Dirac Structures in Open Thermodynamics

Abstract We study aspects of two-dimensional nonlinear sigma models with Wess-Zumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space. Our goal in this paper is twofold. In a first part, we investigate the conditions for consistent gauging of sigma models in the presence of a nonclosed 3-form. In the Abelian case, we find that the target of the gauged theory has the structure of a contact Courant algebroid, twisted by a 3-form and two 2-forms. Gauge invariance constrains the theory to (small) Dirac structures of the contact Courant algebroid. In the non-Abelian case, we draw a similar parallel between the gauged sigma model and certain transitive Courant algebroids and their corresponding Dirac structures. In the second part of the paper, we study two-dimensional sigma models related to Jacobi structures. The latter generalise Poisson and contact geometry in the presence of an additional vector field. We demonstrate that one can construct a sigma model whose gauge symmetry is controlled by a Jacobi structure, and moreover we twist the model by a 3-form. This construction is then the analogue of WZW-Poisson structures for Jacobi manifolds.

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Interconnection and composition of Dirac structures for Lagrange-Dirac systems

IEEE Conference on Decision and Control and European Control Conference ◽

10.1109/cdc.2011.6160480 ◽

2011 ◽

Cited By ~ 2

Author(s):

Henry O. Jacobs ◽

Hiroaki Yoshimura

Keyword(s):

Dirac Structures ◽

Dirac Systems

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On linear dirac structures of order p//sup >/sub -//7

Proceedings. The 8th Russian-Korean International Symposium on Science and Technology, 2004. KORUS 2004. ◽

10.1109/korus.2004.1555566 ◽

2005 ◽

Author(s):

E.M. Gorbatenko ◽

G.M. Matveenko

Keyword(s):

Dirac Structures

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Complexification of Dirac Structures and Generalised Kirchhoff Operators

IFAC-PapersOnLine ◽

10.1016/j.ifacol.2015.10.217 ◽

2015 ◽

Vol 48 (13) ◽

pp. 75-80

Author(s):

Dimitri Jeltsema ◽

Arjan van der Schaft

Keyword(s):

Dirac Structures

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Dirac structures on Hilbert spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171299220972 ◽

1999 ◽

Vol 22 (1) ◽

pp. 97-108 ◽

Cited By ~ 2

Author(s):

A. Parsian ◽

A. Shafei Deh Abad

Keyword(s):

Hilbert Space ◽

Linear Operator ◽

Hilbert Spaces ◽

Smooth Manifold ◽

Real Hilbert Space ◽

Dirac Structure ◽

Dirac Structures ◽

Basic Properties ◽

Hilbert Subspace ◽

Densely Defined

For a real Hilbert space(H,〈,〉), a subspaceL⊂H⊕His said to be a Dirac structure onHif it is maximally isotropic with respect to the pairing〈(x,y),(x′,y′)〉+=(1/2)(〈x,y′〉+〈x′,y〉). By investigating some basic properties of these structures, it is shown that Dirac structures onHare in one-to-one correspondence with isometries onH, and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of [1] yields a Dirac structure on some Hilbert space. The graph of any densely defined skew symmetric linear operator on a Hilbert space is, also, shown to be a Dirac structure. For a Dirac structureLonH, everyz∈His uniquely decomposed asz=p1(l)+p2(l)for somel∈L, wherep1andp2are projections. Whenp1(L)is closed, for any Hilbert subspaceW⊂H, an induced Dirac structure onWis introduced. The latter concept has also been generalized.

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