[Paper Review] Differential Operator Algebras on compact Riemann Surfaces

Martin Schlichenmaier|Open Repository and Bibliography (University of Luxembourg)|Nov 5, 1993

Advanced Topics in AlgebraMathematics12 references18 citations

TL;DR

This paper generalizes the Virasoro and Krichever-Novikov algebras to compact Riemann surfaces of arbitrary genus by constructing algebras of meromorphic differential operators holomorphic outside a finite set of points. It introduces a grading via in/out-point partitions, establishes almost-graded Lie algebra structures, and studies central extensions and semi-infinite wedge representations, yielding higher-genus analogues of conformal field theory algebras with explicit cocycles and Verma module realizations.

ABSTRACT

Invited talk at the International Symposium on Generalized Symmetries in Physics at the Arnold-Sommerfeld-Institute, Clausthal, Germany, July 26 -- July 29, 1993. This talk reviews results on the structure of algebras consisting of meromorphic 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 of lambda - forms 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.

Motivation & Objective

• To generalize the Virasoro algebra from genus zero to compact Riemann surfaces of arbitrary genus.
• To construct algebras of meromorphic differential operators holomorphic outside a finite set of points (in- and out-points).
• To introduce a grading on the algebra and modules of λ-forms via partitioning the pole set into in- and out-points.
• To study central extensions and semi-infinite wedge representations for these algebras.
• To establish higher-genus analogues of untwisted affine Kac-Moody algebras and their representations.

Proposed method

• Define differential operator algebras Dλ(A) on compact Riemann surfaces X with poles only at a finite set A = I ∪ O of in- and out-points.
• Introduce a grading on Dλ(A) and modules of λ-forms using the partition (I,O), leading to almost-graded Lie algebra structures.
• Construct a central extension of the algebra via a local cocycle derived from the residue of a meromorphic differential ρ with prescribed residues and purely imaginary periods.
• Embed the algebra into the Lie algebra of infinite matrices gl(∞) to regularize the action on semi-infinite wedge spaces Hλ(A).
• Use the pullback of the standard cocycle on gl(∞) to define a regularized, centrally extended action on Hλ(A), ensuring compatibility with Clifford algebra structures.
• Define a pairing between left and right semi-infinite forms of weights λ and 1−λ using the pairing (9) on λ-forms.

Experimental results

Research questions

• RQ1 How can the Virasoro algebra be generalized to compact Riemann surfaces of genus g ≥ 1 with multiple punctures for conformal field theory applications?
• RQ2 What grading structure emerges when the pole set is partitioned into in- and out-points, and how does it affect the Lie algebra structure?
• RQ3 What is the form of the central extension for the algebra of differential operators on higher-genus Riemann surfaces?
• RQ4 How can semi-infinite wedge representations be constructed for these algebras, and what role does the almost-graded structure play?
• RQ5 What are the higher-genus analogues of untwisted affine Kac-Moody algebras, and how do they act on semi-infinite forms?

Key findings

• The algebra of differential operators Dλ(A) on a compact Riemann surface with poles at a finite set A = I ∪ O is almost-graded with respect to the partition (I,O), generalizing the Virasoro algebra in genus zero.
• A central extension of Dλ(A) is constructed via a local cocycle, with the central element acting as (−2)(6λ² − 6λ + 1) times the identity on Verma modules.
• Semi-infinite wedge representations Hλ(A) are constructed by regularizing the action of the centrally extended algebra via embedding into gl(∞), with the cocycle pulled back from the standard one on gl(∞).
• The action of the centrally extended algebra D¹(A) on Hλ(A) is compatible with the Clifford algebra structure induced by wedging and contracting with b-c systems.
• For the torus (g=1), explicit calculations are performed for 2-, 3-, and 4-point cases, and degenerations of the Riemann surface are studied using algebraic geometry techniques.
• The construction yields a consistent framework for highest weight representations of centrally extended differential operator algebras on higher-genus Riemann surfaces, extending conformal field theory to higher genus.