L-infinity algebras from multisymplectic geometry

A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the algebraic and geometric structures found in symplectic geometry should naturally arise in 2-plectic geometry. In particular, we showed that just as a symplectic manifold gives a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions. Lie n-algebras are examples of L-infinity algebras: graded vector spaces which are equipped with a collection of skew-symmetric multi-brackets that satisfy a generalized Jacobi identity. Here, we generalize our previous result. Given an n-plectic manifold, we explicitly construct a corresponding Lie n-algebra on a complex consisting of differential forms whose multi-brackets are specified by the n-plectic structure. We also show that any n-plectic manifold gives rise to another kind of algebraic structure known as a differential graded Leibniz algebra. We conclude by describing the similarities between these two structures within the context of an open problem in the theory of strongly homotopy algebras.Comment: 22 pages. To appear in Lett. Math. Phy

Courant algebroids from categorified symplectic geometry

In categorified symplectic geometry, one studies the categorified algebraic and geometric structures that naturally arise on manifolds equipped with a closed nondegenerate (n+1)-form. The case relevant to classical string theory is when n=2 and is called "2-plectic geometry". Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, there is a Lie 2-algebra of observables associated to any 2-plectic manifold. String theory, closed 3-forms and Lie 2-algebras also play important roles in the theory of Courant algebroids. Courant algebroids are vector bundles which generalize the structures found in tangent bundles and quadratic Lie algebras. It is known that a particular kind of Courant algebroid (called an exact Courant algebroid) naturally arises in string theory, and that such an algebroid is classified up to isomorphism by a closed 3-form on the base space, which then induces a Lie 2-algebra structure on the space of global sections. In this paper we begin to establish precise connections between 2-plectic manifolds and Courant algebroids. We prove that any manifold M equipped with a 2-plectic form omega gives an exact Courant algebroid E_omega over M with Severa class [omega], and we construct an embedding of the Lie 2-algebra of observables into the Lie 2-algebra of sections of E_omega. We then show that this embedding identifies the observables as particular infinitesimal symmetries of E_omega which preserve the 2-plectic structure on M.Comment: These preliminary results have been superseded by those given in arXiv:1009.297

The economic consequences of a hung parliament : lessons from February 1974

The British general election on 10 May 2010 delivered Britain’s first hung Parliament since February 1974, and in the run-up, the Conservative Party made much of the economic difficulties Britain faced in the second half of the 1970s in order to try and convince voters that anything other than a Tory vote would risk exposing the nation to the discipline of financial markets. The question of how well equipped an exceptional kind of British government is to deal with exceptional economic circumstances is therefore of paramount importance. This paper argues that the Conservative Party made too much of the impact of the 1974 hung Parliament in precipitating subsequent economic crisis and suggests that as such, there is no reason to assume that the Conservative-Liberal coalition government is ill-equipped to manage British economic affairs in difficult circumstances

Higher U(1)-gerbe connections in geometric prequantization

We promote geometric prequantization to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show fairly generally how there is canonically a tower of higher gauge groupoids and Courant groupoids assigned to a higher prequantization, and establish the corresponding Atiyah sequence as an integrated Kostant-Souriau infinity-group extension of higher Hamiltonian symplectomorphisms by higher quantomorphisms. We also exhibit the infinity-group cocycle which classifies this extension and discuss how its restrictions along Hamiltonian infinity-actions yield higher Heisenberg cocycles. In the special case of higher differential geometry over smooth manifolds we find the L-infinity-algebra extension of Hamiltonian vector fields -- which is the higher Poisson bracket of local observables -- and show that it is equivalent to the construction proposed by the second author in n-plectic geometry. Finally we indicate a list of examples of applications of higher prequantization in the extended geometric quantization of local quantum field theories and specifically in string geometry.Comment: Title changed. Exposition revised. 55 page

Conditions for Parametric and Free-Carrier Oscillation in Silicon Ring Cavities

We model optical parametric oscillation in ring cavities with two-photon absorption, focusing on silicon at 1.55$\mu$m. Oscillation is possible if free-carrier absorption can be mitigated; this can be achieved using carrier sweep-out in a reverse-biased p-i-n junction to reduce the carrier lifetime. By varying the pump power, detuning, and reverse-bias voltage, it is possible to generate frequency combs in cavities with both normal and anomalous dispersion at a wide range of wavelengths including 1.55$\mu$m. Furthermore, a free-carrier self-pulsing instability leads to rich dynamics when the carrier lifetime is sufficiently long.Comment: 7 pages, 12 figures. Presented at 2017 International Topical Meeting on Microwave Photonics. Submitted to Journal of Lightwave Technolog

Homotopy moment maps

Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L-infinity morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such maps. This allows us to easily and explicitly construct a large number of examples. These include results concerning group actions on loop spaces and moduli spaces of flat connections. Relationships are also established with previous work by others in classical field theory, algebroid theory, and dg geometry. Furthermore, we use our theory to geometrically construct various L-infinity algebras as higher central extensions of Lie algebras, in analogy with Kostant's quantization theory. In particular, the so-called `string Lie 2-algebra' arises this way.Comment: Final version will appear in Advances in Mathematics. Results concerning equivariant cohomology strengthened. In particular, we exhibit the explicit relationship between equivariant de Rham cocycles of arbitrary degree and homotopy moment maps. 62 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:1402.0144 by other author