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Constructing Nonabelian (1,0) Hypermultiplet Theories in Six Dimensions

We construct a class of nonabelian superconformal (1,0) hypermultiplet theories in six dimensions by introducing an abelian auxiliary field. The gauge fields of this class of theories are non-dynamical, and this class of theories can be viewed as Chern-Simons-matter theories in 6D.


Introduction
It is known that six is the maximal dimension for a superconformal field theory, and the superalgebra of the superconformal symmetry in 6D is OSp(8|2) or OSp(8|4) [1]. 6D superconformal theories are interested because they can be the candidates of dual gauge theories of multiple D5 or M5-branes [2], or may be used to study the dynamics of compactifications of superstring theory to 6D [3].
In this paper we work on another direction: we construct a class of nonabelian superconformal (1, 0) hypermultiplet theories in 6 dimensions; the gauge fields of the theories are nondynamical. 1 . They can be thought of as a class of Chern-Simons-matter theories in 6D. Classically, this class of theories also have an OSp(8|2) superconformal symmetry. Nevertheless, they should be simpler than the nonabelian (1, 0) tensor multiplet theories in Ref. [4], since they do not involve the self-dual strengths H µνρ associated with the tensor multiplets. On the other hand, it seems that this class of theories cannot be derived by truncating any known nonabelian (1, 0) or (2, 0) tensor multiplet theories. So it may be interesting to study this class of (1, 0) hypermultiplet theories. In particular, it would be interesting to quantize this class of theories and to construct their gravity duals. We shall present the details of construction of this class of theories in the next section.
The above free theory can be generalized to be a class of nonabelian (1, 0) theories. To see this, we propose the following supersymmetry transformations Here C µi is an abelian auxiliary field 2 with scaling dimension −1. The scalar and fermionic fields are in the adjoint representation of the Lie algebra of gauge symmetry; the Lie algebra can be chosen as the ADE type. The covariant derivative is defined as where f np m are the structure constants of the Lie algebra. 2 The role of C µi is similar to that of the auxiliary field C µ a in the LP theory [5], where a is a 3-algebra index (See also [4,10,11]). The geometric meaning of C µ a in loop space was investigated in [12]. It would be interesting to find out the geometric interpretation of C µi .
Let us now examine the closure of superalgebra. We begin by considering the scalar fields. A short calculation gives It is clear that the superalgebra is closed on X i m , up to the gauge transformation [Λ, X i ] m . We now consider the super-variation of spinor fields. After some algebraic steps, we obtain To close the superalgebra, we impose the equations of motion In deriving (2.8), we have used the Fierz identity [4] ( Here χ + satisfies the conditions Γ 012345 χ + = −χ + and Γ 6789 χ + = +χ + . As for the auxiliary field, we have The commutator [Λ, C µi ] vanishes due to that C µi is an abelian field. As a result, we have i.e. C µ is a constant abelian field. Finally, the commutator [δ 1 , δ 2 ]A m µ is given by We see that the second line of (2.13) must be the equations of motion for gauge fields (2.14) Taking a super-variation on the equations of motion for the fermionic fields (2.9), one obtains the equations of motion for the scalar fields In summary, we have the equations of motion: The set of equations (2.16) transform into themselves under the supersymmetry transformations (2.3). Eqs. (2.3) and (2.16) are main results of this work. The superconformal equations of motion for a (2, 0) theory can be developed via twistor techniques [13,14]. It would be very interesting to construct the equations of motion of this paper using a twistor approach.
If we make the replacements