Geometry & Topology

Universal manifold pairings and positivity

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

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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.

Article information

Geom. Topol., Volume 9, Number 4 (2005), 2303-2317.

Received: 25 May 2005
Revised: 2 December 2005
Accepted: 3 December 2005
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R56: Topological quantum field theories 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]
Secondary: 57R80: $h$- and $s$-cobordism 57N05: Topology of $E^2$ , 2-manifolds 57N10: Topology of general 3-manifolds [See also 57Mxx] 57N12: Topology of $E^3$ and $S^3$ [See also 57M40] 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx]

manifold pairing unitary positivity TQFT $s$–cobordism


Freedman, Michael H; Kitaev, Alexei; Nayak, Chetan; Slingerland, Johannes K; Walker, Kevin; Wang, Zhenghan. Universal manifold pairings and positivity. Geom. Topol. 9 (2005), no. 4, 2303--2317. doi:10.2140/gt.2005.9.2303.

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