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[EQUATION] As expected, this is exactly the tangent space to the twisted conjugacy class [MATH] defined by [EQUATION] shown explicitly in Stanciu:1999id |
Figueroa-OFarrill:1999cmq . Hence, the worldvolumes [MATH] of the integrable D-brane configurations lie on twisted conjugacy classes [MATH] of the group [MATH] on which the deformation is based. In the [MATH] -deformed background only the size of the branes (determined by the induced deformed metric) change, as also il... |
. However, the involution condition [MATH] from integrability does not allow these group translation to be arbitrary and in practice there will only be a small number of them, depending on the dimensionality of [MATH] . When the group automorphism [MATH] is taken to be the identity element of [MATH] , the twisted conju... |
Note finally that, using Frobenius’ integrability theorem and [MATH] being an automorphism, the D-brane [MATH] is a submanifold of [MATH] eliminating the possibility of intersecting integrable D-brane configurations Stanciu:1999id |
Figueroa-OFarrill:1999cmq using the methodology outlined in section 3.1 Examples In this section we will apply the above observations explicitly in two simple examples: the [MATH] deformation based on the [MATH] and [MATH] Lie groups. In the compact [MATH] case only regular conjugacy classes exist and we will derive th... |
4.1 The [MATH] deformation For [MATH] the metric-preserving algebra automorphisms form the group of rotations [MATH] while [MATH] is known to be trivial. The D-branes in the [MATH] manifold therefore lie on regular conjugacy classes. In light of this we choose the following convenient parametrisation for the [MATH] gro... |
[EQUATION] such that the [MATH] is given by an [MATH] parametrised by [MATH] and [MATH] fibred over an interval [MATH] . Regular conjugacy classes are distinguished by [MATH] constant and so correspond to this [MATH] . Note that integrability only allows group translations from involutive inner automorphisms that corre... |
Pawelczyk:2000ah Stanciu:1999nx Figueroa-OFarrill:2000gfl We first recall the target space geometry of the [MATH] -deformed theory Sfetsos:2013wia |
[EQUATION] where [MATH] . Here we note that in performing the Gaussian integration to arrive at the [MATH] model a dilaton ( 13 ) is produced. In the metric observe that, and this will be important, that the deformation leaves the [MATH] intact changing only the radius of this sphere as it is fibred over [MATH] |
The integrable boundary condition obtained from ( 38 ) with [MATH] reads, [EQUATION] It is immediately clear that [MATH] obeys a Dirichlet boundary condition and [MATH] obey (generalised) Neumann boundary conditions ( 122 ) which should take the standard form, |
[EQUATION] in which [MATH] (see also appendix ). We first re-express the Neumann boundary condition as, [EQUATION] and making use of the metric restricted to [MATH] we extract the two-form, |
[EQUATION] We can now evaluate the DBI action ( 128 ), [EQUATION] where we absorbed a factor of [MATH] into the constant dilaton [MATH] Naively this would suggest that the only stable D-branes are the ones for which this quantity is minimised i.e. the D0 branes located at [MATH] and [MATH] . However the flux quantisati... |
Gawedzki:1999bq Figueroa-OFarrill:2000lcd Stanciu:2000fz demanded by consistency for the definition of the WZ term in the action ( ) (see also appendix ). Put plainly, the difference in periods of [MATH] over the D2-brane worldvolume [MATH] and [MATH] on an extension [MATH] whose boundary is the D2-brane [MATH] should ... |
[EQUATION] For the case at hand, let us locate the D2 brane at [MATH] and integrate [MATH] over an extension [MATH] . This yields |
[EQUATION] At the same time we have [EQUATION] so that in the combination entering the quantisation condition eq. ( 57 ) all dependence on the deformation parameter [MATH] drops out and one recovers the conventional result for the WZW model; there are, in addition to the D0-branes, stabilised D2’s located at |
[EQUATION] Let us now evaluate the dynamics of fluctuations of this D-brane. For that we need to consider dependence on the target space time coordinate; i.e. we need to introduce an extra time-like dimension to the target space. We will choose a synchronous gauge and let the coordinates of the worldvolume of the D2 be... |
we examine fluctuations of the transverse scalar and worldvolume gauge field in the DBI action, [EQUATION] where , [EQUATION] For direct comparison to Bachas:2000ik |
we reinstate explicit factors of [MATH] . It should be emphasised that the metric [MATH] needs to be pulled back to the worldvolume according to, |
[EQUATION] which induces derivatives for fluctuations in the transverse scalar. To perform this analysis it is helpful to pick a gauge for the antisymmetric field Sfetsos:2013wia |
[EQUATION] Then we make the ansatz for fluctuations, [EQUATION] Now the procedure is to expand the DBI action to quadratic order in the fluctuation and to extract classical equations of motion . The intermediate steps of this calculation are extremely unedifying and made algebraically complicated by the appearance of t... |
[EQUATION] in which [MATH] is the Laplacian on the [MATH] . This operator can be diagonalised in terms of an expansion in spherical harmonics. In the [MATH] sector (i.e. where [MATH] ) we find that the eigenvalues are, |
[EQUATION] These are all positive, hence stable fluctuations of positive mass. Since [MATH] carries no [MATH] -wave as a consequence of flux-quantisation and that the s-wave of [MATH] has a frequency squared of [MATH] – it is not a moduli. It is interesting to notice that the [MATH] -wave triplet (i.e. [MATH] ) acquire... |
4.2 The AdS deformation For the [MATH] algebra it can be shown that the metric-preserving automorphisms form the group [MATH] where [MATH] correspond to the usual rotations and boosts, while [MATH] transformations are obtained from an additional reflection and time-reversal. The metric-preserving outer automorphism gro... |
[EQUATION] The corresponding Lie algebra automorphism [MATH] defined by [MATH] is readily shown to be an involution and therefore defining with ( 38 ) consistent integrable boundary conditions. Note that we might as well represent the non-trivial [MATH] element by a conjugation with, |
[EQUATION] which is connected to [MATH] with an involutive inner automorphism that corresponds in [MATH] with a rotation over [MATH] . Moreover, it is important to emphasise that this inner automorphism is the only allowed group translation leading to integrable D-brane configurations in AdS . We are thus allowed to ma... |
First, let us parametrise the [MATH] in a general way as, [EQUATION] with [MATH] [MATH] , real and with the defining relation [MATH] , making apparent the AdS embedding. |
The regular conjugacy classes (obtained by taking [MATH] the identity) are distinguished by [MATH] constant and substituting this into the defining relation one finds the geometry of the corresponding D-branes (see also Bachas:2000fr |
Stanciu:1999nx Figueroa-OFarrill:2000gfl ). The geometry will depend on the values of [MATH] ; for [MATH] the conjugacy classes correspond to de Sitter D [MATH] -strings, for [MATH] they correspond to [MATH] instantons and for [MATH] to the future- and past light-cone. In the next, we will consider only the former de S... |
, the de Sitter WZW-branes are tachyonic and, as we will shortly touch upon, they will stay so when turning on the deformation. The twisted conjugacy classes ( 49 ) are obtained by conjugation with [MATH] |
[EQUATION] and are thus distinguished by [MATH] constant. The corresponding D-brane configurations are, for any [MATH] , two-dimensional Anti de Sitter D [MATH] -strings (see also Bachas:2000fr |
). Equivalently, when conjugating with [MATH] , twisted conjugacy classes will be distinguished by [MATH] constant corresponding again to AdS D [MATH] -strings. The choice of representation depends on how one wants to analyse these D-brane configurations together with the choice of parametrisation of the [MATH] group e... |
dS D [MATH] -strings A convenient parametrisation to describe the dS D [MATH] -strings is one where we replace in ( 70 ) the elements by, |
[EQUATION] with [MATH] [MATH] and [MATH] (although, they are not good global coordinates). Here the AdS is build up out of fixed [MATH] -slices of dS spacetimes parametrised by [MATH] and [MATH] corresponding to the dS D [MATH] -strings. Note that in this coordinate system we describe D [MATH] -strings that are static ... |
The target space geometry of the [MATH] -deformed AdS is, [EQUATION] with [MATH] . In these coordinates it is obvious that the deformation leaves the dS [MATH] -strings intact but changing the radius with a squashing factor [MATH] Comparing the integrable boundary conditions obtained from ( 38 ) with [MATH] with the bo... |
[EQUATION] See also Alekseev:1998mc Stanciu:2000fz Gawedzki:1999bq for a general formula in terms of the gluing matrix. The induced metric [MATH] on the D [MATH] -brane is obtained simply by enforcing [MATH] = constant in eq. ( 73 ). The DBI action now evaluates to (absorbing a factor [MATH] into the constant dilaton), |
[EQUATION] and, hence, is supercritical. One could equally perform this analysis in the global cylindrical coordinates along the lines of Bachas:2000fr |
to describe dynamical configurations of a circular D [MATH] -string and argue that these are unphysical trajectories. We expect however no new information to be gained. |
Anti de Sitter D [MATH] -strings At first sight a logical coordinate system to describe the AdS D [MATH] -strings seems to be the global AdS coordinates where the elements of ( 70 ) are replaced by, |
[EQUATION] with [MATH] . The twisted conjugacy class ( 71 ) (obtained by conjugation with [MATH] of eq. ( 68 )) lie along fixed [MATH] -slices which correspond to AdS spacetimes parametrised by [MATH] and [MATH] |
In this coordinate system the target space geometry of the [MATH] -deformed theory is, [EQUATION] with [MATH] . One can readily verify that in the WZW [MATH] limit the metric reduces to the obvious slicing of AdS by AdS spacetimes along [MATH] . However, when turning on the deformation this slicing becomes obscure and ... |
[EQUATION] The DBI action simply gives, [EQUATION] where we absorbed a factor of [MATH] into the constant [MATH] . The action is readily minimised when [MATH] where the D [MATH] -string is still of finite size. However, when making appropriate gauge choices for the induced antisymmetric field [MATH] and the [MATH] fiel... |
) given by, [EQUATION] with the integer [MATH] known to be the number of fundamental strings bound to the D [MATH] -string Witten:1995im |
. Similar to the [MATH] example we thus have again, due to a flux quantisation condition, additional locations where the D [MATH] -strings are stabilised, independent of the value of [MATH] . However, contrary to the [MATH] case, this does not descend from topological conditions of the boundary WZW (see also appendix )... |
One could instead also consider the twisted conjugacy class obtained from conjugation by [MATH] of eq. ( 69 ). The corresponding worldvolume is then characterised by [MATH] constant and is obtained from the previous fixed [MATH] -slices by a rotation over [MATH] in the spatial directions. The analysis of these worldvol... |
[EQUATION] Eliminating then the coordinate [MATH] by [MATH] with [MATH] a constant one can identify the gauge-invariant two-form [MATH] to be, |
[EQUATION] with the dilaton factor [MATH] in Poincaré coordinates. The Gauss constraint similarly quantises the constant [MATH] as, |
[EQUATION] where again a factor of [MATH] was absorbed in the constant [MATH] We conclude that from a classical point of view the AdS D [MATH] -strings in the [MATH] background are still physical. Moreover they are stablised in the same manner as in the WZW case, i.e. due to flux quantisation. Semi-classically it would... |
Lee:2001xe ) accompanied with a lifting of zero modes. However, we will not pursue this interesting point here. Relation to generalised T-dualities |
5.1 The non-Abelian T-dual limit In a scaling limit [MATH] the [MATH] -deformation recovers the non-Abelian T-dual of the principal chiral model Sfetsos:2013wia |
. To achieve this limit one expands the group element around the identity, [EQUATION] and takes [MATH] to find, [EQUATION] In this limit the [MATH] -deformed action ( ) becomes the non-Abelian T-dual with respect to the [MATH] action of the PCM ( ), |
[EQUATION] For the case of [MATH] in the parametrisation used in eq. ( 50 ) this limit is achieved by taking [MATH] with [EQUATION] |
The metric becomes [EQUATION] where [MATH] . The first point to note is that in the limit the two-sphere remains intact so one anticipates that the D-branes described previously are preserved. Performing the [MATH] limit procedure on the boundary conditions of eq. ( 52 ) yields, |
[EQUATION] To understand these conditions it is useful to work instead with the following combination of worldsheet derivatives [EQUATION] |
which can be used to construct a Lax connection for the dynamics of the non-Abelian T-dual theory eq. ( 86 ): [EQUATION] In terms of these the boundary conditions of eq. ( 89 ) take a remarkably simple form |
[EQUATION] This property of the boundary conditions holds in general, at least in the case where we set the extra automorphism [MATH] , and follows due to the limit |
[EQUATION] from eq. ( 38 ). Now for the punchline. The non-Abelian T-dual theory described by eq. ( 86 ) is classically equivalent to the principal chiral theory |
[EQUATION] The T-duality transformation rules are non-local in terms of the coordinates of the sigma-models but in terms of world sheet derivatives are a canonical transformation of the form |
[EQUATION] Thus we can immediately conclude that under non-Abelian T-duality the D2-brane described by the boundary condition ( 89 ) results in |
[EQUATION] i.e. a space-filling D3-brane. This analysis agrees exactly with the integrable boundary conditions that we would obtain for the principal chiral model by substituting the Lax ( 91 ) into the result ( 35 ) (note that this holds as the Lax ( 91 ) satisfies eq. ( 32 )) which leads to the boundary conditions, |
[EQUATION] Interestingly, these have the form of type N gluing conditions. In the context of the WZW, these type N gluings eq. ( 138 ) preserve conformal invariance but break the chiral-algebra (and for which a good understanding is still lacking to our knowledge). It would be interesting to relate these observations t... |
5.2 The pseudo-dual limit A second interesting limit described in Georgiou:2016iom is a scaling limit as [MATH] which results in the pseudo-dual Nappi:1979ig |
of the principal chiral model. The pseudo-dual theory is obtained by replacing the currents of the PCM with scalars according to |
[EQUATION] such that the conservation of the currents becomes a trivial consequence of the commutation of partial derivatives. The Bianchi identities and equations of motion written in terms of [MATH] can be obtained from a “dual” action. However this is not a true dualisation Nappi:1979ig |
– even at the classical level the two theories are not related by a canonical transformation and at the quantum level they have striking differences. The PCM is asymptotically free where as the pseudo-dual is not. The PCM is quantum integrable whereas the pseudo-dual displays particle production. Nonetheless it is intr... |
[EQUATION] Evidently in taking this limit one needs to relax the requirement that [MATH] required of the original construction of the [MATH] -model. |
Let us see the effect of this on the boundary conditions in the context of the [MATH] theory. The scaling is quite similar; we define [MATH] and |
[EQUATION] The limit [MATH] can then be taken in the boundary conditions of eq. ( 52 ) resulting simply in [EQUATION] In terms of the PCM variables we recover, as with the non-Abelian T limit, a D3-brane described by |
[EQUATION] 5.3 Poisson-Lie dual interpretation The [MATH] -deformed theory is closely connected to a class of integrable deformations of the principal chiral model known as [MATH] –deformation (also known as Yang-Baxter deformations) Klimcik:2008eq |
Delduc:2013fga . To establish the relation between the [MATH] and [MATH] theories one first performs an analytic continuation of the coordinates parameterising the [MATH] -theory and also of the deformation parameter itself. Performing this analytic continuation in the [MATH] -deformed action of ( ) results in a new (r... |
Sfetsos:2015nya Klimcik:2015gba Hoare:2017ukq . Our goal here is to track this connection through with the boundary conditions considered here. To make this rather technical procedure accessible we first introduce the rudiments of Poisson-Lie technology. |
Poisson-Lie T-duality Klimcik:1995ux is a generalised notion of T-duality between a pair of [MATH] -models on group manifolds [MATH] and [MATH] (with corresponding algebras [MATH] and [MATH] ) that do not enjoy isometries but instead posses a set of currents that are non-commutatively conserved with respect to the dual... |
. The two Poisson-Lie dual sigma-models defined in this way are of the form, [EQUATION] in which [MATH] [MATH] ) are pullbacks of left-invariant one-forms for [MATH] [MATH] ), [MATH] is a constant matrix of freely chosen moduli, and [MATH] [MATH] ) is a matrix formed by the combination of the adjoint action of [MATH] [... |
[EQUATION] where [MATH] and [MATH] resp. are the generators of [MATH] and [MATH] resp. The overall tension of the sigma-models has been introduced for later convenience. What will be useful in our consideration is that the two PL models are canonically equivalent Sfetsos:1996xj |
Sfetsos:1997pi with the canonical transformation defined by, [EQUATION] in which, if we let [MATH] be local coordinates on [MATH] , we define the momentum [MATH] |
In the context of the [MATH] [MATH] connection the relevant Drinfeld double is [MATH] and [MATH] is identified with a Borel sub-algebra coming from the Iwasawa decomposition [MATH] . The action [MATH] is defined on the group manifold [MATH] and is the one obtained by analytic continuation of the [MATH] –deformation. Th... |
[EQUATION] in which [MATH] are right-invariant one forms (pulled back) and [MATH] solves the modified classical Yang-Baxter equation. For the isotropic single parameter deformations considered here [MATH] . This is the integrable [MATH] -deformation Klimcik:2008eq |
Delduc:2013fga For didactic purpose we consider the case of the [MATH] [MATH] -deformation. In the parametrisation used in eq. ( 50 ) this analytic continuation amounts to a mapping between coordinates [MATH] and parameters [MATH] given by, |
[EQUATION] In this case the [MATH] and [MATH] (Bianchi II) and we will work with group elements parametrised by, [EQUATION] Applying the analytic continuation eq. ( 107 ) to the boundary conditions eq. ( 52 ) yields a result that is real (as required to be a consistent boundary condition) and rather elegant when writte... |
[EQUATION] Notice that in these conditions all the complicated dependence on the deformation parameter [MATH] (notwithstanding the factors of [MATH] ) is subsumed into the momenta [MATH] . In this form we can now immediately apply the canonical transformation of eq. ( 105 ) to deduce the corresponding boundary conditio... |
We are now in a position to present the boundary conditions of the [MATH] -deformed principal chiral model obtained in this fashion. In terms of the coordinates themselves the boundary condition takes a rather simple form, |
[EQUATION] For reference the geometry corresponding to the [MATH] -deformed theory reads [EQUATION] Since all the coordinates enjoy (generalised) Neumann boundary condition we are describing here a space-filling brane supported by a worldvolume two-form [MATH] . Making use of the above metric we can readily extract thi... |
[EQUATION] showing that [MATH] It is also illuminating to express the results in terms of right invariant forms , [EQUATION] such that the boundary conditions take the conventional form of a gluing, |
[EQUATION] with [EQUATION] It is easily verified that [MATH] so defined is an algebra automorphism. It is worth emphasising that here the gluing between currents after the generalised duality is again with an overall plus sign (it is of the form of a WZW N-type boundary condition eq. ( 138 )) whereas in the original [M... |
To close the circle we can again relate these boundary conditions to the general integrable boundary condition construction. First we recall that the Lax for the [MATH] -deformed PCM eq. ( 106 ) is given by Klimcik:2008eq |
Delduc:2013fga [EQUATION] Using this we can readily see that boundary condition above is obtained from [EQUATION] and hence is of the form of the integrable boundary condition that one would obtain from eq. ( 31 ) in which the extra automorphism [MATH] and when would choose the freedom to change the deformation paramet... |
[EQUATION] which also gives an integrable condition; this is just [MATH] Conclusions We have seen that integrable boundary conditions of the [MATH] -deformed theory can be obtained by demanding that the monodromy matrix of the Lax connection generates conserved charges even in the presence of a boundary. Rather elegant... |
Armed with the integrable D-branes of the [MATH] -model we were then able to show their connection to D-branes in the PCM and its [MATH] -deformation. First we could track the D-brane boundary condition through to the non-Abelian T-dual point ( [MATH] ) and dualise them to an N-type boundary condition of the PCM, which... |
Let us comment on a few interesting open problems triggered by this study. The concerns and analysis of this paper have been predominantly classical. An important next direction is to make more precise the quantum description corresponding to the boundary conditions considered here. Assuming no Goldschmidt-Witten anoma... |
. A quantum integrable boundary should supplement this bulk S-matrix with a boundary ‘K-matrix’ that obeys a boundary version of the Yang-Baxter equation Cherednik:1985vs |
Sklyanin:1988yz . It will be interesting, and the subject of further investigation, to establish the boundary K-matrix corresponding to integrable boundary conditions found within. Developing this line further, the quantum inverse scattering construction shows how the [MATH] -theory can be quantised on a lattice as a s... |
. It is appealing to establish a match between integrable boundary conditions of such spin chains (studied e.g. in deVega:1993xi |
) and the boundary conditions of the continuum theory we constructed here. Here we have considered just bosonic [MATH] -theory on a group manifold. These [MATH] -deformations have an analogue in the context of symmetric spaces Hollowood:2014rla |
(i.e. deformations of gauged WZW models) which will be of interest to study, with the anticipation that the geometric description of D-branes of Maldacena:2001ky |
Gawedzki:2001ye Fredenhagen:2001kw Stanciu:1997sk persists in the deformed theory. Going further one can consider [MATH] -deformations of theories based on supercosets with applications to the [MATH] superstring Hollowood:2014qma |
Appadu:2015nfa . Here the deformation is expected to be truly marginal and conjectured to correspond to a root-of-unity deformation of the holographic dual gauge theory. The study of the integrable D-branes in this arena also seems profitable. |
One way to introduce fermionic degrees of freedom is by considering supergroups or supercosets as target manifolds as outlined in the previous paragraph. Another way is through the supersymmetrization of the deformed [MATH] -model thereby introducing worldsheet fermions. We expect that the results obtained in this pape... |
. As far as we know, the question whether [MATH] -deformed theories allow for extended supersymmetry, even in the absence of boundaries, has not been addressed yet and forms an interesting open question. |
Acknowledgments DCT is supported by a Royal Society University Research Fellowship Generalised Dualities in String Theory and Holography URF 150185 and in part by STFC grant ST/P00055X/1. This work is supported in part by the “FWO-Vlaanderen” through the project G020714N and one “aspirant” fellowship (SD), and by the V... |
Appendix A General sigma models with boundaries To establish our sigma model conventions we briefly review the necessary basics of bosonic open strings in general curved backgrounds (see for instance Schomerus:2002dc |
). We discard the dilaton in this brief discussion and adapt throughout this paper the open string picture. The string sigma model is a theory of maps [MATH] from the worldsheet [MATH] parametrised by [MATH] to a target space manifold [MATH] parametrised by [MATH] with [MATH] . Considering open strings, the worldsheet ... |
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