Vertex Algebras and 4-Manifold Invariants

Abstract

We propose a way of computing 4-manifold invariants, old and new, as chiral correlation functions in half-twisted 2d N = (0, 2) theories that arise from compactification of fivebranes. Such formulation gives a new interpretation of some known statements about Seiberg-Witten invariants, such as the basic class condition, and gives a prediction for structural properties of the multi-monopole invariants and their non-abelian generalizations. 5 Non-abelian generalizations 51 5.1 Coulomb-branch index and new 4-manifold invariants 54 A Curvature of the canonical connection on the universal bundle 57 A.1 SU (N f ) invariance and the moment map 59 B Topological twist of SQED 60 1.1 Unorthodox invariants of smooth 4-manifolds Searching for new invariants of smooth 4-manifolds, that potentially could go beyond the Seiberg-Witten and Donaldson invariants, it was proposed in [1] to consider a twodimensional quantum field theory T [M 4 , G] as a rather unusual invariant of smooth structures on a 4-manifold M 4 . Specifically, T [M 4, G] is a 2d N = (0, 2) superconformal theory that, apart from M 4 , also depends on a choice of a root system G and is invariant under the Kirby moves. Luckily, conformal field theories in two dimensions exhibit rich mathematical structure which, on the one hand, is rich enough to (potentially) describe the wild world of smooth 4-manifolds and, on the other hand, is rigorous enough to hope for a precise mathematical definition of the invariant T [M 4 , G]. In fact, for many practical purposes and applications in this paper, a mathematically inclined reader can think of T [M 4 , G] as a functor that assigns a vertex operator algebra (VOA) to a smooth 4-manifold M 4 (and a "gauge" group G).Composing it with other functors that assign various quantities to 2d conformal theories, one can obtain more conventional invariants of smooth 4-manifolds:Here, Z can be any invariant of a 2d conformal theory with N = (0, 2) supersymmetry, e.g., its elliptic genus, chiral ring, moduli space of marginal couplings, or central charge. Since 2d theory T [M 4 ; G] is systematically determined by M 4 and invariant under the Kirby moves, all such invariants lead to various 4-manifold invariants; some are simple and some are quite powerful. In particular, it was conjectured in [1] that chiral ring of the theory T [M 4; G] for G = SU (2) or, equivalently, its Q + -cohomology knows about Donaldson invariants of M 4 . One of the main goals in this paper is to present some evidence to this conjecture and to build a bridge between VOA G [M 4 ] and more traditional 4-manifold invariants. Conjecturally, at least for manifolds with b + 2 > 1, one can trade SU (2) Donaldson invariants for Seiberg-Witten invariants defined in a simpler gauge theory, with gauge group G = U (1), which would be too trivial if not for an extra ingredient, the additional spinor fields. We wish to study how these invariants are realized in 2d theory T [M 4 ; G] with G = SU (2) and G = U (1), respectively. In particular, we shall see that in 2d realizati...

G2 holonomy, Taubes’ construction of Seiberg-Witten invariants and superconducting vortices

“…where W is a 7-fold of G 2 holonomy with parallel 3-form Φ 3 [20,21], and H a hyperKähler 4-fold; neither space is supposed to be compact or complete. This geometric compactification of M-theory has been considered in [9]. This geometry preserves two supersymmetries.…”

“…From this alternative point of view we get the 2d TFT on C obtained by twisting the 2d (2, 0) model associated to the 4-fold X, see refs. [9,28]. However the deformation we are interested in seems more naturally described from the perspective of TFT on the space-time X.…”

“…We are thus led to consider FI terms of abelian gauge theories. The FI deformation of topological theory under consideration has also been considered in [9].…”

“…It is natural to expect that the result of the gauging the extra U(1) is related to the residue of the pole in topological amplitudes in the original theory as ǫ i m i → 0. This may have a particularly simple interpretation in the setup [9] which relates the topological amplitudes to correlators of chiral operator insertions on C, as the subtraction needed to remove the singularity for coincident chiral operators. It would be worthwhile developing this further.…”

“…We would then consider M-theory on the 11-dimensional manifold W × T * C and wrap M5 branes around X 4 × C. The low energy, supersymmetric partition function of this theory is naturally captured by N = 2 TQFT of the 4d QFT labeled by the curve C on the 4-manifold X. Precisely this geometric realization of 4d TQFT in M-theory, using the G 2 structure has already been constructed and studied in [9]. Indeed many elements of what we encounter in this paper have been considered there as well.…”