Semiclassical Partition Functions for Gravity with Cosmic Strings

In this paper we describe an approach to construct semiclassical partition functions in gravity which are complete in the sense that they contain a complete description of the differentiable structures of the underlying 4-manifold. In addition, we find our construction naturally includes cosmic strings. We discuss some possible applications of the partition functions in the fields of both quantum gravity and topological string theoryComment: 17 pages, 2 figures, revisions of example

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 Alaska Permanent Fund Dividend and Membership in the State’s Political Community

Despite decades of unmitigated administrative success, the Alaska Permanent Fund Dividend (PFD) is not immune from political and legal controversy. The symbolic and financial importance that Alaskans ascribe to their annual dividend checks has generated disputes between ordinary residents and executive agencies over eligibility. Litigation concerning three dominant status requirements—minimum residency, U.S. citizenship, and felony incarceration—reveal not only the extent to which Alaskans will pursue what they believe to be valid claims on their share of natural resource wealth, but also the limits of full political membership in the state. This Comment frames a sample of the Alaska Supreme Court’s decisions on PFD eligibility in terms of membership in Alaska’s political community. The PFD reflects the Alaska Legislature’s opinion about valid beneficiaries from oil revenues, and the state courts police eligibility at the margin. This Comment therefore argues that the Alaska Supreme Court implicitly determines, on the basis of statutory intent and administrative rule interpretations, “insiders” and “outsiders” within the state’s political community