M2 M5 Giant Graviton Index Relation 2026

For rank $\tilde{N}\in {\mathbb{Z}}_{>0}$, fix the three-dimensional $U(\tilde{N}{)}_{+1}\times U(\tilde{N}{)}_{-1}$ ABJM theory: the subscripts are the two Chern–Simons levels, so $k=1$. Use its cylinder Hilbert space on $S^2$, including the source’s full $U(\tilde{N})$ factors. For dimensionless complex chemical potentials $\tilde{\omega },{\tilde{\mathrm{\Delta }}}_A\in \mathbb{C}$, choose a Poincaré supercharge $\tilde{\mathcal{Q}}$ of the three-dimensional ABJM theory, let $\tilde{\mathcal{S}}={\tilde{\mathcal{Q}}}^{†}$, and choose spin and $SO(8)$ Cartan charges $\tilde{J},{\tilde{Q}}_A$ so that $\{\tilde{\mathcal{Q}},\tilde{\mathcal{S}}\}=\tilde{H}-\tilde{J}-\frac{1}{2}{\sum }_{A=1}^4{\tilde{Q}}_A\mathrm{.}$ Its source-defined index is the Hilbert-space trace ${\mathcal{I}}_{M2}(\tilde{N};\tilde{\omega };{\tilde{\mathrm{\Delta }}}_A)={\mathrm{Tr}}_{{\mathcal{H}}_{M2,\tilde{N}}}\text{\hspace{0.17em}⁣}\left(e^{-\tilde{\beta }\{\tilde{\mathcal{Q}},\tilde{\mathcal{S}}\}}{e}^{-\tilde{\omega }\tilde{J}-{\sum }_{A=1}^4{\tilde{\mathrm{\Delta }}}_A{\tilde{Q}}_A}\right),$ with $\tilde{\omega }-{\sum }_A{\tilde{\mathrm{\Delta }}}_A=2\pi i$; under this constraint the trace is independent of $\tilde{\beta }$ and receives only the $\tilde{\mathcal{Q}}$-cohomology contribution. Here ${\mathcal{H}}_{M2,\tilde{N}}$ is the ABJM cylinder Hilbert space on $S^2$, and the source’s $(-1{)}^F$ sign is encoded in this fugacity convention rather than written as a separate trace factor. For the six-dimensional $U(N)$ $(2,0)$ index, take dimensionless ${\omega }_i,{\mathrm{\Delta }}_I\in \mathbb{C}$, choose $\mathcal{S}={\mathcal{Q}}^{†}$ with $\{\mathcal{Q},\mathcal{S}\}=H-J_1-J_2-J_3-2(Q_1+Q_2)$, and define ${\mathcal{I}}_{M5}(N;{\omega }_i;{\mathrm{\Delta }}_I)={\mathrm{Tr}}_{{\mathcal{H}}_{M5,N}}\text{\hspace{0.17em}⁣}\left(e^{-\beta \{\mathcal{Q},\mathcal{S}\}}{e}^{-{\sum }_{i=1}^3{\omega }_i{J}_i-{\sum }_{I=1}^2{\mathrm{\Delta }}_I{Q}_I}\right),$ where $N\in {\mathbb{Z}}_{>0}$, ${\mathcal{H}}_{M5,N}$ is the cylinder Hilbert space on $S^5$, $J_i$ are the three $SO(6)$ spin Cartan charges, $Q_I$ the two $SO(5)$ R-symmetry Cartan charges, and ${\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2-{\omega }_1-{\omega }_2-{\omega }_3=2\pi i$ makes this trace independent of $\beta$.

For the grand-canonical series, set the rank-zero coefficient ${\mathcal{I}}_{M2}(0)=1$ (the empty theory). The source grand-canonical generating function is ${\mathrm{\Xi }}_{M2}(\mu ;\tilde{\omega };{\tilde{\mathrm{\Delta }}}_A)={\sum }_{\tilde{N}\ge 0}{\mathrm{\Lambda }}^{\tilde{N}}{\mathcal{I}}_{M2}(\tilde{N};\tilde{\omega };{\tilde{\mathrm{\Delta }}}_A)$ with fugacity $\mathrm{\Lambda }=e^{\mu }$. Set $q=e^{-\tilde{\omega }}$ and $u_A=e^{-{\tilde{\mathrm{\Delta }}}_A}$. The $N$-independent BPS-graviton index in $Ad{S}_4\times S^7$ is ${\mathcal{I}}_{\mathrm{\infty }}(\tilde{\omega };{\tilde{\mathrm{\Delta }}}_A)=\mathrm{Pexp}\text{\hspace{0.17em}⁣}[\frac{{\prod }_{A=1}^4(1-q\mathrm{/}{u}_A)}{(1-q{)}^2{\prod }_{A=1}^4(1-u_A)}-\frac{1-q+q^2}{(1-q{)}^2}],\mathrm{Pexp}f:=\exp \text{\hspace{0.17em}⁣}\left({\sum }_{m=1}^{\mathrm{\infty }}\frac{f(q^m,u_1^m,\dots ,u_4^m)}{m}\right)\mathrm{.}$ Here ${\mathrm{Res}}_{\mathrm{\Lambda }={\mathrm{\Lambda }}_0}F$ denotes the complex residue of the meromorphic function $F(\mathrm{\Lambda })$, namely the coefficient of $(\mathrm{\Lambda }-{\mathrm{\Lambda }}_0{)}^{-1}$ in its Laurent expansion; it is a residue in $\mathrm{\Lambda }$, not in $\mu$. The proposed relation is

This formula uses $\tilde{\omega }={\mathrm{\Delta }}_1$ and $({\tilde{\mathrm{\Delta }}}_1,{\tilde{\mathrm{\Delta }}}_2,{\tilde{\mathrm{\Delta }}}_3,{\tilde{\mathrm{\Delta }}}_4)=({\omega }_1,{\omega }_2,{\omega }_3,-{\mathrm{\Delta }}_2)$, which maps the M2 chemical-potential constraint to the M5 constraint. The source’s giant-graviton expansion proposes a meromorphic continuation of ${\mathrm{\Xi }}_{M2}$ in $\mathrm{\Lambda }$ and identifies its poles with brane contributions; the meromorphy and residue extraction used in the displayed relation are part of this conjectural framework, not a theorem about the index series on its full chemical-potential domain. The source first uses the convergent-trace region $\mathrm{Re}({\tilde{\mathrm{\Delta }}}_A)\ge 0$, in particular $\mathrm{Re}({\tilde{\mathrm{\Delta }}}_4)=\mathrm{Re}(-{\mathrm{\Delta }}_2)>0$, and analytically continues to $\mathrm{Re}({\mathrm{\Delta }}_2)>0$ for the M5 trace.

Establish this relation beyond the source’s exact Higgs-branch limit and its first three terms in the combined large-$N$ and six-dimensional Cardy regime. It relates protected indices only; it does not assert equivalence of the complete M2 and M5 quantum field theories.

Chen et al., Abstract, Introduction, equations (1.4)–(1.5), (2.7), §1.1, §2, §3.2 and §4

The source-status review dated 2026-10-30 confirms that Chen et al. label the residue formula a conjecture. They verify it exactly in the stated Higgs-branch limit and show agreement through the first three orders in a combined large-$N$/Cardy comparison: the M2-side large-$N$ prediction is compared with the M5 Cardy expansion. Later Cardy orders, the full chemical-potential domain, and an operator or theory-level M2/M5 equivalence are not established by those checks. Review is limited to the cited version and sections and makes no exhaustive literature claim.