. We also find many interesting characteristic properties that these putative theories satisfy which may behelpful in proving or disproving their existence using other methods.
We derive a family of matrix models which encode solutions to the Seiberg-Witten theory in 4 and 5 dimensions. Partition functions of these matrix models are equal to the corresponding Nekrasov partition functions, and their spectral curves are the Seiberg-Witten curves of the corresponding theories. In consequence of the geometric engineering, the 5-dimensional case provides a novel matrix model formulation of the topological string theory on a wide class of non-compact toric Calabi-Yau manifolds. This approach also unifies and generalizes other matrix models, such as the Eguchi-Yang matrix model, matrix models for bundles over $P^1$, and Chern-Simons matrix models for lens spaces, which arise as various limits of our general result.
We consider N=2 supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli space. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called dessins denfants or childrens drawings on the Riemann sphere. From a mathematical point of view, the dessins are important because the absolute Galois group Gal(bar{Q}/Q) acts faithfully on them. We argue that the relation between the dessins and Seiberg-Witten theory is useful because gauge theory criteria used to distinguish branches of N=1 vacua can lead to mathematical invariants that help to distinguish dessins belonging to different Galois orbits. For instance, we show that the confinement index defined in hep-th/0301006 is a Galois invariant. We further make some conjectures on the relation between Grothendiecks programme of classifying dessins into Galois orbits and the physics problem of classifying phases of N=1 gauge theories.
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