A Nonabelian Particle-Vortex Duality

We define a nonabelian particle-vortex duality as a $3-$dimensional analogue of the usual $2-$dimensional worldsheet nonabelian T-duality. The transformation is defined in the presence of a global $SU(2)$ symmetry and, although derived from a string theoretic setting, we formulate it generally. We then apply it to so-called"semilocal strings"in an $SU(2)_{G}\times U(1)_{L}$ gauge theory, originally discovered in the context of cosmic string physics.


I. INTRODUCTION
Beginning with the remarkable correspondence between the sine-Gordon and massive Thirring models [1], dualities have played a crucial role in the modern understanding of quantum field theories. Indeed, they have been an indispensible tool in the understanding of both strongly coupled systems as well as various nonperturbative problems. This was certainly the case, for instance, for Seiberg and Witten's landmark study of (3+1)-dimensional, N = 2 supersymmetric gauge theory [2,3], where electric-magnetic duality (a generalized form of the usual electric-magnetic duality of Maxwell electrodynamics) that exchanges particles with monopoles, was essential in fully solving the low energy theory. In that (3 + 1)−dimensional case, even though an explicit path integral transformation exists only for the abelian case, the duality is understood as being essentially non-abelian in the sense of acting on the full non-abelian theory.
One duality which has received considerably less attention occurs in (2 + 1)−dimensional gauge theories and exchanges particles with topological solitons, specifically vortices [4]. One possible reason for the dearth of literature on the subject could be that its utility lies primarily in condensed matter systems which, being usually non-relativistic are much less susceptible to the powerful relativistic methods employed in high energy theory. Another is likely the fact that the duality was generally less well-defined than its (3 + 1)− dimensional counterpart. To the best of our knowledge, particle-vortex duality has, until now, only been defined in the context of abelian gauge theories, exhibiting Neilsen-Olesen-like vortices. In [5], this duality was defined as a path integral transformation in a manifestly symmetric way, and embedded into a planar N = 6 Chern-Simons-matter theory commonly known as the ABJM model, which is itself known to be * Electronic address: jeff.murugan@uct.ac.za † Electronic address: nastase@ift.unesp.br dual to the type IIA superstring on an AdS 4 × CP 3 background [6]. In this context, the particle-vortex duality of the boundary field theory was shown to correspond to an electric-magnetic duality in the bulk. As a final point in [5], it was speculated that, based on the structure of the embedding into the ABJM model, it should be possible to define a nonabelian version that would act on the whole non-abelian ABJM model.
In this letter, we show that it is indeed the case that we can define a version of particle-vortex duality that acts on a non-abelian theory, at least in a certain restricted sense. Key to our argument are the recent advances in the study of 2−dimensional non-abelian T-duality acting on the string worldsheet in string theory [7] (see also [8][9][10] for the action of the nonabelian T-duality in supergravity). By generalizing the procedure to (2 + 1)−dimensions, we obtain a non-abelian version of particle-vortex duality that acts on gauge theories with a global SU (2), as well as a local symmetry. Recognizing that this is precisely the set-up for the "semi-local" vortices found in [12] (see also [13,14]) in the context of cosmic strings in the case of a local U (1) symmetry, we explicitly exhibit the action of the nonabelian particle-vortex transformation on these solutions.
The letter is organized as follows. In section 2 we revisit non-abelian T-duality and its relation to the abelian T-duality, extending it in section 3 to three spacetime dimensions, consequently defining a non-abelian particlevortex duality on a general theory which we illustrate with a simple example of a semilocal vortex in section 4. This article should be viewed as a proof-of-principle of a phenomenon with potential application from condensed matter to cosmology, with a longer companion paper to follow in which we will elaborate further on the duality and provide more substantial examples [15].

II. NONABELIAN T-DUALITY
In string theory, abelian T-duality is a symmetry that acts on a compact dimension as an inversion of its ra-dius, R → α ′ /R. First noted at the level of the string spectrum, it was proven to be a symmetry of the perturbative string path integral in [16], where it was defined as a duality transformation of the worldsheet action. Specifically, one writes a constrained first order form for the worldsheet action for the compact direction, with a Lagrange multiplier implementing the constraint that mixed second derivatives of the compact coordinate commute. Then, if instead of eliminating the Lagrange multiplier the original coordinate is integrated out, one obtains a T-dual theory in which the Lagrange multiplier plays the role of a new coordinate. This formulation is very similar in spirit to the abelian particle-vortex duality transformation at the level of the path integral [5].
Initially carried out with commuting abelian isometries, a natural next step was to "nonabelianize" the transformation. This was first accomplished in [7] with the transformation acting on three coordinates transforming under a (global) SU (2) symmetry, obtaining what became known as non-abelian T-duality. In this section we review the procedure.
Consider the string background with metric and B-field and constant dilaton φ = φ 0 . Here, are SU (2) left-invariant 1−forms for the Euler angles (θ, φ, ψ), such that dL i = 1 2 f i jk L j ∧ L k . The angles have the range 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π, 0 ≤ ψ ≤ 4π, and the SU (2) transformations act as Using the normalized Pauli matrices , understood here as a field g(τ, σ) on the string worldsheet, the 1−forms can be rewritten more conveniently as L i ± = −iTr (t i g −1 ∂ ± g). Note that while g is complex, the L i are all real. Then, with the string worldsheet action in this background takes the globally SU (2)−invariant form One can make this invariance local by introducing an SU (2) gauge field A and replacing derivatives with covariant derivatives, Since we don't want to add a new degree of freedom (the gauge field A), we need to impose its triviality as a constraint. A good way of doing that is by requiring the field strength to vanish and enforcing this in the action through a La- In this way we obtain a first order action that acts as a master action for the T-duality. Integrating out the Lagrange multiplier v leads to F +− = 0 which, in the absence of any topological issues, leads to a trivial A, equivalent to A = 0, recovering the original theory.
If instead, we integrate out the gauge field A and gauge fix the SU (2) symmetry, we get A ± in terms of v, and on substituting into the master action, obtain the T-dual action. Explicitly, we first partially integrate the Lagrange multiplier term to where After varying this with respect to A + and A − and solving the resulting equations of motion, we obtain where M ij = E ij + f ij . Finally, substituting A ± back in the master action, produces the T-dual action At the quantum level, i.e. considering the one-loop determinant, the T-duality also modifies the dilaton to

III. PARTICLE-VORTEX DUALITY AS NONABELIAN T-DUALITY IN 3 DIMENSIONS
We now want to generalize the above construction to (2 + 1)−dimensions. Again, it is natural to con-sider the real variables Φ k 0 and L i µ = −iTr [t i g −1 ∂ µ g] , where, as before g(x µ ) ∈ SU (2) is complex. We will first write down a desired master action generalizing the 2−dimensional case, except with Q µi = 0 and Q µν = δ µν . First though, we define the local SU (2) symmetry, which means replacing derivatives with covariant derivatives, The desired master action is then where the gauge field strength is the usual Varying the action with respect to the Lagrange multipliers v i µ leads to F i µν = 0 which, in the absence of any topological issues, leads to a trivial gauge field. Consequently, the choice of A µ = 0 leads toL i µ = L i µ , reducing the action to the pre-dualizing, If instead we first partially integrate the Lagrange multiplier term to where f νij ≡ f ijk v k ν , and gauge fix by setting g = 1, thenL i µ → iTr [t i A µ ] = iA i µ . Subsequent variation of the master action with respect to A i µ gives On substituting A i µ back in the master action (11), we get the particle-vortex dual action Evidently then, we have found a transformation of the path integral in (2 + 1)−dimensional theories of the form (12) that furnishes a non-abelian particle-vortex duality. In order to consider it a genuine particle-vortex duality transformation, we must be able to derive (12) from a more familiar action that admits vortex solutions, couple the theory to a nontrivial gauge field and add a vortex current term to the action. To show that this sequence can be executed, we consider a scalar field Φ in a tensor product representation, obtained from the adjoint representations of two groups, that a priori need not be related to the SU (2) on which particle-vortex duality acts. As an ansatz we take where T a and T A are adjoint matrices transforming under a priori different groups, and F A i are given coefficients (a "background"), out of which we will construct E ij . Normalizing the generators through Tr [T a T b ] = δ ab and Tr [ T A T B ] = δ AB , leads to and consequently, the standard kinetic term for Φ be- where N E ij ≡ F A i F A j , which up to a normalization of Φ 0 is the same as (12). We can now add to this action a potential depending only on Φ a 0 which, as we saw earlier, is untouched by the duality transformation. Thereafter, we need to couple to a gauge field, write a vortex ansatz and add a vortex current to the action. Toward this end, we need a more general ansatz for the scalar.
One simple, if naive, possibility is if F A i is simply F i , i.e. T A is trivial and in which we can write an ansatz with just a common phase, and for which the standard scalar kinetic term becomes Again, we reproduce (12) except with E ij = F i F j now separable. Next, we couple the scalar to an external gauge field, a µ = a m µ T m in a Lie algebra direction not covered by A µ (Tr [A µ T m ] = 0). This amounts to replac-ingL i µ in (11) bỹ and adding a kinetic term of + 1 4 Tr [f 2 µν ], for the external gauge field.
However, for the purposes of writing a vortex ansatz, it is more useful to consider instead a modification that creates a covariant derivative acting on the field Φ. For Φ in the adjoint representation, the normal derivative is Therefore, in effect, the gauge field coupling gives the replacement , (23) to first order. We note that nothing makes it necessary that the gauge field be nonabelian at all. Indeed, if A belongs to the singlet representation, we may write the usual U (1) covariant derivative for Φ without a problem.
We are now ready to consider a vortex ansatz. Assuming azimuthal symmetry, Φ a 0 = Φ a 0 (r) and "vorticial" information about the solution is encoded in its phase where N A is the vortex number and θ is the polar angle on the plane. For a U (1) gauge field, it suffices to simply erase the A index. As in the abelian case, the requirement that D µ Φ → 0 at r → ∞ ensures both a finite energy solution (since the kinetic term |D µ Φ| 2 vanishes at infinity) and the existance of a topological charge (since it implies that A θ dθ is quantized). Of course, having an ansatz doesn't guarantee the existence of a solution.
One needs to show that it is a solution of the equations of motion in a specific model (specified by a particular potential V (Φ a 0 )). In a forthcoming article, we will show explicitly how the duality acts of nonabelian vortices in an SU (2) × U (1) gauge theory that arises, for example, in the low energy limit of N = 2, SU (3) QCD with N f flavors [11].
Finally, with an actual solution at hand we can isolate the vortex contributions to the action in the path integral, and obtain a vortex current term. Similarly to the abelian case in considered at length in [5], where the phase α separates into α smooth + α vortex , with α vortex being the part that contains a topological charge of the vortex, we now replace L i µ with L i µ,smooth + L i µ,vortex . Gauge fixing g = 1, we get L i µ = iA i µ + L i µ,vortex , or rather A i µ → A i µ,smooth + A i µ,vortex . Then, varying the master action (11) with respect to A i µ,smooth gives The associated vortex current term, is obtained from the term linear in A µ . From the vortex ansatz (24), we have where j µ = Φ † ∂ µ Φ − Φ∂ µ Φ † is a U (1) scalar particle current. In other words, the relation (26) expresses a duality between particle and vortex currents, generalizing the ǫ µνρ ∂ ν j ρ = j µ vortex relation from the abelian case, and justifying us calling it a nonabelian particle-vortex duality for the path integral transformation.

IV. AN EXAMPLE: SEMILOCAL VORTICES
To illustrate the above, we now exhibit the duality transformation explicitly for the case of the semilocal (cosmic) strings of [12][13][14]. Defined through the Lagrangian  (2), so for the duality transformation we simply write the ansatz (18) but without Φ = Φ a T a . Here Φ a 0 , a = 1, 2 and L i µ , i = 1, 2, 3, 4 ∈ adj(U (2)) are real, i = 4 corresponds to 1l, thus we see that even though we have 6 real variables, we are constrained to have the same phase for Φ 1 and Φ 2 . That is actually fine, since for the axially symmetric n-vortex ansatz where (r, θ) are polar coordinates on the plane, leads to the condition that at r → ∞, α 2 = α 1 + c, with c a constant. Taking c = 0 (without loss of generality), the vortex solution indeed satisfies the ansatz for the particle-duality transformation in (18). The energy is Bogomolnyi-saturated at critical coupling β ≡ 2λ/e 2 = 1, where the second order equations of motion for Φ and a µ , defining ϕ(r) = (ϕ 1 (r)) 2 + (ϕ 2 (r)) 2 , descend to the first order BPS equations same ones as for the Nielsen-Olesen vortex, thus the same numerical vortex solution is used to constuct this "semilocal string". Making the identification Tr [t i T m ] = δ i m and the embedding a 4 µ = a µ , a 1,2,3 µ = 0 (and A 1,2,3 µ = 0; A 3 µ = 0), we have the master action for the duality (replacingL i µ withL i µ in (11) and adding the kinetic term) where E ij = F i F j and Φ a 0 = vϕ a . As before, varying with respect to v i µ leads to the original action, where the terms on the first line combine to give −(1/2)|D µ Φ| 2 .

V. DISCUSSION
Abelian particle-vortex duality has proven a powerful tool in the understanding of bosonic systems that range from anyonic superconductivity through to cosmic strings. An excellent example of this is illustrated in [17], which utilizes precisely this duality to explain the current-voltage symmetry observed near the critical point of the transition between the Laughlin plateaux and Quantum Hall insulator, a phenomenon not captured in the linear electromagnetic approximation.
As exciting as these developments have been to date, we are today at the birth of a new scientific paradigm with the discovery of topological phases of matter as embodied in, for example, high temperature superconductors and the fractional quantum Hall effect. A key feature of such states of matter is that their quasi-particle excitations are neither fermionic nor bosonic but are best described as nonabelian anyons that obey nonabelian braiding statistics. Certainly since Moore and Read's landmark paper [18] identifying quasiparticle excitations of certain fractional quantum Hall systems which obey nonabelian statistics, nonabelian states of matter have posed an exciting challenge to theoretical physics. Recent technological advances coupled with equally rapid developments in topological field theory have served only to fuel interest in this area and make the study of nonabelian states of matter one of the hottest topics in theoretical condensed matter physics today. It is our hope that the nonabelian particle-vortex duality communicated in this article will develop into as useful a tool to understand these new states of matter as its counterpart did for abelian physics.