数理解析研究所講究録
第 1915 巻 2014 年 89-96

89

Triple Systems and Applications to Gauge
Theories
Matsuo Sato
Department of Natural Science, Faculty of Education, Hirosaki University
Bunkyo-cho 1, Hirosaki, Aomori 036-8560, Japan

1

Introduction

It has been expected that there exists -theory, which unifies string theories. In -theory,some structures of 3-algebras were found recently. First, it was found that field theories
applied with $u(N)\oplus u(N)$ hermitian 3-algebras are the Chern-Simons gauge theories that
describe eﬀective actions of coincident supermembranes [1-5], which are fundamental
objects in -theory. In a certain limit, a novel Higgs mechanism works, where the ChernSimons gauge theories become the Yang-Mills theories that describe eﬀective actions of-branes in string theory. Second, 3-algebra models of -theory themselves have been
proposed and were studied in [6-13].
The hermitian 3-algebras [14-51] are special cases, where $<abc>=-<cba>$ , of
hermitian generalized Jordan triple systems $<abc>[52-74]$ . Therefore, it is natural
to extend the $u(N)\oplus u(N)$ hermitian 3-algebras to more general hermitian generalized
Jordan triple systems. Moreover, it is interesting to find a hermitian generalized Jordan
triple system with which a Chern-Simons field theory reduces to a Yang-Mills theory ina certain limit.
In the following section, we review some results concerning with [75, 76].
$M$

$M$

$N$

$M$

$M$

$D$

2

Definitions

Let us start with a definition of a hermitian generalized Jordan triple systems.

Definition. A triple systemsystems if relations (0) $-(iv)$ satisfy;O)
is a Banach space,$U$

is said to be a hermitian generalized Jordan triple$U$i) $[L(a, b), L(c, d)]=L(<abc>, d)-L(c, <bad>)$ ,ii) $<xyz>is$ $C$ -linear operator on $x,$ and C-anti-linear operator oniii) $<abc>$ continuous with respect to a normthat is, there exists $K>0$ s.t.
$z$

$y,$

$||||$$||<xxx>||\leq K||x||^{3}$ for $al1x\in U.$iv)1

There is a metric $(x, y)$ that satisfies $(L(x, y)z, w)+(z, L(x, y)w)=0$ and$(x, y)=\overline{(y,x)}.$lThis definition is slightly diﬀerent with that in [75, 76].

90

3Generalization of the hermitian 3-algebra
In this section, we extend the $u(N)\oplus u(M)3$ -algebras to a hermitian generalized Jordan
triple system.
Letbe the set of all $N\cross M$ matrices with operation$D_{N,M}^{*}$$<xyz>=x\overline{y}^{T}z-z\overline{y}^{T}x+zx^{T}\overline{y}-\overline{y}x^{T_{Z}}.$

Then
is a hermitian generalized Jordan triple system. In fact, it satisfies theconditions in the previous section with the metric $(x, y):=tr(x\overline{y}^{T})$ . This is an extensionof the $u(N)\oplus u(M)$ hermitian 3-algebras $<xyz>=x\overline{y}^{T}z-z\overline{y}^{T}x.$$D_{N,M}^{*}$

Application to field theory

4

In this section, we apply the hermitian generalized Jordan triple system in the previous
section to a field theory.
We start with
$S= \int d^{3}xtr(-D_{\mu}Z^{A}\overline{D^{\mu}Z_{A}}^{T}$$+L\epsilon^{\mu\nu\lambda}(-A_{\mu\overline{b}c}\partial_{\nu}A_{\lambda\overline{d}a}\overline{T}^{T\overline{d}}[T^{c},\overline{T}^{\overline{b}}, T^{a}]$$+ \frac{2}{3}A_{\mu\overline{d}a}A_{\nu\overline{b}c}A_{\lambda\overline{f}e}[T^{c}, \overline{T}^{\overline{b}}, T^{a}]\overline{[T^{f},\overline{T}^{\overline{e}},T^{d}]}))$

,

where
$D_{\mu}Z^{A}=\partial_{\mu}Z^{A}-A_{\mu\overline{b}a}[T^{a}, \overline{T}^{\overline{b}}, Z^{A}].$andare matter and gauge fields, respectively. runs from 1 to , whereas runs
from to 2. This action is invariant under the transformations generated by the operator
to$L(x, y)-L(y, x)$ . Here, we apply
this action.
The covariant derivative is explicitly written down as$Z^{A}$$A$$A_{\mu}$

$p$

$\mu$

$0$

$[x, \overline{y}, z]$$:=<xyz>=(x\overline{y}^{T}-\overline{y}x^{T})z-z(\overline{y}^{T}x-x^{T}\overline{y})$$D_{\mu}Z^{A}=\partial_{\mu}Z^{A}-iA_{\mu}^{L}Z^{A}+iZ^{A}A_{\mu}^{R},$are real antiand
where$o(M)$$o(N)$
Lie algebras, respectively. The
symmetric matrices, which generate theandaction can be rewritten in a covariant form with respect to $o(N)$ and $o(M)$ and we obtaina Chern-Simons gauge theory,$:=-iA_{\mu\overline{b}a}(T^{a}\overline{T}^{T\overline{b}}-\overline{T}^{\overline{b}}T^{Ta})$$:=-iA_{\mu\overline{b}a}(\overline{T}^{T\overline{b}}T^{a}-T^{Ta}\overline{T}^{\overline{b}})$$A_{\mu}^{L}$$A_{\mu}^{R}$$S = \int d^{3}xtr(-(\partial_{\mu}Z^{A}-iA_{\mu}^{L}Z^{A}+iZ^{A}A_{\mu}^{R})\overline{(\partial_{\mu}Z_{A}-iA_{\mu}^{L}Z_{A}+iZ_{A}A_{\mu}^{R})}^{T}$$+L \epsilon^{\mu\nu\lambda}(\frac{1}{2}(A_{\mu}^{L}\partial_{\nu}A_{\lambda}^{L}-A_{\mu}^{R}\partial_{\nu}A_{\lambda}^{R})+\frac{i}{3}(A_{\mu}^{L}A_{\nu}^{L}A_{\lambda}^{L}-A_{\mu}^{R}A_{\nu}^{R}A_{\lambda}^{R}$

In this action,
tively, whereas$A_{\mu}^{L}$andtransform as adjoint representations of $o(N)$ and $o(M)$ , respectransforms as a bi-fundamental representation of $o(N)\oplus o(M)$ ;$A_{\mu}^{R}$$Z^{A}$$\delta A_{\mu}^{R} = [i\Lambda^{R}, A_{\mu}^{R}]$$\delta A_{\mu}^{L} = [i\Lambda^{L}, A_{\mu}^{L}]$$\delta Z^{A} = i\Lambda^{L}Z^{A}-Z^{A}(i\Lambda^{R})$,91, respecand
are defined in the same way as
andwhere gauge parameters
tively.
Next, let us examine whether the Novel Higgs mechanism works in this theory when
$M=N$ . By redefining the gauge fields as$\Lambda^{L}$$\Lambda^{R}$$A_{\mu}^{R}$$A_{\mu}^{L}$$A_{\mu}^{L} = A_{\mu}+B_{\mu}$$A_{\mu}^{R} = A_{\mu}-B_{\mu},$

we can separate a non-dynamical mode

$B_{\mu}$as$S = \int d^{3}xtr(-(D_{\mu}Z^{A}-i\{B_{\mu}, Z^{A}\})\overline{(D_{\mu}Z^{A}-i\{B_{\mu},Z^{A}\})}^{T}$$+L \epsilon^{\mu\nu\lambda}(B_{\mu}F_{\nu\lambda}+\frac{2i}{3}B_{\mu}B_{\nu}B_{\lambda}))$

,

where
$D_{\mu}Z^{A} = \partial_{\mu}Z^{A}-i[A_{\mu}, Z^{A}],$$F_{\mu\nu} = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+i[A_{\mu}, A_{\nu}].$We divide$Z^{A}$

into two real matrices as$Z^{A}=iX^{A}+X^{p+A},$

and consider fluctuations around a background solution as $X^{p}=vI+\tilde{X}^{p}$ . If we rescale
and
as

$L$

$B_{\mu}$$L = \mathcal{O}(v)$$B_{\mu} = \mathcal{O}(\frac{1}{v})$and use the equation of motion of,$B_{\mu},$$B^{\mu}= \frac{L}{8v^{2}}\epsilon^{\mu\nu\lambda}F_{\nu\lambda}-\frac{1}{2v}D^{\mu}X^{2p}+\mathcal{O}(\frac{1}{v^{2}})$,the action reduces to$S arrow\int d^{3}xtr(-g^{2}F_{\mu\nu}^{2}-(D_{\mu}X^{i})^{2})$in $varrow\infty$ , where
and runs from 1 to 2p-l. Therefore, we conclude that the Novel
Higgs mechanism works in the Chern-Simons gauge theory with the hermitian generalized
Jordan triple system in the previous section with $M=N$ , and we obtain a Yang-Mills theory
in this limit.
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$(\epsilon, \delta)$$f^{*}$