By M. L. Ge, C. N. Yang
Yang C.N., Ge M.L. (eds.) Braid workforce, knot conception, and statistical mechanics (WS, 1989)(ISBN 9971508281)(331s)
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Extra info for Braid group, knot theory, and statistical mechanics
This observation raises an important question: if we start at a given multi critical model, and then flow up to a higher one using Eq. t We can express Eq. (31) more compactly by introducing the vector fields which generate the KdV flows: g1 = I i~O R(i+l) _8_ 1 8u(i)' (32) where the superscript refers to differentiation with respect to x and the U(i) are considered independent. Integrability of the KdV hierarchy depends crucially on the fact that these vector fields commute: (33) We can then write Eq.
First, note that deg Pi-I = 4N 2 / Vi-I, which gives the degree of the conjugate polynomial deg Pf-I = 4N 2/(l- Vi-I)' From deg Py = 4N 2 , one finds that -I v·I = 2 -deg P N2 I = 2Pi + 1 ± 1 -I Vi-I - 1 , (A6) as derived by MacDonald, Aers, and Dharma-wardana. 51 The fractions obtained are identical to those proposed by Haldane 50 and by Halperin24: 1 Vi = - - - - - - - - - - - - (Ti-I 2Pi + 1 + -------'----'----(Ti-2 2Pi - I + ------'---"---- (A7) where (Ti = ± 1 determines whether one descends down the hole (+ 1) or the particle (-1) branch at level i of the hierarchy.
We thank L. Susskind for discussions of this point. 19. See S. Carlip and S. P. de Alwis, lAS pre print, October, 1989, for a discussion of problems with the dilute wormhole approximation in 2 + 1 dimensions. Chapter 3 Supersymmetry and Gauge Invariance in Stochastic Quantization Laurent Baulieu 1. INTRODUCTION Stochastic quantization is an alternative to the Feynman path integral for quantizing a theory. In Ref. 1, Parisi and Wu have suggested applying stochastic quantization to gauge theories.
Braid group, knot theory, and statistical mechanics by M. L. Ge, C. N. Yang