scholarly journals The Lie group of automorphisms of a Courant algebroid and the moduli space of generalized metrics

2019 ◽  
Vol 36 (2) ◽  
pp. 485-536
Author(s):  
Roberto Rubio ◽  
Carl Tipler
Keyword(s):  
Lie Group ◽  
Moduli Space ◽  
Courant Algebroid ◽  
Group Of Automorphisms ◽  
Generalized Metrics

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 Geometric Langlands duality and the equations of Nahm and Bogomolny

2010 ◽  
Vol 140 (4) ◽  
pp. 857-895 ◽  
Author(s):  
Edward Witten
Keyword(s):  
Gauge Theory ◽  
Lie Group ◽  
Moduli Space ◽  
Dual Group ◽  
Langlands Duality ◽  
Geometric Langlands

Geometric Langlands duality relates a representation of a simple Lie group Gv to the cohomology of a certain moduli space associated with the dual group G. In this correspondence, a principal SL2 subgroup of Gv makes an unexpected appearance. This can be explained using gauge theory, as this paper will show, with the help of the equations of Nahm and Bogomolny.

Central Extensions of Loop Groups and Obstruction to Pre-Quantization

2013 ◽  
Vol 56 (1) ◽  
pp. 116-126 ◽  
Author(s):  
Derek Krepski
Keyword(s):  
Riemann Surface ◽  
Lie Group ◽  
Moduli Space ◽  
Explicit Construction ◽  
Loop Groups ◽  
Central Extensions ◽  
Simply Connected

AbstractAn explicit construction of a pre-quantumline bundle for themoduli space of flat G-bundles over a Riemann surface is given, where G is any non-simply connected compact simple Lie group. This work helps to explain a curious coincidence previously observed between Toledano Laredo's work classifying central extensions of loop groups LG and the author's previous work on the obstruction to pre-quantization of the moduli space of flat G-bundles.

Limits of equisymmetric $1$-complex dimensional families of Riemann surfaces

2017 ◽  
Vol 121 (1) ◽  
pp. 26 ◽  
Author(s):  
Antonio F. Costa ◽  
Víctor González-Aguilera
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Group Of Automorphisms ◽  
The Family

We describe the limit surfaces of some equisymmetric $1$-complex dimensional families of Riemann surfaces in the boundary of the Deligne-Mumford compactification of moduli space. We provide a description of such nodal Riemann surfaces in terms of the group of automorphisms defining the family. We apply our method to some known examples.

scholarly journals Generalized Ricci flow on nilpotent Lie groups

2021 ◽  
Vol 33 (4) ◽  
pp. 997-1014
Author(s):  
Fabio Paradiso
Keyword(s):  
Heisenberg Group ◽  
Lie Group ◽  
Ricci Flow ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Geometric Flows ◽  
Courant Algebroid ◽  
New Family ◽  
Lie Brackets ◽  
Simply Connected

Abstract We define solitons for the generalized Ricci flow on an exact Courant algebroid. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.

scholarly journals Poisson-geometric Analogues of Kitaev Models

2021 ◽  
Vol 383 (1) ◽  
pp. 345-400
Author(s):  
Alexander Spies
Keyword(s):  
Lie Group ◽  
Moduli Space ◽  
Symplectic Structure ◽  
Vector Fields ◽  
Lattice Models ◽  
Embedded Graph ◽  
Adjacent Face ◽  
Kitaev Model ◽  
Poisson Data ◽  
Heisenberg Double

AbstractWe define Poisson-geometric analogues of Kitaev’s lattice models. They are obtained from a Kitaev model on an embedded graph $$\Gamma $$ Γ by replacing its Hopf algebraic data with Poisson data for a Poisson-Lie group G. Each edge is assigned a copy of the Heisenberg double $${\mathcal {H}}(G)$$ H ( G ) . Each vertex (face) of $$\Gamma $$ Γ defines a Poisson action of G (of $$G^*$$ G ∗ ) on the product of these Heisenberg doubles. The actions for a vertex and adjacent face form a Poisson action of the double Poisson-Lie group D(G). We define Poisson counterparts of vertex and face operators and relate them via the Poisson bracket to the vector fields generating the actions of D(G). We construct an isomorphism of Poisson D(G)-spaces between this Poisson-geometrical Kitaev model and Fock and Rosly’s Poisson structure for the graph $$\Gamma $$ Γ and the Poisson-Lie group D(G). This decouples the latter and represents it as a product of Heisenberg doubles. It also relates the Poisson-geometrical Kitaev model to the symplectic structure on the moduli space of flat D(G)-bundles on an oriented surface with boundary constructed from $$\Gamma $$ Γ .

scholarly journals Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

2017 ◽  
Vol 25 (2) ◽  
pp. 99-135
Author(s):  
Rory Biggs
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Isometry Group ◽  
Three Dimensional ◽  
Isotropy Subgroup ◽  
Left Translation ◽  
Isometry Groups ◽  
Group Automorphism ◽  
Group Of Automorphisms ◽  
Simply Connected

Abstract We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in the group of automorphisms of the Lie group. It turns out (in both the Riemannian and sub-Riemannian cases) that for most structures any isometry is the composition of a left translation and a Lie group automorphism.

The Lie group of automorphisms of a principle bundle

1989 ◽  
Vol 6 (2) ◽  
pp. 215-235 ◽  
Author(s):  
M.C. Abbati ◽  
R. Cirelli ◽  
A. Mania' ◽  
P. Michor
Keyword(s):  
Lie Group ◽  
Group Of Automorphisms

scholarly journals Octonionic geometry and conformal transformations

2016 ◽  
Vol 13 (07) ◽  
pp. 1650092 ◽  
Author(s):  
Merab Gogberashvili
Keyword(s):  
Cosmological Constant ◽  
Minkowski Space ◽  
Lie Group ◽  
Space Time ◽  
Conformal Transformations ◽  
Group Of Automorphisms ◽  
Minkowski Space Time ◽  
Split Octonions

We describe space-time using split octonions over the reals and use their group of automorphisms, the noncompact form of Cartan’s exceptional Lie group G2, as the main geometrical group of the model. Connections of the G2-rotations of octonionic 8D space with the conformal transformations in 4D Minkowski space-time are studied. It is shown that the dimensional constant needed in these analysis naturally gives the observed value of the cosmological constant.

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