Universal manifold pairings and positivity

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

doi:10.2140/gt.2005.9.2303

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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

DOI: 10.2140/gt.2005.9.2303

arXiv: math.GT/0503054

Abstract

Gluing two manifolds $M_{1}$ and $M_{2}$ with a common boundary $S$ yields a closed manifold $M$ . Extending to formal linear combinations $x = Σ a_{i} M_{i}$ 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 $〈 x, x 〉 \neq 0$ whenever $x \neq 0$ . If this is the case, we call the pairing $p$ positive. The question arises for each dimension $d = 0, 1, 2, \dots$ . 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.

Keywords

manifold pairing, unitary, positivity, TQFT, $s$–cobordism

Mathematical Subject Classification 2000

Primary: 57R56, 53D45

Secondary: 57R80, 57N05, 57N10, 57N12, 57N13

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Publication

Received: 25 May 2005

Revised: 2 December 2005

Accepted: 3 December 2005

Published: 10 December 2005

Proposed: Robion Kirby

Seconded: Peter Teichner, Cameron Gordon

Authors

Michael H Freedman
Microsoft Research 1 Microsoft Way Redmond Washington 98052 USA

Alexei Kitaev
California Institute of Technology Pasadena California 91125 USA

Chetan Nayak
Microsoft Research 1 Microsoft Way Redmond Washington 98052 USA	Department of Physics and Astronomy University of California Los Angeles California 90095-1547 USA

Johannes K Slingerland
Microsoft Research 1 Microsoft Way Redmond Washington 98052 USA

Kevin Walker
Microsoft Research 1 Microsoft Way Redmond Washington 98052 USA

Zhenghan Wang
Department of Mathematics Indiana University Bloomington Indiana 47405 USA

Volume 9, issue 4 (2005)