Universal Manifold Pairings and Positivity

Abstract

Abstract: Gluing two manifolds M1 and M2 with a common boundary S yields a closed manifold M. Extending to formal linear combinations x = ΣaiMi yields a sesquilinear pairing p = 〈 , 〉 with values in (formal linear combinations of) closed manifolds. Topological quantum field theory (TQFT) represents this universal pairing p onto a finite dimensional quotient pairing q with values in C which in physically motivated cases is positive definite. To see if such a “unitary ” TQFT can potentially detect any nontrivial x, we ask if 〈x,x 〉 ̸ = 0 whenever x ̸ = 0. If this is the case, we call the pairing p positive. The question arises for each dimension d = 0,1,2,.... We find p(d) positive for d = 0,1, and 2 and not positive for d ≥ 4. We conjecture that p(3) is also positive. Similar questions may be phrased for (manifold, submanifold) pairs and manifolds with other additional structure. The negative result in dimension = 4 may illuminate the difficulties that have been met by several authors in their attempts to formulate unitary TQFTs for d = 3 + 1. 1