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Universal manifold pairings and positivity

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

Geometry & Topology 9 (2005) 2303–2317

arXiv: math.GT/0503054


Gluing two manifolds M1 and M2 with a common boundary S yields a closed manifold M. Extending to formal linear combinations x = ΣaiMi 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 which in physically motivated cases is positive definite. To see if such a “unitary" TQFT can potentially detect any nontrivial x, we ask if x,x0 whenever x0. 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, 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.

manifold pairing, unitary, positivity, TQFT, $s$–cobordism
Mathematical Subject Classification 2000
Primary: 57R56, 53D45
Secondary: 57R80, 57N05, 57N10, 57N12, 57N13
Forward citations
Received: 25 May 2005
Revised: 2 December 2005
Accepted: 3 December 2005
Published: 10 December 2005
Proposed: Robion Kirby
Seconded: Peter Teichner, Cameron Gordon
Michael H Freedman
Microsoft Research
1 Microsoft Way
Washington 98052
Alexei Kitaev
California Institute of Technology
California 91125
Chetan Nayak
Microsoft Research
1 Microsoft Way
Washington 98052
Department of Physics and Astronomy
University of California
Los Angeles
California 90095-1547
Johannes K Slingerland
Microsoft Research
1 Microsoft Way
Washington 98052
Kevin Walker
Microsoft Research
1 Microsoft Way
Washington 98052
Zhenghan Wang
Department of Mathematics
Indiana University
Indiana 47405