scholarly journals Courant Algebroid Connections and String Effective Actions

2017 ◽  
Author(s):  
B. Jurčo ◽  
J. Visoký
Keyword(s):  
Courant Algebroid

scholarly journals Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Athanasios Chatzistavrakidis ◽  
Grgur Šimunić
Keyword(s):  
Sigma Models ◽  
Sigma Model ◽  
Target Space ◽  
Two Dimensional ◽  
Courant Algebroids ◽  
Dirac Structures ◽  
Courant Algebroid ◽  
Abelian Case ◽  
Jacobi Manifolds ◽  
Jacobi Structure

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.

2-PLECTIC MANIFOLDS AND ITS RELATION WITH COURANT ALGEBROIDS

2012 ◽  
Vol 09 (03) ◽  
pp. 1250015
Author(s):  
M. SHAFIEE
Keyword(s):  
Lie Group ◽  
Lagrangian Submanifolds ◽  
Compact Lie Group ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Hamiltonian Group

In this paper we study the relation between 2-plectic manifolds and Courant algebroids. We establish a relation between 2-Lagrangian submanifolds of 2-plectic manifolds and subbundles of Courant algebroids. Also we show that an action of a compact Lie group G on a 2-plectic manifold (M, ω) can be extended to an action of G on an exact Courant algebroid E over M if and only if G is a subgroup of Hamiltonian group of (M, ω).

scholarly journals Constructions of L∞ Algebras and Their Field Theory Realizations

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Olaf Hohm ◽  
Vladislav Kupriyanov ◽  
Dieter Lüst ◽  
Matthias Traube
Keyword(s):  
Field Theory ◽  
Initial Data ◽  
Vector Space ◽  
Jacobi Identity ◽  
General Theorem ◽  
Commutator Algebra ◽  
Courant Algebroid ◽  
Linear Map ◽  
General Initial Data ◽  
Special Cases

We construct L∞ algebras for general “initial data” given by a vector space equipped with an antisymmetric bracket not necessarily satisfying the Jacobi identity. We prove that any such bracket can be extended to a 2-term L∞ algebra on a graded vector space of twice the dimension, with the 3-bracket being related to the Jacobiator. While these L∞ algebras always exist, they generally do not realize a nontrivial symmetry in a field theory. In order to define L∞ algebras with genuine field theory realizations, we prove the significantly more general theorem that if the Jacobiator takes values in the image of any linear map that defines an ideal there is a 3-term L∞ algebra with a generally nontrivial 4-bracket. We discuss special cases such as the commutator algebra of octonions, its contraction to the “R-flux algebra,” and the Courant algebroid.

scholarly journals CONFORMAL COURANT ALGEBROIDS AND ORIENTIFOLD T-DUALITY

2012 ◽  
Vol 10 (02) ◽  
pp. 1250084 ◽  
Author(s):  
DAVID BARAGLIA
Keyword(s):  
Line Bundle ◽  
Bilinear Form ◽  
Conformal Structure ◽  
Crossed Module ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Circle Bundles ◽  
Twisted Cohomology ◽  
Free Involution ◽  
Special Case

We introduce conformal Courant algebroids, a mild generalization of Courant algebroids in which only a conformal structure rather than a bilinear form is assumed. We introduce exact conformal Courant algebroids and show they are classified by pairs (L, H) with L a flat line bundle and H ∈ H3(M, L) a degree 3 class with coefficients in L. As a special case gerbes for the crossed module (U(1) → ℤ2) can be used to twist TM ⊕ T*M into a conformal Courant algebroid. In the exact case there is a twisted cohomology which is 4-periodic if L2 = 1. The structure of Conformal Courant algebroids on circle bundles leads us to construct a T-duality for orientifolds with free involution. This incarnation of T-duality yields an isomorphism of 4-periodic twisted cohomology. We conjecture that the isomorphism extends to an isomorphism in twisted KR-theory and give some calculations to support this claim.

scholarly journals VB-Courant Algebroids, E-Courant Algebroids and Generalized Geometry

2018 ◽  
Vol 61 (3) ◽  
pp. 588-607 ◽  
Author(s):  
Honglei Lang ◽  
Yunhe Sheng ◽  
Aïssa Wade
Keyword(s):  
Lie Algebra ◽  
Complex Structure ◽  
Contact Structures ◽  
Complex Structures ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Generalized Complex Structures ◽  
Generalized Complex Structure ◽  
Generalized Geometry ◽  
Generalized Complex

AbstractIn this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids, and we construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra gl(V) ⊕ V correspond to complex Lie algebra structures on V.

scholarly journals On LA-Courant Algebroids and Poisson Lie 2-Algebroids

2020 ◽  
Vol 23 (3) ◽  
Author(s):  
M. Jotz Lean
Keyword(s):  
Lie Algebroids ◽  
General Context ◽  
Matched Pairs ◽  
Courant Algebroids ◽  
Dirac Structures ◽  
Courant Algebroid ◽  
The Core ◽  
Precise Sense ◽  
Nondegenerate Case ◽  
New Construction

Abstract This paper reformulates Li-Bland’s definition for LA-Courant algebroids, or Poisson Lie 2-algebroids, in terms of split Lie 2-algebroids and self-dual 2-representations. This definition generalises in a precise sense the characterisation of (decomposed) double Lie algebroids via matched pairs of 2-representations. We use the known geometric examples of LA-Courant algebroids in order to provide new examples of Poisson Lie 2-algebroids, and we explain in this general context Roytenberg’s equivalence of Courant algebroids with symplectic Lie 2-algebroids. We study further the core of an LA-Courant algebroid and we prove that it carries an induced degenerate Courant algebroid structure. In the nondegenerate case, this gives a new construction of a Courant algebroid from the corresponding symplectic Lie 2-algebroid. Finally we completely characterise VB-Dirac and LA-Dirac structures via simpler objects, that we compare to Li-Bland’s pseudo-Dirac structures.

Sign in / Sign up

Export Citation Format

Share Document