The Drinfeld-Sokolov holomorphic bundle and classical W algebras on Riemann surfaces (original) (raw)
1995, Journal of Geometry and Physics
Developing upon the ideas of ref. [6], it is shown how the theory of classical W algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. The basic geometric object is the Drinfeld-Sokolov principal bundle L associated to a simple complex Lie group G equipped with an SL(2, C) subgroup S, whose properties are studied in detail. On a multipunctured Riemann surface, the Drinfeld-Sokolov-Krichever-Novikov spaces are defined, as a generalization of the customary Krichever-Novikov spaces, their properties are analyzed and standard bases are written down. Finally, a WZWN chiral phase space based on the principal bundle L with a KM type Poisson structure is introduced and, by the usual procedure of imposing first class constraints and gauge fixing, a classical W algebra is produced. The compatibility of the construction with the global geometric data is highlighted.
Sign up for access to the world's latest research.
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Related papers
A Krichever-Novikov formulation of classical W algebras on Riemann surfaces
Physics Letters B, 1994
It is shown how the theory of classical W -algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. An intriguing relation between the theory of A 1 embeddings into simple Lie algebras and the holomorphic geometry of Riemann surfaces is exihibited.
W Algebras on Riemann Surfaces
1994
It is shown how the theory of classical W –algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. An intriguing relation between the theory of A 1 embeddings into simple Lie algebras and the holomorphic geometry of Riemann surfaces is exihibited.
2003
This paper continues the same-named article, Part I (math.QA/9812083). We give a global operator approach to the WZWN theory for compact Riemann surfaces of an arbitrary genus g with marked points. Globality means here that we use Krichever-Novikov algebras of gauge and conformal symmetries (i.e. algebras of global symmetries) instead of loop and Virasoro algebras (which are local in this context). The elements of this global approach are described in Part I. In the present paper we give the construction of conformal blocks and the projective flat connection on the bundle constituted by them.
On the Chiral WZNW Phase Space, Exchange r-Matrices and Poisson-Lie Groupoids
1999
This is a review of recent work on the chiral extensions of the WZNW phase space describing both the extensions based on fields with generic monodromy as well as those using Bloch waves with diagonal monodromy. The symplectic form on the extended phase space is inverted in both cases and the chiral WZNW fields are found to satisfy quadratic Poisson bracket relations characterized by monodromy dependent exchange r-matrices. Explicit expressions for the exchange r-matrices in terms of the arbitrary monodromy dependent 2-form appearing in the chiral WZNW symplectic form are given. The exchange r-matrices in the general case are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter (YB) equation and Poisson-Lie (PL) groupoids are constructed that encode this equation analogously as PL groups encode the classical YB equation. For an arbitrary simple Lie group GGG, exchange r-matrices are exhibited that are in one-to-one correspondence with the possible PL structures on GGG and admit them as PL symmetries.
Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras
Russian Mathematical Surveys, 1999
Dedicated t o P r of. S.P.Novikov in honour of his 60-th birthday Abstract. Elements of a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus g are given. Sheaves of representations of a ne Krichever-Novikov algebras over a dense open subset of the moduli space of Riemann surfaces (respectively of smooth, projective complex curves) with N marked points are introduced. It is shown that the tangent space of the moduli space at an arbitrary moduli point is isomorphic to a certain subspace of the Krichever-Novikov vector eld algebra given by the data of the moduli point. This subspace is complementary to the direct sum of the two subspaces containing the vector elds which v anish at the marked points, respectively which are regular at a xed reference point. For each representation of the a ne algebra 3g ; 3 + N equations (@ k + T e k ]) = 0 are given, where the elements fe k g are a basis of the subspace, and T is the Sugawara representation of the centrally extended vector eld algebra. For genus zero one obtains the Knizhnik-Zamolodchikov equations in this way. The coe cients of the equations for genus one are found in terms of Weierstrafunction.
Knizhnik-Zamolodchikov equations for positive genus and Krichever-Novikov algebras
Russian Mathematical Surveys, 2004
We give a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus with marked points. Globality means here that we use Krichever-Novikov algebras of gauge and conformal symmetries (i.e. algebras of global symmetries) instead of loop and Virasoro algebras (which are local in this context). The basic elements of this global approach are described in a previous article of the authors (Russ. Math. Surv., (54)(1)). In the present article we construct the conformal blocks and the projectively flat connection on the bundle constituted by them.
Communications in Mathematical Physics, 2013
We describe of the generalized Drinfeld-Sokolov Hamiltonian reduction for the construction of classical W-algebras within the framework of Poisson vertex algebras. In this context, the gauge group action on the phase space is translated in terms of (the exponential of) a Lie conformal algebra action on the space of functions. Following the ideas of Drinfeld and Sokolov, we then establish under certain sufficient conditions the applicability of the Lenard-Magri scheme of integrability and the existence of the corresponding integrable hierarchy of bi-Hamiltonian equations.
DIFFERENTIAL OPERATOR ALGEBRAS ON COMPACT RIEMANN SURFACES1
1994
This talk reviews results on the structure of algebras consisting of meromor- phic differential operators which are holomorphic outside a finite set of points on compact Riemann surfaces. For each partition into two disjoint subsets of the set of points where poles are allowed, a grading of the algebra and of the modules offorms is introduced. With respect to this grading the Lie structure of the algebra and of the modules are almost graded ones. Central extensions and semi-infinite wedge representations are studied. If one considers only differential operators of degree 1 then these algebras are generalizations of the Virasoro algebra in genus zero, resp. of Krichever Novikov algebras in higher genus.
Zamolodchikov relations and Liouville hierarchy in SL (2, R) k WZNW model
Nuclear Physics B, 2005
We study the connection between Zamolodchikov operator-valued relations in Liouville field theory and in the SL(2, R) k WZNW model. In particular, the classical relations in SL(2, R) k can be formulated as a classical Liouville hierarchy in terms of the isotopic coordinates, and their covariance is easily understood in the framework of the AdS 3 /CF T 2 correspondence.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.