Geometry & Topology, Vol. 9 (2005) Paper no. 53, pages 2303--2317.

Universal manifold pairings and positivity

Michael H Freedman, Alexei Kitaev, Chetan Nayak, Johannes K Slingerland, Kevin Walker and Zhenghan Wang

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=Sum_i(a_i M_i) 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 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,.... 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, eg 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.

Keywords. Manifold pairing, unitary, positivity, TQFT, s-cobordism

AMS subject classification. Primary: 57R56, 53D45. Secondary: 57R80, 57N05, 57N10, 57N12, 57N13.

E-print: arXiv:math.GT/0503054

DOI: 10.2140/gt.2005.9.2305

Submitted to G&T on 25 May 2005. (Revised 2 December 2005.) Paper accepted 3 December 2005. Paper published 10 December 2005.

Notes on file formats

Michael H Freedman, Alexei Kitaev, Chetan Nayak, Johannes K Slingerland, Kevin Walker and Zhenghan Wang
MHF,CN,JKS,KW: Microsoft Research, 1 Microsoft Way, Redmond, WA 98052, USA
AK: California Institute of Technology, Pasadena, CA 91125, USA
CN: Department of Physics and Astronomy, UCLA, CA 90095-1547, USA
ZW: Dept of Mathematics, Indiana University, Bloomington, IN


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