Physics Letters B. New supersymmetric Wilson loops in ABJ(M) theories. V. Cardinali a, L. Griguolo b, G. Martelloni a, D.

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1 Physics Letters B Contents lists available at SciVerse ScienceDirect Physics Letters B New supersymmetric Wilson loops in ABJM theories V. Cardinali a, L. Griguolo b, G. Martelloni a, D. Seminara a, a Dipartimento di Fisica, Università di Firenze and INFN Sezione di Firenze, Via G. Sansone 1, Sesto Fiorentino, Italy b Dipartimento di Fisica, Università di Parma and INFN Gruppo Collegato di Parma, Viale G.P. Usberti 7/A, 3100 Parma, Italy article info abstract Article history: Received 22 September 2012 Received in revised form 5 October 2012 Accepted 21 October 2012 Available online 2 October 2012 Editor: L. Alvarez-Gaumé Keywords: Wilson loops Supersymmetric gauge theories Chern Simons-matter theories We present two new families of Wilson loop operators in N = 6 supersymmetric Chern Simons theory. The first one is defined for an arbitrary contour on the three dimensional space and it resembles the Zarembo construction in N = SYM. The second one involves arbitrary curves on the two dimensional sphere. In both cases one can add certain scalar and fermionic couplings to the Wilson loop so it preserves at least two supercharges. Some previously nown loops, notably the 1/2 BPS circle, belong to this class, but we point out more special cases which were not nown before. They could provide further tests of the gauge/gravity correspondence in the ABJM case and interesting observables, exactly computable by localization techniques Elsevier B.V. All rights reserved. 1. Introduction and results Three-dimensional N = 6 supersymmetric Chern Simons-matter theories with gauge group UN UM [1,2] provide an exciting arena for studying the duality between string theories on asymptotically AdS spaces and conformal field theories. The gravity dual of this theory is M-theory on AdS S 7 /Z, where is the level of the Chern Simons term, or, for large enough, type IIA string theory on AdS CP 3. Wilson loop operators, which in the dual string theory are given by semi-classical string surfaces [3,], can be also defined. The most symmetric string solution preserves half of the supercharges of the vacuum as well as a U1 SL2, R SU3 bosonic symmetry and its dual operator in the field theory has been ingeniously derived in [5] see [6] for an alternative derivation. Other Wilson loop operators, previously constructed in [7 10], preserve only 1/6 of the supercharges and cannot be dual of this classical string see also [11] for a general discussion of 1/2 BPSstatesin ABJM theories. In addition to the gauge fields, the 1/2 BPSWilson loop couples to bilinears of the scalar fields and, crucially, also to the fermionic fields transforming in the bi-fundamental representation of the two gauge groups. The operator is classified by representations of the supergroup UN M and is defined in terms of the holonomy of a superconnection: the analysis presented in [5] * Corresponding author. addresses: cardinali@fi.infn.it V. Cardinali, griguolo@fis.unipr.it L. Griguolo, martelloni@fi.infn.it G. Martelloni, seminara@fi.infn.it D. Seminara. considers loops supported along an infinite straight line and along acircle. For the 1/6 BPS Wilson loop a matrix model, describing its vacuum expectation value, has been derived in [12] and this result carries over to the 1/2 BPS case. The calculation of [12] uses localization with respect to a specific supercharge which is also shared by the 1/2 BPS operator. This Wilson loop is cohomologically equivalent to a very specific choice of the 1/6 BPS loop, constructed with bosonic couplings only, and is therefore also given by a matrix model. Happily it can be calculated for all values of the coupling also beyond the planar approximation [13 15] and, in the strong coupling regime, it matches string computations. In four-dimensional N = super-yang Mills theory the original examples of 1/2 BPS Wilson loops the straight line and the circle [16,17] can be embedded into whole families preserving between 2 and 16 supercharges. The straight line has been generalized by Zarembo [18], the amount of conserved supersymmetry being related to the dimension of the subspace containing the contour. The circular Wilson loop, that can be computed exactly through localization [19], has been instead generalized in [20] to a class of contours living in an S 3 also called DGRT loops. A subset of those operators, preserving 1/8 of the original supersymmetry, are contained in an S 2 and their quantum behavior is described by perturbative [21] two-dimensional Yang Mills theory [20,22 2] a property that is also shared by loop correlators [25 27]. We remar that in N = SYM a general classification of supersymmetric Wilson loops does exist [28,29]. In this Letter we present two new families of BPS Wilson loops operators in ABJM theories, generalizing respectively the straight /$ see front matter 2012 Elsevier B.V. All rights reserved.

2 616 V. Cardinali et al. / Physics Letters B line and the circle constructed in [5]: they can be considered as the analogous to the Zarembo and DGRT loops in three dimensional N = 6 super-chern Simons-matter theories. Remarably we recover within our analysis some BPS configurations that we have introduced in [30], where a generalized cusped Wilson loop has been carefully studied at classical and quantum level see also [31] for a discussion at strong coupling. Our results might be useful in studying the connection, originally proposed in D = by[32], between quar antiquar potential and cusp anomalous dimension [33,3]. Potentially they could also play a role in the exact computation of the elusive function hλ [35 37], as suggested in[38]. 2. Supersymmetry conditions for an arbitrary contour The ey idea exploited in [5] to construct 1/2 BPS lines and circles is to embed the natural UN UM gauge connection present in ABJM theories into a super-connection 1 ia ilτ η I ψ I ψ I η I iâ { A Aμ ẋ μ i with M J I C I C J,   μ ẋ μ i M 1 J I C J C I, belonging to the super-algebra of UN M. In1 the coordinates x μ τ describe the contour along which the loop operator is defined, while the quantities M I J τ, M I J τ, η ατ I and ηi α τ parameterize the possible local couplings. The latter two, in particular, are taen to be Grassmann even quantities even though they transform in the spinor representation of the Lorentz group. We shall focus on operators that possess a local U1 SU3 R-symmetry invariance, since they are those described by semiclassical string surfaces in the dual picture. The R-symmetry structure of the couplings in 1 is thus described by a vector n I τ and its complex conjugate n I, specifying the local embedding of the unbroen SU3 subgroup into SU η α τ I = n Iτ η α τ, η α I τ = ni τ η α τ, M I J τ = p 1 τ δ I J 2p 2τ n J τ n I τ, M I J τ = q 1 τ δ I J 2q 2τ n J τ n I τ. 2 By rescaling the Grassmann even spinors η α and η α, we can always choose n I n I = 1. The next step is to constrain the form of the free functions present in2 by requiring that the Wilson loop defined by 1 is globally supersymmetric. This part of the construction is different from its four-dimensional analog. The usual condition δ susy Lτ = 0 is here too strong and it does not yield any solution for the couplings 2. To obtain non-trivial results, we must replace δ susy Lτ = 0 with the weaer requirement [5,6] δ susy Lτ = D τ G τ G + i{l, G], 3 where the r.h.s. is the super-covariant derivative constructed out of the connection Lτ acting on a super-matrix G in un M. The condition 3 guarantees that the action of the supersymmetry charge translates into an infinitesimal UN M super-gauge transformation for Lτ and thus the traced loop operator is invariant. 2 1 In Minowsi space time, where ψ and ψ are related by complex conjugation, Lτ belongs to un M if η = iη. In Euclidean space, where the reality condition among spinors is lost, we shall deal with its complexification sln M. 2 The stronger condition δ susy Lτ = 0 would imply that both the superholonomy and its trace are invariant. In the text we use the term trace loosely: it might mean the actual trace, or the super-trace or something more exotic see Section.1. Since the supersymmetry transformations of the bosonic fields do not contain derivatives, the super-matrix G in 3 cannot have an arbitrary structure but it has to be anti-diagonal, i.e. 0 g1 G = ḡ 2 0 D τ G = η I ψ I ḡ 2 g 1 ψ I η I Dτ g 1 Dτ ḡ. 2 ḡ 2η I ψ I + ψ I η I g 1 Here the covariant derivative D τ in is constructed out of the dressed bosonic connections A and  D τ g 1 = τ g 1 + iag 1 g 1 Â, D τ ḡ 2 = τ ḡ 2 iḡ 2 A Âḡ 2. 5 Supersymmetry is preserved if there exist two functions g 1 and ḡ 2 such that I: i η Iδ Ψ I = D τ g 1, II: i δψ I η I = D τ ḡ 2, 6a III: η I ψ I ḡ 2 g 1 ψ I η I = δa, IV: ḡ 2 η I ψ I + ψ I η I g 1 = δ A, 6b for a suitable form of the couplings, taing into account the superconformal transformation of the ABJM fields see Appendix A in [5]. The analysis can be performed in full generality and it will be presented in details in a forthcoming paper [39]: herewejust state the main results, eeping trac of their origin. First of all, the reduced spinor couplings η α and η β introduced in 2 are determined by the contour x μ through the relations A: δ β α = 1 2i ηβ η α η α η β and B: ẋμ γ μ α β = l 2i η β η α + η α η β. 7 These conditions originate from I and II in 6a, basically representing the request that derivative terms are taen along the contour. The matrices M and M have the form M J I τ = M J I τ = l δ J K 2n K n J. 8 The constant parameter l can only tae two values, ±1, and the choice specifies the eigenvalues of the matrices M and M: 1, 1, 1, 1 [l = 1] and 1, 1, 1, 1 [l = 1]. The invariance of 7 under the replacement η, η uη, u 1 η is instead related to III and IV in 6b, simply determining the relative scale of the reduced spinor couplings. The SU tensor structure of the preserved supercharge Θ IJ is controlled by a couple of constraints, consisting of the following algebraic relations A: ɛ IJKL η Θ IJ n K = 0 and B: n I η Θ IJ = 0, 9 where the vectors n K and n K are defined in 2. Finally there are two sets of differential conditions A: Θ IJ τ η K ɛ IJKL = 0 and B: Θ IJ τ η I = They ensure that the derivative term in the supersymmetry variation taes the correct form without leaving any unwanted remnant.

3 V. Cardinali et al. / Physics Letters B We remar that all the above conditions are strictly local. To construct an actual supersymmetric Wilson loop, we must provide a family of couplings η, η,n I, n I so that the solution of Eqs. 9 and 10 taes the form of a conformal Killing spinor, i.e. Θ IJ = θ IJ x γ ɛ IJ, 11 where θ IJ and ɛ IJ are constant spinors. The relations 7, 9 and 10 provide a complete set of supersymmetry conditions, but their form is not unique. For instance the requirement 9 is equivalent to the following expansion for the preserved supercharge in terms of the couplings η K and η K : Θ IJ α = l 2i [ η I α h J η J α h I 1 2 ɛ IJKL η Kα m L ], 12 where the Grassmann odd vectors m I and h I are defined from ηγ μ Θ KL n K = ẋμ h L, ɛ IJKL n K Θ IJ γ μ η = ẋμ m L 13 and they obey the orthonormality relations: n I n I = 1, n I hi = n I m I = 0. The explicit form of the gauge functions g 1 and ḡ 2 can be nicely written using these vectors g 1 2 ηi Θ IL C L = hl 2 C L, ḡ 2 ɛijkl η K Θ IJ C L = ml C L Supersymmetric Wilson loops on R 3 We consider first a family of Wilson loops of arbitrary shape, which preserve at least a supercharge of the Poincaré type, i.e. a supercharge with ɛ IJ = 0. These operators can be viewed as the three dimensional companion of the loops discussed by Zarembo in [18]. The BPS straight-line constructed by Druer and Trancanelli in [5] is the simplest example enjoying this property. We start by considering the differential constraints 10, written in a way which is easier to solve: τ hl +η ɛ KL n K = 0, τ m L + n K ɛ IJ η ɛ IJKL = For ɛ IJ = 0 the vectors m I and h I are seen independent of the contour parameter τ. To further proceed we contract 12 and its dual with η α η Θ IJ = l n I h J n J hi and ɛ IJKL η Θ IJ = ln K m L n L m K 16 and we observe that, for a generic contour, these expansions are compatible with a constant Θ IJ if we choose, for instance, the following ansatz for n I and n I : n I = η s I and n I = s I η. 17 Here s α I are four τ -independent spinors and η and η are determined by 7. The normalization condition n I n I = 1isequivalent to the following completeness relation on the spinors s α I and s α I s I β sα I = 1 2i δα β. 18 We plug our ansatz into the algebraic conditions 9 and, after some wor, we can show that for a generic contour they are equivalent to the linear system of equations ɛ IJKL θ IJ γ μ s K = 0 and s I γ μ θ IJ = The relations 19 also ensure that the remaining differential constraints 10 are identically satisfied in the case of Poincaré charges. The general solution of the supersymmetry conditions 19 can be written as follows Θ IJρ = θ IJρ = v J s Iρ v I s Jρ, with v I s Iβ = It is straightforward to chec that the above ansatz solves the conditions 19, in fact si γ μ θ IJ = s I γ μ s I v J v I s I γ μ s J = 1 2i Trγ μ v J = 0, ɛ IJKL θ IJ γ μ s K = 2ɛ IJKL v J s I γ μ s K = The first result follows from the completeness relation 18, while the property s I γ μ s K = s K γ μ s I, which holds for bosonic spinors, is responsible for the second one. Any solution of 19 can be cast into the form 20 and these loops are generically 1/12-BPS [39]. Summarizing we have constructed a family of Wilson loops of arbitrary shape, which are invariant under the Poincaré supercharges 20, with couplings η α I = n I η α = s β I η β η α = is β I 1 + l ẋ γ β η α I = n I η α = η α η β s β I = i 1 + l ẋ γ α M J K = M J K = l δ J K 2n K n J = l δ JK 2is K s J 2il ẋμ s K γ μ s J. Supersymmetric Wilson loops on S 2 α, β s I β, 22a. 22b We propose a second family of Wilson loops, that is defined for an arbitrary curve on the unit sphere S 2 : x μ x μ = 1. We shall consider a deformation of the ansatz 17 n I = r ηu s I and n I = 1 si U 1 η, 23 r where sα I and si α are again four τ -independent spinors obeying the relation 18. Theparameterr is a function of τ and it will become useful when we will solve the differential constraints. The matrix U is the characterizing ingredient of our ansatz: it is an element of SU2 constructed with the coordinates x μ τ of the circuit, namely U = cosα1 + i sin α x μ γ μ, 2 with α free constant parameter. There is a natural connection among U in 2, the tangent vector to the circuit and the invariant one-forms on S 2. In fact if we evaluate the Lie-algebra element τ UU,weobtain τ UU 1 = i sin α cosαẋ λ sin αɛ λμν x μ ẋ ν γ λ, 25 where the r.h.s. is a linear combination of the tangent vector and of the SU2 invariant forms. Let us first focus our attention on the algebraic conditions A in 9. Using the Fierz identity and the explicit form 11 for the Killing spinor, we can rewrite it as follows r ɛ IJMN ζγ μ ζ IJ γ M μ s = 0, 26 2 where we have defined an auxiliary reduced coupling ζ = U 1 η and an auxiliary super-conformal charge IJ = Θ IJ U. We can also

4 618 V. Cardinali et al. / Physics Letters B rearrange the condition B in 9 following the same idea and we find 1 ζγ μ ζ s I γ μ IJ = 0, 27 2r where ζ = U 1 η. Now we notice that Eqs. 26 and 27, for a generic contour, lead to the same conditions 19 discussed in the previous section ɛ IJKL IJ γ μ s K = 0 and IJ γ μ s I = Conversely they possess the same ind of solutions 3 : IJ is the constant spinor θ IJ defined in 20. In other words the preserved supercharges can be parametrized as follows Θ IJ = [ cos α1 + i sin α x μ γ μ ] θ IJ = U θ IJ. 29 The above representation is very useful when we examine the derivative constraints 10: in fact it allows us to easily recognize all the terms which automatically vanish since they are proportional to the two SUSY conditions 19. Using this fact, the first of the two constraints 10 can be easily translated into an ordinary differential equation for the unnown function r ṙ + ilr sin α cos α = 0, 30 which determines the arbitrary function r: r = r 0 e i 2 sin 2αs. 31 Here s is the affine parameter of the curve and r 0 an arbitrary constant. The second differential constraint 10 is identically satisfied and the couplings for this family of supersymmetric Wilson loops are: η β I = i e i 2 lsin 2αs [s I U 1 + l ẋ γ ] β, r 0 [ η β I = ir 0e i 2 lsin 2αs 1 + l ẋ γ [ M J K = M J K = l δ J K 2is K s J 2il cos 2α 2il sin 2α s K γ λ s J ɛ λμν x μ ẋν ] U s I, 32a β ẋ γ s K s J ]. 32b We notice that for α = 0 we recover the couplings 22 and we can consider this class of loops as a deformation of those considered in the previous section. For generic α, the situation is more intricate. Consider, for instance, the structure of the scalar couplings: there is a universal constant sector which is not controlled by α. Then we find a term of the form R J μk ẋ μ, which is the analog of Zarembo coupling in four dimensions. Finally we have a contribution of the type T λ K J ɛ λμν x μ ẋ ν describing the coupling of the scalars to the invariant forms on S 2. This is reminiscent of the Wilson loops on S 3 in D = discussed in [20]. In this picture the value α = 0 corresponds to the decoupling of the forms on S 2. There is a second interesting value of α, i.e. α = π, for which the Zarembo-lie term vanishes and the scalars couple only to the invariant forms. For this value of α we also recover the 1/2 BPS circle discussed in [5]. One is then tempted to identify these operators as the three dimensional companions of the so-called DGRT loops [20]. 3 Since x 2 = 1 it is straightforward to show that IJ on S 2 has still the structure of a conformal Killing spinor..1. Gauge transformation and the construction of the invariant operator In order to construct a gauge-invariant operator we have to discuss the global effect of the super-gauge transformations related to supersymmetry. In general the functions g 1 and ḡ 2 for a closed loop are neither periodic nor anti-periodic. In fact if we tae therangeofτ to be [0, ] and we denote with L the perimeter of the curve, we find the following twisted boundary conditions for the matrix G defined in e i 2 G = sin 2αL 0 G0 0 e i 2 = G0 sin 2αL e i 2 sin 2αL 0 0 e i sin 2αL If we introduce the auxiliary matrix e i T = sin 2αL 0 0 e i sin 2αL, 3 we can easily show that the infinitesimal gauge transformation G obeys the following relation G = T G0T 1, which in turn implies U = T U0T 1, for the finite gauge transformation, U = expig. ThenSTrWT defines a supersymmetric operator STrWT STr U 1 0WUT = STr U 1 0WT T 1 UT = STrWT. 35 In the case of the particular α = π and for the equatorial circle L =, the twist matrix T is iσ 3 which means that we have to tae the trace, as already shown in [5]. The dependence of T on the perimeter of the curve is not a complete surprise. In fact a hint of this result is implicitly contained in the original analysis of [5] for the circle. The authors suggest to use the trace since the gauge functions are anti-periodic. However if we cover the circle twice so doubling its length the gauge functions are now periodic and thus we have to go bac to the super-trace..2. An example: the great α-circle As an example, we shall consider the great circle for generic α: x 1 = cos τ, x 2 = sin τ, x 3 = 0. In this case the vector ɛ λμν x μ ẋ ν is τ -independent and it is simply given by 0, 0, 1. It is convenient to introduce the following parametrization for the constant spinors s α I and s Iα s α I = ūi λ α + v I λ α and s Iα = u I λ α v I λ α, 36 where ū I u I = v I v I = 1 and ū I v I = v I u I = 0, while λ and λ span a basis and they are normalized so that λ α λ β λ β λ α = 1 2i δα β.for instance, we can choose λ and λ tobetheeigenstateofγ 3 and in that case the couplings tae the following form η α I = i [ e i lsin 2ατ 2 cos α l π u I r 0 sin α l π ] 1, v I e iτ ile iτ, 37a η α I = ir 0e i 2 [ū lsin 2ατ I cos α l π e iτ v I sin α l π ] i le iτ, 37b M J J K = M K = l [ δ J K 1 + l sin 2αu K ū J + 1 l sin 2αv K v J l cos 2α u K v J e iτ + v K ū J e iτ ]. 37c

5 V. Cardinali et al. / Physics Letters B For α = l π, we obtain the 1/2-BPS circle [5] while generically we find supercharges 1/6-BPS loops. If we choose x μ τ to be two half-latitudes of the sphere differing of an angle δ, we recover the supersymmetric wedge discussed in [30] for α = π and l = 1. More examples and other features of these loops, such as their perturbative behavior, will be discussed in [39]. Acnowledgements This wor was supported in part by the MIUR-PRIN contract 2009-KHZKRX. We warmly than Flavio Porri for participating to the early stages of the these investigations. References [1] O. Aharony, O. Bergman, D.L. Jafferis, J. Maldacena, JHEP , arxiv: [2] O. Aharony, O. Bergman, D.L. Jafferis, JHEP , arxiv: [3] W.J. Rey, J.T. Yee, Eur. Phys. J. C , arxiv:hep-th/ [] J.M. Maldacena, Phys. Rev. Lett , arxiv:hep-th/ [5] N. Druer, D. Trancanelli, JHEP , arxiv: [6] K.M. Lee, S. Lee, JHEP , arxiv: [7] N. Druer, J. Plefa, D. Young, JHEP , arxiv: [8] B. Chen, J.-B. Wu, Nucl. Phys. B , arxiv: [9] S.J. Rey, T. Suyama, S. Yamaguchi, JHEP , arxiv: [10] M. Fujita, Phys. Rev. D , arxiv: [11] M.M. Sheih-Jabbari, J. Simon, JHEP , arxiv: [12] A. Kapustin, B. Willett, I. Yaaov, JHEP , arxiv: [13] M. Marino, P. Putrov, JHEP , arxiv: [1] N. Druer, M. Marino, P. Putrov, Commun. Math. Phys , arxiv: [15] N. Druer, M. Marino, P. Putrov, JHEP , arxiv: [16] J.K. Ericson, G.W. Semenoff, K. Zarembo, Nucl. Phys. B , hep-th/ [17] N. Druer, D.J. Gross, J. Math. Phys [18] K. Zarembo, Nucl. Phys. B , arxiv:hep-th/ [19] V. Pestun, arxiv: [20] N. Druer, S. Giombi, R. Ricci, D. Trancanelli, JHEP , arxiv: [21] A. Bassetto, L. Griguolo, Phys. Lett. B , arxiv:hep-th/ [22] A. Bassetto, L. Griguolo, F. Pucci, D. Seminara, JHEP , arxiv: [23] D. Young, JHEP , arxiv: [2] V. Pestun, arxiv: [25] S. Giombi, V. Pestun, R. Ricci, JHEP , arxiv: [26] A. Bassetto, L. Griguolo, F. Pucci, D. Seminara, S. Thambyahpillai, D. Young, JHEP , arxiv: [27] A. Bassetto, L. Griguolo, F. Pucci, D. Seminara, S. Thambyahpillai, D. Young, JHEP , arxiv: [28] A. Dymarsy, V. Pestun, JHEP , arxiv: [29] V. Cardinali, L. Griguolo, D. Seminara, JHEP , arxiv: [30] L. Griguolo, D. Marmiroli, G. Martelloni, D. Seminara, arxiv: [31] V. Forini, V.G.M. Puletti, O. Ohlsson Sax, arxiv: [32] N. Druer, V. Forini, JHEP , arxiv: [33] N. Druer, arxiv: [3] D. Correa, J. Maldacena, A. Sever, JHEP , arxiv: [35] T. Nishioa, T. Taayanagi, JHEP , arxiv: [36] D. Gaiotto, S. Giombi, X. Yin, JHEP , arxiv: [37] G. Grignani, T. Harmar, M. Orselli, Nucl. Phys. B , arxiv: [38] D. Correa, J. Henn, J. Maldacena, A. Sever, JHEP , arxiv: [39] V. Cardinali, L. Griguolo, G. Martelloni, D. Seminara, in preparation.