88,157 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symmetries of E_omega which preserve the 2-plectic structure on M.Comment: These preliminary results have been superseded by those given in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
Effective Hamiltonians and dilution effects in kagome and related antiferromagnets
What is the zero-temperature ordering pattern of a Heisenberg antiferromagnet
with large spin length (and possibly small dilution), on the kagome
lattice, or others built from corner-sharing triangles and tetrahedra? First, I
summarize the uses of effective Hamiltonians to resolve the large ground-state
degeneracy, leading to long-range order of the usual kind. Secondly, I discuss
the effects of dilution, in particular to {\it non}-frustration of classical
ground states, in that every simplex of spins is optimally satisfied. Of three
explanations for this, the most complete is Moessner-Chalker
constraint-counting. Quantum zero-point energy may compete with classical
exchange energy in a diluted system, creating frustration and enabling a
spin-glass state. I suggest that the regime of over 97% occupation is
qualitatively different from the more strongly diluted regime.Comment: 11 pages; invited talk at "HFM 2000" (Waterloo, June 2000); submitted
to Can. J. Phy
- …