nLab higher Kac-Moody algebra




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  • In dimension 11, we have that 𝒪(𝔸 1{0})𝔤=𝔤[z,z 1]\mathcal{O}(\mathbb{A}^1-\{0\})\otimes\mathfrak{g} = \mathfrak{g}[z,z^{-1}] and, thus, we can consider its non-trivial central extensions by 22-cocycles. These are the essential ingredients of an (ordinary) Kac-Moody algebra.

  • In higher dimension, Hartogs' extension theorem tells us that 𝒪(𝔸 n{0})𝔤𝒪(𝔸 n)𝔤\mathcal{O}(\mathbb{A}^n-\{0\})\otimes\mathfrak{g} \cong \mathcal{O}(\mathbb{A}^n)\otimes\mathfrak{g}, which means that we do not have any interesting central extension. On the other hand, the punctured affine spaces 𝔸 n{0}\mathbb{A}^n-\{0\} 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 𝒪(𝔸 n{0})\mathcal{O}(\mathbb{A}^n-\{0\}) with the dg-algebra Γ(𝔸 n{0},𝒪)\mathbb{R}\Gamma(\mathbb{A}^{n}-\{0\},\mathcal{O}) of derived sections, which allows interesting central extensions.

This principle can be applied not only to punctured affine spaces 𝔸 n{0}\mathbb{A}^{n}-\{0\}, but also to punctured formal disks 𝔻 n \mathbb{D}_n^\circ.


Faonte-Hennion-Kapranov higher Kac-Moody algebras

Let XX be an nn-dimensional variety over \mathbb{C} and 𝔤\mathfrak{g} a Lie algebra. We can define the higher current algebra by the dg-Lie algebra

𝔤 n=𝔤Γ(𝔻 n ,𝒪 X), \mathfrak{g}_n \;=\; \mathfrak{g}\otimes\mathbb{R}\Gamma(\mathbb{D}_n^\circ,\mathcal{O}_X),

where Γ(𝔻 n ,𝒪 X)\mathbb{R}\Gamma(\mathbb{D}_n^\circ,\mathcal{O}_X) is the commutative dg-algebra of derived global sections of the structure sheaf 𝒪 X\mathcal{O}_X on the punctured formal disk 𝔻 n =Spec([[z 1,,z n]]){0}\mathbb{D}_n^\circ = \mathrm{Spec}(\mathbb{C}[[z_1,\dots,z_n]])-\{0\}.

A higher Kac-Moody algebra 𝔤^ n,Θ\widehat{\mathfrak{g}}_{n,\Theta} is the central extension of the higher current algebra 𝔤 n\mathfrak{g}_n by an invariant polynomial Θ\Theta on 𝔤 n\mathfrak{g}_n of degree (n+1)(n+1).

Gwilliam-Williams higher Kac-Moody factorisation algebras

Let XX be a complex manifold of dimension nn equipped with a holomorphic principal bundle PXP\rightarrow X with structure group GG.

Let 𝔞𝔡(P)P× G𝔤\mathfrak{ad}(P) \coloneqq P\times_G\mathfrak{g} be the adjoint bundle associated to PP. Now, let 𝒜𝒹(P)\mathscr{Ad}(P) be the local Lie algebra whose sections are Ω c 0,*(U,𝔞𝔡(P))\Omega_c^{0,\ast}(U,\mathfrak{ad}(P)) on any open set UXU\subset X and whose differential is the (0,1)(0,1)-connection ¯ P\bar{\partial}_P.

Let Θ\Theta be a degree 11 cocycle in the local Chevalley-Eilenberg cochains CE loc(𝒜𝒹(P))\mathrm{CE}_{\mathrm{loc}}(\mathscr{Ad}(P)), which defines a 11-shifted central extension 𝒜𝒹(P)^ Θ\widehat{\mathscr{Ad}(P)}_\Theta.

The higher Kac-Moody factorization algebra on XX of type Θ\Theta is defined in GW 21 as the twisted enveloping factorization algebra 𝕌 Θ(𝒜𝒹(P))\mathbb{U}_\Theta(\mathscr{Ad}(P)) whose sections are

𝕌 Θ(𝒜𝒹(P))(U)=(Sym(Ω c 0,*(U,𝔞𝔡(P))[1]),¯+d CE+Θ) \mathbb{U}_\Theta\left(\mathscr{Ad}(P)\right)(U) \;=\; \Big(\mathrm{Sym}\left(\Omega_c^{0,\ast}(U,\mathfrak{ad}(P))[1]\right), \; \bar{\partial} + \mathrm{d}_\mathrm{CE} + \Theta \Big)

on any open set UXU\subset X.

Relation between the two

Let r:𝔸 n{0}(0,+)r : \mathbb{A}^{n}_{\mathbb{C}} - \{0\} \rightarrow (0,+\infty) be the radial projection map sending (z 1,,z n)|z 1| 2++|z n| 2(z_1,\dots,z_n)\mapsto \sqrt{|z_1|^2+\dots+|z_n|^2}.

In GW 21 the following map of factorization algebras on the positive reals is constructed:

𝕌(π^ 𝔤,n,Θ):𝕌(Ω c 0,*𝔤^ n,Θ)r *𝕌 Θ(𝒜𝒹(P)), \mathbb{U}(\hat{\pi}_{\mathfrak{g},n,\Theta}) \, : \; \mathbb{U}(\Omega^{0,\ast}_c\otimes\hat{\mathfrak{g}}_{n,\Theta}) \; \longrightarrow \; r_\ast\mathbb{U}_\Theta(\mathscr{Ad}(P)),

where on the left-hand side we have the enveloping factorization algebra which encodes the enveloping A A_\infty -algebra of the FHK 19 higher Kac-Moody algebra 𝔤^ n,Θ\hat{\mathfrak{g}}_{n,\Theta}.

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 𝔤^ n,Θ\hat{\mathfrak{g}}_{n,\Theta} “controls” its corresponding higher Kac-Moody factorization algebra just like an affine Kac-Moody algebra “controls” its corresponding vertex algebra.


Ordinary Kac-Moody algebras

For n=1n=1 and X=𝔸 1X=\mathbb{A}^1_{\mathbb{C}}, one recovers the formal current algebra 𝔤 1=𝔤((z))\mathfrak{g}_1 = \mathfrak{g}((z)), whose central extensions 𝔤˜ 1,Θ\tilde{\mathfrak{g}}_{1,\Theta} are ordinary Kac-Moody algebras.

Affine space 𝔸 n\mathbb{A}^n_{\mathbb{C}}

Consider the affine space X=𝔸 n=Spec([z 1,,z n])X=\mathbb{A}^n_{\mathbb{C}} = \mathrm{Spec}(\mathbb{C}[z_1,\dots,z_n]). The cohomology of the complex of derived global sections Γ(𝔻 n ,𝒪 X)\mathbb{R}\Gamma(\mathbb{D}_n^\circ,\mathcal{O}_X) will be the following:

H i(𝔻 n ,𝒪 X)={[[z 1,,z n]], i=0 z 1 1z n 1[z 1 1,,z n 1], i=n1 0, otherwise \mathrm{H}^i(\mathbb{D}_n^\circ,\mathcal{O}_X) \;=\; \begin{cases}\mathbb{C}[[z_1,\dots,z_n]], & i=0\\z_1^{-1}\cdots z_n^{-1}\mathbb{C}[z_1^{-1},\dots,z_n^{-1}], & i=n-1\\ 0, & \text{otherwise}\end{cases}


Higher Kac-Moody algebras were proposed in

Higher Kac-Moody algebras casted in the language of factorization algebras:

  • Owen Gwilliam, B. R. Williams, Higher Kac-Moody algebras and symmetries of holomorphic field theories, Adv. Theor. Math. Phys. 25 1 (2021) 129-239 [math.QA/1810.06534]

A variant of this construction on the product of a worldsheet and a spectral plane (i.e. another copy of the complex plane \mathbb{C} with marked points which defines any rational Gaudin model):


Last revised on January 4, 2024 at 12:48:30. See the history of this page for a list of all contributions to it.