Nonabelian M5 2 0 Theory

Let $N\ge 2$ coincident parallel M5-branes have a smooth oriented Lorentzian spin six-manifold ${\mathcal{W}}_6$ as worldvolume. Their interacting sector is the type-$A_{N-1}$ $(2,0)$ theory; the full $U(N)$ stack notation used for the index literature includes, in addition, a decoupled free center-of-mass tensor multiplet. The construction problem concerns the interacting sector and requires every full-stack observable or anomaly to state explicitly whether that free multiplet is included.

Construct that interacting six-dimensional quantum theory with specified local operators and Hilbert spaces or equivalent observables, and with a local anomaly polynomial $I_8(T{\mathcal{W}}_6,\mathcal{N})$ for the tangent bundle and rank-five $SO(5{)}_R$ bundle $\mathcal{N}$. Here an anomaly polynomial means a degree-eight invariant characteristic-form polynomial in the tangent- and R-symmetry-bundle curvatures whose descent gives the perturbative local gauge, R-symmetry, and gravitational anomaly of the six-dimensional partition function; a proposed construction must realize the known source-qualified anomaly constraint below in its stated normalization, rather than leaving those coefficients unconstrained. For circle reduction, take ${\mathcal{W}}_6={\mathcal{W}}_5\times S_r^1$ with periodic spin structure, circle radius $r>0$, and energies $E\ll r^{-1}$; the reduced theory must reproduce five-dimensional maximally supersymmetric Yang–Mills, its gauge group, the required Kaluza–Klein sectors, and the coupling-to-$r$ relation in the chosen normalization. A controlled bulk coupling must specify globally defined couplings to the eleven-dimensional metric and $C_3,C_6$ background potentials, their gauge transformations and charge data, and anomaly inflow sufficient to make the combined quantum system well-defined. A single abelian five-brane action or an index conjecture does not supply these construction data.

The source-reported local anomaly constraint is conditional: in the five-brane embedding/inflow setup of Harvey–Minasian–Moore, assume that the eleven-dimensional bulk inflow contributions cancel the total anomaly for a stack of positive integral five-brane charge $Q_5$. In this setup $\mathcal{N}$ is the oriented real rank-five normal bundle and hence the $SO(5{)}_R$ background bundle. Let $p_j(V)$ denote the degree-$4j$ Pontryagin forms of a real bundle $V$, normalized to represent its integral Pontryagin classes, and use the source's anomaly chirality/orientation convention. These are universal degree-eight characteristic polynomials before descent, not eight-forms to integrate over the six-dimensional worldvolume. The single free tensor-multiplet polynomial is $I_8^{(1)}=\frac{1}{48}[p_2(\mathcal{N})-p_2(T{\mathcal{W}}_6)+\frac{1}{4}(p_1(T{\mathcal{W}}_6)-p_1(\mathcal{N}){)}^2]\mathrm{.}$ Under the stated cancellation assumption the full charge-$Q_5$ stack must have $I_8^{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}(Q_5)=Q_5{I}_8^{(1)}+\frac{Q_5^3-Q_5}{24}{p}_2(\mathcal{N})\mathrm{.}$ For the $N$ coincident-brane stack in the construction problem, set $Q_5=N$. The full-stack polynomial includes the free center-of-mass tensor multiplet. With the decoupled center-of-mass splitting specified above, the interacting type-$A_{N-1}$ constraint is therefore $I_8^{A_{N-1}}=I_8^{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}(N)-I_8^{(1)}=(N-1)I_8^{(1)}+\frac{N^3-N}{24}{p}_2(\mathcal{N})\mathrm{.}$ The subtraction is the additive removal of one free tensor multiplet in the same background bundles. These source-qualified local anomaly constraints do not construct the interacting theory, determine its full operator algebra or Hilbert spaces, establish global-anomaly cancellation, or supply globally consistent couplings to arbitrary bulk backgrounds.

Single M5 worldvolume tensor action

Pasti, Sorokin, and Tonin, Abstract and §§1–4

Chen et al., Abstract and Introduction

Douglas, Abstract and Introduction

Harvey, Minasian, and Moore, §2.1, equations (2.5)–(2.6)

The source-status review dated 2026-10-30 confirms that Pasti, Sorokin, and Tonin construct a covariant bosonic action for one abelian M5-brane, with a chiral two-form and auxiliary scalar; it is not an interacting $N>1$ theory. Chen et al. study a conjectured relation between ABJM and the $U(N)$ $(2,0)$ superconformal indices and report checks in selected regimes. Douglas discusses circle compactification to five-dimensional maximally supersymmetric Yang–Mills while leaving the reverse reconstruction question open. Harvey, Minasian, and Moore explicitly assume anomaly cancellation for general $Q_5$ and use the known bulk inflow to predict the local polynomial above; this is a conditional consistency constraint, not a microscopic construction. The interacting polynomial removes the one free center-of-mass tensor already distinguished in this item. These sources do not construct the local interacting type-$A_{N-1}$ theory or its globally consistent bulk coupling. The existence of the source-qualified anomaly constraint does not close this construction problem. This is a targeted source review, not an exhaustive literature claim.