Dimensional oxidization on coset space
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Springer
Received: June 5, Revised: August 27, Accepted: September 28, Published: October 29,
2020 2020 2020 2020
Koichi Harada,a Pei-Ming Ho,b Yutaka Matsuoc and Akimi Watanabea a
Department of Physics, Faculty of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan b Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 106, Taiwan, R.O.C. c Department of Physics & Trans-scale Quantum Science Institute & Mathematics and Informatics Center, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: In the matrix model approaches of string/M theories, one starts from a generic symmetry gl(∞) to reproduce the space-time manifold. In this paper, we consider the generalization in which the space-time manifold emerges from a gauge symmetry algebra which is not necessarily gl(∞). We focus on the second nontrivial example after the toroidal compactification, the coset space G/H, and propose a specific infinite-dimensional symmetry which realizes the geometry. It consists of the gauge-algebra valued functions on the coset and Lorentzian generator pairs associated with the isometry. We show that the 0-dimensional gauge theory with the mass and Chern-Simons terms gives the gauge theory on the coset with scalar fields associated with H. Keywords: Gauge Symmetry, Matrix Models ArXiv ePrint: 2005.13936
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)198
JHEP10(2020)198
Dimensional oxidization on coset space
Contents 1
2 A brief review of the cosets G/H 2.1 Lie algebra decomposition 2.2 Coset representative 2.3 Adjoint representation 2.4 Vielbein and H-connection 2.5 Infinitesimal isometry transformation
2 2 2 3 3 4
3 An infinite dimensional Lie algebra for G/H 3.1 Examples
6 7
ˆ 4 Gauge theory with symmetry algebra C 4.1 Cˆ components as k-valued fields on G/H 4.2
Solution for Z
8 9 10
5 Gauge theory with quadratic and cubic terms 5.1 Modified action 5.2 Gauge invariance
11 11 13
6 Reduction to the conventional gauge theory on the coset: G/H = SU(2)/U(1) = S 2
14
ˆ a into gauge potential and scalar field 7 Decomposition of X 7.1 Algebraic properties of basis 7.2 Transformation properties of basis
15 16 16
8 Gauge theory on coset space from 0-dimension
17
9 Conclusion and discussion
20
A Check of Jacobi identity and invariance of inner product
22
B Coset representation of S 2
23
–i–
JHEP10(2020)198
1 Introduction
1
Introduction
h
i
C TnA , TmB = iF AB C Tn+m + nv GAB ,
h
i
u, TnA = nTnA .
(1.1)
We note that the additional generator u describes the derivative with respect to θ. The center v is necessary to generate the KK-mass. As a straightforward generalization, one may obtain an algebraic description of the toroidal compactification T n from the n-loop algebra.1 In this paper, as the next simplest example, we consider the algebra
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