Free field realization of current superalgebras and bosonization of parafermionic algebras

Kault, Samuel Oliver (2012). Free field realization of current superalgebras and bosonization of parafermionic algebras PhD Thesis, School of Mathematics and Physics, The University of Queensland.

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Author Kault, Samuel Oliver
Thesis Title Free field realization of current superalgebras and bosonization of parafermionic algebras
Formatted title
School, Centre or Institute School of Mathematics and Physics
Institution The University of Queensland
Publication date 2012
Thesis type PhD Thesis
Supervisor Yao-Zhong Zhang
Wen-Li Yang
Jon Links
Total pages 230
Total black and white pages 230
Language eng
Subjects 010501 Algebraic Structures in Mathematical Physics
010505 Mathematical Aspects of Quantum and Conformal Field Theory, Quantum Gravity and String Theory
Formatted abstract
The technique of bosonisation allows complex affine Lie (super) algebras [or current (super) algebras] to be expressed as a composite of more elemental operators. We review the history of free field realization and bosonisation from Wakimoto onwards. We fully bosonize the su(2)k current algebra, and present an explicit bosonic construction for both the positive and negative modes of the su(2)k/u(1) coset parafermion at an arbitrary level k, using Faa di Bruno polynomials of a vector of 2 bosons.

    We present an algebraic proof of these general parafermions by inductively calculating the operator product expansion, and applying a kind of Faa di Bruno ``convolution" to the most singular terms generated. We apply the same techniques to more complicated, graded algebras. Using a particular ordering for the roots of basic Lie superalgebras, we present an explicit differential operator representation for the generators of osp(2r|2n) and sl(r|n). Based on this, we present an explicit free field realization of the corresponding current superalgebra osp(2r|2n)k. The energy momentum tensors of these quite general theories are derived. From this, the full bosonization may also be derived.

    For the specific case of osp(2|2), we focus on the non-standard basis, with two fermions in its simple root system. We present the full bosonization of the first and last terms of the positive modes of the parafermionic coset osp(2|2)k/u2(1), and one term of the negative modes. By extending algebraic convolution of the Faa di Bruno polynomials to less singular terms, we show that we can calculate the full structure constant, and verify its status as a primary field with respect to the energy momentum tensor of the theory. The conformal dimension is also calculated.
Keyword Bosonization
Central charge
Conformal dimension
Conformal field theory
Energy momentum tensor
Faa di Bruno polynomial
Free field
Operator product expansion
Lie algebra

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