1,469 research outputs found
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symmetries of E_omega which preserve the 2-plectic structure on M.Comment: These preliminary results have been superseded by those given in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
Unique Quantum Stress Fields
We have recently developed a geometric formulation of the stress field for an
interacting quantum system within the local density approximation (LDA) of
density functional theory (DFT). We obtain a stress field which is invariant
with respect to choice of energy density. In this paper, we explicitly
demonstrate this uniqueness by deriving the stress field for different energy
densities. We also explain why particular energy densities give expressions for
the stress field that are more tractable than others, thereby lending
themselves more easily to first-principles calculations.Comment: To appear in Proceedings for Fundamental Physics of Ferroelectrics
(2001
Categorified Symplectic Geometry and the String Lie 2-Algebra
Multisymplectic geometry is a generalization of symplectic geometry suitable
for n-dimensional field theories, in which the nondegenerate 2-form of
symplectic geometry is replaced by a nondegenerate (n+1)-form. The case n = 2
is relevant to string theory: we call this 2-plectic geometry. Just as the
Poisson bracket makes the smooth functions on a symplectic manifold into a Lie
algebra, the observables associated to a 2-plectic manifold form a "Lie
2-algebra", which is a categorified version of a Lie algebra. Any compact
simple Lie group G has a canonical 2-plectic structure, so it is natural to
wonder what Lie 2-algebra this example yields. This Lie 2-algebra is
infinite-dimensional, but we show here that the sub-Lie-2-algebra of
left-invariant observables is finite-dimensional, and isomorphic to the already
known "string Lie 2-algebra" associated to G. So, categorified symplectic
geometry gives a geometric construction of the string Lie 2-algebra.Comment: 16 page
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
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