Mirror symmetry and exact solution of 4D $N = 2$ gauge theories: I

Abstract

Using geometric engineering in the context of type II strings, we obtain exact solutions for the moduli space of the Coulomb branch of all N = 2 gauge theories in four dimensions involving products of SU gauge groups with arbitrary number of bi-fundamental matter for chosen pairs, as well as an arbitrary number of fundamental matter for each factor.Asymptotic freedom restricts the possibilities to SU groups with bi-fundamental matter chosen according to ADE or affine ADE Dynkin diagrams. Many of the results can be derived in an elementary way using the self-mirror property of K3. We find that in certain cases the solution of the Coulomb branch for N = 2 gauge theories is given in terms of a three dimensional complex manifold rather than a Riemann surface. We also study new stringy strong coupling fixed points arising from the compactification of higher dimensional theories with tensionless strings and consider applications to three dimensional N = 4 theories.

New results on superconformal quivers

“…Now we have to reconstruct the three block quivers, starting from the four triples of Table 2. This is a bit long but straightforward, so we discuss just the more involved example: (X, Y, Z) = (8,8,16). This is precisely the case where we find quivers not directly related to del Pezzo surfaces.…”

“…In the case of N = 2 supersymmetry it is already known [16,17] that there is a oneto-one correspondence between superconformal quivers and geometries preserving N = 2 supersymmetry. The latter are classified by orbifolds of C 2 ; the well known Mc-Kay correspondence then implies that the quiver diagrams are exactly the Dynkin diagrams of A,D,Ê algebras.…”

“…However we would like to describe these minimal models more explicitly. In order to have a better understanding we define the "relative number of flavors" n F of a node in a chiral quiver to be 16) i.e. the ratio between the total amount of matter charged under the gauge group and the rank of the group.…”

Gravitational corrections in supersymmetric gauge theory and matrix models

“…Using the explicit expressions ∆ = u 2 − Λ 4 , and the explicit relation between u and a worked out in [24], u(a) = 2a 2 …”

“…One of the most successful approaches is geometric engineering, where the gauge theory is appropriately embedded in a local Calabi-Yau compactification of type II string theory. A large and interesting class of supersymmetric field theories can be geometrically engineered in this way, including N = 2 supersymmetric Yang-Mills theory with or without matter [1] [2]. The gauge theory is recovered in a certain singular limit of the Calabi-Yau geometry where the string corrections decouple, and the gauge group and matter content of the field theory depend only on local properties of the Calabi-Yau space near the singularity.…”