Triality in SU(2) Seiberg-Witten theory and Gauss hypergeometric function

In this first of a series of three papers we outline an approach to classifying 4d $\mathcal{N}{=}2$ superconformal field theories at rank 2. The classification of allowed scale invariant $\mathcal{N}=2$ Coulomb branch geometries of dimension (or rank) greater than one is a famous open problem whose solution will greatly constrain the space of $\mathcal{N}{=}2$ superconformal field theories. At rank 2 the problem is equivalent to finding all possible genus 2 Seiberg-Witten cu...

We consider the Seiberg-Witten solution of pure $\mathcal{N} =2$ gauge theory in four dimensions, with gauge group $SU(N)$. A simple exact series expansion for the dependence of the $2 (N-1)$ Seiberg-Witten periods $a_I(u), a_{DI}(u)$ on the $N-1$ Coulomb-branch moduli $u_n$ is obtained around the $\mathbb{Z}_{2N}$-symmetric point of the Coulomb branch, where all $u_n$ vanish. This generalizes earlier results for $N=2$ in terms of hypergeometric functions, and for $N=3$ in te...

We analyze the geometric engineering of the N=2 SU(2) gauge theories with $1\leq N_f\leq 3$ 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 in...

The constraining mathematical structure of the Coulomb branch of four dimensional $\mathcal{N}=2$ supersymmetric theories is discussed. The presentation follows a somewhat different route from other excellent reviews on the subject and it is geared towards using this tool to classify four dimensional $\mathcal{N}=2$ superconformal field theories. This is the writeup of the lectures given at the Winter School "YRISW 2020" to appear in a special issue of JPhysA.

This is the third in a series of papers which outlines an approach to the classification of $\mathcal{N}{=}2$ superconformal field theories at rank 2 via the study of their Coulomb branch geometries. Here we use the fact that the encoding of a Coulomb branch geometry as a Seiberg-Witten curve and 1-form enjoys a large reparametrisation invariance. While there is always a unique way to fix this invariance such that the curve and 1-form are single-valued over the Coulomb branch...

We provide a uniform solution to 4d N=2 gauge theory with a single gauge group G=A,D,E when the one-loop contribution to the beta function from any irreducible component R of the hypermultiplets is less than or equal to half of that of the adjoint representation. The solution is given by a non-compact Calabi-Yau geometry, whose defining equation is built from explicitly known polynomials W_G and X_R, associated respectively to the gauge group G and each irreducible component ...

In this paper we show how hypercomplex function theoretical objects can be used to construct explicitly self-dual SU(2)-Yang-Mills instanton solutions on certain classes of conformally flat 4-manifolds. We use a hypercomplex argument principle to establish a natural link between the fundamental solutions of $D \Delta f = 0$ and the second Chern class of the SU(2) principal bundles over these manifolds. The considered base manifolds of the bundles are not simply-connected, in ...

We identify the spectral curve of pure gauge SU(2) Seiberg-Witten theory with the Weierstrass curve $\mathbbm{C}/L \ni z \mapsto (1,\wp(z),\wp(z)')$ and thereby obtain explicitely a modular form from which the moduli space parameter $u$ and lattice parameters $a$, $a_D$ can be derived in terms of modular respectively theta functions. We further discuss its relationship with the $c=-2$ triplet model conformal field theory.

The properties of the N=2 SUSY gauge theories underlying the Seiberg-Witten hypothesis are discussed. The main ingredients of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential are presented, the invariant sense of these definitions is illustrated. Recently found exact nonperturbative solutions to N=2 SUSY gauge theories are formulated using the methods of the theory of integrable systems and where poss...

We study the Coulomb phase of N=1 SU(2)^3 gauge theory coupled to one trifundamental field, and generalizations thereof. The moduli space of vacua is always one-dimensional with multiple unbroken U(1) fields. We find that the N=1 Seiberg-Witten curve which encodes the U(1) couplings is given by the double cover of a Riemann surface branched at the poles and the zeros of a meromorphic function.