Supersymmetric Yang-Mills theory in D=6 without anti-commuting variables

Supersymmetric Yang-Mills theory is formulated in six dimensions, without the use of anti-commuting variables. This is achieved using a new Nicolai map, to third order in the coupling constant. This is the second such map in six dimensions and highlights a potential ambiguity in the formalism.


Introduction and Notation
Supersymmetric theories may be formulated without the use of anti-commuting variables [1,2]. In this approach, supersymmetric gauge theories are characterized by a Nicolai map -a transformation of the bosonic fields such that the Jacobian determinant of the transformation exactly cancels against the product of the Matthews-Salam-Seiler (MSS) [3] and Faddeev-Popov determinants [4]. The formalism avoids any use of anti-commuting objects thus offering an alternate perspective on the physics of gauge theories.
The map, for Yang-Mills theory, was explicitly constructed to second order in the coupling constant in [2], refined in [5] and derived from a rigorous R-prescription in [6]. It was subsequently shown [7] that this construction holds in all the critical dimensions D = 3, 4, 6, 10 where supersymmetric YangMills theories exist [8]. The map and the framework itself were extended to third order in the coupling constant in [9].
In this paper, we present a stand-alone result -a new map, also to third order in the coupling constant, but valid exclusively in six dimensions. The map presented here, arrived at by trial and error (starting with an educated guess), is simpler than the one in [9] and highlights a potential ambiguity in the formalism.
Supersymmetric gauge theories, in D dimensions, are characterized by the existence of a Nicolai map T g -of the Yang-Mills fields such that • The Yang-Mills action without gauge-fixing terms is mapped to the abelian action where S g [A] = 1 4 dx F a µν F a µν is the Yang-Mills action with gauge coupling g and , the Jacobi determinant of T g is equal to the product of the MSS and FP determinants, order by order in perturbation theory.
• The gauge fixing function A new expression for A ′ a µ (x) up to order g 3 is presented in this paper and shown to satisfy all three requirements above only in D = 6.
We work in Euclidean space using the Landau gauge The results presented below may be adapted to other gauges (the light-cone gauge being of particular interest given potential links to [10]). The free scalar propagator is (✷ ≡ ∂ µ ∂ µ ) The free fermion propagator is (spinor indices suppressed) . In a gauge-field dependent background 2 Result The new result in this paper is the following explicit expression for T g to O(g 3 ).
. It is important to note that this result differs from the one in [9]. All terms above have the base structure ∂CA ∂CA ∂CAA at O(g 3 ), while the result in [9] also includes the structures ∂C ∂CAA ∂CAA, A CA ∂CAA and ∂C ∂(AC) A ∂CAA.
Further, terms that overlap with those in [9], appear here with different coefficients. As a consequence, the expression above is not a subset of the result in [9]. Finally, while the result in [9] was valid in all the critical dimensions, we will see that the result in (8) constitutes a map only in six dimensions.

Checks of the Result
In this section, we prove that expression in (8) satisfies all three requirements, (1), (2) and (3), necessary for it to be a map. The calculations up to O(g 2 ) are identical to those in [7,9], so the focus here will be on O(g 3 ).

Gauge condition
We begin with the third requirement, listed in (3). We need to show that We apply ∂ µ to the terms of order g 3 in (8). This gives us a symmetric ∂ µ ∂ ρ at the beginning of the expression so we eliminate all terms that are anti-symmetric under the exchange µ ↔ ρ and find The first two terms cancel each other under the interchange of µ and ρ. Similarly, the other two terms also cancel out confirming that

Free Action
We now move to the first requirement in (1) which states that the transformed gauge field must satisfy Because of the invariance of the gauge function, we ignore the second term on the l.h.s. and the corresponding term on the r.h.s. of this equation [7]. At third order, (11) has two contributions This expression reads We simplify the r.h.s. to obtain This is further simplified with some re-writing [for example, based on the symmetries a ↔ c and µ ↔ λ]. The r.h.s. simplifes to There is a symmetry to these terms: the ∂CAA blocks are invariant under a cyclic permutation of the Lorentz indices. This motivates re-writing the term as We now find, for the first time in this computation, that for (14) to vanish we need to invoke the Jacobi identity Thus (11) holds up to O(g 3 ).

Jacobians, fermion and ghost determinants
Finally, we turn to (2), the second requirement. This is, in some sense, the most constraining of the three requirements, demanding that the bosonic Jacobian determinant equal the product of the MSS and FP determinants. Again, this check up to O(g 2 ) was performed in [1,7] allowing us to concentrate here on O(g 3 ).
log det . (16) It is this non-trivial requirement which results in a dimensional dependence. We prove that the map in (8) satisfies (16) only for D = 6.

Fermion determinant
To compute the fermion determinant, we need to evaluate the following quantity where the relevant functional matrix reads We use to arrive at the following five independent terms at order g 3 where r represents the number of spinor components.

Ghost determinant
For the ghost determinant, we compute where Up to O(g 3 ) this yields

Bosonic Jacobian
At O(g 3 ) the logarithm of the Jacobian determinant schematically consists of three terms and the final trace involves setting µ = ν, a = b, x = y and integrating over x.
All terms at O(g 3 ) are of the form ∂CA ∂CA ∂CAA. The functional derivative on the very first field, in this structure, vanishes trivially [7]. The functional differentiation of the field in the middle block produces the structure ∂CA ∂C ∂CAA not seen elsewhere. These terms vanish as described in the appendix. Functional differentiation of either field from the last block produces terms with the same structure as those from the fermion and ghost contributions. The table below offers a summary of the various contributions to the Jacobian from (24).

Jacobian table
In the table, colums 2 − 5 capture bosonic contributions, summed up in column 6. Column 7 contains the sums of the fermion and ghost contributions. The detailed breakdown for the bosonic contributions is as follows: Column 2 contains the contributions from O(g) terms when "cubed". Column 3 lists contributions from O(g) × O(g 2 ). Column 4 has contributions from the 9 terms in the bosonic result (first three lines of O(g 3 ) from (8)). In column 5, we present contributions from the next four lines of (8) (12 terms).
The main result is that Columns 6 and 7 are equal only for D = 6.
This completes our proof of (1), (2) and (3). It is curious that we have not had to invoke the gauge condition, which was needed in [9], in this proof.

* * *
We conclude that (8) represents an alternate Nicolai map [12] in six dimensions, up to O(g 3 ), distinct from the map in [9]. This raises the possibility that there exists a dimensiondependent map that differs for each critical dimension. However, we note that the checks to this order for this particular map do not guarantee that this map will work at next/higher order 2 . The result in [9] is different because it is derived from the R-prescription and is limited to O(g 3 ) only because the procedure becomes technically involved at higher orders.
There is a third and rather unlikely outcome: that six dimensions is special for yet unknown reasons. For another curious result within this formalism that singles out six dimensions, see equation (3.10) in [11]. D = 6 is also home to the mysterious N = (2, 0) theory [13] which still lacks a complete Lagrangian description [14].
Line 5 in (8) After differentiating and tracing this reads These three terms vanish by the same arguments that applied to the terms in line 4 of (8).

Second set of terms at O(g 3 )
We have twelve remaining terms in (8). Functional differentiation and trace in the middle block yields These three term vainsh by using ∂ y σ C(y − x) = − ∂ x σ C(x − y) as the first line above is symmetric in ρ, σ while the bracket is anti-symmetric in the same two indices. So this contribution vanishes.