Universal manifold pairings and positivity

Abstract

Gluing two manifolds $M_1$ and $M_2$ with a common boundary $S$ yields a closed manifold $M$. Extending to formal linear combinations $x=\Sigma a_i{M}_i$ yields a sesquilinear pairing $p=\langle ,\rangle$ 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 $\langle x,x\rangle \ne 0$ whenever $x\ne 0$. If this is the case, we call the pairing $p$ positive. The question arises for each dimension $d=0,1,2,\dots$. We find $p\left(d\right)$ positive for $d=0,1,$ and $2$ and not positive for $d=4$. We conjecture that $p\left(3\right)$ is also positive. Similar questions may be phrased for (manifold, submanifold) pairs and manifolds with other additional structure. The results in dimension 4 imply that unitary TQFTs cannot distinguish homotopy equivalent simply connected 4–manifolds, nor can they distinguish smoothly $s$–cobordant 4–manifolds. This may illuminate the difficulties that have been met by several authors in their attempts to formulate unitary TQFTs for $d=3+1$. There is a further physical implication of this paper. Whereas 3–dimensional Chern–Simons theory appears to be well-encoded within 2–dimensional quantum physics, e.g. in the fractional quantum Hall effect, Donaldson–Seiberg–Witten theory cannot be captured by a 3–dimensional quantum system. The positivity of the physical Hilbert spaces means they cannot see null vectors of the universal pairing; such vectors must map to zero.