Abstract
Regularizing a volume preserving diffeomorphism (VPD) is equivalent to a long standing problem, namely regularizing a Nambu–Poisson bracket. In this paper, as a first step toward regularizing VPD, we find general complete independent bases of VPD algebra. Especially, we find a complete independent basis that gives simple structure constants, where three area preserving diffeomorphism algebras are manifest. This implies that an algebra that regularizes a VPD algebra should include three \(u(N)\) Lie algebras.
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1 Introduction
The area preserving diffeomorphism (APD) algebra is regularized by the \(u(N)\) Lie algebra. Actually, a large \(N\) limit of structure constants of \(u(N)\) Lie algebra in the ’t Hooft basis reduces to those of the APD algebra defined on \(T^2\) [1–4]. Because the APD algebra is generated by the Poisson bracket, it is regularized by the Lie bracket of the \(u(N)\) Lie algebra. This structure induces the following: the Heisenberg picture of quantum mechanics reduces to the canonical formalism of classical mechanics in the classical limit. Another application is that one can show that BFSS matrix theory and the IIB matrix model contain the lightcone supermembrane and the type IIB superstring, respectively, by using this regularization [5–7].
On the other hand, regularizing the Nambu–Poisson bracket is a long standing problemFootnote 1 [14–37]. As in the case of APD, the Nambu–Poisson bracket generates a volume preserving diffeomorphism (VPD) algebra. In this paper, as a first step toward regularizing the Nambu–Poisson bracket, we search for several independent bases of the VPD algebra and obtain simple structure constants.
2 General complete independent bases of VPD algebra
VPD is a diffeomorphism \(x^i \rightarrow y^i(x)\) (\(i=1,2,3\)) that satisfies \(\text{ det }\partial _i y^j(x)=1\). Then the infinitesimal transformation \(y^i(x)=x^i+ \delta x^i(x)\) satisfies
Also \(\delta x^i(x) = \epsilon ^{i j k}\partial _j f(x) \partial _k g(x)\) satisfy this equation. Transformations of a scalar field generated by these solutions are given by
This implies that the Nambu–Poisson bracket generates VPD. The transformations
form the VPD algebra.
The APD is a two-dimensional analog of the VPD. The infinitesimal transformations
on \(T^2\) \((I,J=1,2)\), where
are spanned by the generators
which are obtained by substituting \( F(Y)=\mathrm{e}^{iAY}\) into (2.4).
On the other hand, a complete independent basis of VPD cannot be obtained by substituting \( f(x)=\mathrm{e}^{iax}\) and \(g(x)=\mathrm{e}^{ibx}\) into (2.3) because \(\delta x^i(x)\) is a local vector in three dimensions. We need to solve (2.1). In the case of APD, (2.6) are complete independent solutions of (2.5). On \(T^3\), we make a Fourier transformation, \(\delta x^i(x)= \sum _{a} v^i(a)\mathrm{e}^{iax}. \) Equation (2.1) implies
An independent solution of (2.7) is given by
for \(a=(a_1, a_2, a_3)\) (except \(a_1=a_2=0\)), and
for \(a=(0,0,a_3)\).
The corresponding VPD generators are given by
which form the VPD algebra
where
This algebra has a complicated form because the bases (2.10) and (2.11) are complicated.
General independent solutions of (2.7) are given by
where \(a\), \(l(a)\), and \(m(a)\) are all independent for all \(a\). The corresponding generators are given by
where \(\text{ det }(abc):= \epsilon ^{ijk}a_i b_j c_k\).
If we choose \(l=(0,0,1)\) and \(m=(-a_2, a_1,0)\) for \(a=(a_1,a_2,a_3)\) (except \(a_1=a_2=0\)), (2.14) and (2.15) represent (2.8) and (2.10), respectively. If we choose \(l=(0,-\frac{1}{a_3},0)\) and \(m=(\frac{1}{a_3},0,0)\) for \(a=(0,0,a_3)\), (2.14) and (2.15) represent (2.9) and (2.11), respectively.
3 Simple structure constants of VPD algebra
In this section, we search for a complete independent basis that gives more simple structure constants. Although (2.15) for constant \(l\) and \(m\) are not independent in a part of the region of \(a\) where \(a\) is on a plane spanned by \(l\) and \(m\), we can calculate the commutation relations among (2.15) for constant \(l\) and \(m\), and then obtain simple relations:
For example, if we choose \(l=(0,0,1)\) and \(m=(1,0,0)\), we obtain \(v_1=(-a_2, a_1, 0)\), \(v_2=(0, -a_3, a_2)\), and the corresponding generators
In this case, for \(a_2=0\), \(v_1=(0, a_1, 0)\) and \(v_2=(0, -a_3, 0)\) are dependent, and thus \(U_1(a)\) and \(U_2(a)\) are.
Then we choose a step function, \(m=(1,0,0)\) for \(a_2 \ne 0\) and \(m=(0,1,0)\) for \(a_2 = 0\). When \(m=(0,1,0)\) we have \(v_3=(a_3, 0, -a_1)\) and
which is independent of \(U_1(a)\) and \(U_2(a)\) for \(a_2 = 0\). After considering \(v_1=0\) when \(a_1=a_2=0\), we have a complete set of independent generators,
In fact, for each \(a\) there are two independent generators:
Then we obtain the simple structure constants of the VPD algebra,
From (3.8), (3.9), and (3.10), one can see three APD algebras corresponding to the \((x^1, x^2)\), \((x^2, x^3)\), and \((x^3, x^1)\) planes.
4 Conclusion and discussion
In this paper, we found general complete independent bases of the VPD algebra. Especially, we found a complete independent basis that gives simple structure constants where the three APD algebras are manifest. This implies that an algebra that regularizes a VPD algebra should include three \(u(N)\) Lie algebras.
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Acknowledgments
We would like to thank T. Asakawa, K. Hashimoto, N. Ikeda, N. Kamiya, H. Kunitomo, T. Matsuo, S. Moriyama, K. Murakami, J. Nishimura, S. Sasa, P. Schupp, F. Sugino, T. Tada, S. Terashima, S. Watamura, K. Yoshida, and especially H. Kawai and A. Tsuchiya for valuable discussions. This work is supported in part by a Grant-in-Aid for Young Scientists (B) No. 25800122 from JSPS.
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Sato, M. On the structure constants of volume preserving diffeomorphism algebra. Eur. Phys. J. C 74, 2878 (2014). https://doi.org/10.1140/epjc/s10052-014-2878-3
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DOI: https://doi.org/10.1140/epjc/s10052-014-2878-3