On a construction of self-dual gauge fields in seven dimensions

We consider gauge fields associated with a semisimple Malcev algebra. We construct a gauge-invariant Lagrangian and found a solution of modified Yang-Mills equations in seven dimensions.


Introduction
In Ref. [1], the 4d (anti-)self-dual Yang-Mills equations were generalized to the higher-dimensional linear relations where the numerical tensor c mnps is completely antisymmetric and λ = const is a non-zero eigenvalue. It is obvious that these equations lead to the full Yang-Mills equation, via the Bianchi identity. Several anti-self-dual solutions of (1) were found in [2][3][4][5][6]. Conversely, self-dual solutions of (1) are known. However, if we assume that the tensor F mn may take values in a Malcev algebra, then such solution may be found. Interestingly that this solution exactly coincides with the general solution of 7d analogue of the 3d Euler top equation [7]. The 7d model is regarded as a model describing self-dual membran instantons [8].
The paper is organized as follows. Section 2 contains well-known facts about Cayley-Dickson algebras, Malcev algebras, and analytic Moufang loops which we use. In Sections 3 gauge fields associated with a semisimple Malcev algebra are investigated, and a gauge-invariant Lagrangian of the Yang-Mills type is constructed. In Section 4 solutions of equations of motion in seven dimensions are found.

Moufang loops and Malcev algebras
Recall that a loop is a binary system S with an unity element, in which the equations ax = b and ya = b are uniquely solvable for all a, b ∈ S. Moufang loops are distinguished from the class of all loops by the identity (xy)(zx) = (x(yz))x. This paper is concerned with connected analytic Moufang loops; that is, connected analytic manifolds equipped with the Moufang loop structure, in which the binary operations are analytic. Everywhere below for short we use the term "Moufang loop" in place of the term "connected analytic Moufang loop".
Moufang loops are closely associated with alternative algebras that are defined by the identities It is evident from the definition that any associative algebra is alternative. The most important example of a nonassociative alternative algebra is the real Cayley-Dickson algebra. Let us recall its construction (see [9]). Let A be a real linear space equipped with a nondenerate symmetric metric g of signature (8,0) or (4,4). Choose the basis {1, e 1 , ..., e 7 } in A such that g = diag(1, 1, 1, 1, ±1, ±1, ±1, ±1), and define the multiplication where the structure constants c ijk = c ij s g sk are completely antisymmetric and different from 0 only if The multiplication (3) transform A into a linear algebra that is called the Cayley-Dickson algebra.
In general, the commutator algebra A (−) of the alternative algebra A is not a Lie algebra. Instead of the Jacobi identity, A (−) satisfies the identity where J(x, y, z) is so-called Jacobian of the elements x, y, z An anticommutative algebra whose multiplication satisfies the identity (4) is called a Malcev algebra. Everywhere below we use the term "Malcev algebra" in place of the term "finite-dimensional real Malcev algebra". Malcev algebras and alternative algebras are closely associated [10][11][12]. Any non-Lie simple Malcev algebra is isomorphic to the quotient algebra In particular, it follows from here that any semisimple Malcev algebra is embedded in the commutator algebra of an appropriate alternative algebra.
Further, recall that a derivation of a Malcev algebra M is a linear transfor- for all x, y ∈ M. We may write it in an equivalent form where T : M → End(M) is the regular representation of M, such that We denote by Lie(M) the Lie algebra generated by all operators T x . If M is a semisimple Malcev algebra, then Lie(M) is a direct sum of the derivation subalgebra Der(M) = Der(N)⊕Der(J) and vector subspace T (M) ≃ J(M). In addition, the subalgebra Der(N) ≃ N(M) and the subalgebra Der(J) is linearly generated by the operators where x, y ∈ J(M). With the regular representation T we may connect the bilinear form (x, y) = tr(T x T y ). It is clear that the form (x, y) is symmetric (x, y) = (y, x). In addition, it follows from (4) that for any x, y, z ∈ M. The form (x, y) is called a Killing form.
The regular representation of Malcev algebras can be extended. Let M be a Malcev algebra, V be a real vector space, and τ : is a Malcev algebra. In this case V is called a Malcev module. It is known [13][14] that every representation of a semisimple Malcev algebra is completely reducible. Any irreducible Malcev module is either Lie or the regular bimodule for a nonassociative simple Malcev algebra or sl(2)-module of dimension 2 such that τ x =x, wherex is the adjoint matrix to x ∈ sl(2).
Finally, we note that there exists a correspondence between Moufang loops and Malcev algebras, which generalizes the classical Lie correspondence between Lie groups and Lie algebras [15][16][17]. Namely, there exists an unique, to within isomorphism, simply connected Moufang loop S with a given tangent Malcev algebra, and any Moufang loop with the same tangent algebra is isomorphic to the quotient algebra S/N where N is a discrete central normal subgroup of S. Any simply connected semisimple Moufang loop is decomposed in the direct product of a semisimple Lie group and a simple nonassociative Moufang loops each of which is analytically isomorphic to one of the spaces S 7 , S 3 × R 4 , or S 7 × R 7 . Actually, any simply connected simple nonassociative Moufang loop is isomorphic to the loop of elements of norm 1 in the Cayley-Dickson algebra over R or C.
Using the previously mentioned correspondence between Moufang loops and Malcev algebras, we can define a notion of the representation of Moufang loop. Let S be a Moufang loop and M be its tangent Malcev algebra. The representation M → End(V ) of M induces the mapping S → Aut(V ) of the simply connected Moufang loop S, locally isomorphic to S, into the simply connected Lie group G ⊆ Aut(V ). This mapping can extend into the group G ≃ G/ G 0 , where the subgroup G 0 lies in the discrete center of G. Therefore if S ≃ S/ S 0 and S 0 is isomorphic to a subgroup of G 0 , then there exists the mapping S → G induced by the representation of M. We shall call this mapping by a representation of S. In particular, if S is a nonassociative compact simple Moufang loop, then G ≃ Spin (7) and G ≃ SO(7).

Nonassociative gauge fields
Suppose S is a semisimple Moufang loop, M is its tangent Malcev algebra, and A m (x) is a vector field taking values in M and defined in an Euclidean or pseudo-Euclidean space. Further, let be a representation of S, and H be a subgroup of G locally isomorphic to the group Int(S) of inner automorphisms of S. Define the field ψ(x) taking values in V and its covariant derivativê whereÂ m ∈ End(V ) is an image of A m corresponding to the representation (9). As usually, we require thatD m ψ has transformation properties the same as the field ψ, i.e.
whereû(x) is function taking values in the subgroup H.
Consider an infinitesimal gauge transformationû = 1 +ε. It is obvious thatε ∈ Der(M). If the field A m takes values in the Lie center N(M), then the condition (10) defines the usual transformation rule of gauge fields where ε is an isomorphic prototype ofε in N(M). If A m is an element of the ideal J(M) generated by Jacobians, then the situation is different. In this case the operatorε has not a prototype in M. On the other hand, the representation of J(M) induced by the representation of S is regular. Therefore we may use the identity (6) and get the following transformation rule Obviously, the transformation (12) Now we go on to a construction of the gauge-invariant Lagrangian. To this end define tensor fields in M such that their images in End(V ) are transformed by the adjoined representation of G. This request is satisfied if we suppose where c mnps is a completely antisymmetric numerical tensor. Indeed, it can easily be checked that the infinitesimal transformations (11) and (12) induce the transformations whereF mn andĴ m are images of F mn and J m , which are defined by the representation (9). Note that the tensor F mn extend a notion of the Yang-Mills field strength. Conversely, the field J m has not a prototype in the Yang-Mills theory. The field J m is not zero only if M is a non-Lie Malcev algebra. We define the Lagrangian L = 1 8g 2 tr(F mnF mn +Ĵ sÂ s ).
If M is a Lie algebra, then the Lagrangian (15) coincides with the Lagrangian of Yang-Mills and hence it is gauge-invariant. On the other hand, every representation of a semisimple Malcev algebra is completely reducible. Therefore it is enough to prove H-invariance of the Lagrangian (15) only if M is a non-Lie simple Malcev algebra. But in this case the gauge-invariance of the Lagrangian (15) follows from (13) and (14). In this basis the operators T e i are represented in the form where e jk are generators of the Lie algebra Lie(M) ≃ so(7) or so (3,4) with the matrix elements (e jk ) a b = g jb δ a k − g kb δ a j .
Using the representation (16) and the identity (17), we can find the Killing form Substituting (18) in the Lagrangian (15), we easily get the equations of motion Using the definition (5), we prove that every solution of the equations is a solution of the equations of motion (19). then it is easy to prove that the tensors c ijk and c ijkl satisfy the following identities c mni c ps i = g mp g ns − g ms g np + c mnps , c mnij c ps ij = 4g mp g ns − 4g ms g np + 2c mnps .
Note that if the gauge fields are defined in an Euclidean space and M is a compact algebra, then any solution of (20) correspond to a minimum of the functional of action. Indeed, using the identities (8) and (24), we can shown that where the prototype ofÎ m in M has the form Since the variation of field on the boundary of volume vanish in the deduction of equations of motion, we get the required statement. Now we turn to a search of solutions of the equations (20). Suppose that the field F mn is defined in the space M with the metric (2). We choose the ansatz where x 2 = x k x k . Using the identities (21)-(22), we get the following expression of field strength By the identity (23), it follows that the tensor (26) is self-dual The obtained solution may be generalized. We choose the ansatz where depending on u = λ 2 + x 2 the vector B j takes values in M. If the vector B i satisfies the identity then the field strength again is self-dual A particular solution of the equation (28) can find if to compare the right parts of (25) and (27). General solution of (28) was found in the work [7].