Towards a gauge theory interpretation of the real topological string

We consider the real topological string on certain noncompact toric Calabi-Yau three-folds $\mathbb{X}$, in its physical realization describing an orientifold of type IIA on $\mathbb{X}$ with an O4-plane and a single D4-brane stuck on top. The orientifold can be regarded as a new kind of surface operator on the gauge theory with 8 supercharges arising from the singular geometry. We use the M-theory lift of this system to compute the real Gopakumar-Vafa invariants (describing wrapped M2-brane Bogomol’nyi-Prasad-Sommerfield (BPS) states) for diverse geometries. We show that the real topological string amplitudes pick up certain signs across flop transitions, in a well-defined pattern consistent with continuity of the real BPS invariants. We further give some preliminary proposals of an intrinsically gauge theoretical description of the effect of the surface operator in the gauge theory partition function.

Figure 1

Notation for arm and leg lengths of the box $s=(2,1)\in R$.

Figure 2

Refined vertex conventions. We express a leg in the preferred direction by $\Vert$.

Figure 3

Web diagram for a toric singularity engineering pure $\mathrm{SU}(N)$ gauge theory with $K=N-2$.

Figure 4

Web diagram for the conifold.

Figure 5

SU(4) with two involutions.

Figure 6

Commutative diagram for equivariant map.

Figure 7

Web diagram for our orientifolds of the conifold and SU(2) theories.

Figure 8

$\mathrm{SU}(N)$ with $K=N-2$ and involution.

Figure 9

Three involutions for a generic internal leg; notice that each involution requires a specific symmetry to be present.