[Paper Review] Finite-$N$ superconformal index via the AdS/CFT correspondence

Yosuke Imamura|arXiv (Cornell University)|Aug 27, 2021

Black Holes and Theoretical Physics17 references4 citations

TL;DR

This paper proposes a holographic computation of the finite-$N$ superconformal index in ${\cal N}=4$ $U(N)$ SYM using D3-branes wrapped on three-cycles in $S^5$, with contributions from multiple wrappings systematically included. It confirms that finite-$N$ corrections are correctly reproduced by summing over wrapped brane indices, validating the method numerically up to $n=3$ and providing a framework for higher-order terms.

ABSTRACT

We propose a prescription to calculate the superconformal index of the ${\cal N}=4$ $U(N)$ supersymmetric Yang-Mills theory with finite $N$ on the AdS side. The finite $N$ corrections are included as contributions of D3-branes wrapped around three-cycles in $\boldsymbol{S}^5$, which are calculated as the index of the gauge theories realized on the wrapped branes. The single-wrapping contribution has been studied in a previous work, and we further confirm that the inclusion of multiple-wrapping contributions correctly reproduces the higher order terms as far as we have checked numerically.

Motivation & Objective

• To extend the holographic computation of the superconformal index beyond the large-$N$ limit by including finite-$N$ corrections.
• To systematically compute contributions from multiple-wrapping D3-branes on three-cycles in $S^5$.
• To validate the method by numerically confirming that the sum over wrapped brane indices reproduces known higher-order terms in the index expansion.
• To establish a consistent prescription for gauge fugacity integrals and pole selection in the presence of multiple wrappings.

Proposed method

• The superconformal index is decomposed as $\mathcal{I}_{U(N)} = \mathcal{I}_{\rm KK} \sum_{n_1,n_2,n_3=0}^\infty \mathcal{I}_{(n_1,n_2,n_3)}$, where $\mathcal{I}_{\rm KK}$ is the supergravity contribution and $\mathcal{I}_{(n_1,n_2,n_3)}$ is the index of a $U(n_1)\times U(n_2)\times U(n_3)$ quiver gauge theory on wrapped D3-branes.
• Each wrapping number $n_I$ corresponds to D3-branes wrapped on $S_I \subset S^5$ defined by $z_I = 0$, with the worldvolume theory being a quiver with bi-fundamental hypermultiplets.
• The gauge fugacity integrals are evaluated using a contour prescription that selects only positive poles, ensuring cancellation of mixed poles and consistency with the pole selection rule.
• The single-particle index is expanded in $q$, and a cut-off order is determined to ensure numerical accuracy up to desired $q$-orders, accounting for tachyonic terms in the expansion.
• The method uses matrix integral techniques from localization, adapted to finite $N$, with careful treatment of integration contours to avoid spurious contributions.
• The contribution from each $n = n_1+n_2+n_3$ is of order $q^{nN + \delta_n}$ with $\delta_n \geq 0$, and the full sum reproduces the finite-$N$ index.

Experimental results

Research questions

• RQ1 How can finite-$N$ corrections to the superconformal index in ${\cal N}=4$ $U(N)$ SYM be systematically computed using the AdS/CFT correspondence?
• RQ2 What is the role of multiple-wrapping D3-branes on $S^5$ in reproducing higher-order terms in the $q$-expansion of the index?
• RQ3 How can gauge fugacity integrals be consistently evaluated in the presence of multiple wrappings and charged fields?
• RQ4 Can the pole selection rule be rigorously applied to cancel mixed poles and ensure only physical contributions remain?
• RQ5 What cut-off order is required in the single-particle index to achieve accurate numerical results up to a given $q$-order?

Key findings

• The inclusion of multiple-wrapping contributions correctly reproduces the higher-order terms in the superconformal index up to $n=3$, confirming the validity of the method.
• The pole selection rule ensures that only positive poles contribute, with mixed poles canceling out due to symmetric contributions from dual pole structures.
• The $n=1$ sector, corresponding to a $U(1)$ gauge theory with neutral fields, reproduces the correct index up to $\mathcal{O}(q^{2N+4})$ error, consistent with expectations.
• The method successfully computes the index for $n \geq 2$ by evaluating matrix integrals with proper contour choices, avoiding spurious contributions.
• The cut-off order for the single-particle index is determined as $m_{\rm max} = c + (n-1)t$, where $t$ is the order of the leading tachyonic term, ensuring accuracy up to $q^{m_0 + c}$.
• The framework reproduces the analytic result of [12] for the Schur index, confirming consistency with known results in the literature.