Wess-Zumino-Witten Model as a Theory of Free Fields

Abstract

The free field representation or "bosonization" rule1 for Wess-Zumino-Witten model (WZWM) with arbitrary Kac-Moody algebra and arbitrary central charge is discussed. Energy-momentum tensor, arising from Sugawara construction, is quadratic in the fields. In this way, all known formulae for conformal blocks and correlators may be easily reproduced as certain linear combinations of correlators of these free fields. Generalization to conformal blocks on arbitrary Riemann surfaces is straightforward. However, projection rules in the spirit of Ref. 2 are not specified. The special role of βγ systems is emphasized. From the mathematical point of view, the construction involved represents generators of Kac-Moody (KM) algebra in terms of generators of a Heisenberg one. If WZW Lagrangian is considered as d−1 of Kirillov form on an orbit of KM algebra,3 then the free fields of interest (i.e. generators of the Heisenberg algebra) diagonalize Kirillov form and the action. Reduction of KM algebra within the same construction should naturally lead to arbitrary coset models.

Toric Calabi-Yau threefolds as quantum integrable systems. ℛ $$ \mathrm{\mathcal{R}} $$ -matrix and ℛ T T $$ \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} $$ relations

“…However, the generalized Macdonald polynomials in this case remain to be computed. One of possible difficulties on this way is that bosonization involves less trivial free fields a la [135][136][137].…”

О Комбинаторных Разложениях Конформных Блоков

“…4) Эта перенормировка является артефактом неудобной нормировки, изначально выбранной в работах [26], [27], и может быть поглощена перенормировкой скалярного поля φ(z) = √ 2 ϕ(z) (ср., например, нормировки в работах [29] и [30]). …”

“…За пределами свободных полей. Произвольная КТП может быть эф-фективно описана в терминах свободных полей [29], [30], однако число свободных полей в общем случае больше единицы, и должны быть наложены связи (20) на промежуточное состояние. При этом возникает естественный вопрос: что происхо-дит с уравнениями (24)- (26)?…”

A mathematical formalism for the Kondo effect in Wess-Zumino-Witten branes

“…The main question is whether the worldsheet vacua converge in Segal's formalism which we use here. However, the level k WZW model is a subquotient of a free field theory by [22]. For free field theories, it is known that the scaled vertex operators corresponding to worldsheets are smooth (cf.…”