Open problems of Witten's four-dimensional topological gravity: gravitational Donaldson-type invariants, BRST parentage, and the emergence of dynamical gravity

In 1988 Witten formulated a four-dimensional cohomological topological gravity: conformal gravity coupled to an auxiliary multiplet and to a conformal-weight-two scalar, with a local fermionic symmetry whose square closes onto a diffeomorphism, a local Lorentz transformation, and a local conformal transformation. The paper proposed, but left open, the gravitational analogue of the Donaldson polynomials - correlation functions of a Q-closed composite vector field - and closed with five open issues: the underlying system whose BRST gauge fixing produces this theory, the existence of nontrivial four-manifold invariants, a version in which diffeomorphism invariance is spontaneously broken so that dynamical gravity re-emerges, a microscopic realization of the scalar whose expectation value breaks local conformal invariance, and a string-theoretic or other ultraviolet embedding. The core questions remain open: no nontrivial gravitational Donaldson-type invariant has been constructed from this theory, and neither the emergence mechanism nor the ultraviolet embedding is known - despite follow-up work ranging from the 1988 BRST-gauge-fixing derivations and the 1990-1993 attempts at observables to the 2004 assessment and the modern supergravity-twist and topological-early-universe revivals.

Background

In 1988, Witten introduced cohomological (topological) quantum field theory in two companion papers. In the first, he twisted $N=2$ supersymmetric Yang-Mills theory in four dimensions (a gauge theory with two independent supersymmetry charges, that is, two fermionic symmetries mixing bosons and fermions) - twisting means redefining the spins of the fields by mixing the spacetime rotation group with an internal symmetry group so that one supersymmetry charge becomes a scalar - so that a scalar fermionic operator $Q$ emerged, and he showed that correlation functions of $Q$-closed observables reproduce the Donaldson polynomials. BRST symmetry (the acronym embeds the names of its discoverers) is the fermionic symmetry of a gauge-fixed gauge theory whose square closes onto bosonic gauge transformations; a cohomological (Witten-type) topological field theory is a gauge-fixed theory whose action is $Q$-exact - the graded anticommutator of $Q$ with a gauge-fixing fermion $V$ - so that its partition function and the correlation functions of $Q$-closed, non-$Q$-exact observables are independent of the choice of $V$ and of the spacetime metric, and the path integral localizes on the field configurations selected by the bosonic equations hidden inside $V$. Donaldson invariants are built from the moduli space - the space of solutions modulo gauge transformations - of anti-self-dual Yang-Mills connections (instantons) over a smooth, closed, oriented four-manifold; they partially distinguish smooth (differential) structures on homeomorphic four-manifolds and transformed four-dimensional topology in the 1980s.

The source of the present problem is the second companion paper, 'Topological gravity' (Phys. Lett. B 206 (1988) 601), which asks for the gravitational analogue. It is explicitly four-dimensional, formulated on an oriented four-manifold in the $SO(4)≅ SU(2)_L× SU(2)_R$ spinor formalism, with two-component spinor indices $A,B$ (left-handed factor) and $A',B'$ (right-handed factor). The starting point is conformal gravity: the generally covariant theory invariant under local scale transformations, whose action is the square of the Weyl tensor - the trace-free part of the Riemann curvature that measures the failure of the metric to be locally conformally flat. To this, Witten couples a multiplet of auxiliary fields of unusual statistics and nonzero ghost number (ghost number is the conserved charge that counts the net number of ghost fields in a BRST-invariant theory; the statistics are unusual because the commuting, bosonic fields are precisely the ones that carry nonzero ghost number): the commuting ghost-of-ghost vector field $C^{AA'}$ of ghost number 2 and the commuting one-form $B_{AA'}$ of ghost number $-2$, the anticommuting (fermionic) self-dual tensor $χ_{ABCD}$ and vector $λ^{AA'}$, both of ghost number $-1$, the anticommuting tensor $ψ_{AB,A'B'}$ of ghost number $+1$, and - to make the fermion kinetic term conformally invariant - a scalar $Φ$ of conformal weight 2 (transforming with two powers of the local scale factor). The action is the sum of the Weyl-squared action $\mathcal{L}_1$, a term $\mathcal{L}_2$ chosen so that the anticommuting self-dual tensor $χ$ obeys its equation of motion (the fermionic algebra closes on $χ$ only on shell), and a $Q$-exact gauge-fixing term $\mathcal{L}_3=\{Q,Z\}$, where $Z$ is a functional of the tetrad, $B$, $ψ$, and $Φ$ invariant under diffeomorphisms, local Lorentz transformations, and conformal transformations; under the local fermionic symmetry $Q$ the fields shift by the auxiliary fields. The square of this symmetry is not zero but a bosonic local symmetry: $Q^2$ equals an infinitesimal diffeomorphism plus a local Lorentz transformation plus a local conformal (scale) transformation, exactly in the pattern of BRST symmetry, so observables are classified by the cohomology of $Q$ at nonzero ghost number.

The gauge-fixing terms are chosen so that the bosonic field equations force the conformal structure of the metric to be self-dual, that is, the self-dual part of the Weyl tensor to vanish, $W^+=0$ (self-duality is the decomposition of antisymmetric index pairs under $SO(4)$ into self-dual and anti-self-dual parts). The path integral thus localizes on the moduli space of conformally self-dual (half-flat) conformal structures - the gravitational analogue of the instanton moduli space. The paper notes that the composite vector field $c^a = e^a_{AA'} C^{AA'}$ (the tetrad contracted with the ghost-of-ghost field) is $Q$-closed and suggests that 'suitable correlation functions of c^a may serve as analogues of the Donaldson polynomials'. Consistency of the construction requires the scalar $Φ$ to acquire a nonzero vacuum expectation value, and this spontaneously breaks local conformal invariance. The abstract states the two motivations verbatim: 'It may have mathematical applications (gravitational counterpart of Donaldson theory) or physical ones (unbroken phase of general relativity).' The physical reading echoes induced gravity - the idea, reviewed by Adler, that the Einstein-Hilbert action arises as a symmetry-breaking effect in a more fundamental theory: the topological theory, whose fermionic symmetry is so large that all local propagating degrees of freedom cancel, would be an unbroken phase of gravity, from which ordinary gravity re-emerges when diffeomorphism invariance is spontaneously broken down to the global Poincare group.

For context, the two-dimensional sibling construction of Labastida, Pernici, and Witten became the fully solved laboratory for these ideas: Witten identified it with the topological phase of two-dimensional gravity and showed that its observables' correlation functions are intersection numbers on the moduli space of Riemann surfaces, a picture later proved computable by Kontsevich's exact solution via a matrix model (a solvable statistical system of random matrices, whose answer reduces to Airy functions). No comparable resolution exists in four dimensions, and the known obstructions are severe: the localization spaces - moduli spaces of conformally self-dual conformal structures on compact four-manifolds - are extremely rigid (the classification of compact conformally self-dual Einstein four-manifolds quoted in the later literature leaves essentially only the four-sphere, the complex projective plane, K3 surfaces (the simply connected compact complex surfaces with trivial canonical bundle, central examples in four-manifold topology), and flat manifolds), and when nonempty they can fail to be smooth manifolds and can be non-compact, so that only a virtual dimension, computed from the index of a deformation complex via the Atiyah-Singer index theorem (the standard theorem equating the signed count of solutions of an elliptic system of equations with a topological characteristic number), is available in general. It is against this background that the source paper's closing section leaves open the five linked issues that constitute the present problem.

Problem Statement

The problem is to complete the mathematical and physical interpretation of the four-dimensional cohomological topological gravity constructed in Witten's 'Topological gravity' (Phys. Lett. B 206 (1988) 601): the theory whose bosonic sector is conformal gravity (the Weyl-squared action), whose fields include the tetrad $e^{AA'}$, the ghost-of-ghost vector $C^{AA'}$ of ghost number 2, the auxiliary multiplet $B$, $χ$, $ψ$, $λ$, and the conformal-weight-2 scalar $Φ$, and whose local fermionic symmetry $Q$ squares to the sum of an infinitesimal diffeomorphism, a local Lorentz transformation, and a local conformal transformation. The source closes by posing five linked issues; a complete solution closes each of them, by explicit construction or derivation, or by a principled impossibility or triviality result.

(1) Invariants (the mathematical core). Determine whether correlation functions of $Q$-closed observables of this theory - the composite vector field $c^a = e^a_{AA'} C^{AA'}$ and its cohomological descendants, in the source's words 'suitable correlation functions of c^a' - define nontrivial invariants of the smooth structure of compact, oriented four-manifolds: the gravitational analogue of the Donaldson polynomials. Nontrivial means not reducible to the standard invariants (Euler characteristic, signature, and Betti numbers). A nontrivial invariant would have to be definable on the moduli space of the (conformally) self-dual configurations on which the path integral localizes, which would have to carry a well-defined integration cycle (compactness, smoothness, or a virtual-class substitute); or prove that no nontrivial invariant of this type exists for this class of models.

(2) Emergence (the physical core). Determine whether a version of the theory exists in which local diffeomorphism invariance is spontaneously broken down to the global Poincare symmetry, so that local physics re-emerges with propagating, unitary gravitons - the source's twin challenge of finding 'a version in which general covariance is spontaneously broken and dynamical gravity emerges' - including the mechanism and dynamics of the breaking; or show that this cannot occur within this framework.

(3) Microscopic realization of the conformal breaking sector. Construct a concrete elementary or composite realization of the scalar $Φ$ with the required conformal weight and nonzero vacuum expectation value, including its dynamics and stability; or prove that the role of $Φ$ is eliminable in a formulation-equivalent way, so that no new microscopic degree of freedom is needed for the spontaneous breaking of local conformal invariance.

(4) Parent system. Identify a system with higher symmetry whose BRST gauge fixing produces this theory, beyond the formal parent with vanishing Lagrangian exhibited in 1988; or establish that no such more fundamental system exists.

(5) Ultraviolet embedding. Establish whether this topological gravity arises as a low-energy limit of string theory or of another known quantum theory, or provide an equivalent ultraviolet completion.

Items (1)-(3) and (4)-(5) are the mathematical and physical halves of a single objective: deciding, with proof or construction, what this theory is as an invariant-producing and physically interpretable object. Answers that transfer the questions to a neighboring model (two-dimensional topological gravity, twisted-supergravity variants, Ricci-flow-type topological gravity, or the modern topological-early-universe proposals) constitute progress, but do not by themselves close the problem.

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