Dimensional Topological Quantum Field Theory Research Articles

ON OCNEANU'S THEORY OF ASYMPTOTIC INCLUSIONS FOR SUBFACTORS, TOPOLOGICAL QUANTUM FIELD THEORIES AND QUANTUM DOUBLES

A fully detailed account of Ocneanu's theorem is given that the Hilbert space associated to the two-dimensional torus in a Turaev-Viro type (2+1)-dimensional topological quantum field theory arising from a finite depth subfactor N⊂M has a natural basis labeled by certain M∞- M∞ bimodules of the asymptotic inclusion M∨(M'∩M∞)⊂M∞, and moreover that all these bimodules are given by the basic construction from M∨(M'∩M∞)⊂M∞ if the fusion graph is connected. This Hilbert space is an analogue of the space of conformal blocks in conformal field theory. It is also shown that after passing to the asymptotic inclusions we have S- and T-matrices, analogues of the Verlinde identity and Vafa's result on roots of unity. It is explained that the asymptotic inclusions can be regarded as a subfactor analogue of the quantum double construction of Drinfel'd. These claims were announced by Λ. Ocneanu in several talks, but he has not published his proofs, so details are given here along the lines outlined in his talks.

Stochastic and topological gauge theories

We show that the form of the Langevin equations for the stochastic quantization of a Yang-Mills theory in n-dimensional space is dictated by the geometrical structure. The arbitrary Lie algebra zero form, initially introduced by Zwanziger to induce a drift force along gauge orbits, is now defined as an additional gauge field component which permits stochastic time-dependent gauge transformations. The action of the functional integral representation of the Langevin equation is the same as that of an ( n+1)-dimensional topological quantum field theory. Covariance between the stochastic time and the physical space directions is made possible. As an application, the action which generates Donaldson invariants in four-dimensional space is shown to be identical to the supersymmetric action describing the stochastic quantization of the three-dimensional Chern-Simons theory, which generates Jones invariants.