∞-Lie theory

# Contents

## Idea

In the broad meaning a current algebra is the Poisson bracket algebra of conserved currents of a prequantum field theory, or else its quantized version of the corresponding quantum field theory.

If the symmetry corresponding to the conserved currents via Noether's theorem preserves the given Lagrangian only up to a divergence term, then the current algebra is a central extension of the Lie algebra of the underlying symmetries. This effect makes current algebras tend to be subtle and of particular mathematical interest.

Particularly famous is the case of the WZW sigma model field theory for a string propagating on a Lie group $G$. In this case one chiral half of the algebra of currents is the corresponding affine Lie algebra. In parts of the (mathematical) literature it is this special case which meant by default by the term current algebra.

In fact, taking into account that this divergence term itself in general has higher gauge symmetries, current algebras are secretly strong homotopy Lie algebras/Lie n-algebras. This we discuss below.

## As a homotopy Lie algebra

With local prequantum field theory in dimension $d$ formulated as de Donder-Weyl field theory, every local Lagrangian is incarnated as a principal d-form connection $\mathbf{L}$ on some space $X$, such that its curvature form $\omega$ determines by its kernel the tangents to the solutions of the equations of motion. (see cftcht, sections 3.1, 3.2, 3.4).

What is traditionally considered in the literature is the special case of this where $X$ is the (dualized, first) jet bundle of a field bundle over a smooth manifold (see at multisymplectic geometry) and the underlying principal infinity-bundle of $\mathbf{L}$ is trivial, so that $\mathbf{L}$ is incarnated as a globally defined differential d-form. Moreover, this is traditionally expressed as the product $L \, vol$ of a smooth function times a prescribed volume form. This function $L$ then is what in much of the traditional literure is referred to as the Lagrangian of the theory.

A symmetry of the theory which preserves the Lagrangian up to a divergence is then precisely a quantomorphism of $\mathbf{L}$, namely a pair consisting of the symmetry

$\phi \colon X \stackrel{\simeq}{\longrightarrow} X$

together with an equivalence

$\eta \colon \phi^\ast \mathbf{L} \stackrel{\simeq}{\longrightarrow} \mathbf{L} \,.$

When expressing $\mathbf{L}$ as a modulating morphism $\mathbf{L} \colon X \ongrightarrow \mathbf{B}^d U(1)_{conn}$ into the smooth moduli infinity-stack of circle d-bundles with connection then this is an equivalence in the slice (infinity,1)-topos over $\mathbf{B}^d U(1)_{conn}$:

$\mathbf{QuantMorph}(\mathbf{L}) = \left\{ \array{ X && \stackrel{\phi}{\longrightarrow} && X \\ & \searrow &\swArrow_{\eta}& \swarrow \\ && \mathbf{B}^d U(1)_{conn} } \right\} \,.$

(The detailed definition of this quantomorphism d-group involved differential concretification of the naive automorphism infinity-group in the slice.)

Under Lie differentiation this gives the Poisson bracket Lie n-algebra $\mathfrak{Pois}(\mathbf{L})$. In its dg-Lie algebra version spelled out here and restricted to the special case of globally defined Lagrangian form, this has, in degree 0, precisely the Lie bracket of conserved currents as known from traditional literature, for instance (AGIT89, equations (13), (14)). Details are in cftcht, 3.3.

But the $\mathfrak{Pois}(\mathbf{L})$ contains in addition the equivalence between two central terms, coming from higher gauge transformations. Taking this into account, the extension theorem for $\mathfrak{Pois}(X,\omega)$ says that its truncation to a Lie algebra is an extension by $H^{d-1}_{dR}(X)$. This has been informally argued for instance in AGIT 89, p. 8.

## References

General discussion and application to the Green-Schwarz super p-brane sigma models is in

Discussion from an nPOV is in

Revised on February 24, 2015 19:30:28 by Urs Schreiber (195.113.30.252)