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In dimension , we have that and, thus, we can consider its non-trivial central extensions by -cocycles. These are the essential ingredients of an (ordinary) Kac-Moody algebra.
In higher dimension, Hartogs' extension theorem tells us that , which means that we do not have any interesting central extension. On the other hand, the punctured affine spaces have non-trivial higher cohomology.
The idea underpinning the definition of higher Kac-Moody algebras in FHK 19 is that the natural framework to solve this issue and construct a higher-dimensional generalization of Kac-Moody algebras is derived algebraic geometry. This is achieved by replacing the algebra with the dg-algebra of derived sections, which allows interesting central extensions.
This principle can be applied not only to punctured affine spaces , but also to punctured formal disks .
Let be an -dimensional variety over and a Lie algebra. We can define the higher current algebra by the dg-Lie algebra
where is the commutative dg-algebra of derived global sections of the structure sheaf on the punctured formal disk .
A higher Kac-Moody algebra is the central extension of the higher current algebra by an invariant polynomial on of degree .
Let be a complex manifold of dimension equipped with a holomorphic principal bundle with structure group .
Let be the adjoint bundle associated to . Now, let be the local Lie algebra whose sections are on any open set and whose differential is the -connection .
Let be a degree cocycle in the local Chevalley-Eilenberg cochains , which defines a -shifted central extension .
The higher Kac-Moody factorization algebra on of type is defined in GW 21 as the twisted enveloping factorization algebra whose sections are
on any open set .
Let be the radial projection map sending .
In GW 21 the following map of factorization algebras on the positive reals is constructed:
where on the left-hand side we have the enveloping factorization algebra which encodes the enveloping -algebra of the FHK 19 higher Kac-Moody algebra .
This map establishes a relation between derived algebraic geometry and quantum field theory, formulated the language of factorization algebras. In particular, the higher Kac-Moody algebra “controls” its corresponding higher Kac-Moody factorization algebra just like an affine Kac-Moody algebra “controls” its corresponding vertex algebra.
For and , one recovers the formal current algebra , whose central extensions are ordinary Kac-Moody algebras.
Consider the affine space . The cohomology of the complex of derived global sections will be the following:
Higher Kac-Moody algebras were proposed in
G. Faonte, B. Hennion and M. Kapranov, Higher Kac-Moody algebras and moduli spaces of -bundles, Advances in Mathematics, Volume 346, 13 April 2019, pages 389-466, math.AG/1701.01368.
M. Kapranov, Infinite-dimensional (dg) Lie algebras and factorization algebras in algebraic geometry, Japanese Journal of Mathematics volume 16, pages 49–80 (2021) https://doi.org/10.1007/s11537-020-1921-4.
Higher Kac-Moody algebras casted in the language of factorization algebras:
A variant of this construction on the product of a worldsheet and a spectral plane (i.e. another copy of the complex plane with marked points which defines any rational Gaudin model):
Review:
Last revised on January 4, 2024 at 12:48:30. See the history of this page for a list of all contributions to it.