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

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.

On the other hand, it is well known that symmetric bounded domains have one-to-one correspondence to Hermitian Jordan triple systems [12][13][14][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 (, )-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][18][19].) of Hermitian 3-algebras , 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 Mtheory.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 Dbranes in string theory.Moreover, 3-algebra models of Mtheory itself have been proposed and were studied in [63][64][65][66][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.
Here, we are concerned with algebras and triple systems that are finite or infinite over a complex number field, unless 2 Advances in High Energy Physics otherwise specified.We refer to [68][69][70][71][72] for nonassociative algebras and to [73][74][75] for Lie superalgebras, for example.

Definitions and Preamble
A triple system  is a vector space over a field  of characteristic ̸ = 2, 3 with a trilinear map  ⊗  ⊗  → .In this paper, we are concerned with triple systems over the complex number C, and we denote the trilinear product by ⟨⟩ ∈  for , ,  ∈ , assuming that ⟨⟩ is C-linear on  and  and C-antilinear on .Also we use the notations of two operations (, ) ∈ End  and (, ) ∈ End  with respect to the triple systems [1,2], where (, ) = ⟨⟩ and (, ) = ⟨⟩ − ⟨⟩ ( = ±1).(iv) ⟨⟩ is continuous with respect to a norm, ‖‖; that is, there exists  > 0 such that Furthermore, a * -(, )-Freudenthal-Kantor triple system is said to be Hermitian if it satisfies the following condition.
Let  be a * -(, )-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 of  if ⟨⟩ = ,  ∈ .
(3) Definition 3. It is said to be a strong bitripotent of  if a pair ( 1 ,  2 ) of tripotents satisfies the relations, and other products are zero.
Definition 4. It is said to be a bitripotent of  if a pair ( 1 ,  2 ) of tripotents satisfies the relations and other products are zero, where From now on, we will consider a finite-dimensional triple system equipped with a tripotent, unless otherwise specified.
By a standard calculation,  may be simultaneously diagonalized.Therefore, there exists a basis  1 , . . .,   of  such that   ∈ R  for all  ∈ .Hence, from (  ,   ) ∈ , we have Furthermore, after sign changes, we may assume that  > 0.
On the other hand, if  ̸ = , we have These imply that   and   are tripotents or bitripotents.This completes the proof.
We define odd powers of  inductively as follows: By using this theorem, we have the following.
These expressions may also be written as . . .
For the HFKTS , we can define a norm ‖‖ as follows: where   are tripotents or bitripotents.Note that
We recall conditions (i) and (ii) in Section 2, which are equivalent to the following conditions (i)  and (ii) We write  =  =  =  (if  is a tripotent element, ⟨⟩ = ), and then we obtain Summarizing the results, we have the following.
Example 1.Let  * , be the set of all  ×  matrices with the operation where   and x denote the transpose and conjugation of , respectively.Then  * , 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 () ⊕ () Hermitian 3-algebra (the metric of the Hermitian 3algebra is defined as (, ) = tr ( † ), which is different from our Definition 1(v))⟨⟩ =  ȳ  − ȳ  , which is a basis for the effective action of multiple membranes in M-theory.
One of the tripotents is given by where  is the  ×  identity matrix ( ≤ , ).Because any element can be decomposed as the Peirce decomposition is given by As in Theorem 5, we can expand any element as  = Σ(    +   √ −1  ), where   denotes that element (, ) is 1 and other elements are zero, and   and √ −1  are tripotents, that is, Example 2. Let  * ,2 be the set of all  × 2 matrices with the operation where  = ( 1 ,  2 ),   is an  ×  matrix and () = ( 2 , − 1 ).
In this example, we can show that  = (  ,   ), where  ̸ =  are tripotents and any element can be expanded by using part of them.
Example 3. Let  * , be the set of all  ×  matrices with the operation Then  * , is a * -generalized Jordan triple system.In fact, it satisfies conditions (0), (i) with  = −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.

Application to a Field Theory
In this section, we apply a * -generalized Jordan triple system to a field theory.
We start with where and   are matter and gauge fields, respectively. runs from 1 to , whereas  runs from 0 to 2.   ā  satisfies   ā  = −  b  .This action is invariant under the transformations generated by the operator (, ).This action describes the bosonic parts of the effective actions of supermembranes in M-theory if appropriate potential terms of   are added and a Lorentzian Lie 3-algebra or a Hermitian 3-algebra is applied.Here, we apply the * -generalized Jordan triple system [, , ] := ⟨⟩ = ( ȳ  − ȳ   ) − ( ȳ   −   ȳ ) in Example 3 to this action.
The covariant derivative is explicitly written as The action can be rewritten in a covariant form with respect to () and () and we obtain a Chern-Simons gauge theory: In this action,    and    transform as adjoint representations of () and (), respectively, whereas   transforms as a bifundamental representation of () ⊕ (), where gauge parameters Λ  and Λ  are defined in the same way as    and    , respectively.Next, let us examine whether the novel Higgs mechanism works in this theory when  = .By redefining the gauge fields as we can separate a nondynamical mode   as where We divide   into two real matrices as and consider fluctuations around a background solution as   = V + X .If we rescale  and   as we obtain By using the equation of motion of   , the action reduces to as V → ∞, where  = /V and  run from 1 to 2 − 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  = , and we obtain a Yang-Mills theory in this limit.

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 ⟨⟩ can be regarded as left and right actions on  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 areapreserving diffeomorphism (APD) algebra defined by a Poisson bracket on a torus is regularized by the () algebra in the 't Hooft base.On the other hand, the APD algebra defined on a sphere is regularized by the () algebra in another base.That is, regularizations of infinite-dimensional algebras depend on the basis of the corresponding finite algebras.Although Hermitian 3-algebras (() ⊕ () and (2)⊕(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 (() ⊕ () to Example 1 and (2) ⊕ (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 3algebra is regularized by a Hermitian 3-algebra.