Abstract
We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1, 0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0, 1) supersymmetry and, possibly, a residual flavor symmetry. The equivariant topological Witten genus of this 2d theory then produces a new invariant of the 4-manifold equipped with a principle bundle, valued in the ring of equivariant weakly holomorphic (topological) modular forms. We describe basic properties of this map and present a few simple examples. As a byproduct, we obtain some new results on ’t Hooft anomalies of 6d (1, 0) theories and a better understanding of the relation between 2d (0, 1) theories and TMF spectra.
Highlights
The existence of non-trivial superconformal theories in six dimensions has been one of the major discoveries of the past few decades in string theory
We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1, 0) theories on 4-manifolds with flavor symmetry backgrounds
In a sense the new invariants we associate to four manifolds are extensions of Donaldson’s invariants: if we reverse the order of compactification and first compactify on T 2 and on the four-manifold, the theory has N = 2 supersymmetry in four dimensions
Summary
The existence of non-trivial superconformal theories in six dimensions has been one of the major discoveries of the past few decades in string theory. In a sense the new invariants we associate to four manifolds are extensions of Donaldson’s invariants: if we reverse the order of compactification and first compactify on T 2 and on the four-manifold, the theory has N = 2 supersymmetry in four dimensions It differs from the usual twist studied by Witten [20], in that it includes extra degrees of freedom coming from six dimensions. These extra fields would lead to a modular partition function instead of what one has in the case of Donaldson theory. In appendix C, we look back at the “big picture” and ask how far, beyond the existent set of invariants, can 6d (1, 0) theories take us in exploring the wild world of smooth 4-manifolds
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