From Diffeomorphisms to Dark Energy?

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2467-2474 ◽  
Author(s):  
Vincent G. J. Rodgers ◽  
Takeshi Yasuda
Keyword(s):  
Dark Energy ◽  
Lie Algebras ◽  
Virasoro Algebra ◽  
Two Dimensions ◽  
Coadjoint Orbits ◽  
Coadjoint Representation ◽  
Natural Setting ◽  
Affine Lie Algebras ◽  
Yang Mills ◽  
Geometric Action

There are two physical actions that have a natural setting in terms of the coadjoint representation of the algebra of diffeomorphisms and of affine Lie algebras. One is the usual geometric action that comes from coadjoint orbits. The other action lives on the phase space that is transverse to the orbits and are called transverse actions, where Yang-Mills theory in two dimensions is an example. Here we show that the transverse action associated with the Virasoro algebra might contain clues for a theory for dark energy. These actions might also suggests a mechanism for symmetry changing.

scholarly journals Principal subspaces of higher-level standard $\widehat{\mathfrak{sl}(n)}$-modules

2015 ◽  
Vol 26 (08) ◽  
pp. 1550053 ◽  
Author(s):  
Christopher Sadowski
Keyword(s):  
Lie Algebras ◽  
Operator Algebras ◽  
Intertwining Operators ◽  
Natural Setting ◽  
Affine Lie Algebras ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Defining Relations ◽  
Standard Formula ◽  
Exact Sequences

Using completions of certain universal enveloping algebras, we provide a natural setting for families of defining relations for the principal subspaces of standard modules for untwisted affine Lie algebras. We also use the theory of vertex operator algebras and intertwining operators to construct exact sequences among principal subspaces of certain standard [Formula: see text]-modules, n ≥ 3. As a consequence, we obtain the multigraded dimensions of the principal subspaces W(k1Λ1 + k2Λ2) and W(kn-2Λn-2 + kn-1Λn-1). This generalizes earlier work by Calinescu on principal subspaces of standard [Formula: see text]-modules.

scholarly journals SELF-DUAL YANG–MILLS: SYMMETRIES AND MODULI SPACE

1999 ◽  
Vol 11 (09) ◽  
pp. 1091-1149 ◽  
Author(s):  
A. D. POPOV
Keyword(s):  
Dimensional Space ◽  
Solution Space ◽  
Virasoro Algebra ◽  
Simons Theory ◽  
Affine Lie Algebras ◽  
Field Equations ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Conformal Field ◽  
Chern Simons

Geometry of the solution space of the self-dual Yang–Mills (SDYM) equations in Euclidean four-dimensional space is studied. Combining the twistor and group-theoretic approaches, we describe the full infinite-dimensional symmetry group of the SDYM equations and its action on the space of local solutions to the field equations. It is argued that owing to the relation to a holomorphic analogue of the Chern–Simons theory, the SDYM theory may be as solvable as 2D rational conformal field theories, and successful nonperturbative quantization may be developed. An algebra acting on the space of self-dual conformal structures on a 4-space (an analogue of the Virasoro algebra) and an algebra acting on the space of self-dual connections (an analogue of affine Lie algebras) are described. Relations to problems of topological and N=2 strings are briefly discussed.

scholarly journals GENERAL COORDINATE TRANSFORMATIONS AS THE ORIGINS OF DARK ENERGY

2007 ◽  
Vol 22 (04) ◽  
pp. 749-776 ◽  
Author(s):  
VINCENT G. J. RODGERS ◽  
TAKESHI YASUDA
Keyword(s):  
Dark Energy ◽  
Scalar Field ◽  
Virasoro Algebra ◽  
Field Theories ◽  
One Dimension ◽  
Coordinate Transformations ◽  
Moody Algebra ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Phantom Dark Energy

In this note we demonstrate that the algebra associated with coordinate transformations might contain the origins of a scalar field that can behave as an inflaton and/or a source for dark energy. We will call this particular scalar field the diffeomorphism scalar field. In one dimension, the algebra of coordinate transformations is the Virasoro algebra while the algebra of gauge transformations is the Kac–Moody algebra. An interesting representation of these algebras corresponds to certain field theories that have meaning in any dimension. In particular, the so-called Kac–Moody sector corresponds to Yang–Mills theories and the Virasoro sector corresponds to the diffeomorphism field theory that contains the scalar field and a rank-two symmetric, traceless tensor. We will focus on the contributions of the diffeomorphism scalar field to cosmology. We show that this scalar field can, qualitatively, act as a phantom dark energy, an inflaton, a dark matter source, and the cosmological constant Λ.

Foliations Formed by Generic Coadjoint Orbits of a Class of Real Seven-Dimensional Solvable Lie Groups

2021 ◽  
Vol 61 ◽  
pp. 79-104
Author(s):  
Tuyen Nguyen ◽  
◽  
Vu Le
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Topological Classification ◽  
Coadjoint Orbits ◽  
Coadjoint Representation ◽  
Nilpotent Lie Algebra ◽  
The Family ◽  
Measurable Foliation ◽  
Simply Connected

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to seven-dimensional Lie algebras such that their nilradical is a five-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ given in Table~\ref{tab1}. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.

scholarly journals Exact extended supersymmetry on a lattice: Twisted N=2 super-Yang–Mills in two dimensions

2006 ◽  
Vol 633 (4-5) ◽  
pp. 645-652 ◽  
Author(s):  
Alessandro D'Adda ◽  
Issaku Kanamori ◽  
Noboru Kawamoto ◽  
Kazuhiro Nagata
Keyword(s):  
Extended Supersymmetry ◽  
Two Dimensions ◽  
Yang Mills

A class of non-standard modules for affine Lie algebras

1987 ◽  
Vol 196 (3) ◽  
pp. 303-313 ◽  
Author(s):  
Nolan R. Wallach
Keyword(s):  
Lie Algebras ◽  
Affine Lie Algebras
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