arXiv subjects

Christopher L. Rogers

Publications and source records attributed to Christopher L. Rogers.

27 records · Page 2Linked to original sources

arXiv2012

A higher Chern-Weil derivation of AKSZ sigma-models

Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. Kotov and Strobl have observed that this naturally generalizes from Lie algebras to dg-manifolds and dg-bundles and that the Chern-Simons action functional associated this way to an $n$-symplectic manifold is the action functional of the AKSZ $σ$-model whose target space is the given $n$-symplectic manifold (examples of this are the Poisson sigma-model or the Courant sigma-model, including ordinary Chern-Simons theory, or higher dimensional abelian Chern-Simons theory). Here we show how, within the framework of the higher Chern-Weil theory in smooth infinity-groupoids, this result can be naturally recovered and enhanced to a morphism of higher stacks, the same way as ordinary Chern-Simons theory is enhanced to a morphism from the stack of principal G-bundles with connections to the 3-stack of line 3-bundles with connections.

math-ph↗

arXiv2011

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.

math.DG↗

arXiv2011

Higher Symplectic Geometry

We consider generalizations of symplectic manifolds called n-plectic manifolds. A manifold is n-plectic if it is equipped with a closed, nondegenerate form of degree n+1. We show that higher structures arise on these manifolds which can be understood as the categorified or homotopy analogues of important structures studied in symplectic geometry and geometric quantization. Just as a symplectic manifold gives a Poisson algebra of functions, we show that any n-plectic manifold gives a Lie n-algebra containing certain differential forms which we call Hamiltonian. Lie n-algebras are examples of strongly homotopy Lie algebras. They consist of an n-term chain complex equipped with a collection of skew-symmetric multi-brackets that satisfy a generalized Jacobi identity. We then develop the machinery necessary to geometrically quantize n-plectic manifolds. In particular, just as a prequantized symplectic manifold is equipped with a principal U(1)-bundle with connection, a prequantized 2-plectic manifold is equipped with a U(1)-gerbe with 2-connection. A gerbe is a categorified sheaf, or stack, which generalizes the notion of a principal bundle. Furthermore, over any 2-plectic manifold there is a vector bundle equipped with extra structure called a Courant algebroid. This bundle is the 2-plectic analogue of the Atiyah algebroid over a prequantized symplectic manifold. Its space of global sections also forms a Lie 2-algebra, which we use to prequantize the Lie 2-algebra of Hamiltonian forms. Finally, we introduce the 2-plectic analogue of the Bohr-Sommerfeld variety associated to a real polarization, and use this to geometrically quantize 2-plectic manifolds. The output of this procedure is a category of quantum states. We consider a particular example in which the objects of this category can be identified with representations of the Lie group SU(2).

math-ph↗

arXiv2010

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.

math-ph↗

arXiv2009

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.

math-ph↗

arXiv2008

Categorified Symplectic Geometry and the Classical String

A Lie 2-algebra is a "categorified" version of a Lie algebra: that is, a category equipped with structures analogous those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an "n-plectic manifold": a finite-dimensional manifold equipped with a closed nondegenerate (n+1)-form. Here we consider the case n = 2. For any 2-plectic manifold, we construct a Lie 2-algebra of observables. We then explain how this Lie 2-algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2-plectic structure for the string.

math-ph↗

arXiv2001

A Geometric Formulation of Quantum Stress Fields

We present a derivation of the stress field for an interacting quantum system within the framework of local density functional theory. The formulation is geometric in nature and exploits the relationship between the strain tensor field and Riemannian metric tensor field. Within this formulation, we demonstrate that the stress field is unique up to a single ambiguous parameter. The ambiguity is due to the non-unique dependence of the kinetic energy on the metric tensor. To illustrate this formalism, we compute the pressure field for two phases of solid molecular hydrogen. Furthermore, we demonstrate that qualitative results obtained by interpreting the hydrogen pressure field are not influenced by the presence of the kinetic ambiguity.

cond-mat.mtrl-sci↗

arXiv2001

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.

cond-mat.mtrl-sci↗

arXiv2001

Geometric Formulation of Unique Quantum Stress Fields

We present a derivation of the stress field for an interacting quantum system within the framework of local density functional theory. The formulation is geometric in nature and exploits the relationship between the strain tensor field and Riemannian metric tensor field. The resultant expression obtained for the stress field is gauge-invariant with respect to choice of energy density, and therefore provides a unique, well-defined quantity. To illustrate this formalism, we compute the pressure field for two phases of solid molecular hydrogen.

cond-mat.mtrl-sci↗