Supersymmetric Duality in Superloop Space

In this paper we constructed superloop space duality for a four dimensional supersymmetric Yang-Mills theory with $\mathcal{N} =1$ supersymmetry. This duality reduces to the ordinary loop space duality for the ordinary Yang-Mills theory. It also reduces to the Hodge duality for an abelian gauge theory. Furthermore, the electric charges, which are the sources in the original theory, appear as monopoles in the dual theory. Whereas, the magnetic charges, which appear as monopoles in the original theory, become sources in the dual theory.


Introduction
An important concept in the electromagnetism is the existence of the Hodge duality. The symmetry and topological concepts inherent in field theories have been analysed using this duality [1]- [4]. In fact, this duality has been thoroughly studied and many interesting physical consequences arising from this duality have also been analysed [5]- [13]. It is known that electrodynamics is dual under Hodge star operation, * F µν = −ǫ µντ ρ F µν /2. This is because the field tensor for pure electrodynamics, F µν = ∂ ν A µ − ∂ µ A ν , satisfies, ∂ ν F µν = 0. This field tensor also satisfies the Bianchi identity, ∂ ν * F µν = 0. This field equation for pure electrodynamics can be interpreted as the Bianchi identity for * F µν , because the Hodge star operation is reflexive, * ( * F µν ) = −F µν . So, we can express * F µν , in terms of a dual potential,Ã, such that * F µν = ∂ νÃµ − ∂ µÃν . It has also been known that the existence of magnetic monopoles is equivalent to (electric) charge quantization which in turn is equivalent to the electromagnetic gauge group being compact (i. e. U (1)) [14]. However, a non-abelian version of this duality and its consequences for non-abelian monopoles can only be analysed in the framework of loop space [15]- [16].
For Yang-Mills theory, the field tensor, It also satisfies the Bianchi identity, D ν * F µν = 0. However, now this does not imply the existence of a dual potential because the covariant derivative in the Bianchi identity involves the potential A µ and not some dual potentialÃ µ , appropriate to * F µν = 0. In fact, it has been demonstrated that in certain cases no such solution for such a dual potential exist even for the ordinary Yang-Mills theory [15]- [16]. Thus, the Yang-Mills theory is not dual under the Hodge star operation. However, it is possible to construct a generalized duality transformation for the ordinary Yang-Mills theory in the loop space, such that for the abelian case, it reduces to the Hodge star operation [17]- [18]. This duality has been used for studding 't Hooft's order-disorder parameters [19]. For any two spatial loops C and C ′ with the linking number n between them, and the gauge symmetry generated by the gauge group su(N ), the order-disorder parameters satisfy, A(C)B(C ′ ) = B(C ′ )A(C)exp(2πin/N ). The magnetic flux through C is measured by A(C). So, it also creates an electric flux along C and is thus expressed in terms of the potential A µ . However, B(C) measures the electric flux through C, and thus creates magnetic flux along C. So, it can only be expressed in terms of the dual potentialÃ µ [20]- [21]. A Dualized Standard Model has also been constructed using this duality [20]- [26]. In the Standard Model the fermions of the same type but different generations have widely different masses. The CKM matrix is also not an identity matrix and the off-diagonal elements of the CKM matrix vary in different magnitude [27]. These facts can be explained using the Dualized Standard Model [28]- [29]. In fact, even the Neutrino oscillations [30], and the Lepton transmutations [31], have been studied in the Dualized Standard Model. Polyakov loops have been used for deriving this duality in non-abelian gauge theories [32]. In mathematical language Polyakov loops are the holonomies of closed loops in space-time. In fact, in the physics literature they are called Dirac phase factors. Even though they are defined via parameterized loops in space-time, they are independent of the parameterization chosen. They are gauge group-valued functions of the infinite-dimensional loop space. The main difference between a Polyakov loop and a Wilson loop is that in the Wilson loop a trace is taken and no such trace is taken in the Polyakov loop [32]. Thus, the Polyakov loops are by definition elements of the gauge group. Polyakov loops for three and four dimensional supersymmetric Yang-Mills theories with N = 1 supersymmetry have already been analysed [33]- [34]. In this paper we will derive a supersymmetric duality for the four dimensional supersymmetric Yang-Mills theory in the Wess-Zumino gauge.

Superloop Space
In four dimensional gauge theories with N = 1 supersymmetry, we can construct a covariant derivative [35]. Furthermore, the Bianchi identity can now be written Thus, again for a supersymmetric Yang-Mills theory, no dual potential can be constructed. However, as it is possible to derive a duality for the ordinary Yang-Mills theory in loop space, we will derive a duality a supersymmetric Yang-Mills theory in superloop space formalism. We will derive our results in Wess-Zumino gauge, and impose the constraint F aȧ = F ab = Fȧ˙b = 0. Now can we define ξ(s) = (σ µ ξ µ (s)) aȧ θ a θȧ + ξ a (s)θ a + ξȧ(s)θȧ [35]. The superloop can now be parameterized by ξ A = (ξ aȧ , ξ a , ξȧ), along a curve C, where ξ A (0) = ξ A (2π) is a fixed point on this curve. The space of all such superfunctions parameterizes the superloop space. A functional on this superloop space can be constructed as [34] Φ here P s denotes ordering in s increasing from right to left and the derivative in s is taken from below. This loop space variable is a scalar superfield from the supersymmetric point of view, and can be projected to component superloops.
, which in Wess-Zumino gauge is given by We can also define the parallel transport from a point ξ(s 1 ) to ξ(s 2 ) along path parametrized by ξ as where δ A (s) = δ/δξ A (s) = (δ/δξ ab (s), δ/δξ a (s)). Here we first followed a path to s and then turn backwards along the same path. Thus, the phase factor for the segment of the superloop beyond s did not contribute and H AB (ξ(s)) was obtained because of the infinitesimal circuit generated at s. It is convenient at this stage to define a functional curl and a functional divergence for these superloop space variables as We first note that where iΘ(s − s ′ ) is the Heavisde function.

Duality
For ordinary gauge theories, it is possible to construct a duality using loop space formalism, such that it reduces to the Hodge star operation for the abelian case [17]- [18]. In this section we will further generalize this duality from a ordinary Yang-Mills theory to a supersymmetric Yang-Mills theory. In order to achieve this we define a new variableẼ A [η|t] which is dual to E A [ξ|s]. Here η is another parameter loop which is parameterized by t, η(t) = η a (t)θ a + (γ µ η µ (t)) ab θ a θ b . This dual variable is constructed as follows, where N is a normalization constant. Here ω[η(t)] is a local rotational matrix which accounts for transforming the quantities from a direct frame to the dual frame. In the integral E C [ξ|s] depends on a little segment from s − to s + , such that the limit ǫ → 0 is taken only after integration, where ǫ = s + − s − . As we may need to calculate the loop derivative ofẼ A [η|t], so we regardẼ A [η|t] as a segmental quantity depending on a segment from t − to t + and only after differentiation the limit ǫ ′ → 0 is taken, where ǫ ′ = t + − t − . This limit is taken before the limit ǫ → 0 for the integral. Thus, we can take ǫ ′ < ǫ, and the δ-function now ensures that ξ(s) coincides from s = t − to s = t + with η(t).
After the limit is taken and the segment shrinks to a point, we have E A [η|t] → H AB (η(t))dη B (t)/dt. HereH AB can be constructed from a dual potential. Thus, this superloop space duality implies the existence of a dual potential If we let the segmental width ofẼ µ [η|t] go to zero, then we can write Here we first do the integration before taking the limit to zero. Thus, in the abelian case, when we take the the limit ǫ → 0, we obtain [18] Now identifyingF µν with * F µν , we obtain the Hodge star operation for ordinary electrodynamics. Thus, for the ordinary abelian gauge theory, this duality reduces to the usual Hodge duality.

Sources and Monopoles
We will shown in this section that this duality in the superloop space transforms the electric charges, which are the sources in the original theory, into monopoles in the dual theory. It also transforms the magnetic charges, which are monopoles in the original theory, into sources in the dual theory. In order to prove this result, it is useful to first show that this duality is invertible. This can be demonstrated by first defining E A [ζ|u] as, where ζ B (u) is a new loop parameterized by u. Now we define A A [ζ(u)] as Thus, we obtain, Now identifying ζ(u) with ξ(s), we obtain the desired result that this duality is invertible.
The color electric charge is the source term in the supersymmetric Yang-Mills theory. Thus, it can be defined as the non-vanishing of ∇ C H BC . Alternately, it can also be defined as the non-vanishing of divF [ξ|s]. . Now as η(t) coincides with ξ(s) from s = t − to s = t + , so we can write Here we have performed the integration by parts with respect to Dξ. This expression can be simplified to the following expression, where Now if divE[ξ|s] = 0, then (curlẼ[η|t]) AB = 0. As the duality is invertible, we can also show that if divẼ[ξ|s] = 0, then (curlE[η|t]) AB = 0. So, an electric charge which is a source in the original theory appears as a monopole in the dual theory, and magnetic charge which is a source in the dual theory appears as a monopole in the original theory.

Conclusion
In this paper we have analysed a four dimensional pure Yang-Mills theory with N = 1 supersymmetry in superloop space formalism. We have constructed a generalized duality in superloop space, for this theory. Under this generalized duality transformation the electric charges which appear as sources in the original theory become monopoles in the dual theory. Furthermore, the magnetic charges which appear monopoles in the original theory become sources in the dual theory. This duality reduces to the ordinary loop space duality for ordinary Yang-Mills theory. As the loop space duality for ordinary Yang-Mills theory reduces to the Hodge star operation in the abelian case, so, this generalized duality transformation also reduces to the Hodge star operation for ordinary electrodynamics.
It may be noted that the existence of a duality for ordinary Yang-Mills theory has many interesting physical consequences [21]- [29]. It will be interesting to construct a supersymmetric version of these results using the results of this paper. Thus, the results of this paper can be used to construct a supersymmetric Dualized Standard Model. It will also be interesting to analyses the phenomenological consequences of this model. In particular, we expect to have a dual symmetry corresponding to the super-gauge symmetries of the supersymmetric Standard Model. It will also be interesting to generalize the results of this paper to theories with greater amount of supersymmetry. The results obtained in this paper can also be used for analysing monopoles in the ABJM theory [36]. It may be noted that the supersymmetry of the ABJM theory is expected to get enhanced because of monopole operators [37]- [38]. Thus, the formalism developed in this paper can be find application in the supersymmetry enhancement of the ABJM theory.