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Universal manifold pairings and positivity. (English) Zbl 1129.57035

Summary: Gluing two manifolds \(M_1\) and \(M_2\) with a common boundary \(S\) yields a closed manifold \(M\). Extending to formal linear combinations \(x=\sum_i(a_i M_i)\) 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 \(\mathbb C\) which in physically motivated cases is positive definite. To see if such a “unitary” TQFT can potentially detect any nontrivial \(x\), we ask if \(\langle x,x\rangle\) is non-zero whenever \(x\) is non-zero. If this is the case, we call the pairing \(p\) positive. The question arises for each dimension \(d=0,1,2,\dots\). We find \(p(d)\) 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 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.

MSC:

57R56 Topological quantum field theories (aspects of differential topology)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
57R80 \(h\)- and \(s\)-cobordism
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57N12 Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
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