Surface operators from $M$-strings

It has been found that surface operators have a significant role in Alday-Gaiotto-Tachikawa (AGT) relation. This duality is an outstanding consequence of $M$-theory, but it is actually encoded into the brane web for which the topological string can work. From this viewpoint, the surface defect in AGT relation is geometrically engineered as a toric brane realization. Also, there is a class of the brane configuration in $M$-theory called $M$-strings which can be translated into the language of the topological string. In this work, we propose a new $M$-string configuration which can realize AGT relation in the presence of the surface defect by utilizing the geometric transition in the refined topological string.

Figure 1

The left side is the original $M$-strings where multiple M2s (horizontal) are suspended between two parallel M5s (vertical). The $M$-string is expressed by a torus on the M5. The right side is our new proposal for the $M$-strings in the presence of an additional M5 intersecting with the original M5.

Figure 2

${\mathcal{T}}_{2,1}$ and its degeneration limit to ${\mathcal{T}}_{1,1}$ with a degenerate operator (a lower cross). The web diagrams on the left and the right of the upper line correspond to the ${\widehat{\mathrm{A}}}_1$ quiver gauge theory and the $\mathcal{N}=2^{*}$ SU(2) gauge theory with a surface operator, respectively.

Figure 3

The degenerate limit of the Kähler parameters.

Figure 4

The geometric transition for the mergence of a toric brane corresponding to a surface operator.

Figure 5

The restriction on the web diagram to get the nontrivial partition function.

Figure 6

The $M$-strings. The blue sheets and the green sheets are the M5-branes and the M2-branes, respectively. Their boundaries, which are denoted by the dark green lines, are the $M$-strings. We set the background geometry to be the ${\mathrm{A}}_{N-1}$ ALE space.

Figure 7

The duality chain for $M=2$ and $N=2$. The $M$-strings set up is in the panel (a). This is dual to the $(p,q)$ five brane web in the panel (b) and D1-D5 brane on orbifold in (c). The red sheets are the D5 branes which correspond to the ${\mathrm{A}}_{N-1}$ ALE space.

Figure 8

The web diagram with a Lagrangian brane.

Figure 9

The $M$-theory brane system including the surface operator. The surface operator becomes a single M5-brane (M5’ in the table). Although the ${\mathrm{A}}_0$ ALE space is flat, we describe this space in order to connect easily to the type IIB theory.

Figure 10

The duality chain as the open topological string extension. The purple sheet in the panel (a) is an additional M5-brane (M5’) which is dual to the Lagrange submanifold denoted by dashed line in the panel (b), and the D5-brane (D5’) in the panel (c).

Figure 11

The assignment of the Kähler parameters on the web diagram for $M=N=2$ which engineers the fundamental surface operator in the degenerate limit.

Figure 12

The building block to compute the partition function. The preferred direction is taken to be horizontal (blue).

Figure 13

The Young diagram. We define ${\mu }_i$ as (a), ${\mu }_j^t$ as (b), and the coordinates $(i,j)$ as (c). ${\mu }_i$ is the number of boxes in the $i$th horizontal line. ${\mu }_j^t$ is the number of boxes in the $j$th vertical line.

Figure 14

The pictorial description of the refined topological vertex.