Geometry & Topology
- Geom. Topol.
- Volume 9, Number 4 (2005), 2303-2317.
Universal manifold pairings and positivity
Gluing two manifolds and with a common boundary yields a closed manifold . Extending to formal linear combinations yields a sesquilinear pairing with values in (formal linear combinations of) closed manifolds. Topological quantum field theory (TQFT) represents this universal pairing onto a finite dimensional quotient pairing with values in which in physically motivated cases is positive definite. To see if such a “unitary" TQFT can potentially detect any nontrivial , we ask if whenever . If this is the case, we call the pairing positive. The question arises for each dimension . We find positive for and and not positive for . We conjecture that 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 –cobordant 4–manifolds. This may illuminate the difficulties that have been met by several authors in their attempts to formulate unitary TQFTs for . 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.
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
Permanent link to this document
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]
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. https://projecteuclid.org/euclid.gt/1513799681