Freedman, Michael H. and Kitaev, Alexei and Nayak, Chetan and Slingerland, Johannes K. and Walker, Kevin and Wang, Zhenghan (2005) Universal manifold pairings and positivity. Geometry and Topology, 9 (53). pp. 2303-2317. ISSN 1465-3060 http://resolver.caltech.edu/CaltechAUTHORS:FREgt05
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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.
|Additional Information:||Submitted to G&T on 25 May 2005. (Revised 2 December 2005.) Paper accepted 3 December 2005. Paper published 10 December 2005. E-print: arXiv:math.GT/0503054|
|Subject Keywords:||Manifold pairing, unitary, positivity, TQFT, s-cobordism|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Archive Administrator|
|Deposited On:||02 Jan 2006|
|Last Modified:||26 Dec 2012 08:43|
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