The Virtue of Defects in 4D Gauge Theories and 2D CFTs

Abstract

We advance a correspondence between the topological defect operators in Liouville and Toda conformal field theories - which we construct - and loop operators and domain wall operators in four dimensional N=2 supersymmetric gauge theories on S^4. Our computation of the correlation functions in Liouville/Toda theory in the presence of topological defect operators, which are supported on curves on the Riemann surface, yields the exact answer for the partition function of four dimensional gauge theories in the presence of various walls and loop operators; results which we can quantitatively substantiate with an independent gauge theory analysis. As an interesting outcome of this work for two dimensional conformal field theories, we prove that topological defect operators and the Verlinde loop operators are different descriptions of the same operators.

Figures (13)

• Figure 1: Two curves on a genus-2 Riemann surface with one puncture. The curve $p$ separates the surface into two regions on which two distinct CFTs can be defined, joined by a topological defect. The curve $p'$ does not split the surface into two disconnected surfaces, but rather to a single surface with two boundaries. One can define a topological defect identifying two copies of the same CFT along $p'$.
• Figure 2: Locally a topological defect can always be represented as separating two CFTs on the cylinder. This is equivalent to a boundary condition for the tensor product of the two CFTs.
• Figure 3: Two examples of topological defect webs on a genus two surface with one puncture. Both have a pair of topological defect junctions. In $(a)$ they combine the defects with representations $\mu_1$, $\mu_2^*$, $\mu_3$ and $\mu_1^*$, $\mu_2$, $\mu_3^*$. In $(b)$ they combine $\mu_1$, $\mu_2$, $\mu_3^*$ and $\mu_1^*$, $\mu_2^*$, $\mu_3$.
• Figure 4: Two different pants decomposition of the surface $C_{2,1}$ of genus 2 with one puncture and the two associated trivalent graphs.
• Figure 5: The construction of the Verlinde loop operator. $(a)$ Two extra punctures carrying conjugate representations are inserted such that the channel marked by the red dashed line carries the identity representation. $(b)$ One of the operators transverses the surface along a specified path. $(c)$ When it returns to its original position we again project on to the identity representation in the channel separating the two punctures from the rest of the surface. $(d)$ If the punctures are removed we are left with the original surface with only the "memory" of the path and a representation, which label the Verlinde loop operator.