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The Noether map as an L-$\infty$-algebra central extension of variational symmetries by higher topological conserved currents

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Igor Khavkine
Wed, 2016-10-05 10:30 - 12:00
MPIM Lecture Hall

I will report on joint work in progress with Urs Schreiber. It is well-known that, upon quantization, the algebra of classical symmetries may acquire a central extension (sometimes known as an anomaly) when implemented via the commutators of conserved charge operators, with anomalies playing an important role, e.g., in current algebras. While this central extension may be of purely quantum origin, it is possible for such an extension to arise already at the classical level (a classical anomaly), when the symmetry algebra is implemented via the Poisson brackets of conserved charges associated with symmetries by Noether's theorem. Thus, to disentangle the two effects, it is important to study the classical anomaly separately. In an $n$-dimensional Lagrangian Field Theory, conserved charges are obtained by integrating local conserved currents (on-shell closed $(n-1)$-forms) and their Poisson brackets can be naturally lifted to the conserved currents themselves (the Dickey bracket). The Noether map is then a Lie algebra homomorphism from conserved currents to the symmetries of the Lagrangian, which makes manifest a central extension of the algebra of symmetries by its kernel, namely topological conserved currents. The charges of topological conserved currents happen to have the property of being constant on homotopy classes of field configurations. However, as noted long ago by de Azcárraga et al., this central extension does not exhaust the classical anomaly because it does not take into account charges obtained by integrating over lower than $(n-1)$-dimensional surfaces, with integrands represented by higher conserved current} (on-shell closed forms of degree lower than $n-1$). By appealing to modern extensions of Noether's theorem and the treatment of higher conserved currents via the variational bicomplex on infinite jets, we can unite both ordinary and higher conserved currents into an L-$\infty$-algebra and extend the Noether map to an L-$\infty$-algebra homomorphism, whose kernel is the abelian L-$\infty$-algebra of all topological currents, thus exhibiting the classical anomaly as an L-$\infty$-algebra central extension of the symmetries of the Lagrangian by topological currents. Along the way, we implement infinite jet bundles and the variational bicomplex in the category of higher smooth stacks, which has the immediate benefit of giving a global interpretation to field theories that are only locally Lagrangian and conservation laws that are only locally defined, as well as establishing a foundation for extending the above results to ordinary and higher gauge theories.

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