Confinement in 3d $$ \mathcal{N} $$ = 2 Spin(N) gauge theories with vector and spinor matters

Abstract

We present various confinement phases in three-dimensional N = 2 Spin(N) gauge theories with vector and spinor matters. The quantum Coulomb branch in the moduli space of vacua is drastically modified when the rank of the gauge group and the matter contents are changed. In many examples, the Coulomb branch is one-or twodimensional but its interpretation varies. In some examples, the Coulomb branch becomes three-dimensional and we need to introduce a "dressed" Coulomb branch operator.

3d $Spin(N)$ Seiberg dualities

“…However, almost all the (classical) flat directions are quantum-mechanically unstable and lifted because a monopoleinstanton could generate a runaway potential for the Coulomb branch [16,17,36]. In so(N) gauge theories, only a few directions are non-perturbatively allowed and become exactly flat [26,27,[37][38][39]. In the present case, there is a single flat direction: The electric Coulomb branch operator, which is denoted by Y SO (5) , corresponds to the spontaneous gauge symmetry breaking [40][41][42] so (7) → so(5) × u (1) (2.1)…”

“…Next, we study the Coulomb moduli space of the Spin(8) gauge theory. For the description of the s-confinement (the F = 5 case), the Coulomb branch was studied in [38] (see also [26,27]). When the Coulomb branch operator Y SO(6) obtains an expectation value, the gauge group is spontaneously broken as…”

On s-confinement in 3d $\mathcal{N}=2$ gauge theories with anti-symmetric tensors

“…This theory is equivalent to the 3d N = 2 Spin(6) gauge theory with three vectors and two spinors. The Coulomb branch of the Spin(N) theory was studied in [7,12,13,16]. The corresponding 4d theory was studied in [17,18] and we can derive the 4d result from a 3d perspective.…”

“…This is equivalent to the 3d N = 2 Spin(5) gauge theory with two vectors and two spinors. For the Coulomb branch of the Spin(N) theory, see [7,12,13,16].…”

“…The s-confinement phase in 4d N = 1 supersymmetric gauge theories was found in [1] and then classified in [3]. The 3d s-confinement phase was studied, for example, in [4][5][6][7][8][9][10][11][12][13].…”