Mirrors and Phases of N=4 in D=3

Abstract

3d mirror symmetry for was first proposed in [1] as a duality between certain pairs of generally different 3d N=4 theories at the infrared limit. Infinite sequences of new mirror pairs and strong field theoretic evidence for them were found in [2]. It is a nonperturbative duality that in particular equates certain quantities receiving large quantum corrections with some that are determined entirely classically. Naturally one asks whether this can be the consequence of some string dualities. There are several different approaches [1, 2, 3, 4, 5, 6, 7], each of which has it own advantage and is related to the others by some sequences of dualities. In this talk I will review one that is particularly intuitive [4, 5]. The global R-symmetry of N=4 theories appears as geometric rotations; R-symmetry breaking part of the moduli space of vacua and the parameters of the field theory are realized as the moduli space of D-branes configurations; mirror symmetry itself is implemented by the S duality of type IIB string theory [4]. This construction allows us to engineer a large class of theories and find their mirror duals [5]. Since then it has been generalized to 3d N=2 theories [8, 9]. Similar ideas of constructing field theories in 4d, reviewed in Eguchi’s lecture at this school [10], have also been very fruitful¹. In this talk, I will focus on 3d N=4 theories. After reviewing the rules for “model building” via “brane engineering,” I will show how the mirror pairs emerge from this prescription. As an unexpected reward we can predict an infinite number of 3d field theories without conventional Lagrangian descriptions. Some of them are dual to ordinary Lagrangian theories via mirror symmetry, but the rest are not; yet they can be smoothly connected in the moduli space of brane configurations.