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. 23032317. ISSN 14653060. 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 nonzero whenever x is nonzero. 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 4manifolds, nor can they distinguish smoothly scobordant 4manifolds. 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 3dimensional ChernSimons theory appears to be wellencoded within 2dimensional quantum physics, eg in the fractional quantum Hall effect, DonaldsonSeibergWitten theory cannot be captured by a 3dimensional 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. Eprint: arXiv:math.GT/0503054 
Subject Keywords:  Manifold pairing, unitary, positivity, TQFT, scobordism 
Issue or Number:  53 
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:  18 Apr 2017 22:15 
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