Fiber-Base Duality and Global Symmetry Enhancement

TL;DR

The paper demonstrates that 5D $ ext{N}=1$ Nekrasov partition functions encode enhanced UV global symmetries through fiber-base duality, which acts as a symmetry when combined with manifest flavor symmetries. By introducing invariant Coulomb moduli $ ilde{A}$, the holomorphic part of the Nekrasov partition function expands into characters of the corresponding $E_{N_f+1}$ (for SU(2)) or $SU(N)^2 imes SU(M)^2$ (for higher-rank linear quivers) symmetry groups, confirming the UV enhancements predicted by Seiberg and realized in brane engineering. The analysis blends 7-brane monodromies, topological string techniques (refined vertices and flop transitions), and duality maps between dual frames to provide a unified framework for symmetry enhancement across SU(2) and SU(N) quivers, including explicit expansions to several orders in the invariant modulus. The work highlights the role of the effective coupling and perturbative vs. topological-string contributions in maintaining invariance under the enhanced symmetry and lays groundwork for extending these ideas to broader toric geometries and dualities.

Abstract

We show that the 5D Nekrasov partition functions enjoy the enhanced global symmetry of the UV fixed point. The fiber-base duality is responsible for the global symmetry enhancement. For $SU(2)$ with $N_f\leq 7$ flavors the fiber-base symmetry together with the manifest flavor $SO(2N_f)$ symmetry generate the $E_{N_f+1}$ global symmetry, while in the higher rank case the manifest global symmetry of the two dual theories related by the fiber-base duality map generate the symmetry enhancement. The symmetry enhancement at the level of the partition function is manifest once we chose an appropriate reparametrization for the Coulomb moduli.

Figures (21)

• Figure 1: The 5-brane web configuration that gives $\text{SU}(2)$ gauge theory with 4 flavors (a). We can translate it into the 5-brane probe of the 7-brane configuration ${\bf X}_{(1,0)}^4{\bf X}_{(1,-1)}{\bf X}_{(1,1)}{\bf X}_{(1,-1)}{\bf X}_{(1,1)}$ as (b). Then (c) is a maximally collapsed configuration that describes the UV fixed point.
• Figure 2: The web diagram for $\text{SU}(3)$$N_f=6$ SQCD. All external legs are regularized by 7-branes.
• Figure 3: Moving 7-branes inside and reordering them yields the middle configuration. The maximally enhanced symmetry at the UV fixed point is realized by colliding the $(0,1)$ 5-branes, ${\textrm{\bf{X}}}_{(1,0)}^6$ 7-branes, and the two 5-branes attached to the green 7-branes $\textrm{\bf{X}}_{(3,-1)}^2$.
• Figure 4: The web diagram for $\text{SU}(N)$$N_f=2N$ SQCD (Left). By moving $(0,1)$ and lower $(0,1)$ 7-branes inside the 5-brane loop and reordering them yields the diagram in the right hand side. The blue circles are $(N,-1)$ 7-branes.
• Figure 5: The web diagram for $\text{SU}(N)^{M-1}$ linear quiver (Left). By moving the $(1,0)$ and the $(0,1)$ 7-branes inside the 5-brane multi-loop, we obtain the right hand side. The shape of 5-brane multi-loop is actually warped nontrivially because of the background metric coming from 7-branes.