We study asymptotics of the d = 4, 𝒩 = 1 superconformal index for toric quiver gauge theories. Using graph-theoretic and algebraic factorization techniques, we obtain a cycle expansion for the large−N index in terms of the R-charge-weighted adjacency matrix. Applying saddle-point techniques at the on-shell R-charges, we determine the asymptotic degeneracy in the univariate specialization for <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <msub> <mover accent="true"> <mi>A</mi> <mo stretchy="true">̂</mo> </mover> <mi>m</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$ {\hat{A}}_m $</EquationSource> </InlineEquation>, and along the main diagonal for the bivariate index for 𝒩 = 4 and <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <msub> <mover accent="true"> <mi>A</mi> <mo stretchy="true">̂</mo> </mover> <mn>3</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$ {\hat{A}}_3 $</EquationSource> </InlineEquation>. In these cases we find <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <mo>ln</mo> <mfenced close="|" open="|"> <msub> <mi>c</mi> <mi>n</mi> </msub> </mfenced> <mo>∼</mo> <mi>γ</mi> <msup> <mi>n</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> <mo>+</mo> <mi>β</mi> <mo>ln</mo> <mi>n</mi> <mo>+</mo> <mi>α</mi> </math></EquationSource> <EquationSource Format="TEX">$ \ln \left|{c}_n\right|\sim \gamma {n}^{\frac{1}{2}}+\beta \ln n+\alpha $</EquationSource> </InlineEquation> (Hardy-Ramanujan type). We also identify polynomial growth for dP3, Y3,3 and Yp,0, and give numerical evidence for γ in further Yp,p examples. Finally, we generalize Murthy’s giant graviton expansion via the Hubbard-Stratonovich transformation and Borodin-Okounkov formula to multi-matrix models relevant for quivers.

Quiver superconformal index and giant gravitons: asymptotics and expansions

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2026年7月8日

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Souradeep Purkayastha,
Zishen Qu,
Ali Zahabi

摘要

We study asymptotics of the d = 4, 𝒩 = 1 superconformal index for toric quiver gauge theories. Using graph-theoretic and algebraic factorization techniques, we obtain a cycle expansion for the large−N index in terms of the R-charge-weighted adjacency matrix. Applying saddle-point techniques at the on-shell R-charges, we determine the asymptotic degeneracy in the univariate specialization for ${\hat{A}}_{m}$ $ {\hat{A}}_m $ , and along the main diagonal for the bivariate index for 𝒩 = 4 and ${\hat{A}}_{3}$ $ {\hat{A}}_3 $ . In these cases we find $ln |c_{n}| \sim γ n^{\frac{1}{2}} + β ln n + α$ $ \ln \left|{c}_n\right|\sim \gamma {n}^{\frac{1}{2}}+\beta \ln n+\alpha $ (Hardy-Ramanujan type). We also identify polynomial growth for dP3, Y^3,3 and Y^p,0, and give numerical evidence for γ in further Y^p,p examples. Finally, we generalize Murthy’s giant graviton expansion via the Hubbard-Stratonovich transformation and Borodin-Okounkov formula to multi-matrix models relevant for quivers.