# Download Problems of Modern Quantum Field Theory: Invited Lectures of by V. G. Drinfeld (auth.), Professor Aleksandr A. Belavin, PDF

By V. G. Drinfeld (auth.), Professor Aleksandr A. Belavin, Professor Anatolii U. Klimyk, Professor Aleksandr B. Zamolodchikov (eds.)

This quantity comprises the invited lectures of a college on sleek quantum box idea held at Alushta, USSR, in may perhaps 1989. the advance of this topic, together with string theories trying to version undemanding debris, is heavily interwoven with glossy mathematical physics. The lectures offered by means of specialists within the box supply an summary of the examine pursued in several branches of this speedily evolving box and draw recognition to specific interconnections and difficulties. themes lined comprise: geometrical quantization and finite dimension results in conformal box idea; quasi-Hopf, Kac-Moody present and Lie super-algebras; quantum teams; Wess-Zumino-Witten versions; Nizhnik-Zamolodchikov equations; non-archimedian strings; string dynamics; KdV and KP (super) equations and calculations on (super-) riemannian surfaces; 2nd Ising version and second electron movement on surfaces in exterior magnetic fields.

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**Additional resources for Problems of Modern Quantum Field Theory: Invited Lectures of the Spring School held in Alushta USSR, April 24 – May 5, 1989**

**Sample text**

I) .... + ~ T (j+ ~I) (27) The path integral over bosonic Wx-pair may be reduced to the case considered above, if we notice the following equality: JD'WD'X 50 ex~ Wax - ~ ~ Wx } JD'bD'e e~ bae - ~ ~ be } = 1 (28) where be is a (l,O)-grassmannian pair. s. of (28) can be easily bosonized and is given by (26) with r=l and ordinary theta function with odd characteristic (because of the zero mode). Therefore the result for the path integral (25) can be written as a linear combination of 9" r-, k+C y (~I't') -------1) ('t') (29) r In fact we know in the one-loop case what linear combination of the expressions like (29) should be chosen.

Physically, this construction means that path integrals are averaged over the set of all orbits. More exactly, the parameter which labels the Virasoro coadjoint orbit becomes a quantum-mechanical dynamical variable. The geometrical action, defined in our previous papers (more accurately, its trivial generalization) plays in this construction a crucial role. It defines the symplectic structure on the model space. Using the technique developed in [1,2,3] it is possible to introduce the "Darboux" variables for this symplectic form both in finite-dimensional case and in the Virasoro case (in the Kac-Moody group case - only for the quantum-mechanical part).

9. , Integrable models in 1+1 dimensional quantum field theory. In: Les Houches Lectures 1982, Elsevier, Amsterdam, 1984. , 151 (1982) 61-119. ICM-86, Berkley, 1987, 798-820. Faddeev L. D. Reshetikhin N. , Takhtaj an L. , Algebra and Analysis, 1 (1989) 178-206. Maj id S. Construction of non-commutative and noncommutati ve groups by Hopf algebra bicrossproduct. Preprint HUTMP-87/B216-L, 1988, 1-53. Yu. Representations of the algebra Uq (s1 (2) ), q -orthogonal polynomials and invariants of links.