Codimension-Two Defects and SYM on Orbifolds

Abstract

We study $U(N)$ SYM theories on spaces with orbifold singularities via an equivalent description in terms of gauge theories on smooth manifolds with insertions of Gukov-Witten and twist defects. The combined effect of the defects is to render the fields multivalued with respect to rotations around the support of the defects. This motivates a relation with theories on branched covers, for which the multivaluedness has a geometric interpretation. We compute the partition function of the theory with defects on a patch and use it as a building block to compute partition functions on several closed spaces with conical singularities.

Figures (4)

• Figure 1: Left-hand side: $p$ copies of the theory on $\mathbb{C}$, with a codimension one defect $\mathcal{D}$ implementing the branching structure. Right-hand side: theory for the collection $\{\phi_k\}_k$ of all fields on a single sheet $\mathbb{C}$, with a twist defect $\mathcal{T}$ implementing multivaluedness.
• Figure 2: Sewing of Young diagrams $Y^i_{k,l}$ into a single diagram $Y^i$ for $p_1=3$ and $p_2=2$ (figure adapted from Nishioka:2016guu).
• Figure 3: The four-sphere with two defects, $(\xi,\mathcal{T}_1)$ and $(\zeta,\mathcal{T}_2)$, supported on two-spheres that intersect at the poles (figure adapted from Gomis:2016ljm).
• Figure 4: Left-hand side: singularity structure of $S^5_{\boldsymbol{ p}}$ and $\widehat{S}^5_{\boldsymbol{p}}$. The $p_i$ represent either orbifold singularities or branching indices. Right-hand side: the refined orbifold theory on $S^5$ with the insertion of Gukov-Witten defects $\xi,\mathcal{T}_1$ on $D_1$, $\zeta,\mathcal{T}_2$ on $D_2$ and $\chi,\mathcal{T}_3$ on $D_3$. The intersection of each defect is along an $S^1$, for example $D_1\cap D_3=S^1_{(1)}$.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3