Generalized Current Algebras

• Generalized current algebras are advanced algebraic structures that extend classical current algebras by incorporating higher contractions via a differential graded Lie algebra (DGLA) framework.
• They enable nontrivial central extensions classified by invariant homogeneous polynomials, leading to higher-order cocycles and connections to quantum anomalies.
• The functorial construction bridges algebraic formulations with topological applications, influencing the study of principal bundles, twisted equivariant cohomology, and anomaly theory.

Generalized current algebras are advanced algebraic structures underlying the symmetry principles and anomalies in field theory, representation theory, and differential geometry. They extend classical current algebras by incorporating higher algebraic structures, central extensions classified by invariant polynomials, and connections to topological and geometric obstructions. Their precise formulation unifies twisted equivariant cohomology, anomaly theory, and the topology of principal bundles.

1. Differential Graded Lie Algebra Construction and Resolution

The standard algebraic model for equivariant cohomology is the cone $C\mathfrak{g}$, generated by Lie derivatives $L(x)$ (degree $0$) and contractions $I(x)$ (degree $-1$), with $x \in \mathfrak{g}$, and governed by Cartan’s formula $[d, I(x)] = L(x)$. In the generalized construction, $C\mathfrak{g}$ is replaced by a differential graded Lie algebra (DGLA) $D\mathfrak{g}$, which resolves the usual current algebraic structure. In $D\mathfrak{g}$, the contraction operators are promoted from a commuting set to a free Lie superalgebra, allowing for higher contractions $I(x^k)$, $k \geq 1$, and assembling all such contractions into

$i(x) = I(x) + I(x^2) + I(x^3) + \dots$

with the differential

$d\,i(x) = \frac{1}{2}[i(x), i(x)] + L(x).$

This recursive structure ensures that at each degree $k$,

$$d\,I(x^k) = \text{(combinatorial bracket in contractions)} + (\text{Lie derivative and higher corrections})$$

and in low degrees, this relation reduces to $d I(x) = L(x)$ and $d I(x^2) = [I(x), I(x)]$ (up to normalization). This "resolution" creates a significantly richer DGLA than $C\mathfrak{g}$, capable of encoding higher equivariant and "twisted" cohomology.

2. Central Extensions, Invariant Polynomials, and Anomalies

A critical feature of $D\mathfrak{g}$ is that, unlike $C\mathfrak{g}$, it admits a broad class of nontrivial central extensions, classified by invariant homogeneous polynomials $p \in (S^n\mathfrak{g}^*)^G$ (i.e., $G$-invariant symmetric polynomials of degree $n$ on $\mathfrak{g}$). For instance, for $p$ of degree $2$ (Killing form type), the extension encodes familiar centrally extended current algebras:

$[I(x), I(y)] = -2p(x, y)c$

where $c$ is central. For $p$ of degree $n \ge 3$, central extensions emerge at higher levels,

$d\,I(x^3) = [I(x), I(x^2)] + p(x^3)c,$

resulting in so-called Mickelsson–Faddeev–Shatashvili (FMS) cocycles that describe anomalies in higher-dimensional current algebras. Such extensions cannot be captured by $C\mathfrak{g}$ alone and underscore the necessity of the full $D\mathfrak{g}$ structure for modeling nontrivial topologies and higher anomalies.

Table: Central Extensions in $D\mathfrak{g}$

Degree of $p$	Modified bracket (schematic)	Geometric/topological significance
$2$	$[I(x), I(y)] = -2p(x, y)c$	Kac–Moody central extension
$n\geq 3$	$d\,I(x^n) = [...] + p(x^n)c$	FMS cocycle (higher anomaly, higher class)

3. Functorial Construction and Twisted Cohomology

Given a DGLA $A$ (such as $D\mathfrak{g}$ or its central extension $D_p\mathfrak{g}$), the general current algebra construction proceeds by associating to a manifold $M$ the Lie algebra of "currents" $C.A(M, A)$ or $S.A(M, A)$, essentially built from $(\Omega(M) \otimes A)$ and the DGLA structure of $A$. When $A = D\mathfrak{g}$, these current algebras generalize the classical current algebra of $\mathfrak{g}$-valued functions on $M$. For nontrivial $p$, the resulting Lie algebras naturally encode both standard and centrally extended (FMS) current algebras, providing a unified algebraic setting for the paper of twisted equivariant cohomology and higher (anomalous) extensions.

4. Topological Applications: Principal Bundles, Lifting Problems, and Chern–Weil Theory

The algebraic developments have significant topological implications. For a principal $G$-bundle over $M$, $G$ integrating $\mathfrak{g}$, the current algebra functor applied to $D\mathfrak{g}$ and its extensions produces groups $DG(M)$ and $D_pG(M)$ acting on the space of connections. Every principal $G$-bundle corresponds to a $DG(M)$-torsor, and the question of whether this torsor lifts to a $D_pG(M)$-torsor is controlled entirely by the vanishing of the Chern–Weil class $cw(p) \in H^{2n}(M)$ associated to the invariant polynomial $p$. That is, the obstruction to such a lift is topological, and its nontriviality reflects the presence of quantum anomalies in the field-theoretic context.

This framework generalizes the classical picture of characteristic classes controlling the topology of bundles and provides a precise algebraic–topological dictionary for interpreting anomalies via central extensions.

5. Unification of Higher Anomalies and Twisted Equivariant Cohomology

By resolving the classical cone construction and extending it to $D\mathfrak{g}$, this approach produces a comprehensive model for twisted or higher equivariant cohomology. The free Lie superalgebra organization of contractions captures higher-degree relations that are invisible in the traditional Cartan model, and the classification of central extensions by invariant polynomials naturally subsumes both standard Kac–Moody and higher FMS cocycles. This structure is tightly connected to anomaly theory in quantum field theory, where, for instance, the failure to lift a bundle corresponds to the presence of an obstruction (quantum anomaly) encoded in a characteristic class.

6. Summary and the Unified Perspective

The $D\mathfrak{g}$ resolution and its central extensions provide a technically robust algebraic backbone for the theory of generalized current algebras. Key novel insights include:

• Contractions $I(x)$ are freely generated, yielding higher contractions $I(x^n)$ organized in a free Lie superalgebra, leading to a DGLA $D\mathfrak{g}$ with a recursively defined differential.
• Central extensions of $D\mathfrak{g}$ correspond one-to-one with invariant homogeneous polynomials $p$, producing nontrivial higher cocycles (FMS) that reflect higher anomalies.
• The functorial current algebra construction $C.A(M, D\mathfrak{g})$ naturally generalizes both classical current algebras and their anomaly-prone extensions.
• In a geometric/topological context, principal $G$-bundles over $M$ give rise to $DG(M)$-torsors, with the existence of a lift to $D_pG(M)$-torsors characterized by the vanishing of $cw(p)$, thereby directly relating algebraic anomalies to topological obstructions.

This synthesis substantially enriches the interplay between algebraic topology, operator algebras, and quantum field theory by providing a cohesive resolution and classification of current algebras, their anomalies, and their topological manifestations (Alekseev et al., 2010).