Loop and surface operators in N=2 gauge theory and Liouville modular geometry

Abstract

Recently, a duality between Liouville theory and four dimensional N=2 gauge theory has been uncovered by some of the authors. We consider the role of extended objects in gauge theory, surface operators and line operators, under this correspondence. We map such objects to specific operators in Liouville theory. We employ this connection to compute the expectation value of general supersymmetric 't Hooft-Wilson line operators in a variety of N=2 gauge theories.

Figures (10)

• Figure 1: Surface operators are supported on a surface ${\mathcal{S}}$ in ${\mathbb R}^4$ (shown on the left part of the figure) and are localized at a point $z$ in $C$ (on the right). Similarly, line operators extend along an (open or closed) curve ${\mathcal{C}}$ in ${\mathbb R}^4$ and wrap a 1-cycle $\gamma$ in $C$.
• Figure 3: The hemispherical stereographic projection of $S^4$ onto two copies of $\mathbb{R}^4$. It reflects the factorization of the instanton sum on $S^4$ into two "chiral" halves, given by the $\mathbb{R}^4$ contribution of instantons localized near the north and south pole. Surface operators on $S^4$ similarly factorize into a two "open" surface operators, a north and a south half, glued together at the equator.
• Figure 4: A Wilson-'t Hooft loop is labeled by a closed path $\gamma$ on $C$, and a surface operator is specified by a location $z$ on $C$. When the loop acts on a surface operator on $S^4$, it shifts the relative location of the upper- and lower-half via a monodromy operation associated with the closed path $\gamma$. The vertical direction indicates a 'time coordinate' $t$ on $S^4$, defined such that the equator gets mapped to $t=0$ and the north and south pole to $t=\pm \infty$.
• Figure 5: The brane construction of ${\cal N}=2$ super Yang-Mills theory with a half-BPS surface operator in type IIA string theory $(a)$ and its M-theory lift $(b)$.
• Figure 6: $(a)$ The brane construction of ${\cal N}=2$ super Yang-Mills theory with a half-BPS surface operator (shown on Figure \ref{['branefig']}$a$) where we introduced an extra NS5$'$-brane. Now the D2-brane can end on the NS5$'$-brane, thus, having a finite extent in the $x^7$ direction. $(b)$ The M-theory lift of the type IIA brane configuration on part $(a)$ of the figure.