# Hermitian(ϵ,δ)-Freudenthal-Kantor Triple Systems and Certain Applications of *-Generalized Jordan Triple Systems to Field Theory

Authors: Noriaki Kamiya; Matsuo Sato
Journal: Advances in High Energy Physics
 (2014)

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## Abstract

We define Hermitian(ϵ,δ)-Freudenthal-Kantor triple systems and prove a structure theorem. We also give some examples of triple systems that are generalizations of the u(N)⊕u(M) and sp(2N)⊕u(1) Hermitian 3-algebras. We apply a *-generalized Jordan triple system to a field theory and obtain a Chern-Simons gauge theory. We find that the novel Higgs mechanism works, where the Chern-Simons gauge theory reduces to a Yang-Mills theory in a certain limit.


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## Body

## 1. Introduction

The structure theory of(ϵ,δ)-Freudenthal-Kantor triple systems of finite dimensions, which contain the concept of Jordan triple systems, has been studied in [1–11].On the other hand, it is well known that symmetric bounded domains have one-to-one correspondence to Hermitian Jordan triple systems [12–15], for which a certain trace form is positive definite Hermitian. Moreover, structure theorems of Hermitian generalized Jordan triple systems of the second-order can be found in [16]. Hence, as a generalization of these concepts, it is interesting to investigate the structure theory of Hermitian (ϵ,δ)-Freudenthal-Kantor triple systems (HFKTSs). In this generalization, Hermitian generalized Jordan triple systems of the second-order or the so-called Kantor triple systems are included as Hermitian (-1,1)-Freudenthal-Kantor triple systems.From the viewpoint of string theory, HFKTSs are generalizations (A different generalization, the so-called differential crossed module, was extensively investigated in [17–19].) of Hermitian 3-algebras [20–62], which have played crucial roles in M-theory. The field theories applied with Hermitian 3-algebras are the Chern-Simons gauge theories that describe the effective actions of membranes in M-theory. In a certain limit, the novel Higgs mechanism works, where the Chern-Simons theories become the Yang-Mills theories that describe the effective actions of D-branes in string theory. Moreover, 3-algebra models of M-theory itself have been proposed and were studied in [63–67]. Therefore, it is interesting to apply HFKTSs to field theories. It is particularly important to study whether the novel Higgs mechanism works in a field theory with an HFKTS.The Hermitian 3-algebras are special cases, whereK(a,b)=0 or, equivalently, 〈abc〉=-〈cba〉, of Hermitian (-1,-1)-Freudenthal-Kantor triple systems of the second order. Also, the Hermitian 3-algebras are classified into u(N)⊕u(M) and sp(2N)⊕u(1) Hermitian 3-algebras [34, 36, 37, 43, 67]. Therefore, it is natural to extend these triple systems to more general Hermitian (-1,-1)-Freudenthal-Kantor triple systems or Hermitian generalized Jordan triple systems.Here, we are concerned with algebras and triple systems that are finite or infinite over a complex number field, unless otherwise specified. We refer to [68–72] for nonassociative algebras and to [73–75] for Lie superalgebras, for example.


## 2. Definitions and Preamble

A triple systemU is a vector space over a field F of characteristic ≠2,3 with a trilinear map U⊗U⊗U→U. In this paper, we are concerned with triple systems over the complex number C, and we denote the trilinear product by 〈xyz〉∈U for x,y,z∈U, assuming that 〈xyz〉 is C-linear on x and z and C-antilinear on y. Also we use the notations of two operations L(x,y)∈ End U and K(x,y)∈ End U with respect to the triple systems [1, 2], where L(x,y)z=〈xyz〉 and K(x,y)z=〈xzy〉-δ〈yzx〉(δ=±1).Definition 1.
A triple systemU is said to be a *-(ϵ,δ)-Freudenthal-Kantor triple system if the following relations (0)–(iv) are satisfied:(0)
U is a Banach space,(i)
[L(x,y),L(z,w)]=L(〈xyz〉,w)+εL(z,〈yxw〉), where [A,B]=AB-BA, A,B∈EndU, and ε=±1,(ii)
K(〈xyz〉,w)+K(z,〈xyw〉)+δK(x,K(z,w)y)=0,(iii)
〈xyz〉 is a C-linear operator on x and z and a C-antilinear operator on y,(iv)
〈xyz〉 is continuous with respect to a norm, ∥∥; that is, there exists K>0 such that
(1)∥〈xxx〉∥≤K∥x∥3∀x∈U.
Furthermore, a *-(ϵ,δ)-Freudenthal-Kantor triple system is said to be Hermitian if it satisfies the following condition.(v)
All operatorsL(x,y) are positive Hermitian operators with a Hermitian metric
(2)(x,y)=tr⁡L(x,y);
that is, (L(x,y)z,w)=(z,L*(x,y)w) and (x,y)=(y,x)¯.This definition is a generalization of the concept of known(ϵ,δ)-Freudenthal-Kantor triple systems (see [1–10]) to that of Hermitian triple systems. Note that there are many simple Lie algebras (the case of δ=1) and simple Lie superalgebras (the case of δ=-1) constructed from (ϵ,δ)-Freudenthal-Kantor triple systems [4, 7, 8].LetU be a *-(ϵ,δ)-Freudenthal-Kantor triple system. Then we may define the notations of tripotent and bitripotent as follows.Definition 2.
It is said to be a tripotent ofU if
(3)〈ccc〉=c,c∈U.Definition 3.
It is said to be a strong bitripotent ofU if a pair (c1,c2) of tripotents satisfies the relations,
(4)〈c1c1c2〉=-12c2,〈c2c2c1〉=-12c1,
and other products are zero.Definition 4.
It is said to be a bitripotent ofU if a pair (c1,c2) of tripotents satisfies the relations
(5)〈c1c1c2〉=αc2,〈c2c2c1〉=αc1,〈c1c2c1〉=βc2,〈c2c1c2〉=βc1,〈c2c1c1〉=γc2,〈c1c2c2〉=γc1,
and other products are zero, where
(6)α2+β2+γ2≠0,α,β,γ∈R(realnumber).From now on, we will consider a finite-dimensional triple system equipped with a tripotent, unless otherwise specified.Following [16], theorems concerning tripotents (Theorems 5 and 6) are proved for Hermitian generalized Jordan triple systems. Actually, the proofs of [16] are valid in Hermitian (-1,δ)-Freudenthal-Kantor triple systems because they are independent of δ. Here, we show the theorems below for completeness.Theorem 5.
LetU be a Hermitian (-1,δ)-Freudenthal-Kantor triple system. If W⊂U is flat (i.e., L(x,y)=L(y,x) for all x,y∈W), then we have the decomposition
(7)W=Re1⊕⋯⊕Ren,
where ei are tripotents or bitripotents.Proof.
LetS⊂EndR(W) be theR-linear span of all L(x,x)|W, where x∈W.
From(8)2L(x,y)z=L(x+y,x+y)z-L(x,x)z-L(y,y)z,
it follows that L(x,y) belongs to S for all x,y∈W. Choose a scalar product, (x,y), for example, (x,y):=traceL(x,y), satisfying L(x,y)*=L(y,x).
By taking the real part of(x,y) and restricting W, we obtain a Euclidean scalar product on W such that S consists of self-adjoint transformations. Furthermore, S is commutative since [L(x,x),L(y,y)]=L(〈xxy〉,y)-L(y,〈xxy〉)=0.
By a standard calculation,S may be simultaneously diagonalized. Therefore, there exists a basis e1,…,en of W such that fei∈Rei for all f∈S. Hence, from L(ei,ei)∈S, we have
(9)〈eieiei〉=λei,λ∈R.
Furthermore, after sign changes, we may assume that λ>0.
Indeed, ifλ<0, we write ei′=-ei, then λei=-λei′ such that -λ is positive.
Replacingei by (λ)-1ei, we may assume that ei are tripotents.
On the other hand, ifi≠j, we have
(10)〈eieiej〉=L(ei,ei)ej∈Sej⊆Rej.
These imply thatei and ej are tripotents or bitripotents.
This completes the proof.We define odd powers ofx inductively as follows:
(11)x(3):=〈xxx〉,x(2n+1):=〈xx(2n-1)x〉.By using this theorem, we have the following.Theorem 6.
LetU be a Hermitian (-1,δ)-Freudenthal-Kantor triple system. Then every x∈U can be written uniquely as
(12)x=λ1e1+λ2e2+⋯+λnen,
where ei are tripotents or bitripotents and λi satisfy
(13)0<λ1≤λ2≤⋯≤λn.Proof.
Applying Theorem5 to the subspace W spanned by the powers of x, from definition (i), we have 0=[L(x,x),L(x,x)]=L(〈xxx〉,x)-L(x,〈xxx〉) for x∈W, and thus L(x,〈xxx〉)=L(〈xxx〉,x). Hence, W is flat.
Without loss of generality, we can assume thatW=U. This implies that any element x∈U can be represented by x=μ1e1+μ2e2+⋯+μnen, with real numbers μi, after permutations, and we may assume that 0<μ1≤μ2⋯≤μn, where ei are tripotents or bitripotents.
Furthermore, becauseei are tripotents or bitripotents, the powers of x are
(14)x(2k+1)=μ1(2k+1)e1+μ2(2k+1)e2+⋯+μn(2k+1)en.
That is, we obtain(15)x(3)=〈xxx〉=μ(3)e1+μ2(3)e1+μ2(3)e2⋯+μn(3)en,x(5)=〈x〈xxx〉x〉=μ1(5)e1+μ2(5)e2+⋯μn(5)en,⋮x(2n-1)=〈xx(2n-3)x〉=μ1(2n-1)e1+μ2(2n-1)e2+⋯+μn(2n-1)en,
where μi(k)(i=1,…,n,k=3,5,…,2n-1) are polynomials in μ1,…,μn with their coefficients in R.
These expressions may also be written as(16)(μ1⋯μnμ1(3)⋯μn(3)⋮μ1(2n-1)⋯μn(2n-1))(e1e2⋮en)=(xx(3)⋮x(2n-1)).
This completes the proof.For the HFKTSU, we can define a norm ∥∥ as follows:
(17)∥x∥=max⁡|λi|,ifx=Σλiei∈U,
where ei are tripotents or bitripotents. Note that
(18)∥x∥≥0,∥x+y∥≤∥x∥+∥y∥.


## 3. Peirce Decomposition

In this section, we briefly consider the Peirce decomposition of a *-(-1,-1)-Freudenthal-Kantor triple system equipped with the tripotent 〈ccc〉=c.We recall conditions (i) and (ii) in Section2, which are equivalent to the following conditions (i)′and (ii)′:
(i)′[L(x,y),L(z,w)]=L(〈xyz〉,w)+εL(z,〈yxw〉),(ii)′K(K(x,y)z,w)-L(w,z)K(x,y)+εK(x,y)L(z,w)=0.From condition(i)′ with ε=-1, we have
(i)′′L(c,c)R(c,c)=R(c,c)L(c,c),
where we define R(x,y)z=〈zxy〉. From condition (ii)′ with ε=-1, we have
(19)K(K(x,y)z,w)-L(w,z)K(x,y)-K(x,y)L(z,w)=0.
We write x=z=w=c (if c is a tripotent element, 〈ccc〉=c), and then we obtain
(20)K(K(c,y)c,c)c-L(c,c)K(c,y)c-K(c,y)L(c,c)c=0.Whenδ=-1, we obtain
(ii)′′L(c,c)R(c,c)+R(c,c)2-L(c,c)-R(c,c)=0.From conditions(i)′′ and (ii)′′, we have
(21)(R(c,c)-Id)(R(c,c)+L(c,c))=0,L(c,c)R(c,c)=R(c,c)L(c,c).Summarizing the results, we have the following.Theorem 7.
LetU be a *-(-1,-1)-Freudenthal-Kantor triple system. Then, we have the following decomposition with respect to a tripotent c (i.e., 〈ccc〉=c):
(22)U=U1(c)⊕U1/2(c)⊕U0(c),
where
(23)U1(c)={x∣(L(c,c)+R(c,c))x=0,(R(c,c)-Id)x≠0((c))},U1/2(c)={x∣(L(c,c)+R(c,c))x≠0,(R(c,c)-Id)x=0((c))},U0(c)={x∣(L(c,c)+R(c,c))x=0,(R(c,c)-Id)x=0((c))}.


## 4. Examples

Hermitian 3-algebras are classified intou(N)⊕u(M) and sp(2N)⊕u(1) Hermitian 3-algebras [34, 36, 37, 43, 67]. In this section, we extend these 3-algebras to Hermitian (-1,-1)-Freudenthal-Kantor triple systems and *-generalized Jordan triple systems.Example 1.
LetDN,M* be the set of all N×M matrices with the operation
(24)〈xyz〉=xy-Tz-zy-Tx+zxTy-,
where xT and x- denote the transpose and conjugation of x, respectively.
ThenDN,M* is a Hermitian (-1,-1)-Freudenthal-Kantor triple system. In fact, it satisfies conditions (0), (i), (ii), (iii), (iv), and (v) in Section 2. This is an extension of the u(N)⊕u(M) Hermitian 3-algebra (the metric of the Hermitian 3-algebra is defined as (x,y)=tr⁡(xy†), which is different from our Definition 1(v))〈xyz〉=xy-Tz-zy-Tx, which is a basis for the effective action of multiple membranes in M-theory.
One of the tripotents is given by(25)c=(Id000),
where Id is the n×n identity matrix (n≤N,M). Because any element can be decomposed as
(26)x=(ABCD)=(12(A-AT)B0D)+(12(A+AT)0C0),
the Peirce decomposition is given by
(27)U1(c)={(12(A-AT)B0D)},U1/2(c)={(12(A+AT)0C0)},U0(c)=0.
As in Theorem5, we can expand any element as x=Σ(λijEij+μij-1Eij), where Eij denotes that element (i,j) is 1 and other elements are zero, and Eij and -1Eij are tripotents, that is, 〈EijEijEij〉=Eij and 〈-1Eij-1Eij-1Eij〉=-1Eij.Example 2.
LetSn,2k* be the set of all n×2k matrices with the operation
(28)〈XYZ〉=X(Y-)TZ-Z(Y-)TX+Zϕ(X)Tϕ(Y-),
where X=(x1,x2),  xi is an n × k matrix and ϕ(X)=(x2,-x1).
ThenSn,2k* is a Hermitian (-1,-1)-Freudenthal-Kantor triple system. In fact, it satisfies conditions (0), (i), (ii), (iii), (iv), and (v) in Section 2. This triple system with k=1 reduces to the sp(2N)⊕u(1) Hermitian 3-algebra.
In this example, we can show thate=(Eij,Elj), where i≠l are tripotents and any element can be expanded by using part of them.Example 3.
LetDN,M* be the set of all N×M matrices with the operation
(29)〈xyz〉=xy-Tz-zy-Tx+zxTy--y-xTz.
ThenDN,M* is a *-generalized Jordan triple system. In fact, it satisfies conditions (0), (i) with ϵ=-1, (iii), and (iv) in Section 2 but does not satisfy (ii) and (v). This is an extension of the triple system in Example 1.


## 5. Application to a Field Theory

In this section, we apply a *-generalized Jordan triple system to a field theory.We start with(30)S=∫‍d3xtr⁡(d3xtrQQQQQR×[Tc,T¯b-,Ta][Tf,T¯e-,Td]¯)-DμZADμZA¯T+Lϵμνλd3xtrQQQ×(-Aμb-c∂νAλd-aT¯Td-[Tc,T¯b-,Ta]d3xtrQQQQQR+23Aμd-aAνb-cAλf-ed3xtrQQQQQR×[Tc,T¯b-,Ta][Tf,T¯e-,Td]¯)),
where
(31)DμZA=∂μZA-Aμb-a[Ta,T¯b-,ZA].ZA and Aμ are matter and gauge fields, respectively. A runs from 1 to p, whereas μ runs from 0 to 2. Aμa-b satisfies Aμa-b¯=-Aμb-a. This action is invariant under the transformations generated by the operator L(x,y). This action describes the bosonic parts of the effective actions of supermembranes in M-theory if appropriate potential terms of ZA are added and a Lorentzian Lie 3-algebra or a Hermitian 3-algebra is applied. Here, we apply the *-generalized Jordan triple system [x,y¯,z]:=〈xyz〉=(xy-T-y-xT)z-z(y-Tx-xTy-) in Example 3 to this action.The covariant derivative is explicitly written as(32)DμZA=∂μZA-iAμLZA+iZAAμR,
where AμR:=-iAμb-a(T¯Tb-Ta-TTaT¯b-) and AμL:=-iAμb-a(TaT¯Tb--T¯b-TTa) are real antisymmetric matrices, which generate the o(N) and o(M) Lie algebras, respectively. The action can be rewritten in a covariant form with respect to o(N) and o(M) and we obtain a Chern-Simons gauge theory:
(33)S=∫‍d3xtr⁡(12-(∂μZA-iAμLZA+iZAAμR)d3xtrQQs×(∂μZA-iAμLZA+iZAAμR)¯Td3xtrQQs+Lϵμνλ(12(AμL∂νAλL-AμR∂νAλR)d3xtrQQs+i3(AμLAνLAλL-AμRAνRAλR)12)).
In this action, AμL and AμR transform as adjoint representations of o(N) and o(M), respectively, whereas ZA transforms as a bifundamental representation of o(N)⊕o(M),
(34)δAμR=[iΛR,AμR],δAμL=[iΛL,AμL],δZA=iΛLZA-ZA(iΛR),
where gauge parameters ΛR and ΛL are defined in the same way as AμR and AμL, respectively.Next, let us examine whether the novel Higgs mechanism works in this theory whenM=N. By redefining the gauge fields as
(35)AμL=Aμ+Bμ,AμR=Aμ-Bμ,
we can separate a nondynamical mode Bμ as
(36)S=∫‍d3xtr⁡(2i3-(DμZA-i{Bμ,ZA})d3xtrQQ×(DμZA-i{Bμ,ZA})¯Td3xtrQQ+Lϵμνλ(BμFνλ+2i3BμBνBλ)),
where
(37){Bμ,ZA}=BμZA+ZABμ,DμZA=∂μZA-i[Aμ,ZA],Fμν=∂μAν-∂νAμ+i[Aμ,Aν].
We divide ZA into two real matrices as
(38)ZA=iXA+Xp+A,
and consider fluctuations around a background solution as Xp=vI+X~p. If we rescale L and Bμ as
(39)L=𝒪(v),Bμ=𝒪(1v),
we obtain
(40)S=∫‍d3xtr⁡(-DμZA(DμZA)¯T-4BμDμX2pvd3xtrQQs-4BμBμv2+LϵμνλBμFνλ(ZA)¯T)+𝒪(1v).
By using the equation of motion of Bμ,
(41)Bμ=L8v2ϵμνλFνλ-12vDμX2p+𝒪(1v2),
the action reduces to
(42)S⟶∫‍d3xtr⁡(-g2Fμν2-(DμXi)2)
as v→∞, where g=L/v and i run from 1 to 2p-1. Therefore, we conclude that the novel Higgs mechanism works in the Chern-Simons gauge theory with the *-generalized Jordan triple system in Example 3 with M=N, and we obtain a Yang-Mills theory in this limit.


## 6. Concluding Remarks

For our triple systems, we emphasized in this paper that there exists a generalized concept of Hermitian 3-algebras, which have played crucial roles in M-theory. In particular, we find the novel Higgs mechanism also works in the generalized concept.HFKTSs and *-generalized Jordan triple systems〈xyz〉 can be regarded as left and right actions on z by two Lie algebras, as one can see in the examples. For the novel Higgs mechanism, it is necessary that the two Lie algebras are the same because we need to define summation between the Lie algebras as in Section 5. Thus, the novel Higgs mechanism works only when Example 3 is applied among the examples in this paper.Our principal physical motivation for generalizing the Hermitian 3-algebras to HKFTSs is to regularize the Nambu 3-algebra, which is defined by a Nambu bracket. The area-preserving diffeomorphism (APD) algebra defined by a Poisson bracket on a torus is regularized by theu(N) algebra in the 't Hooft base. On the other hand, the APD algebra defined on a sphere is regularized by the u(N) algebra in another base. That is, regularizations of infinite-dimensional algebras depend on the basis of the corresponding finite algebras. Although Hermitian 3-algebras (u(N)⊕u(M) and sp(2N)⊕u(1) Hermitian 3-algebras) are strong candidates for the regularization of the Nambu 3-algebra because of their large symmetries and their relations to M2-branes, it is not clear how to choose a base for the regularization. To study this systematically, we generalized the Hermitian 3-algebras to HFKTSs (u(N)⊕u(M) to Example 1 and sp(2N)⊕u(1) to Example 2), and we found a nilpotent basis for them. The next step is to generalize the Nambu 3-algebra to HFKTSs and to find a corresponding nilpotent base. Then by restricting the HFKTSs to the 3-algebras, we may prove that the Nambu 3-algebra is regularized by a Hermitian 3-algebra.



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