JHEP10(2008)026 - β s on an ations of the uality, shift, October 1, 2008 October 7, 2008 September 3, 2008 September 26, 2008 ry, D-branes, als can be analyzed by Accepted: Published: mutative, dipole and Revised: Received: ndary conditions. The TsT g some features of the deformed ng theory duals of gauge theories e Bruxelles & Published by IOP Publishing for SISSA = 6 Chern-Simons-matter theory living on multiple M2-brane N Gauge-gravity correspondence, Supersymmetric gauge theo We present general formulae for the TsT transformation (T-d = 4 Super Yang-Mills as well as new backgrounds dual to deform [email protected] N Physique et Th´eorique Universit´eLibre d Math´ematique, International Solvay Institutes, CP 231, 1050 Bruxelles,E-mail: Belgium SISSA 2008 c recently proposed T-duality) of type IItransformation string provides a backgrounds systematic and procedure to openwith find string stri deformed bou productsusing of transformed D-brane fields probes. in Astheories, the examples we illustratin lagrangian, consider and the the known du backgrounds dual to non-com deformed orbifold. Keywords: Emiliano Imeroni Abstract:

M-Theory. On deformed gauge theoriesstring/M-theory and duals their JHEP10(2008)026 1 22 26 27 28 29 30 31 15 18 19 20 29 10 14 (1.1) ty duals 3 eories by replacing the e product of fields in the D-brane probes embedded in y the “star” product: , there is a systematic procedure fg , ) ity) [4] that can be applied in order g 1 p he gauge/gravity correspondence [1 – f 2 p − g 2 p f 1 p ( iπγ e – 1 – = f ⋆ g → fg = 4 super Yang-Mills = 6 Chern-Simons-matter theory 20 N N -deformation -deformation β β These deformed gauge theories are obtained from ordinary th 7.2 3-parameter deformation 7.3 Non-commutative deformation 7.4 Dipole deformation A.1 Supergravity fields A.2 T-duality A.3 D-brane action 7.1 6.2 3-parameter deformation 6.3 Non-commutative deformation 6.4 Dipole deformation 6.1 A. Conventions and useful formulae ordinary point-wise product of two fields in the lagrangian b 3]. This is also due to the fact that, on the string theory side 1. Introduction Deformed field theories, thatlagrangian, constitute arise an interesting from generalization of a t new definition ofcalled the th “TsT transformation” (T-duality, shift, T-dual to derive the dualthe supergravity corresponding solutions backgrounds. as well as analyze 2. Gauge lagrangians with deformed3. products and their TsT gravi transformation: space-time perspective4. TsT transformation: world-sheet perspective5. D-branes and TsT 6. Example 1: 6 8 Contents 1. Introduction 7. Example 2: JHEP10(2008)026 (1.2) . . 1 ϕ ong is a real parameter. , followed by a shift 1 γ etric cycles of Calabi- tions, in particular in ϕ framework as emerging compactified space-time o-torus [5 – 7]. Different ing theory duals, obtained and eral formulae for the TsT eful. The most well-known of non-commutative gauge 2 – 15], which are non-local ink is particularly relevant If we parameterize the two eet fields are given in equa- ontext of ordinary confining iven in equations (3.2), (3.3) g s of (super)conformal gauge ave been thoroughly studied dimensional (but not neces- sults is the following: the for- ming that the gravity dual of 8]) and of the corresponding chnique to obtain the gravity een the gauge and string sides product and the gauge theory eing interesting in themselves, n a supergravity solution of a etry, the gravity description of e a general perspective on how nsformation of the string world- and erspective that allowed Lunin and f . ) redefinition of the complex structure R γτ , τ 1+ = – 2 – = 4 Super Yang-Mills theory considered in [9], γ τ N -deformation) was derived in [4]. Another example, → β τ can lead to different, and often less “exotic”, deformations i p ), the transformation consists of a T-duality along in the T-dual background, and finally by another T-duality al 2 1 , ϕ 1 γϕ ϕ are appropriately chosen charges of the fields + i 2 p ϕ In fact, deformations of the type (1.1) can be seen in a unified A quick reference for the reader interested in our general re Our results can be immediately applied to a plethora of situa Gauge theories with deformed products of fields and their str Another often studied case is the one of “dipole” theories [1 → = 1 gauge theories realized on D-branes wrapped on supersymm 2 from deformations of ordinarysarily irrelevant) Yang-Mills gauge-invariant theory operators. byMaldacena It higher- to is find the this string p the dual of original, the undeformed, deformation gauge [4]. theorythe has Assu deformation a just two-torus consists isom in the following SL(2 of the two-torus: For instance, if wedirections, take the these product charges (1.1) to willafter be reduce the to momenta deformation the along usual willchoices two Moyal for be the defined charges on a non-commutative tw where This transformation can beduals seen of as the a deformed solution-generatingtype gauge te II theories theory, (1.1). ittorus In reduces by particular, to ( o a simple “TsT” transformation. ϕ contexts where the TsT transformationexample has been is shown probably to the betheories, us one such of as exactly marginal the deformation deformations of with or without thein knowledge of the the literature. transformationtransformation (1.2), The of h any purpose type II of background,sheet this including coordinate the paper tra fields is (theopen to study string present of boundary gen which conditions. wasD-brane In initiated probes particular, in behave we [ under willas the giv D-brane transformation, probes which often we provideof an th the invaluable “bridge” correspondence. betw mulae for the TsT transformation ofand a (3.4) type II of background section aretion g 3, (4.1) while of formulae section for 4. the TsT of world-sh whose gravity dual (for the so-called as we have said in the beginning, is provided by gravity duals theories, such as the ones in [10, 11]. theories living on an ordinarydipole commutative deformations space. have Besides been b shownN to be useful also in the c JHEP10(2008)026 to fg = 6 from (2.1) N gravity 5 S × 5 ty duals AdS e IIA and eleven- -deformations. In the β used [18] in the case of fields in the lagrangian, mensions that has been viewing the introduction -deformations. In partic- ed in the context of non- with disentangling gauge t features of the deformed β des on the cycle [17]. The nd its consequences on the ns, we have chosen a couple e background. The two final orld-volume actions, such as ful formulae in the appendix. IB supergravity backgrounds ic y deformations of the ceed in section 5 by studying le and der start with superconformal e features of the transformed volves a light-cone direction of orbifold singularity [24]. Once the gauge/string theory corre- nt general formulae for the TsT -sheet coordinate fields and the k rmations of the recently proposed Z / fg , 4 = 4 Super Yang-Mills theory in four ) g 1 C p N f 2 p − g 2 p f 1 p ( – 3 – iπγ e = , we replace the ordinary point-wise local product g f ⋆ g and = 4 Super Yang-Mills and its well-known f N -deformation we will show some interesting results arising M-theory M2-branes at a β -deformation we will make use of transformed D-brane probes = 1 SQCD with a large number of fundamental flavors [19, 20]. β N N Besides presenting general formulae for TsT transformatio Dipole-type TsT deformations have also recently been appli By studying the second example we will instead derive new typ This paper is organized as follows. We start in section 2 by re = 6 superconformal Chern-Simons-matter theory in three-di relativistic AdS/CFT [21 – 23], wherethe the gravity solution. transformation in of examples that we hopetheories and will their shed gravity light duals. ongauge Both some theories, the of of examples the which we we relevan consi consider non-commutative, dipo first example, we start from conjectured to live on again, in the case of the theory effect from spuriousgeneral effects TsT due to formulae the wesolutions Kaluza-Klein of present mo the here combined can typethe II also solutions supergravity dual for and to D-brane instance w be Yau manifolds (such as in [16]), where the deformation helps ular, in the case of the dual in order to derive,dual in to a its unified perspective, non-commutative, dipole known type and I (non)supersymmetr see the effect of the deformation on the vacuadimensional of supergravity the solutions, which theory. are dualN to defo the study of D-branes in the transformed background. of the deformed “star”string product theory side in of the gauge duality.transformation theory In of lagrangians sections 3 type a and II 4corresponding closed we prese string open backgrounds, string world boundaryhow conditions, D-branes then behave we along pro thesections transformation of are a devoted very simpl totheories: concrete in section examples 6, illustrating we som consider deformations of by a generic “star” product, which is defined as: dimensions and their stringABJM duals, theory. while We outline in our section conventions and 7 collect a we few stud use 2. Gauge lagrangians with deformed products andWe their want to gravi considerbecause gauge they theories will with leadspondence. deformed us Given products to two of interesting fields generalizations of JHEP10(2008)026 , . π M i for Q (2.4) (2.2) (2.3) (2.5) (2.6) µ L = M M i . p ) f . . . πγQ 2 − πγQ )) + non-commutative = x − , but we can obtain ( heory, commutative x g x is to introduce some (the expression above ( 1 hose coordinates obey uct (2.6) is called the to be the momentum, Mµ ories, can be shown to g ∂ . g ) ) L 1 g d on circles of radius 2 neralize this deformation tion  x ch is non-local and breaks p n-local despite living on a ( f (1) conserved charge under f L and 2 y U is a constant vector. In this tail three very different cases. 2 1 πγQ ∂ f = πγ . µ x 2 + + −

fg . L ) ) (1) conserved charges, − ) x x g 1 y ( x U ( Q  ( = f g f 2 g g ) Q 2 =  12 x is an index distinguishing the various ∂ − g ( θ y ) g 2 f L , where x = Q ” ( M 1 2 x f 1 1 (

f M ∂ Q ) 1 ( ∂y − y ∂ M ( 2 12 ( x ∂ g iπγ is an arbitrary real parameter. What are the – 4 – Q ∂x ) iθ  e x − f γ ( charges to be the conserved momenta along two iπγ = 2 = ∂ f ]= i ∂y − ” 2 p M 2 )= 1 ) ∂ ∂ p ∂y x ,x x ∂x f f ⋆ g 1 ( )( “ Q x g [ . ) − iπγ x ij ∂ ( ∂x − f ⋆ g iθ f e g ( Q = “ ]= )= j πγ x e ,x )( i x )= x f ⋆ g ( )( are appropriate conserved charges of the fields i f ⋆ g p ( -deformation [9] and, in the case of superconformal field the case we can rewrite the product as: the various fields spanned by the index the star product becomes: more general deformations by introducing “dipole vectors” This is called dipole deformationcommutative [12 space-time. – 15], Here and we is have shifted clearly a no single direc This can betwo-torus recognized whose as coordinates the satisfy: appropriate Moyal product for a fields of the theory). If we moreover take the first charge which the fields have charge This deformation then yields a non-commutative theory,Lorentz whi invariance and causality. Ofto course obtain we can a easily theory ge livingthe on relations a [ larger non-commutative space w so that the star product (2.1) reduces to: In this case,and we local, see since that the thephases only in deformation effect the yields of interactions. anβ the ordinary The deformed t theory product arising (2.6) from the prod space-time directions, that we think of as being compactifie In this case, the deformed product (2.1) becomes: 2. In the second case, let us suppose that there is a global 3. In the last case, let us suppose there are two global 1. In the first case, we take the should be regarded as schematic) and where the properties of the deformed theory? Let us consider in more de JHEP10(2008)026 ves (2.7) our analysis one of ions to the world-volume he string theory point nd Maldacena [4], who potential, as we will show ed to different directions. In -volume theories living on an treat them in a unified ia deformations induced by e general form (2.1) is that the D-brane where the gauge 1 2 ϕ ϕ 2 ) transformation y e example of the exactly marginal R RMED , 2 1 ϕ γτ y E O βDEF τ LE 1+ E In the case of supersymmetric theories, one I O DIP 2 V I 1 x = ϕ T A 1 ) two-torus corresponds to a “TsT” transforma- – 5 – γ 2 τ 2 , ϕ ϕ 1 -deformation is complex. In this paper, we will limit oursel MUT MM → 1 x β 1 . ϕ τ ϕ γ CO N = O N β p D parameter of the iσ + indicate respectively longitudinal and transverse direct γ is a real number, • = β β and The TsT transformation yields different theories when appli − -brane that hosts the gauge theory to be deformed. p be an exactly marginal deformation. can see the deformation (2.6) aslater acting in directly examples. on the super As we mentioned in the introduction, from the point of view of The transformation (2.7) of the ( The general 1 -deformation shows) gauge invariant operators, so that we c of the D the figure, the nicest things aboutthey can deformations be of seen gaugehigher-dimensional as theories (but arising not of from necessarily th ordinary irrelevant,β Yang-Mills as theory th v way. Let us thenof try view, when to we see thinkD-branes what of of these the the deformations gauge type theories meanhave II as shown from string. that being t the the The deformation world answer amounts was to found the SL(2 by Lunin a Figure 1: to the case where tion of the ten-dimensional type II background generated by on the modulus of a two-torus present in the geometry. JHEP10(2008)026 , - 3 β − t is p (3.1) c ∧ db , antisym- + φ 1 − p . dc µν (1) isometries of b (1) global charges U = , that we denote as U + γ p , dilaton × f ne. In this case, the ansverse to the brane. µν tion will result in a µν y the D-brane will then g g (1) global charge and we (1) es a useful tool to study properties, depending on ate into local momenta of = U string duality, and this is U µν e r-horizon limit, if we are in the “TsT transformation” of a type mounts to a T-duality along the in figure 1: , 1 with parameter ˜ ϕ α γ , and finally to another T-duality along ϕ 1 2 are two commuting + , 2 γϕ ˜ ϕ + = 1 → 2 – 6 – , going back to the type IIA(B) theory. We call α ϕ 2 1 . ˜ 1 ϕ ϕ ϕ → for again, hoping this does not cause any confusion. 2 . The type IIA(B) solution becomes a type IIB(A) α 1 ϕ 1 ϕ ϕ ϕ are related to the world-volume directions of the D-brane. I i and modified Ramond-Ramond field strengths , then a shift ϕ 1 µν ϕ b is an arbitrary real parameter. γ , consists of three steps: is even in type IIA and odd in type IIB. We define TsT γ ) p 2 In this case the transverse direction will correspond to the charges that correspond tothe the gauge torus theory. directions Applying will (2.7)yield to transl the the non-commutative geometry deformation generated b (2.2). land on a dipole deformation (2.4). deformed theory (2.6). for the fields of the world-volume theory, and the transforma solution with T-dual coordinate ˜ where the final T-dual coordinate Finding general formulae for the TsT transformation provid We start with type IIA(B) supergravity solution with metric Assume that the coordinates , ϕ 2. The torus has one direction along the brane and the other tr 1. The torus is entirely part of the world-volume of the D-bra 3. The torus is transverse to the brane. In this case there are 2. Shift the coordinates in the new solution as follows: 3. Perform another T-duality along ˜ 1. Perform a T-duality along . The resulting deformed gauge theory can then have different 1 1 ϕ context of the gauge/string duality).isometry The direction transformation a theory under consideration lives (or the corresponding nea how the coordinates ϕ clear that there will be three possibilities, as summarized metric tensor field all of these deformationswhere from our attention the turns point to of in view the following of section. the3. gauge/ TsT transformation: space-time perspective In this section we presentII explicit closed general string formulae for background. the where the solution. Then the TsT transformation along ( JHEP10(2008)026 , ] 2 (3.7) (3.6) (3.4) (3.2) (3.3) (3.5) ϕ ][ , 1 , and Φ of ϕ [ ]      2  µν b ϕ    ][ ν ν B . 1 ∧ 2 1 µν ] ϕ e e b e [ + 2  ϕ 2 ∧ b ][ µν 22 12 µ product operation 1 b e e e ∧ G ϕ [ ∧ , b 1 .  st way to express the 21 11 µ = ] 1 b ∧ 1 e e 2 e f ϕ e also been written else- − ∧ 1 6 1 µν    ][ standard T-duality rules, f b ession that is valid degree 1 E of the undeformed solution 2 1 ϕ + . The shift may change the [ ∧ !) det b # φ 2 + itions, the transformation will b 12 22 2 . f ∧ e e e b field strengths appearing in the γ , y 2 1 b 1 ] ∧ ∧ and + b 11 21 2 − ∧ q + 3 p e e ϕ is by means of the general formula: f f ∧ α 3 ][ b µν ,

1 3 f b ··· !# e ] ϕ + q ∧ 1 − 2 1 [ ν 2 ] p ∧ α X 2 µν 5 4 ϕ b det ) e e " f 1 f C + ][ p 2 1  f ∧ γ b 1 ω ∧ γ ϕ + 1 2 21 µ [ γ 1 ] ∧ e e 6 + f b H f + + 5 = ( b

)+  f + e ∧ b b – 7 – 1 + 21 3 γ 2 − p ∧ + ∧ ∧ e p f f det [ F q α 7 3 1 + f − f f γ f + ··· − b 1)-form whose components are given by (A.10): = 1  4 q 12 α + + + ∧ ! p f γ − ) e [ X ] ν 5 1 3 2 ( F 1 y p µν f f f γ f [ + e e = γ p = + + = = ω 2 B − ( 12 µ 2 4 1 e , but the transformation is guaranteed to give a new solution e e 1 B B f f α F ∧

( ϕ ∧ ∧ = = q 1 2 = F B F B det F ∧ " q ∧ + M 1 X γ 2 3 F F F − 2 1 + , µν + -form and gives a ( 4 φ e 2 p F B e      ∧ 3 M M is even (odd) in type IIA(B), and the anticommuting interior F q = = + Let us then derive the TsT formulae (general expressions hav As discussed in [4], TsT is a solution-generating technique Repeating the process in the R-R sector, we find that the easie 5 µν acts on a 2Φ e ] F E y [ the TsT-transformed solution can be obtained from of the supergravity equations ofalso motion. not Under generate certain any cond new singularities. where in different forms,summarized in see appendix for A, we instance can [25]). show that the By NS-NS using fields the while in type IIA string theory we get: periodicities of the angles as follows: where · where we have defined the quantity: new R-R modified field strengths Formula (3.4) must of courseby be degree of understood the as differentialaction a forms. of symbolic More type expr explicitly, IIB for we the have: JHEP10(2008)026 TsT γ ) (4.3) (4.2) (4.1) (3.8) 2 . , ϕ 1 ν ϕ ϕ τ , and denote ∂ ) where again γ i )] ν er 1 , ϕ rive the transfor- 2 B er TsT. As in the , µ , ϕ 2 , 1 f ] ϕ 2 µ µ − of the world-sheet coor- ϕ e that under ( t in terms of R-R poten- ϕ ϕ ][ ν boundary conditions: y. The effects of the TsT = ( 1 κ κ 2 ϕ ∂ ∂ [ e are interested in studying µ B , fields in (4.1), derived using µ µ ormed backgrounds, in order [8], while studies on D-brane that are relevant for the open # µ ϕ 1 2 undary conditions too. Before -sheet fermions: the complete b 1 ro subscript. ne extended along the direction e f G G e D-brane world-volume action, = 0 ( ∧ βκ βκ γ into ǫ ǫ q ν (0) c µ + ϕ αβ αβ ϕ τ q µν ∂ γη γη X ≡ f ) " + − µ + µν γ µ µ f X + ϕ ϕ µν + α α b B e ∂ ∂ [ – 8 – µν µ µ ∧ 1 2 b − ( q ν c -field of the TsT-transformed background. For sim- γB γB − ϕ σ B q + − ∂ X ν (0) i 2 1 ϕ )] ϕ ϕ ϕ = ν σ α α α 1 ∂ B ∂ ∂ ∂ ν (0) G e µν µ ϕ = = = g 2 ∧ τ f ∂ q ) i (0) 2 (0) 1 (0) C − ϕ ϕ ϕ µν ν α α α q 2 f ∂ ∂ ∂ X G +        µ 1 are the metric and f µν ( b ( B γ − + are the directions along which the TsT acts with real paramet is the gauge field strength on the world-volume. In order to de and ν (0) 2 µν ϕ µν ϕ G G f σ ∂ in the original background satisfies (generalized) Neumann We wish to use (4.1) to understand how D-branes transform und Using (3.2) and the T-duality rules in the appendix, we deriv We split the world-sheet coordinate fields With a suitable gauge choice, formula (3.4) can also be recas = [ and µν g µ (0) 1 expressions can be found in [28]. case of a T-dualityT-duality in of flat the closed space, string the sigmastring, transformed model, and coordinate we are can the therefore same usethe ones (4.1) transformation, an to open study string open in stringϕ the bo presence of a D-bra plicity, we have limited ourselves to the case without world where mation of these boundary conditions we use (4.1) to compute: where the fields transform as: the original fields, before the TsT transformation, with a ze which is particularly usefulas for we computations will involving see th in the following. 4. TsT transformation: world-sheet perspective Let us now study thedinate TsT fields. transformation In from theopen particular, point string as of boundary we view conditions haveto and already use D-branes mentioned, in them w TsT-transf astransformation a on tool the in world-sheet havetransformations the been can context first be of studied found the in for gauge/string instance dualit in [26, 27]. tials: ϕ JHEP10(2008)026 n in 2 (0) and ϕ (4.6) (4.7) (4.8) (4.4) (4.5) d: it µ (0) ϕ with 1 (0) . ϕ and /n ersely, one n 1 (0) ϕ = 1 sT-transformed γ ake sense for generic undary conditions in o the direction l world-volume field on nto the same Dirichlet boundary conditions: -brane transverse to the itions along s reduced to zero size. If p does not obey (4.8)? Using ust be rational [4]: γ , we see that in order for the ty is for the D-brane to sit at ple wrappings of the brane on oundary conditions: F = 0 = 0 , a D , = 0 . In this case, (4.3) tells us in 1 2 with a world-volume gauge field: ϕ ϕ TsT γ 2 π τ τ ) µν = 0 ϕ 2 ∂ ∂ f ν 1 1 γ γ ϕ . , ϕ τ 1 − + and . ∂ 1 γ ϕ ν ν 1 = 0 n m µν ϕ ϕ + 2) brane, we should have ϕ = τ τ B p ∂ ∂ µ (0) = – 9 – 12 ν ν ϕ − 1 2 γ F τ ν B B ∂ ϕ , since the flux (4.7) along the two-torus must obey σ − − γ yields, after the TsT transformation, Dirichlet bound- ∂ ν ν ) shrinks. 1 2 ϕ ϕ µν − σ σ γ G , ϕ ∂ ∂ , and thus a brane with two fewer longitudinal directions. 1 -brane we started from in the undeformed background has ν − ν 2 p 1 2 ϕ ϕ = G G ( 12 f and 1 ϕ in the TsT-transformed background, unless the conditions o is the field strength on the brane in the undeformed backgroun in the deformed background, that must be computed from (4.3) µ +2)-brane wrapped on the two-torus transformed by TsT. Conv µν ϕ F p f -brane to expand onto a D( p ) torus will generically be mapped onto the same brane in the T 2 are both of Dirichlet type, in which case the resulting set of It is important to point out that the conditions (4.6) do not m Hence the general result is that, under ( What happens to a D-brane transverse to the two-torus if The simplest case one can consider is the one where the initia An open string in the presence of a brane which is transverse t , ϕ 1 2 (0) ϕ undeformed D ary conditions along values of the deformation parameter a quantization condition. Reinstating the factors of 2 and gauge field-strength can for example see that starting with Neumann boundary cond does not match background only if it is placed at the points where the torus i We stress that ( describes instead Neumann conditions along turned on. This means the D turned into a D( the equations we have derived,a we point see where that the the two-torus only ( possibili a D-brane in the undeformed background is zero, integer. We can generalizethe this torus, condition so by that allowing in multi conclusion the deformation parameter m that Neumann boundary conditionsthe deformed are background mapped (with onto zero gauge Neumann field): bo the undeformed background will instead satisfy Dirichlet b We see from (4.1)conditions along that these boundary conditions are mapped o ϕ JHEP10(2008)026 g (5.1) 1 γ 2 2 2 ϕ ϕ ϕ = 1 1 1 parameterize 12 ϕ ϕ ϕ 2 F urned on. This ϕ + 2)-brane wrapped p and 1 1 , ϕ ϕ 1 2 ϕ -brane that extends along , the transformed D-brane ) to the simplest possible he behaviour of D-branes p γ 2 2 2 /gravity duals - it will be e one has to compute the we present a summary in ϕ ϕ dings in a TsT-transformed dϕ e TsT of flat space. transformation to this simple ( correspondence, that we will n. This sets the stage for the t change its dimension after the 2 2 icitly. r TsT γ + ) = 0): a D 2 2 ) b 1 . The angles , ϕ . For generic 9 1 , 1 dϕ ϕ /γ -brane in the TsT-transformed background. ( is quantized) onto a D( R p 2 1 r γ ⊂ =1 T T T + 4 s s s T T T 12 2 2 R F – 10 – dr + 2 1 1 1 dr ϕ ϕ + 5 2 2 , ϕ ϕ 2 1 is rational, there will moreover be the possibility of havin γ dx = 2 ds ) gets mapped onto the same D 2 , ϕ = 0 1 2 2 2 ϕ 2 ϕ ϕ ϕ T ) 1 1 1 TsT transformation of D-branes (with initial g ϕ ϕ ϕ + 2) brane wrapped on the torus with a world-volume flux (4.7) t det( -brane transverse to the two-torus expands (if p In the next section, we begin to study, in a very simple case, t Write the metric of ten dimensional flat Minkowski space as: p Figure 2: on the torus with world-volume field strength the two-torus ( must instead sit attransformation. a point where the torus shrinks and doesthe no deformation parameter A D a D( two obviously contractible circles. Let us apply a ( reproduces in much morefigure 2. generality the Of analysisresulting course, of boundary there conditions [27], will by be and using more (4.1) complicated and (4.3) cases expl wher 5. D-branes and TsT Before we get to concrete examples of D-brane probes in gauge and their world-volume actions alongmore the complicated TsT examples, transformatio relevantexamine for in the sections 6 gauge/gravity and 7. our focus incase the in next order sections tobackground. - analyze Let general let us properties us then of start for D-brane as now embed in [4] limit (see ourselves also [26]) with th where we have chosen polar coordinates in JHEP10(2008)026 .
(5.6) (5.5) (5.7) (5.3) (5.2) (5.4) 2 ) 2 . ϕ
τ 2 ∂ ) ( 2 2 2 r ϕ is generic, the τ + ∂ γ ( 2 2 2 ) , r 1 , , orld-line coordinate 1 ϕ ) ) ˆ + apping the two-torus C 0 τ 2 mbedding: 2 ) -brane probe is mainly ∂ x ) 2 s of the previous section in the background (5.2). ( ( Z 1 i in polar coordinates. 2 1 imensional world-volume, s r ϕ ϕ dϕ 4 1 2. g τ ( e torus. If eometry of the ambient flat R 2 2 ∂ tion, but it will be forced to = s we see that the geometry of ( − r led by the contribution of the + i 2 1 d in this case and we get: ) , such as the background gener-  r + 2 ready possesses all the properties ) ab 2 + 2 ) ˆ , ϕ B r 1 2 . τ ) ) 0 2 2 + 2 ∂ dϕ r r x ( τ 2 ( 1 ab 2 1 i r ∂ + ( r ˆ r G 2 ( 2 turned on, but that this is possible only if γ  ) = + ( 1 , M i r 2 /γ 2 τ ) -brane along the transformation, applying the det -field and dilaton onto the world-volume of a M + 1 ∂ p dϕ r – 11 – 2 2 B − = 1 =1+ τ , r ∧ ∂ + ( 2 ) dr 1 r 0 i 1 ϕ − Φ 1 + x x + ( − ϕ ( τ dϕ 2 1 i ) part reads the metric of i M F ∂ 2 i 2 x i x r dr τ x 2 1 , ϕ dτ e = τ ∂ r i + i i r ∂ -deformation, since the coordinates where the TsT acts are x ( 5 Z , , M τ β 2 1 s γ ∂ 1 − g M − dx − 1 τ, x − M = = = = 1 = 2 r B 2Φ 0 q D0 ds e x dτ S dτ Z s Z 1 g s 1 g − − = = ) with a world-volume field , in the starting flat space-time. Choosing the static gauge e 2 τ Let us see this explicitly by studying D0 and D2-brane probes When we pass to the theory after the deformation, the analysi We now want to study the fate of a D Let us then briefly study the action of a D0-brane probe, with w (0) D0 (TsT) , ϕ D0 = S 1 is appropriately quantized to ensure flux quantization on th S 0 ϕ D0-brane embedded as in (5.4) we easily get: γ D0-brane will not changesit its at dimension a along point the where transforma the torus shrinks, asComputing summarized the in pull-back figure of the metric, background. The formulae in section (3) are straightforwar Neglecting the constant term (thatated in a by more the realistic supersymmetric setup D0-braneR-R itself, part), would by be expanding cancel thethe action moduli in of derivatives of the thespace-time. D0-brane field theory In reproduces, particular as the ( usual, the g where hats denote pull-backsreduces to: of the bulk fields onto the one-d shows that the( D0-brane will tend to expand into a D2-brane wr we wish to show,relevant we for concentrate on the a case D0-brane. of the Notice that a D0 expertise we got in section 4. For simplicity, and since it al where σ transverse to the world-volume. The world-volume action (A.15) JHEP10(2008)026 4 ). 2 R , .
(5.9) (5.8) , ϕ 2 (5.10) (5.13) (5.15) (5.12) (5.11) (5.14)

2 . This ) 2 r 2 2 ) ( φ R 1 , τ A ↔ ∂ ×  τ ( ) ) 2 ∂ 2 2 1 F ( r R 2 2 , r , , ϕ + + ) son lines on the 1 0
+ 2 ˆ r : the moduli space 2 B x ) 2 ) ( ( γ 1 ) i 2 2 φ . r ∧ ϕ τ ) A 1 τ ter 0 ∂ τ = ˆ ∂ ( istic setups). The brane C x ∂ ( i 2 1 auge theory results when ( ( 2 2 r . In this case we can study 2 ion shrinks. If we then put 2 + 1 will see in the next section. r e instability of the configu- ion: heory does not “see” the r 3 A + 0 + ˆ , r C 2 2 es in transformed backgrounds ∂ is crucial for the cancellations ) 2 m/n )  γ ) 0 2 , 2 = r x γ = , + r M Z ( τ 1 2 i τ γ 2 γ ∂ 02 ) φ x ∂ τ 2 = = 0 r − = + ( τ + 1 γ 2 + ( i ∂ r 2 = , F i is imposed on us by the transformation  ) + ) x 2 1 ab 0 +( τ r 1 12 2 F x τ ∂ , x ) ( F i γ ∂ 1 1 2 + x r M− σ τ A τ – 12 – ab 0 ∂ + ( is quantized, ∂ , A = = ˆ ∂ i B , we get: γ 2 x 2 2 γ +( − = φ τ 12 + i σ ∂ 1 F x i 01 = ab =0 or τ = 1, or equivalently x q , ϕ ˆ + ∂ 1 τ 1 G i and 1 r ∂  A dτ x M 12 σ 1 τ ˆ σ B ∂ Z − = det /γ , F s 1 1 1 g − − 1 q − = 1 r 0 q Φ , ϕ = 12 − , reduces to: 0 σ dσ F 1 3 σ − d = 0 we get again the same action with the exchange ( Z σ e ) ) sector anymore, but that the moduli space is reduced to = s 3 (TsT) i D0 s 2 g Z d 1 0 r g S 2 2 x , ϕ 1 γ γ π i τ Z r are allowed fluctuations on the brane, exciting periodic Wil 2 − − τ 2 = = (4 = − -deformations for generic values of the deformation parame A 2 β = τ (TsT) The moduli space is enlarged when D2 (TsT) D2 D2 and S = 0 the action above reduces to: S S 1 1 is a quite generic featuredual of to moduli spaces on D-brane theori reduces to a sumthe of background complex in lines. question is This dual to is some in gauge accordance theory, as with we g part in the ( a D2-brane probe embedded in (5.2) as: Notice that this means that the moduli space of the D0-brane t with world-volume gauge field strength: that led us to (5.13) to happen. If we now define: Notice that the combination ration (and would notthus be has cancelled to by sit the at R-R points part where in more real Now we see an angle-dependent potential term that signals th while if we put where where the torus along whichr we performed the TsT transformat A and integrate the action over two-torus, while the constant magnetic flux rules we have derived. With this embedding, the D2-brane act JHEP10(2008)026 integer) n with of (5.15) that (in round. The same /n of the deformation i rations. Suppose we φ = 1 generated by the brane m/n γ . Is the moduli space in following general perspec- ation going to be qualita- ks, or will expand onto a ive us information on the = just described. Of course, at the results for the gauge per we will limit ourselves te much more attention to /γ relevant to the TsT of the kground was wrapped along ole deformations, because in n, and when will it instead γ us where the transformation he undeformed background! the gauge coupling constant ation becomes more interest- kground? iving us new results? In other ion will be realized in the case probe computation will yield (0) D0 e to the two-torus. After TsT, n of the probe in the deformed unds dual to non-commutative be read on probes that are not es S = (TsT) D2 S , and the new scalars i ϕ ), then the TsT transformation will leave the 2 – 13 – , ϕ 1 ϕ here? The identifications (5.14) show that there is a 4 factor, precisely coincides with the action (5.6) of a D0- R orbifold. This is also a feature which we will see to be quite /γ n Z between the periodic Wilson lines on the torus, that have the . We therefore see that the moduli space is reduced from the × γ n Z π/n = 1 quiver gauge theories). as the original coordinates N π ) sector again the full i , φ i r In this respect, it is clear that the most interesting situat The results of this section and of the previous one hint to the Now suppose that we perform a TsT transformation of the backg Before we turn to more examples, let us pause for a few conside -deformations (2.6), rather than in non-commutative or dip β -deformations, while just presenting the deformed backgro brane in the undeformed flat space-time (5.1): and dipole deformations. which, apart from an overall 1 the ( difference by a factor of same period 2 such case our D-brane probeacts, will and be will transverse thus toone exhibit the can the two-tor envision more different gauge variedthe theory behaviour D-branes quantities we that generating have will theto background this but, kind since ofβ in probes, this in the pa following examples we will devo the simplest case where we have only one wrapped D2-brane and original flat space to a have instead period 2 of background is likely to being the when same. the On D-brane the beforethe contrary, the the brane transformation situ is will transvers eitherhigher be dimensional brane stuck wrapped at ontheory a the will point torus. be where This also means the quite th different torus from shrin the ones obtained in t brane untouched and the result of the evaluation of the actio parameter (see alsogravity for duals instance of the analysis of [29] that is general in TsT-transformed backgrounds with rational valu tive: if the D-brane probeone that or was used more in the directions undeformed of bac the torus ( want to use a D-braneitself probe (or embedded its in the near-horizonrelevant background information limit). that on is the Byand dual construction, the gauge theta-angle. such theory, a for instance D-brane probe in thecorresponding transformed deformed background gauge is theory. supposedtively When different to is from the the g one new inwords, comput the undeformed when background, will g thereproduce probe the same be results sensitive as to in the the original transformatio undeformed bac JHEP10(2008)026 5 S = 4 (6.1) (6.2) (6.3) (6.4) (6.5) × 5 N ume form = 1). For pleteness. AdS s l d the corre- = 4 SYM, reads: tiplet of , 3 N θ . dφ sin ∧ ality. We will consider ow TsT transformed D- f flat space, we pass to of the correspondence. 2 α making use of the formulae by: dr , (in units where dφ unds we are going to present ∧ , ∧ , N 3 = sin s of the deformed gauge theories 1 , s ) , 2 3 ) dx = 1 dφ πg 5
Φ 2 i 5 S 3 ∧ ∧ 2 S µ ω solution. As we said at the end of sec- Φ 2 = 1 notation, it contains three chiral = 4 1 5 ds dθ i + 4 dx Φ S X N 5 ∧ + R θ, µ , ∧ × − 5 5 1 2 3 AdS 2 -deformation. We will therefore start in sub- cos r Φ θdα ω β dr dx 2 AdS , 2 ( α AdS 4 ∧
Φ – 14 – ds + cos 2 i 1 0 R 3 , θ 4 2 , 2 1 dφ dx = sin R − 2 i is given by 3 sin dx µ r 2 5 2 = 1 = = α − r S + 2 5 3 Φ = Tr (Φ 2 i e F = = ds sin 5 5 W dµ and α 5 α, µ AdS 2 AdS 3 =1 ω i X ds AdS = = cos = 4 super Yang-Mills -deformation, before briefly presenting other cases for com space we will use Poincar´ecoordinates, with metric and vol = cos 5 5 β solution of type IIB supergravity, which is dual to 2 S S 1 5 = 4 Super Yang-Mills theory in four dimensions and of its ω N µ 5 ds S N AdS × subject to the superpotential: 5 i is the flat Minkowski metric in four dimensions. The metric an 3 , 2 1 AdS dx The Let us start with the undeformed gauge theory. The vector mul We will now study deformations of the that we will keep in mind, in order to deform it later. After having studied D-brane probes in the TsT deformation o our first simpledeformations example of in the context of the gauge/string du superfields Φ SYM can be decomposed in such a way that, in The common radius of 6. Example 1: gravity dual. This isare mostly already review known, material but we and willwe the rederive them backgro presented in in a unified section waybrane by 3, probes then can we be will used give to some make contact examples between on the h two sides where sponding volume form we will use for the five-sphere are given the unit radius given by: where we have chosen the parameterization: tion 5, the D-brane probes weare will particularly use interesting to in elucidate propertie the case of the section 6.1 with the JHEP10(2008)026 5 , 2
ϕ S 2 3 = 4 , (6.7) (6.8) (6.9) i 3 1 N and θdϕ 2 ) 2 1 dϕ dϕ 2 ϕ . ∧ sin ∧ dϕ 2 2 θ 3 ) 1 2 + dϕ dϕ 1 ) ) , dϕ cos θ θ 1 dϕ 2 2 α − ϕ 4 3 + − cos sin 3 obtain the gravity dual ) (6.6) 3 dϕ sin θ α 2 ( ϕ 2 dϕ 2 α θ Φ ( 2 2 3 on parameter is real). The θ = cos 2 Φ 3 2 1 sin cos und reads: 2 cos α = 4 Super Yang-Mills. ined by replacing the ordinary ) and modifying it as follows: 2 2 Φ α cos − along two directions dϕ γ N 2 θ α 5 ∧ iπγ 2 3 1 2 S − 0 +9ˆ 2sin 1 2 , φ Φ − ) e 2 = 4 Super Yang-Mills [9]. The defor- sin 2 − ) 2 × + sin dϕ 1 θ ϕ − α ) 5 2 N + sin 2 2 dϕ θ 3 (1) groups are the ones listed in table 1. ) 2 + dϕ 2 2 2 Φ +1 +1 Φ U ) − 1 sin 2 − 2 AdS 3 ϕ dϕ 3 sin Φ α +cos 1 1 1 θ dϕ 2 + 0 + θ dϕ dϕ Φ − Φ 2 ( ( 3 1 2 – 15 – : − transformation on the solution (6.2) with the θ α 5 ϕ 3 iπγ 2 2 cos dϕ 1 2 +sin sin S e = , ϕ ϕ θ α θ TsT γ dϕ + sin 3 2 ) ( 2 2 2 3 2 (1) (1) α α + cos -deformation of dϕ = 4 SYM by the star product (2.6), where the charges 2 2 U U cos cos under the two , ϕ dϕ 2 β = Tr ( ( 1 ∧ i , α α + sin θ N γ cos ϕ 2 ) 2 , φ 2 α 5 αdθ , + sin W (1) factors in table 1 [4]. This can be achieved through the 2 θdθ 2 S 2 (1) charges of the chiral fields of ϕ U  cos M ω -deformed theory is an exactly marginal deformation of U → 5 α cos − f 2 S β + + (cos + (sin (cos M × 2 θ 3 h 2 W + sin ds ϕ + α 2 (1) 2 5 sin + = 1 superconformal theory [9]. = U αdθ 5 dα +sin α 2 1 sin N 4 AdS φ = ω 2 AdS ( M 5 sin 2 4 ds 2 S 4 + sin  , R 2 ds ˆ γR 4 2 Table 1: γR R dα M − − -deformation = = = = = 3ˆ = β From the point of view of geometry, it is clear that in order to We can now perform the ( 5 2 2 5 B f S 2 2Φ F C ds e ds of the three chiral superfields Φ mation is obtained by starting from the superpotential (6.1 we have to perform a TsT transformation of SYM preserving (remember we limit ourselvesdeformed to superpotential the (6.6) case can also whereproducts be the in seen deformati as the being lagrangian obta of One can show that the We consider the the well-known 6.1 following change of coordinates on the so that we can rewrite the metric (6.4) as: corresponding to the two metric in the form (6.8). The resulting transformed backgro JHEP10(2008)026 one (6.13) (6.10) (6.14) (6.12) (6.11) 3) and , e action, 2 , 1 , , , since one can , # = 0 4 , 2 . What happens r  3 a
 4 ( ˆ i 1 C B R a . ∧ dφ dφ x eformed gauge theory: = ) 2 metric way by reverting ∧ θ i ˆ = a point where the torus C 2 X 3 of static D-brane probes. (0) 4 a + ˆ  σ C . tion will “read” the moduli . ion 5, we have to study D3 dφ sin 2 3 4 e theory statement that the s a concrete example of what ) copies of 2 1 ndent potential that we need R θ ˆ µ 2 1 1) C µ 2 2 3 2 µ 2 3  N τ µ 2 3 − µ 2 1 µ 2 , cos factor is preserved by the trans- Z = 4 SYM. µ / + ) 3 1 α + 5 3 3 τ − N 2 M 2 3 = 0, is: 2 µ dφ + dφ γ 2 2 M F AdS (  ∧ + µ 4 + ˆ + sin ab 2 2
+ ˆ B α 2 i σ r background, 2 2 dφ dφ 2 4 2 3 µ + 5 dφ d 2 1 + µ 2 i S 2 2 – 16 – µ ab 1 µ ( (cos Z µ background can be traced to the backreaction of a ˆ 2 × G α 2 dφ 5 γ  M + 5 ( 2 π S N , 2 2 + ∧ × ) sin det 2 i − dφ 5 AdS 5 2 S − γ and: ∧ =1+ˆ , as encoded in the general formula (3.8). The DBI part = θdθ dµ ω 1 1 r γ in (6.7). The change of coordinates yields: (0) AdS 4 2 − Φ M cos i 3 ˆ dφ =1 TsT D3 C − R i X φ 2 2 θ + S M =1+ˆ µ = 5 = + 2 1 1 σ e sin 5 µ 4 γ ˆ − B d α AdS . We immediately see that the Wess-Zumino part reduces to the 4 ∧ 1 ω M 2 AdS -dependent factor, so that the result is: Z ( M − 2 γ 2 and the coordinates on the deformed sphere fixed. The D3-bran 3 ) 4 ds sin ˆ C s τ " , 2 r R g − ˆ γR 4 2 3 + π R γR M − − 4 = ˆ C = = ˆ = = = D3 2 2 5 = (8 S B 2Φ F C 3 ds e τ We can rewrite the transformed background (6.9) in a more sym Let us start with a D3-brane. We choose the static gauge We are now ready to study some features of the duality for the d -deformation is an exactly marginal deformation of where we have defined ˆ where This is the Lunin-Maldacena solution [4]. The to the original coordinates Looking at equation (6.12) weto see cancel to that make there the is probewe an supersymmetric have angle-depe and seen stable. in This i general in section 4: the D3-brane has to sit at formation, which is theβ gravity dual counterpart of the gaug of a D3-brane in the undeformed after the deformation? Asand in D5-brane the probes simple in flat the space background case (6.11). of sect verify that instead contains a we will study theThe vacuum origin structure of as the determined undeformed by the study once we also choose the world-volume gauge field where: for now we keep space of the theory, that is just a symmetrized product of stack of D3-branes, so embedding a D3-brane probe in the solu JHEP10(2008)026 (6.15) (6.16) (6.20) (6.21) (6.17) (6.18) (6.19) = 0 can . B  ) ∧ [30]. In this , F B + , = 0 m/n ˆ ) B . a 1 ( . ) x n (6.15) to find two = Φ , a . ( ∧ i , 2 3 x 5 γ ) D5-brane probes, that .  ( F Φ σ ϕ  5 b F 2 π
-deformed theory. These ∧ .11) where A 2 = = β + ) + ) a = 0 = erm equations derived from 2 3 1 ∂ ˆ 2 written in polar coordinates ˆ B φ B , the abelian moduli space is Φ ( a 2 = ∧ γ 1 , , θ , θ ∂ , ϕ ∧ R 5 2 ( Φ a 2 2 ) π π 2 ˆ = 1. There are three possibilities 2 C for the a r is rational, ˆ , ϕ C x , = = + 4 = 0) ( 1 2 C γ + M σ θ 4 α α 2 α ˆ ) C   . This translates in the gauge theory = 0 = = , F r )+  3 a = 0. Repeating the probe computation by 3 ) 1 φ F ∂ a ab to depend on the world-volume coordinates Φ ( α x ) + ( F 1 i + ( ˆ = 0 = 0 = 0 ( 4 B Φ φ , ϕ 1 2 + ˆ b and B 2 1 3 A ∧ ( – 17 – 3) µ µ µ a σ ab , θ 2 + 4 , 4 ∂ ˆ ∧ ) and B φ ˆ 2 1 d = = = C a 4 , θ = Φ x 1 3 2 ˆ + 1 C 3 + ( , Z , 4 r a 6 α Φ ab 2 + ˆ ˆ C π , µ , µ , µ G N 6 = 0 = ( 2 ˆ  , h C a ) where the TsT transformation has been performed. We − ( 2  = 1 = 1 = 1 Z det = = 0 3 2 1 5 , ϕ , α Z τ µ µ µ 1 2 − ) 5 /γ , F a ϕ τ Φ = 4 SYM to three copies of = is generic. TsT r D3 x 3 a ( Φ + S γ Φ = 1 p r σ N − b (iii) (ii) (i) WZ D 45 = = S , with polar angles + F σ e 2 for a r and 3 6 x R 3 d Φ 2 πγ C 2 i Z Φ − 5 τ e − = b = ). We can repeat the computation for the cases (ii) and (iii) i However, new branches of vacua open up when 1 D5 and expanding the result for slowly varying fields we get: S a r, φ allowing the transverse coordinates σ Let us for instance choose case (i) above, where touched by the TsT transformation shrinks, so that The moduli space of the theory on this probe is simply We start from the Wess-Zumino part, that in our background (6 be written as: ( more copies of as the statement that, for generic deformation parameter for this to happen: reduced from branches of the modulithe space superpotential are (6.6): the solution of the three F-t where case, we knowextend from along our the previous torus analysis ( that we can consider then consider the following D5-brane embedding: and world-volume field strength: The D5-brane action is given by: JHEP10(2008)026 B ∧ 2 (6.23) (6.22) C and fi- + solution. 2 ) 4 1 = 4 SYM -deformed be dual to 5 4 TsT γ is rational, rsymmetry, C β ) S A 3 N a γ × , φ 5 γ∂ 2 zed product as in φ − 5 , reduces to the one AdS ound (6.11), and in . A 2 2 tween D5-brane dual a γ ϕ factor and write down ) of the general expres- θ forget however that the γ∂ a perties of the ∂ r of + and . ( formation of vanishes, while flat space metric expressed 3 mations, first derived in [8]. on the five-sphere (6.4) and α  1 ϕ r the study of the theory on i B 1 2 ϕ a . The resulting abelian moduli tuations, studied exhaustively φ ∂ ∧ of the gauge theory [27]. Other -deformed theory. γ − , ( 4 β θ 2 4 o angles on the sphere by a factor to find: C / 2 + sin 1 orbifold of the undeformed moduli 5 + 2 γA  σ ) 6 n cos − α ), that are stable when Z C of the undeformed α  2 a 2 = 2 ∂ ) × and ( , ϕ 2 5 (0) 4  n 4 1 A 2 C σ ϕ Z a r +sin 2 transformation, followed by ( + ) γ∂ – 18 – 4 2 , ϕ ) − A 5 3 r a 3 TsT γ a ) ϕ ∂ 2 γA γ∂ a ( ∂  + , φ ( = 4 1 3 θ 1 1 r φ ϕ 2 ϕ a ∂ sin ( 1+ α α 2 2  4 of the rational deformation parameter σ r + cos + sin 4 . The resulting type IIB supergravity solution, that should n d 2 Z TsT γ ) 2 1 = 1 supersymmetry. If we allow for a complete breaking of supe π N 2 , φ 3 N 1 γ φ − . The full moduli space is then obtained by means of a symmetri 3 = C Specifically, we perform a ( Now we can compute the determinant of the Dirac-Born-Infeld One may consider other probes in the Lunin-Maldacena backgr [34, 27, 35]. In that case, an explicit map was constructed be TsT D5 S 3 nally by ( 6.2 3-parameter deformation In the previous subsection,preserving we have consideredthough, a we can marginal find de aIn more fact, general 3-parameter we family can of form defor three distinct two-tori from the angles Now, either by direct computation or more easily with the aid sion (3.8), we see that we can choose a gauge where reduces once again to the four-form potential Hence the Wess-Zumino part of the action, integrated along we recognize in theas expansion of a the cone square overidentifications root the (6.23) in have undeformed modified (6.22) five-sphere the the (6.8). periodicitiesof the of denominator We tw should not If we now introduce new scalar fields: of a D3-brane in the undeformed background divided by a facto perform successive TsT transformations on all of them. the full action. We integrate the result along particular giant gravitons haveS been shown to be relevant fo gauge theory [30 – 33]. space in these additional branches is then a space the undeformed case. This is in accordance with the known pro giant gravitons, wrapped on the two-torus ( and rotating expectation values ininteresting the objects additional branches are D7-branes,in whose [36], world-volume correspond fluc to mesonic excitations of the flavored JHEP10(2008)026 2 x (6.24) (6.29) (6.28) (6.25) (6.26) (6.27) and 1 , x ,  such that the 5 ) , 1 2 S # ds dφ 2 TsT γ )  ∧ + i 2 3 2 ansformed solution is 2 ,x dφ = 4 Super Yang-Mills. r i 1 dφ wo-torus: dr ˆ γ 2 1 x . N µ ) he coordinates i 2 3 2 1 )+ X als to non-commutative and µ own backgrounds in order to 2 µ ve , 2 ) 2 3  ) 2 γ µ 2 3 3 hes of the theory. For instance (6.24) by setting all deformation 2 2 µ + ˆ dx γ 2 2 dφ 3 µ 3 = 4 SYM, reads: + ˆ 2 1 + ( γ dφ . 2 3 µ 2 N + ˆ 4 . µ ) ∧ 2 2 1 r -deformation, but also in this case there 2 M 2 2 12 µ β 2 γ 1 dx + dφ πγ . iθ dφ γ 2 2 ((
2 3 γ 2 i µ + ˆ − 2 2 ]= M + ˆ 2 2 2 dφ µ = =1+ˆ – 19 – 2 i µ 1 1 + 2 1 ,x µ 1 γ 2 12 1 µ , dφ ) − θ 2 3 x 1 + ˆ 3 ) M [ γ γ 5 , 2 , M (ˆ S dx + 2 . We will not repeat here probe computations analogous ) ω ∧ dφ 2 i γ 5 dx S and + ( , ∧ + = ˆ dµ ω 2 3 ∧ 5 1 γ θdθ =1+(ˆ ) 3 2 1 0 γ 1 M dx dφ 3 R =1 AdS − 2 2 i cos dx X dx = ˆ ∧ + ω 4 ( µ θ = 0 r 2 2 1 5 + M − 2 γ γ µ M ( 5 3 dx ( sin 2 R AdS 4 = ˆ γ r 4 α (ˆ r ω 2 AdS 1 ,  M 2 R ( 4 γ ˆ γ 4 2 4 M ds 2 − γR − M R , " sin R is given by: R 4 2 2 = = ˆ = = = is related to the non-commutativity parameter in (6.26) by: space (6.3), and to perform a TsT transformation ( M − R − R 2 5 2 M 5 B γ 2Φ F = = = = = C ds e 2 5 2 B 2Φ F C AdS ds e We then apply the formulae in section 3. The resulting TsT-tr a non-supersymmetric but marginal deformation of where now where we have defined ˆ Suppose we want to put the gauge theory on a non-commutative t The supersymmetric solution (6.11) of [4]parameters is equal, recovered ˆ from to the ones we have done for the supersymmetric The discussion in section 2 then instructs us to compactify t are D5-branes indicating theD5-brane existence dual of giant additional gravitons branc were considered in [27]. 6.3 Non-commutative deformation In this and in the nextshow subsection, how we our wish general to framework rederivedipole correctly some theories. accounts kn for Let the us du start with the dual to non-commutati of the given by: parameter JHEP10(2008)026 , 1 A has TsT γ ) 5 (6.31) (6.30) 1 F ! , φ
3 2 i x in (6.2). dφ 5 2 i singularity in S µ k Z + / . 2 i i 4 φ C dµ e could have derived. ger, Lambert [37 – 39] ents. Some studies of formation ( hree-dimensional super- 3 =2 i mations by performing a X ng a TsT transformation er to the vast literature. t it easily in a single step. eformations, that are akin ]. ransformation since we noticed in section 5, the = 4 SYM, reads: + ony, Bergman, Jafferis and e gauge theory and along a 2 1 ) and the two Chern-Simons r new insight in the case where ume. So for additional studies N as the dual of non-commutative N dφ ( 2 1 U µ and a coordinate of . 5 × M 2 1 ) µ + 2 N r AdS 2 1 ( 2 M2-branes probing a U γ dµ N + 2 2 – 20 – r =1+ˆ dr = 6 superconformal Chern-Simons-matter theory 1 and: − . The matter part comprises two chiral superfield N )+ γ k 2 2 M ) − R 3 , = dx ) ( 5 and γ being a linear combination of the angles S k M ω ϕ , + + 2 1 2 5 , µ 2 1 = 6 Chern-Simons-matter theory 2 r AdS dx 2 ( ω N 2 ( R are respectively a coordinate of r 4 1

, M R φ ˆ γ 4 2 R M − − transformation, and = = = = 3 2 5 Φ B x TsT γ e F ) ds The resulting solution, dual to a dipole deformation of This is of course not the most symmetric dipole deformation w The gauge group of the ABJM theory is Their work was the starting point of a long series of developm , ϕ 3 (1) transverse isometry. As an example, we perform the trans x where Super Yang-Mills, but our TsTWe will technology not has analyze allowed the usD-brane theory to probes in we ge more are interested detail in herethe do because, TsT not give transformation as any acts particula completelyof along non-commutative the SYM world-vol and its gravity dual we6.4 refer Dipole the read deformation The gravity dual ofalong the a dipole direction theory longitudinalU is to the obtained D-brane by performi supporting th The solution (6.28) is the same solution obtained in [10, 11] Notice that the R-R part of the solution is untouched by this t In an analogous way,( one can write down different dipole defor no components mixing the AdS and sphere parts. where we have again defined ˆ and conjectured it to be the theory living on We now turn to our second example. Motivated by the work of Bag and Gustavsson [40, 41],conformal which theories opened the and wayMaldacena their to (ABJM) relation the constructed study with an of M2-branes, t Ahar factors have opposite levels 7. Example 2: M-theory [24]. the gravity dual of theto the ABJM spirit theory, and of in our work particular in of this its section d can be found in [42 – 58 JHEP10(2008)026 the and (7.1) (7.4) (7.2) (7.3) 5 k k ≪ , = 6 Chern- N
, 2 , N  ) ≪ 2 2 2 dϕ 2 2 k θ dϕ dϕ  2 2 ∧ θ ) cos 2 2 2 the gravity dual is the dϕ ξ 2 dθ dϕ 2 5 θ , 2 2 k θ θ description depends on the ) + sin e four-dimensional superpo- 1 rder to deform them via TsT cos 2 2 larity [59]. As we said, this is ) representation. There is a B t, which in this case is: ≫ . sin + sin cos 2 1 2 dr , dθ ξ N 1 ( , A − 2 ∧ − N ξ 2 solution of type IIA supergravity. dϕ ¯ 1 N 2 1 fixed 2 1 B sin 3 θ 1 dϕ dx dϕ , 1 2 sin 1 A k 1 N CP θ 2 ∧ 1 4 θ 2 cos − − 1 r ) representation, and two chiral superfield . The gravity dual of the ξ × = dr 1 2 2 ¯ cos 4 N N dx )+ B + , dϕ 2 1 2 2 1 + cos ∧ 2 , , N ≫ ∧ A 0 2 1 dϕ AdS + 1 – 21 – 1 1 + cos dψ  k , λ B θ 3 dx dx 1 dθ (2 2 2 2 dψ = 6 supersymmetry in three dimensions 1 dψ 2 CP r r A ∧ θ )  ξ N ds ξ = = →∞ . In particular, for 2 2 sin Tr ( 4 4 + sin k + ξdξ ξ 2 1 4 π sin . sin k 2 4 AdS 2 AdS 4 sin ξ dθ − N , k ω ( and 2 AdS ξ 2 ds = cos ξ ξ AdS 2 2 ds 2 1 N ω cos 1 W 4 3 − cos + cos ,  − 8 (cos R solution of type IIA supergravity can be written in terms of 1 4 2 3 3 3 3  k 3 k 2 k R R dξ − k + CP ======solution of eleven dimensional supergravity, while for 1 3 4 2 × 2Φ F F C 2 IIA k as (see for instance [60, 61, 45]): 2 CP 4 e transforming in the anti-bifundamental ( Z ds ds / 2 we will use the Poincar´epatch, so: 7 kN AdS B 2 4 S transforming in the bifundamental ( π 2 × A 4 The theory is weakly coupled when The AdS and = 32 1 B Simons-matter theory is, as usual, valid in the ’t Hooft limi and quartic superpotential given by: which can also betential seen of as the the theory three-dimensionala living version superconformal on of D3-branes theory th preserving at a conifold singu It was shown inrange [24] of that the the parameters appropriate dual gravitational For AdS We now review the tentransformations and in eleven-dimensional solutions, the in following o subsections. appropriate description is in term of the R JHEP10(2008)026 and (7.6) (7.7) (7.5) i A tions that -deformed β ntals . , ) . 2 ) 2 ϕ 1  2 ϕ 1 − − 2 ormed theory. ψ B own in table 2. Using 2 2 can proceed with the ψ − y k + A . We choose the latter 2 k y 2 ( i avity dual. ( ϕ ed counterpart, as we will B i -deformation. − gger-Lambert theory, have 1 e e een the coordinates we have β (1) = SO(2) R-symmetry of M theory. As in the case of , 1 2 A , we see that the deformation 2 ostly concentrate on the type 2 , θ U θ 2 k and 2 Z 2 1 2 rounds generated by M2-branes, ) / : 1 0 7 1 sin sin B iπγ/ + 2 S ϕ e C ξ ξ , ds − + ψ 2 1 2 1 0 2 R B − , dy B = sin = cos ( 4 2 + 2 2 2 1 A 4 2 1 2 k 1 0 AdS A − ω B + 2 AdS 1 , B 3 3 , A ) A – 22 – 8 ds R appearing in the superpotential (7.1) under the 2 1 1 2 ) 2 2 CP 2 3 0 2 ϕ 1 i A 4 2 + ϕ R − ds + B iπγ/ + 2 ψ 2 = = = ψ 1 2 − − e ϕ ϕ 4 y k + k and  2 11 k y ( Z G i / ( i (1) (1) i 7 ds − Tr e (1) charges of the chiral fields of the ABJM theory. 2 S A U U e 1 2 U π 2 2 k θ isometry will correspond to the ds θ 4 × ψ = cos cos solution of eleven dimensional supergravity is given by: (1) γ ξ ξ k U = 4 SYM that we have studied in subsection 6.1, the W Z are quantities given in the type IIA solution (7.3). / 1 N 7 -deformation of the ABJM theory. The obvious isometry direc → C = sin = cos S β and the supergravity fields that correspond to the bifundame 1 1 W × A B k 4 and Table 2: is the Minkowski metric in three dimensions. Z 3 / 2 7 , 2 1 AdS 2 CP S = 2 supersymmetry (in three dimensions) preserved by the def ds dx (1) factors we have chosen can be derived from (7.6) and are sh -deformation β The charges of the fields Having formulated the gauge theory and its gravity duals, we It will be useful for what follows to recall the relation betw The U N in (7.1). The embedding equations can be written as [45]: = 4 SYM, the case we will study with the most care is the i the corresponding deformed product (2.6)results in in the the case following at hand modification of the superpotential the two, while the remaining two we can use to perform the TsT transformation are Let us study the 7.1 study of their deformations. Generaland deformations of their backg interpretation frombeen the derived point and of carefully studied viewIIA in of solution [42, the 43]. (7.3) dual Here andN Ba we interpret will the m results for the dual ABJ ABJM theory has ashow richer in structure the of following vacua than by making its use undeform of D-brane probes in the gr As in the case of used on where where B JHEP10(2008)026 - , β = 2  (7.8) (7.9) 2 (7.10) . N dϕ
, 1 ∧ θ 2 1 2
dϕ 2 dϕ cos ∧
dθ 2 1 2 1 background (7.3) θ θ ∧ 2 3 ity dual of the 2 ) dϕ
1 dϕ factor is untouched. 2 2 2 ∧ 4 sin ∧ CP dθ cos dθ dϕ ξ 2 , ∧ × 2 2 θ dψ ∧ dψ 2 θ 4  1 AdS 2 1 2 2 θ  he following solution, that ∧ dψ upergravity rather than in 2 , matching the dual 8) that reads: dθ sin +sin ∧ dϕ sin )) AdS cos ξ 2 2 dϕ ∧ ∧ θ θ 2 ) 2 2 dξ at the M 2 θ θ 2 2 2 2 dψ 2 θ + dθ dϕ sin cos ∧ cos 2 2 2 2 +sin sin ξ θ 1 θ sin 2 2 1 θ ξ 2 θ dθ 1 dξ , ( 2 2 2 θ 2 sin − ξ cos θ sin  ξ 1 2 sin sin k sin cos − 2 − ξ sin Z , factor: 1 2 1 ξ dϕ 1 sin / 1 1 2 θ 2 1 7 3 3 sin θ θ ^ 4 1 θ 2 + 2 S dϕ 2 1 2 dψ – 23 – 1 1 sin CP ds 2 θ sin sin cos ξ 2 θ +cos sin − )+ dϕ , ξ ξ 2 2 1 3 2 ξ 2 1 transformation to the 1 R 2 3 2 θ ∧ 2 ) 2 dϕ , + 1 + sin dϕ cos + cos 1 sin 4 1 C , TsT γ θ γ )) dψ = 4 SYM, we see that the fact that the deformation is ∧ sin ξ ) θ (cos 1 2 +cos 2 dψ 2 1 1 3 + ξ 2 dψ  θ 2 C 2 AdS θ 3  N 2 θ + 4ˆ , ϕ dθ (2 sin 2 ξ ξ 2 sin dy 4 cos + g 1 ds 1 2 CP cos 2 ( 2 ∧ θ sin 2 ξ ϕ γ 1 sin 4 2 M sin ds ξ 4 dy R θ AdS k sin M sin ( 1 2 sin 2 + ξ ω +  ξ θ ξdξ + 4ˆ ξ sin ∧ ξ 2 2 1 + 4 3 2 2 4 2 ξ cos 2 sin / 3 3 1 B sin 4 dθ ξ sin 8 ( R sin -deformed ( cos − 2 AdS cos g ξ AdS 2 CP 2 M 3 d cos ξ ξ β factor and a deformed 3 ω cos γ 2 2 ds 4 2 1 1 M − ds 4 (ˆ + , R M and: cos cos 4 1 d 3 k M − + = = = cos cos 3 2 1 + γ  − 8 M 2 M R 8 2 3 4 k k 4 1 2 R AdS ˆ 3 γ γ  3 3 3 2 11 γ 4 Z R  k G / k R R k − − dξ + ˆ + × − 7 ds ^ = 2 S ======γ =1+ˆ ds 3 2 4 1 B 2Φ F F If we are in the range of parameter where the appropriate grav Let us then apply the ( 2 IIA g − 2 CP e ds ds M marginal translates in the gravity solution into the fact th deformed Chern-Simons theorytype lives IIA, in we can eleven compute dimensional the s eleven dimensional uplift of (7. supersymmetric three-dimensional gauge theory. where ˆ The solution (7.8) can be seen to preserve four supercharges As in the case of of type IIA supergravity.contains an After some computation, one gets t JHEP10(2008)026 , ) 2, 2 , .  = 4 1 , ϕ !# 2 ,  -type 1 ) (7.15) (7.16) (7.13) (7.14) (7.12) (7.11)
 β 1 ϕ N α y ϕ = 0 a a a k , ∂ ∂ a ∂ 1 2 ( π , , )) 1 ϕ a  θ on-shell. We a F = x 1 ∂ 2 1 2 ϕ + F θ θ a ˆ -deformed ∂ B = 1 ( β explicit example in + sin θ on the world-volume, ∧ 2 ives, and introduce a 1 ) 1 1 θ ab ˆ + 2 cos d by means of D2-brane ded along θ C se of nd, after integrating the where the torus ( te some of their structure F a 2 brane in the static gauge, ) ∂ + +2 cos α e of the theory, we will need 3 2 a (iv) ˆ . + ( C  ∂ ( 2 y ) nal scalar. The D2-brane action: a Z k is not rational, will feel a static + ( . = 0 ∂ ab 2 2 , 2 # 2 τ F γ ) ( 1 bc  2 + 6 F ϕ becomes: + = 0 k 1 a  R 2 2 8 2 ∂ ϕ ) ρ ab θ π, θ a 1  F ∂ ϕ 2 = + ( = = 1 a r θ 2 + r ∂ 1 1 y dF , ) θ θ + , 1 were constructed and analyzed in [4, 42, 43]. ab – 24 – θ Z 2 α +( cos ˆ 7 ) a B 2 2 r ) ∂ S τ = enforces the Bianchi identity for 1 a ( abc + ǫ θ y ∂ (vi) (iii) × k y 2 ( a 3 ab 2 4 ∂ 4 r  k ρ ˆ , ( G 3 out by using its equation of motion. The result reads: 2 4 2  1

R + r AdS F r 2 π/ − ) det + ρ 1 = 1+ 2 a − are quantities of the type IIA solution (7.8). Notice that ) − ∂ ξ r s ( r r a "  π , M ∂ Φ 3 ( σ − 3 = " (ii) d σ r 2 and σ 3 σ e 3 d 3 Z d B d 3 , = 1: , , θ Z 1 Z R 4 Z 3 2 C 3 2 τ M R , 8 τ = 0 = 0 R 2 3 16 − 2 = 0. We also allow for a non-zero gauge field strength 1 − τ ξ τ g θ 2 CP ξ = − = − ds = (i) = D2 From our general observations in sections 4 and 5 and from the Having derived the deformed solutions, we now try to elucida TsT D2 (v) S S TsT D2 TsT D2 S section 6, by now we know very well that a D2-brane probe embed SYM. The moduli space ofprobes the in undeformed type theory IIA, can or be M2-brane analyze probes in M-theory. that with the change of coordinates potential that will force the brane to sit at one of the points deformations of non-orbifolded which is the only allowed static configuration when can now regard theterm (7.14) action by as parts, a integrate function of the field-strength a where shrinks and S We now expand theLagrange multiplier: square root up to quadratic order in derivat so that the equation of motion for by using D-brane probes, as we did in subsection 6.1 for the ca with the choices we have made reduces to: the reason being that, ifto our dualize the aim three-dimensional is gauge to field get into an the additio full moduli spac Let us consider case (i)then in fix some detail. As usual, we embed the JHEP10(2008)026 . f 1 /γ ϕ (7.19) (7.20) (7.18) (7.17) → = 1 ) 2 2 . 2 ϕ ϕ k 1 d from the Z ϕ required for − , / F = 0, the final and integrate 4 π 1 2 2 ϕ  2 R . ) θ , , ρ ) + a F = = x ψ ( + 1 = 0 = 0 other cases in (7.11) r 1 θ , ( 2 ˆ 1 ϕ B to zero. As we can see 1 2 ( a θ , A B , as in (5.14) and (6.23). i 1 ring side we should then 1 ∂ -deformed ABJM theory 4 ∧ ates B 1 β γ B A σ → ) he F-term equations arising ) with a flux tion of our D2-brane probe 1 2 π/n hen study a D4-brane probe − 2 ξ F = A B instead of the 4 -deformed ABJM theory, for t and 2  = , ϕ + one the six cases in (7.11) and (iii) with β i 1 iπγ iπγ 4 ab ˆ e e B ϕ a A π/k F ( , ϕ − − ∧ + 3 2 1 is 4 1 σ A B ab ˆ α C 1 1 ˆ B , F = 1 2 B A ) , with period 2 1 1 2 + a i , and with a world-volume field strength: is rational in the , is made up of six copies of x a ϕ )+ ab ( γ x 2 γ ˆ F , A , B G , ϕ ϕ  – 25 – a + 2) ∂ , ˆ = 0 = 0 B 1 γ 1 det ( 2 1 , − = A B ∧ 2 2 3 3 = 0 r a ˆ B A C Φ 1 2 a ( − A B + a 5 σ ˆ σ e iπγ iπγ C has been chosen in such a way that we have already traded the ) space is the one of a round three-sphere expressed in terms o 5 e e , F d  = F 1 , α γ − − a 1 Z Z 1 2 x = 4 4 A B , ϕ . τ τ 2 2 1 k 34 − θ + B A Z 2 1 / space. As usual, the nonabelian moduli space will be obtaine = 4 A B k R Z D4 , F / S 4 -deformed superpotential (7.7), ab R β F 6= 0 are solved by setting two out of the four fields The computation of the analogous D2-brane probe in any of the We expect new branches to arise when This is in agreement with gauge theory expectations, since t γ The D4-brane action: a sphere, so the fullis moduli the orbifold space read by the world-volume ac Euler angles. However, the periodicity of the angle precisely reduces to (7.16). For instance, if we choose case Wilson lines on the torus with periodic scalars The parameterization of The metric in the ( We then conclude that the moduli space of the abelian generic values of the deformation parameter when we expand the square root, perform the change of coordin result becomes identical to (7.16) if we make the replacemen under consideration too. Asbe we able are by to now find used a to see, D4-brane on wrapped the on st the two-torus ( abelian one by means of a symmetrized product. turned on. Putting togetherembedded all we as: have learnt until now, we t with all the other coordinates depending on from (7.6), each onespans of an the six possibilities corresponds to from the for JHEP10(2008)026

2 2 ) ) 2 2 ϕ ϕ (7.22) (7.21) a a ∂ ∂ ( ( 2 2 θ θ 2 2 . + sin + sin 2 2 . The moduli space ) )  2 2 2 2 2 2 ϕ θ θ ) orbifold of the moduli a a   2 2 2 ∂ ∂ n ϕ ( ( in the undeformed ABJM ϕ ϕ mputation we did above, a n fact, an M2-brane probe Z and a a ∂ alization). The branches of on here, due to the known y finding a three-parameter ξ ξ ∂ ∂ 1 2 odic scalar, as we did above her cases we have considered × 2 2 2 2 ϕ ϕ hes of the moduli space of the θ θ ) nd references therein. n 1 , see (7.5)), namely the moduli sin sin Z k C cos cos 4 4 1 1 Z 2 / ) 2 2 1 1 7 + ( ξ + + a S 1 − −

∂ 2 2 ϕ 1 1 ) ) a 1 1 ϕ ϕ ∂ + ( a a ϕ ϕ 1 ∂ ∂ 2 a a ϕ 1 1 ) ) ∂ ∂ . In order to obtain the full moduli space, we -deformed three dimensional Chern-Simons- θ θ 1 ( ( is rational. At first sight, the metric looks like ab would instead be described by M5-branes that β 1 1 C 2 /γ F θ θ γ ) γ – 26 – ( cos cos 2 2 ξ 2 (0) D2 6 , a + ( 1 1 2 2 k S ∂ R 2 ( ψ + +  a =   + sin + sin ∂ bc 2 2 ψ ψ 2 2 ψ a a ρ ρ F ) ) ) ∂ ∂ a 1 1 TsT D4 1 ) + + θ θ   S 1 C a a 2 2 ) with a self-dual two-form turned on on their world-volume. ξ ξ , ∂ ∂ ) ) C ( ( ( γ 2 2 (realized as a cone over ρ ρ ,y + ( a a 2 k ξ ξ ∂ ∂ y abc sin sin ( ( ǫ Z a 2 2 , ϕ ξ ξ   / 3 ∂ 1 ( 4 2 2 r σ σ ϕ 4 cos cos 3 3 3 2 C 1 d d 4 k 4 R 1 1 Z Z + + + cos + cos + + 3 3 , becomes: R R 4 4 4 2 2 σ τ τ 1 1 γ γ − − and 3 = = σ It would be instructive to repeat the analysis in M-theory. I TsT TsT D4 D4 -deformed ABJM theory that arise when S S over The action (7.22) gives usβ the metric on the additional branc need to dualize thein three-dimensional the gauge case field of onto the a D2-brane. peri The result reads: which is the same actionfactor we divided would by have obtained a for factor a of D2-brane describing the space space of the abelian undeformedso theory. far, However, we as have in to the take ot into account the new periodicities of difficulties with the M5-brane action, see for instance7.2 [62] 3-parameter a deformation We can generalize the gravity dual of the However, we will not enter into the details of this computati in these additional branches will then be, once again, a matter theory that we have found in the previous subsection b space of the undeformed theory. computation goes throughyielding precisely the as same the resultthe (of D2-brane moduli course probe space without that co the emerge need for of rational any du wrap the three-torus ( JHEP10(2008)026 , 2 2 TsT γ θ , , . We ) 2 5 (7.23) (7.24) 1  2 S . cos dϕ 1  × dϕ ψ, ϕ 2 θ
5 ∧ ) θ 2 1 ∧ 2 1 dψ sin AdS dϕ 3 and ( sin ξ γ dϕ ∧ ξ 2  ) 2 + ˆ 2 1 1 2 2 TsT γ θ 2 dψ 1 ) sin dϕ ) cos dϕ ∧ 2 dϕ 2 2 , ψ dϕ cos ∧ ) 2 γ ion of the ABJM three- , 2 θ 2 )) 1 2 , γ ∧ 2 2 2 ϕ γ θ  ) θ dψ 1 2ˆ 2  +2ˆ 2 + ˆ 2 θ , ( 2 2 2 sin dψ 1 θ − dψ θ 2 1 dϕ 1 sin 2 3 . We can also write the eleven- 3 θ 2 ) dϕ sin section (6.2) for θ TsT γ dϕ γ ξ M the supersymmetric one studied 2 γ θ 2 sin ) ξ 1 cos ∧ 2 sin 2 ξ γ 2 + ˆ + ves the AdS directions and is thus 3 = ˆ 2 dϕ ξ cos (ˆ sin 2 cos 2 γ 2 2 2 , ϕ 3 2 3 ξ θ dθ ∧ 1 γ 1 2 )sin γ 2 2 dθ )sin dϕ 1 2 +(ˆ ϕ θ ( 1 2 γ 2 ) cos − ξ θ +(ˆ cos )sin γ θ dθ 2 2ˆ 1 1 1 2 θ 2 sin ) cos θ γ − ∧ + ˆ − 2 2ˆ cos ξ 2 dϕ 1 1 sin 1 2 1 γ sin 2 1 1 cos θ θ = 0 and ˆ − 4 1 θ 1 γ dθ 2 sin dϕ 2 dϕ θ 2 2ˆ 1 1 γ sin 1 + 2ˆ ξ θ ∧ γ 2 – 27 – θ cos γ − 2 2 cos 2 sin 1 )+ (ˆ 3 3 θ = ˆ 1 2 2 1 + 2ˆ dξ 2 1 sin γ γ cos θ θ − 2 (ˆ cos 1 3 3 3 θ 2 2 1 dϕ γ γ 2 + ˆ cos γ γ + cos 1 + (ˆ − ) 3 (ˆ  , ((ˆ θ 2 sin sin dϕ γ sin ξ 2 4 1 1 γ 2 dψ (ˆ 2 1 3 dψ  1 2 ∧ θ 1 2 ˆ 3  γ θ + + 1 (2 sin +2ˆ 2   ξ g sin 2 CP 2 ξ ξ 2 ∧ dθ θ sin ξ 2 2 sin M 1 ds ξ 2 θ ξ sin 3 + sin sin + cos 4 ξdξ ξ 2 cos 1 3 ξ ξ 4 sin 2 4 sin γ 2 2 sin ξ sin dθ ξ M ξ ( 2 2 AdS cos ξ 3 +(ˆ ( AdS ξ 4 cos d ω 2 ds 3 , cos 3 M cos 1 4 8 cos 3 R 2 R 1 k cos +  − 8 M R 1 4 2 M + 4 cos M 3 − − 3 3 3 = 1 + cos  k k R R − − dξ k 1 + + − ======3 4 2 M B 2Φ F F 2 IIA g 2 CP e ds ds family of non-supersymmetric deformations, as we did in sub where: then perform three successive TsT transformation, ( Of course, this non-supersymmetric deformation reducesin to the previous subsection by putting ˆ dimensional uplift in complete analogy with (7.10). 7.3 Non-commutative deformation We will not spend manydimensional theory, words since, on on the the non-commutative string deformat side, it only invol and the final result reads: JHEP10(2008)026 , (7.25) (7.26) (7.27) 
, 2 0 but let us dϕ dx 2 2 θ 3 ϕ 8 , cos  ˆ γRr 2 ξ )  and 2 2 2  2 − 2 1 , ,
dϕ dϕ ϕ 2
2 2 dϕ  2 , θ 2 θ +sin 3 der consideration is: θ dϕ ψ 2 1 tter theory put on a non- 2 dϕ 2 CP 2 θ cos dϕ θ cos ds 1 2 1 direction to perform the TsT. and: θ 2 1 -dimensional solutions of [42]. + sin cos 3 − cos 2 2 γ ξ + 4 − 3 ξ 1 k cos 2 2 dθ 1 4 CP R 2 2  ξ ( r dϕ 2 ξ 2 dr transformation. Again, more general 2 1 . dϕ = πiγ . 2 r . We can proceed in the same way as 1 θ 4 dr 2 + sin 5 θ γ r + sin 1 S )+ sin − 2 TsT γ 1 +cos 2 cos ) γ 4 1 ) )+ cos × = dϕ and one 2 1 2 2 1 dϕ , ψ ) 5 dψ 1 2 4 1 θ 2 ) 2 12 + dx θ )+ ξ x + 2 1 iθ 2 dx =1+ˆ – 28 – cos AdS ( + ( AdS dψ cos 1 dϕ ξ dψ sin 2  1 ξ ]= − 2 ) M  θ 2 2 − 1 ∧ 2 ξ + ξ + M ,x 2 0 2 dx 2 2 1 , in order to get the gravity dual of a dipole deformation ) x (( + cos 5 dx 1 [ , sin + cos ξdx S  + sin 2 (cos ξ 2 M dx 2 dψ 1 2 ∧ × ) dψ dx + 5 ) 2 ξ sin dθ + ( γr ξ ( 2 2 ∧ cos ξ ) with: ) ξ 2 2 dx 1 2 + ˆ 2 0 ) 2 AdS sin 0 ∧ M sin dx ,x dx 1 cos 4 1 − ( dx cos + 2 r − ( x ξ − 2 4 1 dx r 3 ξ ( − 2 3 2 , ( 2 R dξ + 2 k , r M R r  4 3 k  8 M M (cos 3 r  M for simplicity, and perform a ( M (cos ˆ γ 3 3 3 k 3 k is given in (7.3), and we have defined ˆ R ˆ γ 3 3 3 k k 4 2 R k R R ψ − 3 k 4 2 k R R − + = 6 theory we must choose one = = = = = 2 CP = = = = 1 3 B ds N 1 2Φ C C 2 IIA B 2Φ e C 2 IIA The resulting type IIA solution, dual to the dipole theory un e ds ds very similar to the deformation (6.28) of where The solution (7.25) will becommutative torus dual ( to the ABJM Chern-Simons-ma just choose backgrounds can be constructed, see for instance the eleven 7.4 Dipole deformation As we did in the case of The latter direction can be a combination of the three angles in subsection 6.3 to find the solution: of our JHEP10(2008)026 ,  (A.2) (A.3) (A.4) (A.1) ) (7.28) (7.29) 2 ,  dϕ H , ∧ ⋆ 1  ∧ 3 dϕ F 2 H . The self-duality θ ∧ p 2Φ − H − cos 10 e ∧ ndamentale Collective 1 F θ 4 ⋆ illuminating discussions. C Φ+ = ogramme MRTN-CT-2004- d oughout the paper. Notice − mmon NS-NS part reads: , n Institut Interuniversitaire p ⋆ cesco Bigazzi, Frank Ferrari, 5 ] F )+cos , Asad Naqvi, Nemani Surya- ∧ 4 2 F F ξ . ⋆ Φ 2 = 6 ABJM Chern-Simons-matter dϕ d ∧ ∧ 2 ersity Attraction Poles Programme . are defined as: 4 5 θ 2Φ sin 3 N p F F − − ξ F e explicit for completeness, while in the p 1 2 ∧ 2 8 C s l +cos + B − cos 1 ∧  3 2 − F r H dϕ Z 4 ⋆ 2 1 2 F γ θ + ∧ ⋆ 1 κ – 29 – p 3 4 ∧ F F (cos 4 − = + F ∧ R = 1. Let us start with the bosonic part of the type IIB p =1+4ˆ 1 + s , while the R-R parts read respectively: F F 1 2Φ l dψ 2 ⋆ − ( − 1 F dB ξ ∧ ⋆ M 2 dx 1 = G e ∧ 5 are defined via Hodge duality, F ∧ 2 sin  , is not taken into account by the type IIB action and has to be H 0 F ξ det 5 [ Z 2 p> dx F − 2 ⋆ Z  and 1 κ √ cos 2 4 = 4 ∧ x 1 κ and the modified field strengths s 5 2 l 4 − 10 M s 1 r d F g γ − dx and: = = 2 2ˆ p 3 / Z γ r 7 8 3 + 3 2 k (R) dC (R) π IIA IIB 4 R 1 κ R S S 2 = = 8 = = p = F κ γ 3 C This concludes our study of deformations of the (NS) II S where ˆ where Higher rank forms with main text we work in units where where theory. Acknowledgments It is a pleasure toChethan thank Krishnan, Riccardo Argurio, Stanislav Sujay Kuperstein,narayana Ashok, Carlo Fran and Maccaferri in particularThis Carlos work N´unez is for advice supported(grant and 2.4655.07). by many the It Belgiandes is Sciences Fonds (grant also Nucl´eaires 4.4505.86), de supported the la Interuniv (Belgian in Recherche Science part Policy) Fo by and the the005104 European Belgia (in Commission association FP6 pr with V. U. Brussels). A. Conventions and useful formulae A.1 Supergravity fields This appendix details thethat notation in this and appendix conventions we used keep thr the string length of the five-form, imposed on-shell. and type IIA supergravity actions in the string frame. The co JHEP10(2008)026 is ] y [ , and p (A.8) (A.6) (A.7) (A.5) (A.9) (A.10) (A.11) (A.12) ω , yy 2Φ , α,β,... e G , we get type  11 4 = R ′ G 2Φ ∧ 4 , we decompose it as: p G , e ω ∧ NS-NS and R-R antisym- , given by: components, while 3 j . βy of radius und fields. Denote the di- 2 , an anticommuting operator of ι ) A B ∧ 1 e duality of course exchanges dy 10 1 3 b -form . αy C x p . y ∧ − B dy . 1 + s ] l 4 yy − y − 3 + p product 10 G 1[ / G ] α 1 ⋆ y − βy [ s ··· p dx dy , 1 g ∧ ( B G C α 3 4 ∧ ) / = αy ] p = + G y p . [ 4Φ ω 2 G  b p e 10 ω ∧ − and the remaining directions by Z + = ( 1 being the eleven-dimensional Planck length. If + 2 dx 1 y − – 30 – αβ , b 1 κ , l p − p p , as: 2 IIA ∧ l p 4 s ¯ b G ω α C l α 3 , does not contain any ds s b , 2 ··· + = 3 = ¯ g dx H p + / 1 / 9 α ∧ ] p α and p y yy + = αy ) l ′ αβ j 2Φ ω ] . Metric and dilaton transform as: j 4 dx 4 y G G − y G R [ 11 +1[ + π e F p p 2 R ω = ¯ / B C ( = = , G 7 det j 4 = = − 2 11 yy αy ∧···∧ ′ G ′ p = 2 √ G B ds 1 B C x α 11 = 11 κ dx d p ′ αy α Z ··· and 1 2 11 3 α 1 κ , G ω ! 2 1 dA p 1. yy 1 = − = G = 4 11 p = 1)-form whose components are given by: S G ω ′ yy − The action of eleven-dimensional supergravity is given by: This is of course just a short-hand notation for the interior G p 2 a ( form degree where primed fields are thetype ones IIA after and the IIB). transformationmetric We (th want tensor to fields write in the terms transformations of of differential the forms. Given a Finally, we define two one-forms, The T-duality rules for the NS-NS and R-R potentials are then The reduction ansatz for the fields is given by: we compactify eleven-dimensional supergravity on a circle A.2 T-duality We now summarize therection T-duality along rules, which starting the with T-duality acts backgro by assume that no field depends on where IIA supergravity with the identifications: where ¯ JHEP10(2008)026 ]. actor 62 (A.15) (A.13) (A.14) Adv. , (1999) 032 , 09 F + (1998) 253 ˆ B e global symmetry is the world-sheet 2 JHEP hep-th/9711200 ∧ , Contemp. Math. αβ (1) q , η ˆ U C × q ty acts on the coordinate j , X , ]. (1) ]. µ ∧ ]. Z U b X p +1)-dimensional world-volume τ α p ∧ (1999) 1113] [ ∂ ] 1 + y and of the modified field strengths µ = +1. 38 1[  B − H ab p τσ ǫ − F F µ Gauge theory correlators from non-critical + Adv. Theor. Math. Phys. + hep-th/9711162 X , b db , hep-th/0502086 ab κ Noncommutative geometry and matrix theory: hep-th/9802109 ˆ ∂ ∧ ∧ B 1 1 j µ + − – 31 – G p − ¯ ab F b βκ ˆ G ǫ ∧ +  (1998) 003 [ ] Int. J. Theor. Phys. αβ (2005) 033 [ y dj (1998) 105 [ η -brane in the string frame which is consistent with the 02 the direction along which the T-duality is performed and Deforming field theories with det p + 05 +1[ limit of superconformal field theories and supergravity = 1 . p − Yang-Mills for noncommutative two-tori ] , hats denote pull-backs of the bulk fields onto the world- 1 ¯ i y H F X N [ String theory and noncommutative geometry ˜ r σ X B 428 H Φ = = ) denote the world-volume coordinates, JHEP α JHEP ′ − , ′ ∂ p = , (1998) 231 [ F H τ,σ 2 σ e db , the integrals are performed on the ( is the gauge field strength on the brane (we have reabsorbed a f = ( +1 ]. ]. The large- +1 p F p d α s l Phys. Lett. s 1 g , Z p Anti-de Sitter space and holography p ) τ π (2 − = = into its definition). p p the T-dual coordinate. We have: 2 τ D s 1 hep-th/9802150 hep-th/9908142 S [ Theor. Math. Phys. and their gravity duals (1987) 237. string theory compactification on tori [ At the level of the world-sheet string sigma-model, T-duali ˜ πl defined in (A.4). They read: X p [4] O. Lunin and J.M. Maldacena, [5] A. Connes and M.A. Rieffel, [1] J.M. Maldacena, [2] S.S. Gubser, I.R. Klebanov and A.M. Polyakov, [6] A. Connes, M.R. Douglas and A.S. Schwarz, [3] E. Witten, [7] N. Seiberg and E. Witten, of 2 References It is useful to express the T-duality rules in terms of where spanned by the coordinates F and we also note that volume, and finally fields as follows. Denote with by where the indices A.3 D-brane action The world-volume action of a D above supergravity actions and T-duality rules reads: metric and the antisymmetric symbol is defined as JHEP10(2008)026 , 09 , l , Phys. , 10 , . ]. 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