Topological vertex formalism with O5-plane

TL;DR

The paper extends the topological vertex formalism to Type IIB $(p,q)$ 5-brane webs with an O5-plane by introducing a rule that pairs intersecting branes with identical Young diagrams and a novel edge factor, justified via brane reflection. This framework is applied to compute the unrefined Nekrasov partition functions for $Sp(1)$ with $N_f=0,1,8$, reproducing known results and, in the $N_f=8$ case, yielding the E-string partition function on a circle through a periodic strip with Theta-type contributions. The approach is validated by detailed checks up to 10 instantons for $N_f=0,1$ and by matching the six-dimensional E-string elliptic genus for $N_f=8$, demonstrating the method's consistency and utility. The work suggests extensions to higher rank Sp$(N)$–SU$(N+1)$ dualities, S-dual ON-planes, and other gauge theories (e.g., SO$(M)$ with various matter), and points toward refining the vertex to the refined case.

Abstract

We propose new topological vertex formalism for Type IIB $(p,q)$ 5-brane web with an O5-plane. We apply our proposal to 5d $\mathcal{N}=1$ Sp(1) gauge theory with $N_f=0,1,8$ flavors to compute the topological string partition functions and check the agreement with the known results. Especially for the $N_f=8$ case, which corresponds to E-string theory on a circle, we obtain a new, yet simple, expression of the partition function with two Young diagram sum.

Figures (5)

• Figure 1: (a) 5-brane web diagram for five-dimensional $\mathcal{N}=1$ Sp($N$) gauge theory. (b) 5-brane web diagram for Sp($N$) with $\theta=0$ (for odd $N$) or with $\theta=\pi$ (for even $N$) after the general flop transition, where Sp($N$) theory description is not valid. Instead it describes five-dimensional $\mathcal{N}=1$ SU($N+1$) gauge theory with $\kappa=N+3$.
• Figure 2: (a) New rule for topological vertex formalism including an O5-plane. (b) Interpretation of the rule in terms of the reflected image by an O5-plane.
• Figure 3: (a) $N_f=0$ case where physical parameters are expressed in accordance with the SU(2) parametrization: $Q_F= A^2$ accounts the Coulomb branch modulus, while $Q_{B_2}Q_F=\mathfrak q$ accounts for the instanton factor, and they satisfy $Q_{B_1}= Q_{B_2}Q_{F}^4$. (b) A partial brane configuration connected to the reflected image by an O5-plane.
• Figure 4: (a) A $N_f=1$ configuration with an O5$^-$-plane. (b) A brane configuration connected by a reflected image by an O5-plane.
• Figure 5: A periodic $(p,q)$ brane configuration with two O5-planes for Sp(1) theory with $N_f=8$ flavors, where the periodicity is given as the instanton factor squared $\mathfrak{q}^2$. The Kähler parameters can be easily read off from the positions $x_I,y_i$, for instance, $Q_{B_1} =\mathfrak{q}^2 x_1^{-2}$ and $Q_{B_2} = x_2^2$. Each D5-brane in the middle associated with Young diagrams $\mu_1$ and $\mu_2$ is glued respectively.