Exploring the twisted sector of $\mathbb{Z}_{L}$ orbifolds: Matching $α'$-corrections to localisation

TL;DR

The paper analyzes type IIB string theory on AdS5×S5/ℤL to understand how twisted-sector observables in the dual N=2 quiver gauge theories encode α'^3 corrections. It compares a geometric resolution approach, yielding a 6d effective action for twisted modes, with a direct string-theoretic analysis of twisted-sector amplitudes in the flat orbifold, showing that resonances of twisted states modify the low-energy corrections beyond universal ζ(3) and produce twist-dependent polygamma terms. The results explain why naive reductions from 10d α'^3 terms do not reproduce localisation results for generic L and demonstrate how twisted string amplitudes capture the correct structure, including a clear link to the localisation data. The work also discusses the long-quiver limit (L→∞) and potential emergent dimensions, highlighting the need for a fuller twisted-sector amplitude program to fully determine the 6d effective action in AdS5×S5/ℤL.

Abstract

We consider type IIB string theory on $\mathrm{AdS}_5\times S^5/\mathbb{Z}_{L}$ orbifold spaces with generic $L$. Recent localisation results in the dual 4d $\mathcal{N}=2$ circular quiver gauge theories provide us with strong coupling expansions of certain correlators involving twisted half-BPS operators. To leading order, these results have been matched to an effective theory for massless twisted string states, which can be constructed by resolving the orbifold singularity and considering localised supergravity modes on the resolution cycles. Applying this reasoning to subleading order in strong coupling, we observe that for $L\neq 2,3,4,6$, a naive reduction of the 10d $(α')^3$-correction does not result in the correct coefficients to match the localisation result. We explain this mismatch by the appearance of twisted sector resonances in string amplitudes involving external twisted sector states. We perform the low-energy expansion of a ``twisted'' Virasoro-Shapiro amplitude and recover the expected coefficients, suggesting that the orbifold resolution and the low-energy expansion can not be interchanged directly. Finally, we comment on the long-quiver limit, $L\to\infty$, in the context of the low-energy effective action.

Figures (4)

• Figure 1: Diagrammatic representation of the circular quiver theory. Each node represents an $\mathcal{N}=2$ vector multiplet in the adjoint representation of $\mathrm{SU}(N)$. Each line connecting adjacent nodes gives rise to an $\mathcal{N}=2$ hypermultiplet in the bifundamental representation of $\mathrm{SU}(N_{i})\otimes \mathrm{SU}(N_{i+1})$. In the case of $L=2$, an additional $\mathrm{SU}(2)$ symmetry arises among the two hypermultiplets connecting the only two gauge nodes.
• Figure 2: The resolution of the orbifold singularity provides an interpretation for the massless twisted states. We denote the characteristic scale of the resolution by $a$ and sending it to $0$ we recover the orbifold theory. Similarly, $\alpha'$ controls the mass of excited string states, which become infinitely heavy at $\alpha'\to0$, allowing us to integrate them out of the low-energy effective supergravity theory.
• Figure 3: Depiction of the resolution space \ref{['GH']} as an $S^{1}$ fibration over $\mathbb{R}^{3}$ for a generic configuration of instantons at $\vec{x}_{i},\vec{x}_{i\pm 1}\in\mathbb{R}^{3}$. The two-cycle $\Sigma_{i}$ is spanned by the fiber and a curve connecting $\vec{x}_{i},\,\vec{x}_{i+1}$.
• Figure 4: Numerical integration (left) showing $\sigma$ profile of \ref{['eq:largeLintegrand']} for $a=1$ and $\sigma'=0$. The analytic plot (right) is obtained after transforming the kernel \ref{['eq:continousKernel']} according to \ref{['eq:contcoordsrelation']} for the same values of $a$ and $\sigma'$. In this case the functions are simply $\tfrac{1+\sigma}{2}$ for $\sigma\in[-1,0]$ and $\tfrac{1-\sigma}{2}$ for $\sigma\in[0,1]$. Remarkably, these match exactly. We have checked this matching for various different values of $\sigma$ and $\sigma'$.