Extending the quantum-foam gauge-theory construction of topological strings to compact Calabi-Yau threefolds and the full Donaldson-Thomas dictionary

The topological A-model on a non-compact toric Calabi-Yau threefold is solved at all genera by the topological vertex and, equivalently, by a melting-crystal sum. Iqbal, Nekrasov, Okounkov and Vafa ('Quantum foam and topological strings', JHEP 2008) interpreted this sum as the partition function of a maximally supersymmetric, topologically twisted $U(1)$ gauge theory on a noncommutative threefold, fluctuating over discrete Kaehler geometries — a concrete model of spacetime 'quantum foam' — with gluing data (framing factors and sign factors) arising from fluctuation-determinant calculations. The paper left open: the extension of the construction to compact Calabi-Yau threefolds such as the quintic, where no effective technique existed for integrating the Donaldson-Thomas class over the moduli space of ideal sheaves; the explicit identification of the gauge-theory class with the Donaldson-Thomas class; and the six-dimensional analogue of the full Seiberg-Witten geometry, including monopole equations. Subsequent developments: the toric gauge-theory/Donaldson-Thomas dictionary is now a theorem, and the compact quintic numbers have been obtained via mirror symmetry with the Gromov-Witten/Pairs correspondence proven for the quintic, but the extension of the vertex/gauge-theory construction itself to compact threefolds, the direct effective integration of the Donaldson-Thomas class there, and the six-dimensional Seiberg-Witten geometry remain open.

Background

The topological A-model is a two-dimensional topological string theory whose target space is a Calabi-Yau threefold, that is, a compact or non-compact complex Kaehler threefold with trivial canonical bundle. Its genus-$g$ amplitudes (labeled by the number $g$ of handles of the string worldsheet) are generating functions of the Gromov-Witten invariants, which count holomorphic curves in the target. For a non-compact toric Calabi-Yau threefold (a threefold whose complex geometry is encoded by a polyhedral 'toric diagram' and which carries a holomorphic $(\mathbb{C}^*)^3$-symmetry with fixed loci), Aganagic, Klemm, Mariño and Vafa constructed the topological vertex: a universal building block $C_{\lambda\mu\nu}$ labeled by a triple of Young diagrams (the same combinatorial objects that index irreducible representations of the symmetric group) such that gluing copies of the vertex along the edges of the toric diagram reproduces the complete all-genus A-model partition function. Okounkov, Reshetikhin and Vafa re-derived the same amplitudes as a statistical-mechanical sum over melting crystals, that is, over three-dimensional Young diagrams (plane partitions) with fixed asymptotics.

Iqbal, Nekrasov, Okounkov and Vafa gave this duality a target-space interpretation: the crystal sum is the partition function of a maximally supersymmetric, topologically twisted $U(1)$ gauge theory (twisting is a modification of the supersymmetry algebra under which a metric-independent scalar supercharge survives, so that the path integral computes topological information) on a noncommutative deformation of the threefold (a deformation in which the algebra of functions on the threefold becomes noncommutative), evaluated by equivariant localization — the technique that reduces integrals over a moduli space to sums over the fixed points of a torus action — on its instanton moduli spaces (spaces of solutions of the gauge theory's soliton equations), and the individual crystal configurations represent fluctuating discrete Kaehler geometries — a concrete model of Wheeler-style spacetime 'quantum foam' at area scales of order the string coupling, with a classical limit shape emerging at large scales. Two structural features of the construction matter here. First, the gluing rules carry framing factors (leg-dependent powers of the string coupling) and ± sign factors derived from calculations of fluctuation determinants (the one-loop factors obtained by linearizing the theory around each configuration), with the sign fixed by a mod-2 parity computation. Second, the sum must include 'non-geometric excitations': three-dimensional diagrams whose complement in the toric base is not convex, and two-dimensional diagrams whose complement is not concave, which do not correspond to honest blowup geometries of the threefold — a blowup being the elementary surgery of algebraic geometry that replaces a point by the projective space of directions through it — and are postulated as contributions to the quantum-gravity path integral; the paper identifies them with excitations of its gauge theory.

On the mathematical side, curve counting by sheaves is Donaldson-Thomas theory. An ideal sheaf on a threefold $X$ is a rank-one torsion-free sheaf with trivial first Chern class; its moduli space (equivalently a Hilbert scheme of curves) is in general highly singular, but it carries a virtual fundamental class, a canonical zero-dimensional homology cycle produced by obstruction theory even where the moduli space itself is singular. Integrating the appropriate characteristic class (the paper's 'Donaldson-Thomas class') over this cycle yields integer invariants. For toric threefolds, Maulik, Nekrasov, Okounkov and Pandharipande verified by equivariant localization that the Donaldson-Thomas partition function reproduces the topological-vertex/crystal answer, and conjectured in general the Gromov-Witten/Donaldson-Thomas correspondence — the equality of the Gromov-Witten and Donaldson-Thomas partition functions, up to a change of variables — later proven for all toric threefolds. Behrend subsequently showed that Donaldson-Thomas type invariants equal weighted Euler characteristics of their moduli spaces (Euler characteristics weighted by a canonical constructible integer-valued function, the Behrend function), giving the signs in these counts an intrinsic microlocal-geometric meaning.

The organizing analogy is four-dimensional. Nekrasov's evaluation of the Seiberg-Witten prepotential (the holomorphic function encoding the low-energy effective action) of four-dimensional $N=2$ supersymmetric gauge theory (a gauge theory with an extended supersymmetry algebra) as an equivariant integral over instanton moduli spaces on $\mathbb{R}^4$ is the exact lower-dimensional analogue of the topological vertex. What the four-dimensional story has, and the six-dimensional one lacks, is the 'full Seiberg-Witten geometry': the spectral geometry of the Coulomb branch (the moduli space of vacua in which the gauge group is broken to its maximal torus) together with the Seiberg-Witten monopole equations, a differential system whose solution moduli on a four-manifold define invariants. The source paper asks for the corresponding six-dimensional object for the topologically twisted $U(1)$ theory whose partition function the crystal/vertex sums compute. The whole picture is fully developed only for non-compact toric threefolds; the benchmark compact target is the quintic, the degree-five hypersurface in complex projective four-space $\mathbb{P}^4$ and the archetypal compact Calabi-Yau threefold.

Problem Statement

The problem, posed in Section 7 of 'Quantum foam and topological strings', is to extend the quantum-foam gauge-theory construction from non-compact toric Calabi-Yau threefolds to general and in particular compact Calabi-Yau threefolds, such as the quintic hypersurface in $\mathbb{P}^4$, and in doing so to complete the dictionary with Donaldson-Thomas theory. Concretely:

(1) Compact Calabi-Yau extension with effective integration. Provide an effective technique for computing the integrals of the Donaldson-Thomas class over the virtual fundamental cycle of the moduli space of ideal sheaves on a compact Calabi-Yau threefold — at the time of the paper none was known except for the trivial second Chern class (maps to a point) — by extending the gauge-theory/topological-vertex construction itself.

(2) Donaldson-Thomas class identification. Identify explicitly the class predicted by the gauge theory with the mathematical Donaldson-Thomas class. For toric threefolds the match is verified by equivariant localization; the paper asks to 'relate more explicitly this class with that predicted by our gauge theory'.

(3) Six-dimensional Seiberg-Witten geometry. Construct the analogue, for the six-dimensional topologically twisted $U(1)$ theory on Calabi-Yau threefolds, of the full Seiberg-Witten geometry of four-dimensional $N=2$ theories, 'including the monopole equations'.

These components serve one objective: making the quantum-foam gauge-theory picture of the topological string a mathematically precise and computationally effective theory beyond the local toric case. The effective-integration core of (1) — a construction whose outputs are the Donaldson-Thomas invariants of compact Calabi-Yau threefolds, obtained within the gauge-theoretic/vertex formalism itself — is the main open problem the source poses, and (2) and (3) are the related open problems it raises alongside it.

References

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