## Geometry & Topology

### Universal manifold pairings and positivity

#### Abstract

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 $ℂ$ which in physically motivated cases is positive definite. To see if such a “unitary" TQFT can potentially detect any nontrivial $x$, we ask if $〈x,x〉≠0$ whenever $x≠0$. 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

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

Dates
Revised: 2 December 2005
Accepted: 3 December 2005
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799681

Digital Object Identifier
doi:10.2140/gt.2005.9.2303

Mathematical Reviews number (MathSciNet)
MR2209373

Zentralblatt MATH identifier
1129.57035

#### Citation

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

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