Setting the quantum integrand of M-theory on non-orientable (pin+) spacetimes and general evaluation of the global phases of the effective action

The 1996 source paper established the shifted quantization law $[G/2\pi]-\lambda/2\in H^4(Y,\mathbb Z)$ for the M-theory C-field and showed that the Chern-Simons couplings of eleven-dimensional supergravity are well-defined up to a $G$-independent, manifold-dependent sign that cancels the sign ambiguity of the Rarita-Schwinger Pfaffian, while explicitly deferring the $\eta$-invariant refinement of that sign and giving only a mod-2 sketch of unorientable spacetimes. Later literature resolved most of this program: Diaconescu-Moore-Witten (2000) and Freed-Moore (2006) gave a complete manifold-by-manifold phase prescription (a "setting of the quantum integrand") on orientable spin manifolds, and Freed-Hopkins (2021) proved that all anomalies cancel on non-orientable pin+ manifolds carrying the $w_1$-twisted integral lift of $w_4$. Two pieces of the original program survive as open. First, on pin+ "mc-manifolds" a canonical setting of the quantum integrand has not been constructed: settings form a torsor (a family of equivalent choices, none distinguished) over the group of 11-dimensional invertible mc topological field theories, which Freed and Hopkins conjecture (Conjecture 7.2) to be $\mathbb Z/2\cong\pi_{11}M_{mc}$, detected by the mod-2 index of the pin+ Dirac operator. Second, even in the orientable case there is no general method for evaluating the phase - the Dai-Freed $\eta$-invariants of the Rarita-Schwinger and $E_8$-coupled Dirac operators together with the cubic refinement of the C-field Chern-Simons functional - on arbitrary compact spacetimes, and the effect of summing over torsion C-field fluxes on the cubic refinement has, per the 2022 survey of Bah-Freed-Moore-Nekrasov-Razamat-Schafer-Nameki, not been carried out.

Background

M-theory is the conjectured quantum theory whose low-energy limit is eleven-dimensional supergravity. Besides the metric and the gravitino, its bosonic fields include a three-form gauge potential $C$ with four-form field strength $G=dC$ (the "C-field" or "G-flux"), the three-form analog of the electromagnetic potential; its charged objects are M2-branes (membranes coupling via $\int C$ over their three-dimensional worldvolume) and M5-branes. As with Dirac's quantization of electric charge, one expects the fluxes to be quantized. For spacetime $Y$ a spin manifold (a spin structure, the lift of the structure group of the tangent bundle from $SO(n)$ to its double cover $Spin(n)$, is required because the theory contains fermions), define $\lambda(Y)=p_1(Y)/2$, half the first Pontryagin class; on a spin manifold $p_1/2$ is canonically an integral cohomology class. The source paper's central discovery is that the correct law is not $[G/2\pi]\in H^4(Y,\mathbb Z)$ but the shifted law $[G/2\pi]-\lambda/2\in H^4(Y,\mathbb Z)$: flux quantization receives a gravitational correction. The shift was derived twice - from the relation of M-theory to the $E_8\times E_8$ heterotic string, in which $E_8$ gauge fields propagate on boundaries of spacetime and the $E_8$ characteristic class $w(V)$ (represented by $\mathrm{tr}F\wedge F/16\pi^2$) is an arbitrary integral class, and from worldvolume anomalies of macroscopic membranes, whose fermion path integral changes sign by $(-1)^{\int_D\lambda}$ as the membrane traverses a cycle $D$.

The effective action of eleven-dimensional supergravity contains Chern-Simons-type couplings, schematically $I_M=\int_Y C\wedge G\wedge G$ plus the one-loop gravitational correction $\int_Y C\wedge I_8(R)$. Because $G$ can have nontrivial periods, $C$ is not a global differential form, and such couplings are defined by extending $G$ over a twelve-dimensional spin manifold $Z$ with $\partial Z=Y$. Since the coupling is cubic in the C-field, assigning it a value as a function of the flux class amounts to choosing a cubic refinement: a function on the lattice of flux classes whose finite differences recover the triple cup-product pairing on $H^4(Y,\mathbb Z)$, just as a quadratic refinement of a bilinear form recovers the form from first differences. The naive classical normalization fails by a factor of six; the repair is index-theoretic. For a closed spin twelve-manifold $Q$ one has $I_Q/2\pi=i(E_8)/2+i(RS)/4$, where $i(E_8)$ is the index of the Dirac operator coupled to an $E_8$ bundle with characteristic class $a=[G/2\pi]+\lambda/2$, and $i(RS)$ is the index of the Rarita-Schwinger operator (the operator governing the spin-3/2 gravitino field, realized as a Dirac operator coupled to $TQ-4\mathcal O$, with the trivial-line subtraction accounting for gauge-fixing ghosts). Since $i(E_8)/2$ is integral but $i(RS)/4$ is in general only half-integral, $I_Q/2\pi$ is defined only modulo a $G$-independent half-integer. The source paper showed that precisely this residual cancels the sign ambiguity of the Rarita-Schwinger determinant: in $8k+3$ dimensions Dirac-type operators have real spectrum and the fermion path integral (a Pfaffian, the square root of the determinant) is real but has no natural sign, so the well-defined object is $e^{iI_M}\cdot\det D$. However, the overall manifold-dependent sign convention was left unfixed: the paper's footnote 6 states that a more precise definition using the $\eta$-invariant - the Atiyah-Patodi-Singer spectral-asymmetry invariant $\eta(0)$ of a self-adjoint operator, which enters index formulas on manifolds with boundary through the APS theorem - would be needed to compare M-theory on different manifolds, for instance when analyzing gravitational instantons, and is left to more precise treatments. No scheme for explicitly evaluating $I_M$ and the matching fermion phases on general compact examples was given.

Unorientable spacetimes were treated only in outline. Since M-theory conserves parity (equivalently, after Wick rotation, time-reversal symmetry), its spacetime may be non-orientable, in which case the fermions require a pin structure - the analog of a spin structure on an unoriented manifold, with pin+ the variant selected in eleven dimensions because time-reversal squares to $(-1)^F$ on fermions. Observing that in the orientable case $\lambda\equiv w_4$ modulo 2, where $w_4$ is the fourth Stiefel-Whitney class (a mod-2 characteristic class of the tangent bundle), the source paper recast the shifted law mod 2 as $\int_D G/2\pi=\frac12\int_D w_4$ and checked it against the half-integral flux sourced by an $\mathbb R^5/\mathbb Z_2$ orbifold singularity, leaving a full account of the non-orientable case open.

The modern framework for these questions is the problem, named by Freed and Moore, of "setting the quantum integrand": choosing the phase of the functional-integral integrand after integrating out fermions, which anomaly freedom alone does not determine. On orientable spin manifolds the $E_8$ model of the C-field (representing it via $E_8$ gauge bundles, refined into differential cohomology, the cohomology theory whose degree-4 classes model quantized abelian gauge fields with connection) sets the integrand by geometric index theory. On non-orientable manifolds the C-field is instead a $w_1$-twisted differential cohomology class lifting $w_4$ ($w_1$ is the first Stiefel-Whitney class, the obstruction to orientability; the twist by the orientation bundle reflects that $C$ and $G$ are odd under orientation reversal). A pin+ manifold equipped with such a lift is called an mc-manifold, and this is the arena on which Wick-rotated M-theory exists. Anomaly cancellation statements in this setting are statements about invertible twelve-dimensional topological field theories (topological field theories that assign a one-dimensional state space, hence only a phase, to each closed manifold), which are classified by bordism groups (bordism being the equivalence relation of two manifolds jointly bounding a higher-dimensional one) of the relevant structure; Freed and Hopkins proved cancellation by computing generators of the twelve-dimensional mc bordism group and evaluating the Rarita-Schwinger $\eta$-invariant and the C-field cubic form on each. Their proof shows the anomaly is trivializable but does not choose a trivialization - and the possible choices form a torsor (a family of options on which the group of invertible theories acts freely and transitively, so that all options are equivalent but none is distinguished) whose structure is only conjecturally known.

Problem Statement

The problem is the remaining core of the program deferred in "On flux quantization in M-theory and the effective action" (arXiv:hep-th/9609122), after the orientable spin-case phase prescription and the pin+ anomaly-cancellation theorem were completed by later work:

Give a complete global definition of the phase of the low-energy M-theory effective action - a setting of the quantum integrand - valid on arbitrary compact eleven-dimensional spacetime topologies, together with usable general evaluations. Concretely:

(1) The non-orientable case. On pin+ eleven-manifolds whose C-field is quantized by the $w_1$-twisted integral lift of $w_4$ (mc-manifolds), anomaly cancellation is proven, but the quantum integrand is not set. Construct, or prove impossible, a canonical reflection-positive setting (one compatible with unitarity of the Wick-rotated, time-reversal-symmetric theory) of the quantum integrand of M-theory on mc-manifolds. This requires determining the torsor of settings - the space of possible settings, on which the group of invertible eleven-dimensional mc topological field theories acts freely and transitively - i.e. computing that group (required in either branch): Freed-Hopkins Conjecture 7.2 asserts that the eleven-dimensional mc-bordism group $\pi_{11}M_{mc}$ (the eleventh homotopy group of the Thom spectrum whose homotopy groups are the mc-bordism groups) is $\mathbb Z/2$, with its generator detected by the mod-2 index of the pin+ Dirac operator (the number of zero modes of that operator, counted modulo two); and, when a setting is constructed, it requires identifying the physical content, if any, of the residual $\mathbb Z/2$ choice. The answer should fix the manifold-dependent sign conventions globally across spacetime topologies, including the comparison between different manifolds (the gravitational-instanton-type comparisons that motivated the source paper's deferral of the eta-invariant refinement).

(2) General evaluation of the phases. Develop general methods to evaluate the global phase of the M-theory effective action on arbitrary compact examples: the Dai-Freed $\eta$-invariant phases of the Rarita-Schwinger operator and of the $E_8$-coupled Dirac operator (the Pfaffian phases these acquire from $\eta$-invariants on a bounding twelve-manifold, per the Dai-Freed construction), together with the cubic refinement of the C-field Chern-Simons functional (the function of the flux class whose finite differences recover the triple cup-product pairing), including a full account of the effect of summing over torsion C-field fluxes on the cubic refinement, which is currently uncomputed.

The common objective is that the M-theory partition function be a well-defined complex number - not merely an anomaly-free object - on every compact spacetime topology on which the theory exists, orientable or not. Target (1) completes, in the non-orientable regime, the manifold-by-manifold eta-invariant sign prescription the source paper deferred; target (2) is the general form of the source paper's case-by-case congruence computations. The two targets are independent: the general methods of (2) do not by themselves resolve the setting problem of (1), nor does a solution to (1) resolve (2).

References

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