Celebratio Mathematica — Freedman

Topology in dimension 3

Works connected to Zhenghan Wang

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article M. H. Freedman, K. Walker, and Z. Wang: “Quantum \( \mathit{SU}(2) \) faithfully detects mapping class groups modulo center,” Geom. Topol. 6 (2002), pp. 523–539. MR 1943758 Zbl 1037.57024 ArXiv math.GT/0209150

The Jones–Witten theory gives rise to representations of the (extended) mapping class group of any closed surface \( Y \) indexed by a semi-simple Lie group \( G \) and a level \( k \). In the case \( G = \mathit{SU}(2) \) these representations (denoted \( V_A(Y) \)) have a particularly simple description in terms of the Kauffman skein modules with parameter \( A \) a primitive \( 4r \)-th root of unity (\( r = k + 2 \)). In each of these representations (as well as the general \( G \) case), Dehn twists act as transformations of finite order, so none represents the mapping class group \( \mathcal{M}(Y) \) faithfully. However, taken together, the quantum \( \mathit{SU}(2) \) representations are faithful on non-central elements of \( \mathcal{M}(Y) \). (Note that \( \mathcal{M}(Y) \) has non-trivial center only if \( Y \) is a sphere with 0, 1, or 2 punctures, a torus with 0, 1, or 2 punctures, or the closed surface of genus \( = 2 \).) Specifically, for a non-central \( h\in\mathcal{M}(Y) \) there is an \( r_0(h) \) such that if \( r \geq r_0(h) \) and \( A \) is a primitive \( 4r \)-th root of unity then \( h \) acts projectively nontrivially on \( V_A(Y) \). Jones’ [1987] original representation \( \rho_n \) of the braid groups \( B_n \), sometimes called the generic \( q \)-analog-\( \mathit{SU}(2) \)-representation, is not known to be faithful. However, we show that any braid \( h \neq \mathrm{id} \in B_n \) admits a cabling \( c = c_1,\dots,c_n \) so that \( \rho_N(c(h)) \neq\mathrm{id} \), \( N = c_1 + \dots +c_n \).

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Zhenghan Wang	Related

article M. H. Freedman, A. Kitaev, C. Nayak, J. K. Slingerland, K. Walker, and Z. Wang: “Universal manifold pairings and positivity,” Geom. Topol. 9 (2005), pp. 2303–2317. MR 2209373 Zbl 1129.57035 ArXiv math/0503054

Gluing two manifolds \( M_1 \) and \( M_2 \) with a common boundary \( S \) yields a closed manifold \( M \). Extending to formal linear combinations \( x = \sum a_iM_i \) yields a sesquilinear pairing \( p = \langle\,\cdot\,,\cdot\,\rangle \) 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 \( \mathbb{C} \) which in physically motivated cases is positive definite. To see if such a “unitary” TQFT can potentially detect any nontrivial \( x \), we ask if \( \langle x,x\rangle\neq 0 \) 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.

Zhenghan Wang	Related
Johannes K. Slingerland	Related
Kevin Walker	Related
Chetan Nayak	Related
Alexei Kitaev	Related