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Invariants of 3-Manifolds from Conformal Field Theory Sylvain E. Cappell *'**, Ronnie Lee* and Edward Y. Miller* 0. Introduction In 1989, Reshetikhin and Turaev [RT1], [RT2] constructed a family of invariants of 3- manifolds based upon the theory of quantum groups of Uq(sl(n)). Subsequently, Kirby- Melvin [KM], Ko-Smolinsky [KS], and Lickorish [L], provided different verifications of these invariants in certain special cases of Uq (sl(2)). In retrospect, the proofs discussed by these authors are all based upon a combinatorial approach to 3-manifold theory

Chapter 6 Lax integrable systems and conformal field theory This chapter brings together the material from all the previous chapters. We address here the following problem: given a Lax integrable system of the type discussed above, construct a unitary projective representation of the corresponding Lie algebra of Hamiltonian vector fields. For the Lax equations in question, we pro- pose a way to represent Hamiltonian vector fields by covariant derivatives with re- spect to the Knizhnik–Zamolodchikov connection. This is a Dirac-type prequantiza- tion from the

Ternary codes and Zß-orbifold constructions of conformal field theories P. S. Montague Abstract We describe a pair of constructions of Eisenstein lattices from ternary codes, and a corresponding pair of constructions of conformal field theories from lattices which turn out to have a string theoretic interpretation. These are found to interconnect in a similar way to results for binary codes, which led to a generalisation of the triality structure relevant in the construction of the Monster module. We therefore make some comments regarding a series of

A Conformal Field Theory of Extrinsic Geometry of 2-d Surfaces * K . S . V i swana than a and R. P a r t h a s a r a t h y b a Department of Physics, Simon Fraser University, Burnaby V5A 1S6, B.C., Canada b Institute of Mathematical Sciences, Madras 600113, India Z. Naturforsch. 52a, 97-104 (1997); received July 19, 1996 In the description of the extrinsic geometry of the string world sheet regarded as a conformal immersion of a 2-d surface in R3 it was previously shown that, restricting to surfaces with h yJg = 1, where h is the mean scalar curvature and

noteworthy that such Hermitian forms are intimately related with the correlation functions of non-diagonal type in €sl2-conformal field theory. Keywords. Selberg type integral, connection problem, connection coefficient, q-Racah polynomial, twisted homology, monodromy-invariant Hermitian form, conformal field theory, correlation functions of non-diagonal type. 2010 Mathematics Subject Classification. 34M40, 33C60, 33D45, 34M35. Introduction A Selberg type integralZ Y 1i<jm .tj ti / g Y 1im 1jn .ti zj / j dt1 dtm; (I.1) where g and j are complex numbers and is a

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) proof of this important result was given 1989 by Fialowski in an unpublished note. It is based on cumbersome calculations. Compared to the original proof the presented one is quite elegant and considerably simpler. Keywords. Witt algebra, Virasoro algebra, Lie algebra cohomology, deformations of algebras, rigidity, conformal field theory. 2010 Mathematics Subject Classification. Primary 17B56; secondary 17B68, 17B65, 17B66, 14D15, 81R10, 81T40. 1 Introduction The simplest nontrivial infinite dimensional Lie algebras are the Witt algebra and its central extension the

vertex operator algebra extending V G {V^{G}} called the orbifold of V with respect to G and H (Theorem 5.16 ). In Section 6 we show that Schellekens’ classification of V 1 {V_{1}} -structures of meromorphic conformal field theories of central charge 24 is a rigourous theorem on vertex operator algebras. Next we recall some results on lattice vertex algebras. In the last section we apply our results to construct 5 new holomorphic vertex operator algebras of central charge 24 as orbifolds of Niemeier lattices. We show that they have V 1 {V_{1}} -structures

gears and apply bootstrap techniques to the interesting case of conformal field theories in the presence of a boundary. In particular, we estimate surface critical exponents for the O(N) universality classes. Keywords: CFT, supersymmetry Mathematics Subject Classification 2010: 81T40, 83C47, 81Q60, 82B27 1 The bootstrap philosophy The “bootstrap” has been a recurring dream in theoretical physics. It is the ambi- tious aspiration that, starting from a few basic spectral assumptions, symmetries and general consistency requirements (such as unitarity and crossing) will