Paper published by Geometry and Topology

The following paper has been published:

Geometry & Topology, Volume 9\ (2005) Paper no. 53, pages 2305--2319

URL:
http://www.maths.warwick.ac.uk/gt/GTVol9/paper53.abs.html

DOI: 10.2140/gt.2005.9.2305

Title:
Universal manifold pairings and positivity

Author(s):
Michael H Freedman, Alexei Kitaev, Chetan Nayak, Johannes K Slingerland, Kevin Walker and Zhenghan Wang

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=Sum_i(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> is non-zero
whenever x is non-zero. 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, eg 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.

AMS Classification Numbers. Primary: 57R56, 53D45
Secondary: 57R80, 57N05, 57N10, 57N12, 57N13

Keywords:
Manifold pairing, unitary, positivity, TQFT, s-cobordism

Received: 25 May 2005
Revised: 2 December 2005
Accepted: 3 December 2005
Published: 10 December 2005

Proposed: Robion Kirby
Seconded: Peter Teichner, Cameron Gordon

Author(s) address(es):
MHF,CN,JKS,KW: Microsoft Research, 1 Microsoft Way, Redmond, WA 98052, USA
AK: California Institute of Technology, Pasadena, CA 91125, USA
CN: Department of Physics and Astronomy, UCLA, CA 90095-1547, USA
ZW: Dept of Mathematics, Indiana University, Bloomington, IN

Email: michaelf@xxxxxxxxxxxxx, kitaev@xxxxxxxxxxxxxxx, nayak@xxxxxxxxxxxxxxxx,
joost@xxxxxxxxxxxxx, kwalker@xxxxxxxxxxxxx, zhewang@xxxxxxxxxxx

.