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

$$\begin{aligned} \partial _i \delta x^i(x)=0. \end{aligned}$$
(2.1)

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

$$\begin{aligned} \delta Z(x)&\equiv \delta x^i(x) \partial _i Z(x) \nonumber \\&= \epsilon ^{i j k} \partial _i f(x) \partial _j g(x) \partial _k Z(x) \nonumber \\&= \{ f(x), g(x), Z(x)\}. \end{aligned}$$
(2.2)

This implies that the Nambu–Poisson bracket generates VPD. The transformations

$$\begin{aligned} \delta =\delta x^i(x) \partial _i=\epsilon ^{i j k} \partial _i f(x) \partial _j g(x) \partial _k \end{aligned}$$
(2.3)

form the VPD algebra.

The APD is a two-dimensional analog of the VPD. The infinitesimal transformations

$$\begin{aligned} \delta =\delta X^I(Y) \partial _I=\epsilon ^{I J} \partial _I F(Y)\partial _J \end{aligned}$$
(2.4)

on \(T^2\) \((I,J=1,2)\), where

$$\begin{aligned} \partial _I \delta X^I(X)=0, \end{aligned}$$
(2.5)

are spanned by the generators

$$\begin{aligned} \delta (A)=i\mathrm{e}^{iAY}\epsilon ^{I J} A_I \partial _J, \end{aligned}$$
(2.6)

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

$$\begin{aligned} a_i v^i(a)=0. \end{aligned}$$
(2.7)

An independent solution of (2.7) is given by

$$\begin{aligned} \bar{v}_1&= (-a_2, a_1, 0) ,\nonumber \\ \bar{v}_2&= a \times \bar{v}_1=(-a_1 a_3, -a_2 a_3, a_1^2+a_2^2), \end{aligned}$$
(2.8)

for \(a=(a_1, a_2, a_3)\) (except \(a_1=a_2=0\)), and

$$\begin{aligned} \bar{v}_1^{'}&= (1,0,0) ,\nonumber \\ \bar{v}_2^{'}&= (0,1,0), \end{aligned}$$
(2.9)

for \(a=(0,0,a_3)\).

The corresponding VPD generators are given by

$$\begin{aligned}&S_1(a)=\mathrm{e}^{iax}\bar{v}_1^i \partial _i= \mathrm{e}^{iax}(-a_2 \partial _1 + a_1 \partial _2) ,\nonumber \\&S_2(a)=\mathrm{e}^{iax}\bar{v}_2^i \partial _i= \mathrm{e}^{iax}(-a_1 a_3 \partial _1 - a_2 a_3 \partial _2 \nonumber \\&\qquad \qquad + (a_1^2+a_2^2) \partial _3),\end{aligned}$$
(2.10)
$$\begin{aligned}&S_1'(0,0,a_3)=\mathrm{e}^{ia_3x^3}\bar{v}_1^{'i} \partial _i= \mathrm{e}^{ia_3x^3}\partial _1,\nonumber \\&S_2'(0,0,a_3)=\mathrm{e}^{ia_3x^3}\bar{v}_2^{'i} \partial _i= \mathrm{e}^{ia_3x^3}\partial _2, \end{aligned}$$
(2.11)

which form the VPD algebra

$$\begin{aligned}&[S_1(a), S_1(b)]=i (a_1 b_2 - a_2 b_1) S_1(a+b) ,\nonumber \\&[S_2(a), S_2(b)] = i \alpha S_1(a+b)+ i \beta S_2(a+b), \nonumber \\&[S_1(a), S_2(b)]=i \gamma S_1(a+b)+ i \delta S_2(a+b), \nonumber \\&[S_1(a), S_1'(0,0,b_3)]=-i a_1 S_1(a+b) ,\nonumber \\&[S_2(a), S_1'(0,0,b_3)]=-i a_2 b_3 S_1(a+b)-ia_1 S_2(a+b) ,\nonumber \\&[S_1(a), S_2'(0,0,b_3)]=-i a_2 S_1(a+b) ,\nonumber \\&[S_2(a), S_2'(0,0,b_3)]=i a_1 b_3 S_1(a+b)-ia_2 S_2(a+b), \nonumber \\&[S_1'(0,0,a_3), S_1'(0,0,b_3)]=[S_2'(0,0,a_3), S_2'(0,0,b_3)]\nonumber \\&\quad =[S_1'(0,0,a_3), S_2'(0,0,b_3)]=0, \end{aligned}$$
(2.12)

where

$$\begin{aligned} \alpha&= \frac{1}{(a_1+b_1)^2+(a_2+b_2)^2}(a_2 b_1-a_1 b_2)\big (\big (a_1^2+a_2^2\big )b_3^2 \nonumber \\&+ \big (b_1^2+b_2^2\big ) a_3^2-2 a_3 b_3(a_1b_1+a_2 b_2)\big ),\nonumber \\ \beta&= \frac{1}{(a_1+b_1)^2+(a_2+b_2)^2}\big (\big (a_1^2+a_2^2\big )b_3((a_1+b_1)b_1\nonumber \\&+(a_2+b_2)b_2) \nonumber \\&- \big (b_1^2+b_2^2\big )a_3((a_1+b_1)a_1+(a_2+b_2)a_2)\big ) ,\nonumber \\ \gamma&= \frac{1}{a_1+b_1}\left( \frac{1}{ (a_1+b_1)^2+(a_2+b_2)^2}\big (b_1^2+b_2^2\big )(a_1 b_2\nonumber \right. \\&-a_2 b_1)(a_2+b_2)(a_3+b_3) \nonumber \\&-b_2b_3(a_1b_2-a_2b_1)-a_1\big (-b_3(a_1b_1+a_2b_2)\nonumber \\&\left. +a_3\big (b_1^2+b_2^2\big )\big )\right) ,\nonumber \\ \delta&= \frac{1}{(a_1+b_1)^2+(a_2+b_2)^2} \big (b_1^2+b_2^2\big )(a_1 b_2-a_2 b_1).\nonumber \\ \end{aligned}$$
(2.13)

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

$$\begin{aligned} v_1^i&= \epsilon ^{ijk}a_j l_k(a) ,\nonumber \\ v_2^i&= \epsilon ^{ijk}a_j m_k(a), \end{aligned}$$
(2.14)

where \(a\), \(l(a)\), and \(m(a)\) are all independent for all \(a\). The corresponding generators are given by

$$\begin{aligned} T_1(a)&:= \mathrm{e}^{i a \cdot x} \text{ det }(l a \partial ) ,\nonumber \\ T_2(a)&:= \mathrm{e}^{i a \cdot x} \text{ det }(m a \partial ), \end{aligned}$$
(2.15)

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:

$$\begin{aligned}&[T_1(a), T_1(b)]=i \text{ det }(lab) T_1(a+b) ,\end{aligned}$$
(3.1)
$$\begin{aligned}&[T_2(a), T_2(b)]=i \text{ det }(mab) T_2(a+b), \end{aligned}$$
(3.2)
$$\begin{aligned}&[T_1(a), T_2(b)]=i \frac{1}{\text{ det }(lm(a+b))}(\text{ det }(mab)\text{ det }(lma)\nonumber \\&\quad \times T_1(a+b) +\text{ det }(lab)\text{ det }(lmb)T_2(a+b)). \end{aligned}$$
(3.3)

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

$$\begin{aligned} U_1(a)&= \mathrm{e}^{iax}(-a_2 \partial _1 + a_1 \partial _2) ,\nonumber \\ U_2(a)&= \mathrm{e}^{iax}(- a_3 \partial _2 + a_2 \partial _3). \end{aligned}$$
(3.4)

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

$$\begin{aligned} U_3(a)=\mathrm{e}^{iax}(a_3 \partial _1 - a_1 \partial _3), \end{aligned}$$
(3.5)

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,

$$\begin{aligned}&U_1(a) \text{(except } a_1=a_2=0) , \nonumber \\&U_2(a) \text{(except } a_2=0) ,\nonumber \\&U_2(0,0,a_3) ,\nonumber \\&U_3(a_1, 0, a_3). \end{aligned}$$
(3.6)

In fact, for each \(a\) there are two independent generators:

$$\begin{aligned}&U_1(a) \text{ and } U_2(a) \text{ for } a_2 \ne 0,\nonumber \\&U_1(a) \text{ and } U_3(a) \text{ for } a_2=0 \hbox { and } a_1 \ne 0,\nonumber \\&U_2(a) \text{ and } U_3(a) \text{ for } a_2=a_1=0. \end{aligned}$$
(3.7)

Then we obtain the simple structure constants of the VPD algebra,

$$\begin{aligned}&[U_1(a), U_1(b)]=i (a_1 b_2 - a_2 b_1) U_1(a+b) ,\end{aligned}$$
(3.8)
$$\begin{aligned}&[U_2(a), U_2(b)]=i (a_2 b_3 - a_3 b_2) U_2(a+b) ,\end{aligned}$$
(3.9)
$$\begin{aligned}&[U_3(a_1, 0, a_3), U_3(b_1, 0, b_3)] \nonumber \\&\quad =i (a_3 b_1 - a_1 b_3) U_3(a_1+b_1, 0, a_3+b_3) ,\end{aligned}$$
(3.10)
$$\begin{aligned}&[U_1(a), U_2(b)]=i \frac{1}{a_2 +b_2} (a_2(a_2 b_3 - a_3 b_2) U_1(a+b) \nonumber \\&\quad +b_2(a_1 b_2 - a_2 b_1) U_2(a+b)) , \end{aligned}$$
(3.11)
$$\begin{aligned}&[U_1(a), U_3(b_1, 0, b_3) ]=i ((-a_1b_3+a_3b_1+b_1b_3)\nonumber \\&\quad \times U_1(a+b)+b_1^2U_2(a+b)) , \end{aligned}$$
(3.12)
$$\begin{aligned}&[U_1(a), U_3(b_1, 0, b_3)]=i (-b_3^2 U_1(a+b)\nonumber \\&\quad +(a_3b_1-b_3a_1-b_1b_3)U_2(a+b)). \end{aligned}$$
(3.13)

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.