Leading low-energy effective action in $6D$, ${\cal N}=(1,1)$ SYM theory

We elaborate on the low-energy effective action of $6D,\,{\cal N}=(1,1)$ supersymmetric Yang-Mills (SYM) theory in the ${\cal N}=(1,0)$ harmonic superspace formulation. The theory is described in terms of analytic ${\cal N}=(1,0)$ gauge superfield $V^{++}$ and analytic $\omega$-hypermultiplet, both in the adjoint representation of gauge group. The effective action is defined in the framework of the background superfield method ensuring the manifest gauge invariance along with manifest ${\cal N}=(1,0)$ supersymmetry. We calculate leading contribution to the one-loop effective action using the on-shell background superfields corresponding to the option when gauge group $SU(N)$ is broken to $SU(N-1)\times U(1)\subset SU(N)$. In the bosonic sector the effective action involves the structure $\sim \frac{F^4}{X^2}$, where $F^4$ is a monomial of the fourth degree in an abelian field strength $F_{MN}$ and $X$ stands for the scalar fields from the $\omega$-hypermultiplet. It is manifestly demonstrated that the expectation values of the hypermultiplet scalar fields play the role of a natural infrared cutoff.


Introduction
The low-energy effective action plays an important role in supersymmetric gauge theories, providing a link between superstring/brane theory and quantum field theory. On the one hand, such an effective action can be calculated in the quantum field theory setting and, on the other, it can be derived within the brane stuff. As a result, the low-energy effective action allows one, in principle, to describe the low-energy string effects by methods of quantum field theory and vice versa (see reviews [1][2][3]).
It is known that D3-branes are related to 4D, N = 4 SYM theory (see, e.g., [4,5]). Interaction of D3-branes is described in abelian bosonic sector by the Born-Infeld action, with the leading low-energy correction of the form ∼ F 4 X 4 , where F 4 is a structure of fourth degree in an abelian field strength F mn and X stands for the scalar fields of 4D, N = 4 gauge (vector) multiplet. The one-loop calculation of such an effective action in the Coulomb branch of N = 4 SYM theory, both in the component approach and in terms of N = 1, 2 superfields, has been performed in ref. [6][7][8][9][10][11][12][13][14][15]. The complete N = 4 structure of the one-loop low-energy effective action has been established in [16,17]. The two-loop contributions to the low-energy effective actions of N = 4 SYM theory have been studied in [18,19]. The structure of the low-energy effective action in the mixed Coulomb -Higgs branch was a subject of ref. [20]. A review of the results related to the calculations of low-energy effective actions in four-dimensional extended supersymmetric gauge theories can be found, e.g., in [1,2].
Another interesting class of the extended objects in superstring/brane theory is presented by D5-branes (see e.g., [4,5]). These objects are related to 6D, N = (1, 1) SYM theory likewise D3branes are related to 4D, N = 4 SYM theory. Similarly to the D3-brane case, the interaction of D5-branes is described by the 6D Born-Infeld action [21] (see [22][23][24][25][26][27] for aspects of the Born-Infeld action in diverse dimensions). Since D5-brane is related to 6D, N = (1, 1) SYM theory, it is natural to expect that the D5-brane interaction in the low-energy domain can be calculated on the basis of the low-energy quantum effective action of this theory.
In the present paper we study the quantum aspects of 6D, N = (1, 1) SYM theory. It is the maximally extended supersymmetric gauge model in six dimensions, and it involves eight left-handed and eight right-handed supercharges. An equal number of spinors with mutuallyopposite chiralities guarantees the absence of chiral anomaly in the theory. From the point of view of 6D, N = (1, 0) supersymmetry, the model is built on a gauge (vector) multiplet and a hypermultiplet. Respectively, the bosonic sector of the model includes a real vector gauge field and two complex (or four real) scalar fields.
Here we develop a method to determine the one-loop effective action in general 6D, N = (1, 1) SYM theory and to calculate the leading low-energy contributions to it. To preserve as many manifest supersymmetries as possible we use the harmonic superspace approach [40,41]. The theory under consideration is formulated in terms of N = (1, 0) harmonic superfields describing the gauge multiplet and the hypermultiplet. Therefore it possesses the manifest N = (1, 0) supersymmetry and, in addition, a non-manifest (hidden) on-shell N = (0, 1) supersymmetry mixing N = (1, 0) gauge multiplet and hypermultiplet. These supersymmetries close on the total on-shell N = (1, 1) supersymmetry. Such a formulation of N = (1, 1) SYM theory was described in detail in the paper [42] (see also ref. [43,44]) 1 . An essential difference of our consideration here is the use of the so called "ω-form" of the hypermultiplet (see below).
The theory under consideration is quantized in the framework of N = (1, 0) supersymmetric background field method [37,38]. In this method, the effective action depends on the background superfields of 6D, N = (1, 0) gauge multiplet and hypermultiplet. By construction, it exhibits manifest gauge invariance under the classical gauge transformations and N = (1, 0) supersymmetry. To calculate the one-loop effective action we make use of the superfield propertime technique [47], which ensures the manifest gauge invariance and N = (1, 0) supersymmetry at all steps of calculation. The low-energy effective action is obtained, when we impose the restriction that both the background superfield strength and the background hypermultiplet are space-time-independent. The leading low-energy approximation amounts to keeping those terms in the effective action which are of the lowest order in the superfield strength. We also assume that the background superfields satisfy the classical equations of motion, that guarantees the gauge independence of the effective action.
We consider the case when gauge symmetry SU(N) is broken to SU(N −1)×U(1) ⊂ SU(N). Technically, this means that background superfields align through the fixed generator of Cartan subalgebra of SU(N), which corresponds to an abelian subgroup U(1). In this case the effective action of the theory depends only on the abelian vector multiplet and hypermultiplet. In the bosonic sector we find out the effective action for the single real U(1) gauge field and four real scalar fields. The same number of bosonic world-volume degrees of freedom is needed to describe a single D5-brane in six dimensions [48].
The paper is organized as follows. In section 2 we formulate an arbitrary 6D, N = (1, 1) SYM theory in terms of N = (1, 0) harmonic superfields representing the gauge multiplet and the hypermultiplet. Unlike the majority of the previous papers on effective action in 4D and 6D harmonic superspaces, we prefer to work with ω-form of the hypermultiplet. Such a formulation has certain merits over the more accustomed formulation in terms of q-hypermultiplet. Although the ω-and q-descriptions of the hypermultiplet are classically equivalent [41], and this equivalency apparently extends to the exact quantum theory, the approximate schemes for calculating the quantum effective action in terms of these superfields can be different. Besides, the ω-hypermultiplet possesses an advantage of being real, i.e. carrying no external U(1) charges. This property essentially simplifies the construction of the super-invariants in the ω-representation. In section 3, besides giving details of the 6D, N = (1, 1) SYM action in terms of the harmonic superfields V ++ and ω, we derive the transformations of the hidden on-shell N = (0, 1) supersymmetry which mixes the superfields V ++ and ω and leaves the action invariant. In section 3 we develop a procedure of constructing the one-loop effective action which depends on both the gauge and the hypermultiplet background superfields. Also we demonstrate advantages of the ω-hypermultiplet formulation and construct some on-shell invariants depending on V ++ and ω superfields. Analogous invariants have never been con-structed in terms of q-hypermultiplet. Section 4 describes the calculation of leading low-energy contributions to the one-loop effective action. To this end, we fix the background superfields by requiring them to be space-time independent and to satisfy the classical equations of motion. We consider the case of background superfields breaking SU(N) gauge group of the original Lagrangian to SU(N − 1) × U(1). In this case the effective action depends on an abelian U(1) gauge superfield. It is explicitly demonstrated that the ω-hypermultiplet acts as an infrared regulator securing the absence of the infrared singularities in the low-energy effective action. The last section contains a brief summary of the results obtained and a list of some problems for the future study.

The model and conventions
We consider the formulation of 6D, N = (1, 1) SYM theory in terms of 6D, N = (1, 0) harmonic analytic superfields V ++ and ω, which represent the gauge multiplet and the hypermultiplet 2 . The action of N = (1, 1) SYM theory is written as The action (2.1) is invariant under the infinitesimal gauge transformations where Λ(ζ, u) = Λ(ζ, u) is a real analytic gauge parameter.
Besides the analytic gauge connection V ++ we introduce a non-analytic one V −− as a solution of the zero curvature condition [41]

5)
2 These superfields satisfy the Grassmann harmonic analyticity conditions D + a V ++ = 0 and D + a ω = 0, where possessing the useful off-shell properties Introducing an analytic superfield F ++ , we can write the classical equations of motion corresponding to the action (2.1) as The N = (1, 0) superfield action (2.1) enjoys the additional N = (0, 1) supersymmetry where A = 1, 2 is the Pauli-Gürsey SU (2) index. To check this, one first derives, using (2.9) and (2.10), the N = (0, 1) transformation law of ∇ ++ ω Then one varies the classical action (2.1) with respect to (2.9) and (2.11) In the first integral, we pass to the integration over the analytic subspace and use the explicit form of the variations (2.9) and (2.11) The last two terms in (2.13) are the total harmonic derivative ∇ ++ due to the properties of F ++ and W +a and so they vanish under dζ (−4) . The first two terms cancel each other after integration by parts with respect to the harmonic derivative ∇ ++ and using the properties ∇ ++ ǫ −A = ǫ −A and ∇ ++ u − A = u + A . Finally, the term tr ∇ ++ ω[ω, ∇ ++ ω] vanishes due to the cyclic property of trace.
The zero curvature condition (2.4) allows one to express the transformation of the nonanalytic gauge connection δV −− through δV ++ 14) and to define the transformation low of the gauge superfield strength W +a under the hidden supersymmetry where Note that, while deriving (2.15), we essentially used the ω-hypermultiplet equation of motion (∇ ++ ) 2 ω = 0 and some its consequences.

One-loop effective action in the background field method
The background field method for 4D, N = 2 gauge theories in the harmonic superspace was worked out in [9]. It was generalized to 6D theories in our recent works [37,38]). Following these techniques, we represent the original superfields V ++ and ω as a sum of the "background" superfields V ++ , Ω and the "quantum" ones v ++ , ω , and then expand the action in a power series with respect to the quantum fields. The one-loop contribution to the effective action Γ (1) for the model (2.1) is given by where where η M N = diag(1, −1, −1, −1, −1, −1) is the six-dimensional Minkowski metric, M, N = 0, .., 5, and ∇ M = ∂ M + iA M is the background-dependent vector supercovariant derivative (see [42] for details).
In the action (3.2) the background superfields V ++ and Ω are analytic but unconstrained otherwise. The gauge group of the theory (2.1) is assumed to be SU(N). For the further consideration, we will also assume that the background fields V ++ and Ω align in a fixed direction in the Cartan subalgebra of su(N) where H ia a fixed generator in the Cartan subalgebra generating some abelian subgroup U(1) 3 . Our choice of the background corresponds to the spontaneous symmetry breaking SU(N) → SU(N − 1) × U(1). We have to note that the pair of the background superfields (V ++ , Ω) forms an abelian vector N = (1, 1) multiplet which, in the bosonic sector, contains a single real gauge vector field A M (x) and four real scalars φ(x) and φ (ij) (x) , i, j = 1, 2, where φ and φ (ij) are the scalar components of Ω hypermultiplet [41]. The abelian vector field and four scalars in six-dimensional space-time describe just the bosonic world-volume degrees of freedom of a single D5-brane [4,5].
The classical equations of motions (2.8) for the background superfields V ++ and Ω are free In that follows we assume that the background superfields solve the classical equation of motion (3.7). We will also consider the background slowly varying in space-time, i.e. assume that Finally we are left with an abelian background analytic superfields V ++ and Ω, which satisfy the classical equation of motion (3.7) and the conditions (3.8). Under these assertions the gauge superfield strength W +a is analytic 4 , D + a W +b = δ b a F ++ = 0. For further analysis it is convenient to use the N = (0, 1) transformation for gauge superfield strength W +a (2.15). In the case of the slowly varying abelian on-shell background superfields the hidden N = (0, 1) supersymmetry transformations (2.10) and (2.15) have the very simple form These transformation rules follow from the abelian version of the transformations (2.10), (2.15) in which one should take into account the conditions (3.8) and (3.7). It is worth to point out that these conditions on their own are covariant under N = (0, 1) supersymmetry.
In conclusion of this section, let us consider the simplest N = (1, 1) invariants which can be constructed out of the abelian analytic superfields W +a and Ω under the assumptions (3.7) and (3.8). It is evident that the following gauge-invariant action (3.10) where (W + ) 4 = − 1 24 ε abcd W +a W +b W +c W +d and F (fΩ) is an arbitrary function of Ω, is invariant under the transformation (3.9) due to the nilpotency condition (W + ) 5 ≡ 0. For our further consideration, of the main interest is the choice which corresponds to F = 1 f 2 Ω 2 in (3.10). The coefficient c in (3.11) cannot be fixed only on the symmetry grounds and should be calculated in the framework of the quantum field theory. In the next section we will find it from the calculation of the leading low-energy contribution to the effective action of the theory (2.1).

Leading low-energy contributions to one-loop effective action
We choose the Cartan-Weyl basis for the SU(N) gauge group generators, so that the quantum superfield v ++ has the decomposition where E α is the generator corresponding to the root α normalized as tr (E α E −β ) = δ αβ and H i are the Cartan subalgebra generators, In this case the background covariant d'Alembertian (3.5) under the conditions (3.7) acts on the quantum superfield v ++ as where we have introduced the operator The one-loop effective action (3.2) for the background superfields V ++ and Ω subjected to the conditions (3.7) and (3.8) thus reads where we have defined ∇ ++ H = D ++ + α H V ++ and A (±) (Ω) = Ω∇ ++ H ± 3 2 (D ++ Ω). The first term in the first line of the expression (4.5) is the contribution from the gauge multiplet, while the second one is the total contribution from the hypermultiplet. The first term in the second line comes from Det 1/2 ⌢ in (4.5), while the second and the third ones are contributions from the ghost action (3.4). We use the standard definition for the functional trace over harmonic superspace in (4.5) Here δ (q,4−q) A (1|2) is an analytic delta-function [41] and O (q,4−q) (ζ 1 , u 1 |ζ 2 , u 2 ) is the kernel of an operator acting in the space of analytic superfields with the harmonic U(1) charge q.
As the next step we rewrite the contribution from Det 1/2 ⌢ as (4.6) Hence the one-loop contribution to effective action (4.5) is divided as and where Π (2,2) T (ζ 1 , u 1 ; ζ 2 , u 2 ) is the projector on the space of covariantly analytic transverse superfields Π (2,2) The Green function G (0,0) (ζ 1 , u 1 ; ζ 2 , u 2 ) satisfies the equation and it can be given explicitly [41] as (4.14) By explicit calculation one can show that in our case of the on-shell background the projector (4.12) acquires the simple form We substitute the expression (4.15) for Π (2,2) T in the one-loop contribution Γ (1) lead (4.11) and take the coincident-harmonic points limit u 2 → u 1 . We see that only the third term in (4.15) survives in this limit. Thus we have As expected, the leading low-energy contribution (4.17) to the effective action in the model (2.1) is just the N = (1, 1) invariant I 1 (3.11). The coefficient c now takes the precise value The expression fore c is similar to that in the four-dimensional N = 4 SYM theory (see, e.g., [19] and references therein). In the bosonic sector the effective action (4.17) has the structure where F 4 = F M N F M N F P Q F P Q − 4F N M F M R F RS F SN and F M N is the abelian gauge field strength.

Conclusions
In this paper we have studied the quantum aspects of the six-dimensional N = (1, 1) super-Yang-Mills theory. We formulated the model in 6D, N = (1, 0) harmonic superspace in terms of N = (1, 0) gauge multiplet and ω-hypermultiplet, all being in the adjoint representation of gauge group SU(N). By construction, the theory possesses the manifest N = (1, 0) supersymmetry and an additional non-manifest N = (0, 1) one.
We studied the effective action of 6D, N = (1, 0) SYM theory in the framework of the background field method. There was considered the special case of the slowly varying background superfields which break the initial gauge symmetry SU(N) down to SU(N −1)×U(1) ⊂ SU(N) and are subject to the free classical equations of motion. We provided a general analysis of possible N = (1, 1) invariants which can be constructed out of the background gauge superfield strength and hypermultiplet. We argued that one of these invariants can be treated as the leading low-energy contribution to the one-loop effective action of N = (1, 1) SYM theory.
It is instructive to compare our results on the one-loop low-energy effective action in 6D, N = (1, 1) SYM theory with the recent activity on calculating the one-loop on-shell amplitudes in the same theory [49][50][51]. The four-point amplitude agrees with the F 4 component term in the effective action. However, unlike the amplitudes, the result for the effective action in our paper was derived in a closed superfield form. The four-point amplitude in [49][50][51] does not allow to directly restore this effective action. The point is that the effective action (4.17) contains the hypermultiplet whose scalar serves as a natural infrared regulator. The amplitudes were calculated in the gauge multiplet sector only, that is not sufficient for deriving the true full effective potential.
We have to note that even in the one-loop approximation there might exist more complicated contributions to the effective action, which can be expected on the basis of a general analysis. One can consider, e.g., a 6D analog of supersymmetric Heisenberg-Euler type effective action in N = (1, 1) SYM theory. Also, it should be noticed that we studied only those contributions to the effective action which contain no harmonic derivatives of the hypermultiplet superfield. The contributions involving such derivatives were beyond the scope of our consideration. It would be interesting to analyze such contributions and to consider a more general class of the background superfields. Namely, it is tempting to find a way to consider the complete effective action with the whole dependence on the harmonic derivatives of the background hypermultiplet included. In this way we expect to obtain the complete N = (1, 1) supersymmetric quantum effective action possessing both explicit N = (1, 0) and hidden N = (0, 1) supersymmetries. We hope to address these issues in the forthcoming works.