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Universal manifold pairings and positivity

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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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.


Item Type:Article
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
Record Number:CaltechAUTHORS:FREgt05
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:FREgt05
Alternative URL:http://dx.doi.org/10.2140/gt.2005.9.2305
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1172
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:02 Jan 2006
Last Modified:26 Dec 2012 08:43

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