An M2/M5 Duality from the Giant Graviton Expansion

Technical Summary: An M2/M5 Duality from the Giant Graviton Expansion

Problem Statement
The paper addresses the relationship between two distinct superconformal field theories (SCFTs) arising in M-theory: the three-dimensional $\mathcal{N}=8$ ABJM theory describing the worldvolume of $N$ parallel M2-branes, and the six-dimensional $\mathcal{N}=(2,0)$ theory describing the worldvolume of $N$ parallel M5-branes. While the AdS/CFT correspondence relates these theories to gravity in $AdS_4 \times S^7$ and $AdS_7 \times S^4$ respectively, a direct, quantitative duality relating their exact field-theoretic observables remains elusive. Specifically, the authors investigate the superconformal indices ($I_{M2}$ and $I_{M5}$), which count $1/16$-BPS states. The central problem is to establish a precise mathematical relation between the index of the 3D theory and that of the 6D theory, leveraging the "giant graviton expansion" of the M2-brane index.

Methodology
The authors employ a combination of exact field-theoretic techniques, large $N$ asymptotic analysis, and holographic reasoning:

1. Giant Graviton Expansion: The starting point is the conjecture by Imamura that the M2-brane superconformal index $I_{M2}$ can be expanded as a sum over configurations of M5-branes wrapping 5-cycles in $S^7$. This expansion interprets $I_{M2}$ as counting BPS gravitons dressed by M5-brane excitations.
2. Grand Canonical Ensemble: The authors define the grand canonical superconformal index $\Xi_{M2}(\mu)$ as the generating function of $I_{M2}$ over the rank $\tilde{N}$. They demonstrate that $\Xi_{M2}$ is a meromorphic function of the fugacity $\Lambda = e^\mu$, where the poles correspond to specific M5-brane configurations.
3. Residue Extraction: By analyzing the pole structure of $\Xi_{M2}$, the authors isolate the contribution corresponding to a single stack of $N$ M5-branes wrapping a single $S^5$. They propose that the residue of $\Xi_{M2}$ at this specific pole is directly proportional to the M5-brane superconformal index $I_{M5}$.
4. Asymptotic Verification: To test this conjecture, the authors perform checks in three distinct regimes:
   • The Higgs Branch Limit: A simplifying limit where chemical potentials are taken to infinity, reducing the indices to Hilbert series.
   • Leading Large $N$ Behavior: Using the known $N^{3/2}$ scaling of the M2-brane index to derive the expected $N^3$ scaling of the M5-brane index.
   • Cardy Expansion: A high-temperature (small chemical potential) expansion of the 6D index, comparing subleading $1/N$ corrections derived from the M2-brane side against known results for the 6D theory.

Key Contributions and Results

• The Main Conjecture (Equation 1.4): The paper proposes a precise duality relation:
  $$I_{M5}(N; \omega_i; \Delta_I) = - \frac{e^{\Delta_2 N}}{I_\infty(\Delta_1; \omega_1, \omega_2, \omega_3, -\Delta_2)} \text{Res}_{\Lambda=e^{-\Delta_2 N}} \Xi_{M2}(\mu; \Delta_1; \omega_1, \omega_2, \omega_3, -\Delta_2)$$
  Here, $I_\infty$ represents the index of BPS graviton excitations in $AdS_4 \times S^7$, and the chemical potentials are identified via a specific mapping between the 3D and 6D symmetries. This relation effectively extracts the 6D index from the grand canonical 3D index via a residue operation.

• Verification in the Higgs Branch Limit: The authors explicitly verify the conjecture in the limit where $\omega_1, \omega_2, \Delta_1 \to \infty$. In this regime, $I_{M2}$ becomes the Hilbert series of the Higgs branch (symmetric product of $\mathbb{C}^2$). The residue calculation yields a product formula that matches the known "twisted limit" of the 6D $(2,0)$ index, providing an exact, non-perturbative check.

• Large $N$ and Cardy Regime Matching:

  • Using the leading large $\tilde{N}$ asymptotic of $I_{M2}$, the authors recover the expected $N^3$ leading behavior of $\log I_{M5}$.
  • Crucially, they utilize a conjectured closed-form expression for $I_{M2}$ valid to all perturbative orders in $1/\tilde{N}$ (involving the Airy function). This allows them to compute subleading corrections.
  • They derive an asymptotic expansion for $\log I_{M5}$ containing terms of order $N^3$ and $N$:
    $$\log I_{M5} \sim \alpha(\omega_i; \Delta_I)(N^3 - 1) + \beta(\omega_i; \Delta_I)(N - 1) + O(\log N)$$
  • The coefficients $\alpha$ and $\beta$ are explicitly calculated. The authors show that these match the first three orders of the independent Cardy expansion of the 6D index derived in previous literature (specifically Ref. [8]).

• Connection to Thermal Anomaly Polynomial: The paper identifies the coefficients $\alpha$ and $\beta$ with the equivariant integral of the "thermal anomaly polynomial" $P^T_8$. The conjecture implies that the $N^3$ and $N$ terms in the Cardy expansion of the 6D index are exactly fixed by this integral, extending previous results which only established this match up to order $\epsilon^{-1}$ in the Cardy parameter. The authors argue that the validity of their duality implies that the fourth and fifth orders in the Cardy expansion are also fixed by this polynomial.

Significance and Claims
The paper claims to establish a "precise relation" between the superconformal indices of the 3D ABJM and 6D $(2,0)$ theories, rooted in the giant graviton expansion of M-theory. The significance lies in:

1. Predictive Power: Assuming the conjecture holds, the well-understood perturbative expansion of the 3D index (via the Airy function) provides new, exact predictions for the subleading $N$ terms of the 6D index, which are difficult to compute directly in six dimensions.
2. Holographic Interpretation: The results suggest a deep connection between the grand canonical ensemble of the M2-brane theory and the canonical ensemble of the M5-brane theory. The residue operation acts as a bridge, extracting the M5-brane spectrum from the M2-brane grand canonical partition function.
3. Anomaly Polynomial Link: The work strengthens the conjecture that the Cardy limit of the 6D index is governed by the thermal anomaly polynomial, suggesting that the match holds to all orders in the Cardy expansion for the $N^3$ and $N$ terms, with corrections potentially being exponentially suppressed.

The authors remain modest regarding the scope, noting that their verification relies on existing literature results for the indices and specific asymptotic limits. They do not claim to have proven the duality non-perturbatively for all parameters but rather provide strong evidence through exact matches in specific limits and consistency with known asymptotic behaviors. They also note that the extension to higher Chern-Simons levels ($k>1$) and the interpretation of the $S^3_b$ partition function are left for future work.