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We can keep track of the creation and annihilation of limit sets at a bifurcation by “dressing” its curve with symbols at its sides. Table 3 shows our convention, in which triangles represent nodes, loops represent cycles and squares represent saddle points, and the filling indicates their repulsion. For example, a fil... |
This notation is well suited for representing codimension 2 bifurcations as junctions of several codimension 1 curves. The possible codimension 2 bifurcations are shown in Fig. and can occur generically in two-dimensional parameter spaces Kuznetsov |
A simple, yet powerful constraint we can apply to the set of bifurcation curves of a system is that whenever several such curves meet at one point in parameter space, all their dressings must match. Every symbol that comes “in” through one of the curves must go “out” through another, and on the same side of the bifurca... |
Examples In this section we illustrate how the tools we introduced can be used to help guide a reconstruction of bifurcation diagrams and phase portraits. |
As an example suppose that, from simulations or experiments, a system is known to have a low-dimensional behavior and that, at three different sets of parameters 1, 2 and 3, it has been observed to be respectively stationary, or to oscillate, or to have a coexistence of these two attractors. Moreover, large perturbatio... |
The first step is to make complete phase portraits compatible with these sets of attractors. For the first set of parameters, it should have a stable node, for the second, a stable limit cycle, and for the third both, so the diagrams must contain these symbols and eventually other unstable limit sets. Since large pertu... |
Let us study the first scenario and suppose that the system has an unstable cycle at the third set of parameters (diagram 3a in Fig. ). There will be regions of parameter space, thus, in which diagrams 1, 2 or 3a apply. This case is simple enough that every diagram can be reached from any of the other two by crossing a... |
One possibility, then, would be that every region is connected to the other two in parameter space, separated by those codimension 1 bifurcations. Indeed, the three bifurcation curves can meet in a generalized Hopf codimension 2 bifurcation, and the bifurcation diagram would look like the one seen in Fig. d. Or one of ... |
From the bifurcation diagram it is possible to predict distinct features of the dynamics of the system. From the diagram in Fig. d, for instance, the period of the limit cycle is expected to be finite everywhere in parameter space, and its amplitude should become arbitrarily small near the Hopf bifurcation. There, a sm... |
Let us consider the second possibility, in which the third set of parameters corresponds to a diagram like 3b in Fig. . Now, diagrams 1 and 2 can be connected through a supercritical Hopf as before and 2 and 3b through a saddle-node, but in going from diagram 1 to 3b a stable cycle, an unstable node and a repulsion [MA... |
For example, the Wilson-Cowan oscillator describes the mean activities of two coupled populations of neurons, one excitatory and the other inhibitory. A simplified version is given by Hoppensteadt & Izhikevich |
[EQUATION] where [MATH] are the mean activities of the excitatory and inhibitory populations respectively, [MATH] are the couplings between both populations and [MATH] are the external inputs. [MATH] is a sigmoidal function, e.g. [MATH] . For instance, if we set [MATH] [MATH] [MATH] [MATH] , the resulting bifurcation d... |
Conclusions In this work we have treated the problem of constructing phase portraits and bifurcation diagrams of two-dimensional nonlinear systems with a diagrammatic approach. |
We introduced a class of diagrams to represent the qualitative features of phase portraits of structurally stable, globally attracting (or repelling) two-dimensional dynamical systems. The diagrams emphasize the robust, topological characteristics of the limit sets of the system, and explicitly discard its quantitative... |
Smooth transitions between diagrams give rise naturally to all codimension 1 bifurcations of planar systems (with the exception of heteroclinic connections, which are ignored in our description). We introduced the notion of repulsion of a limit set, an additive quantity that is conserved in all bifurcations, and can be... |
We also developed the representation of codimension 1 bifurcations in a two-dimensional parameter space, by adding dressing symbols to the curves. Apart from describing the type of bifurcation, the dressing makes explicit the orientation of a bifurcation curve, i.e. to which side of the curve are the involved limit set... |
Acknowledgements This work was supported by CONICET, ANCyT, UBA, and NIH through R01-DC-012859 and R01-DC-006876. Generating sequences of bifurcations |
We propose to find sequences of bifurcations that lead from a given phase portrait to another, in a way that is similar to finding contributions to the transition amplitude of a scattering process in Quantum Field Theory (QFT). Our elements will be the limit sets (the “particles” in QFT), the codimension 1 bifurcations... |
The idea is as follows. We can represent the limit sets of planar systems with different types of directed lines, as shown in Table A.1. The direction of the arrow indicates the sign of the repulsion, positive by a left arrow and negative by a right arrow. |
The limits sets can meet in codimension 1 bifurcations, that are represented by vertices, i.e. particular junctions of lines. These are shown in Table A.2. There, the horizontal axis represents the bifurcation parameter [MATH] , and the vertical axis the spatial distribution of the limit sets. The interpretation is str... |
A sequence of bifurcations with parameter [MATH] is represented by a FL diagram containing several vertices. The lines present at each value of [MATH] give the succession of limit sets at each stage. The possible sequences of bifurcations between two phase portraits are given by all FL diagrams whose external lines cor... |
We will illustrate the procedure with the transition between regions 1 and 3b in the example considered in Fig. . The first step is to identify the initial and final limit sets, which will be the external lines of the FL diagrams. In a diagram representing a transition from region 3 to 1, these would look as shown in F... |
The next step is to find, in an orderly manner, well-formed FL diagrams featuring these external lines. As argued above, it is reasonable to look at the diagrams with smaller number of vertices first. In this particular case, the initial and final lines cannot be linked by a single bifurcation, a minimum of two must be... |
Note that in case (a) in Fig. , the saddle-node and Hopf bifurcations involve different limit sets, so they may occur in either order. Thus, in the general case these bifurcation curves could intersect in parameter space, as shown in the right panel. Diagram (a) can be interpreted as a path in parameter space that goes... |
On the contrary, in (b) both bifurcations involve the same node, and the Hopf bifurcation must occur before the saddle-node. The path in parameter space would traverse region 4 again but, unlike case (a), the node that survives is now the one on branch [MATH] . Since these nodes represent different states of the system... |
Similarly, in (c) both bifurcations involve the same saddle point and the homoclinic must occur first. The final region is again [MATH] since the surviving node is the one from branch [MATH] |
The final step is to combine the information from the three diagrams of Fig. in a single bifurcation diagram. We see that region 3 should be adjacent to regions 2, 4 and 5. In turn, regions 2 and 4 limit with 1, and regions 4 and 5 with [MATH] . The resulting bifurcation diagram is shown in Fig. , where we have obtaine... |
# Source: arxiv 1806.10712 # Title: D-branes in $λ$-deformations # Sections: all # Downloaded: 2026-03-02T08:53:35.565242+00:00 D-branes in [MATH] -deformations |
Abstract We show that the geometric interpretation of D-branes in WZW models as twisted conjugacy classes persists in the [MATH] –deformed theory. We obtain such configurations by demanding that a monodromy matrix constructed from the Lax connection of the [MATH] –deformed theory continues to produce conserved charges ... |
Introduction The study of two-dimensional quantum field theories with boundaries has rich physical and mathematical significance. In the context of open string theory, the boundary conditions describe D-branes, an essential non-perturbative ingredient of string theory. More generally 2d conformal field theories with bo... |
Cardy:1989ir . Another interesting class of theories comes from integrable systems ; here one is lead to ask (both classically and quantum mechanically) what boundary conditions can be implemented such that the integrability of the system is preserved. The present paper will in some way consider both contexts; we aim t... |
Our primary motivation will be that of string theory; i.e. we are interested in understanding the D-branes admitted by a given curved closed string background. In general this is a very demanding problem since one would like to have a precise CFT formulation, see e.g. Schomerus:2002dc |
. A simple example is provided by the Wess-Zumino-Witten (WZW) model Witten:1983ar describing closed strings propagating in a group manifold supported by Neveu-Schwarz flux. Whilst BCFT can be employed here to give an algebraic description of the D-brane Kato:1996nu |
our interest will lie in the elegant geometric picture developed through a number of works Alekseev:1998mc Felder:1999ka Stanciu:1999id |
Figueroa-OFarrill:1999cmq . By identifying possible gluing conditions at the boundary it is determined that D-branes are described by (twisted) conjugacy classes. For example in the [MATH] WZW model one finds two D0-branes and a further [MATH] D2-branes that are blown up to wrap the conjugacy classes described by [MATH... |
When an explicit CFT description is unknown and one has just a non-linear sigma-model describing strings in a curved background, giving a precise description of D-branes is challenging. However in a special circumstance, namely when the sigma-model is integrable, progress can be made. One can seek boundary conditions t... |
for local charges and the construction of a non-local Yangian [MATH] Yangian structure starts with Luscher:1977uq ). In this work we will focus our attention on the boundary conditions that preserve some portion of the tower of non-local charges obtained from a monodromy matrix. We anticipate that the same boundary con... |
Sklyanin:1988yz and has been used in a variety of contexts including the identification of integrable boundary conditions for strings in bosonic sigma models Mann:2006rh |
, in Green-Schwarz sigma models Dekel:2011ja , for the [MATH] sigma model Corrigan:1996nt Aniceto:2017jor , the principal chiral model Delius:2001he |
Gombor:2018ppd , open spin chains (e.g. deVega:1993xi Arnaudon:2004sd although the literature is vast) and affine Toda field theories Bowcock:1995vp |
. For methods based on the conservation of local spin charges see e.g. Ghoshal:1993tm MacKay:2001bh MacKay:2004rz Moriconi:2001xz |
The present manuscript will seek to make a bridge between the two above ways of determining boundary conditions. The [MATH] -deformed WZW model introduced by Sfetsos in Sfetsos:2013wia |
provides an ideal arena to do this. At a classical level the [MATH] -deformation is an integrable 1+1 dimensional field theory that depends on a parameter [MATH] . At [MATH] the WZW model is recovered while in a scaling limit as [MATH] one find the non-Abelian T-dual to the principal chiral model. For generic values of... |
A further motivation for the present study comes from duality. To fully establish a duality one would like to have access to its action on both perturbative and non-perturbative degrees of freedom. Consider conventional Abelian target space duality in the absence of any NS two-form potential: here the interpretation is... |
or Poisson-Lie Klimcik:1995ux , such an understanding is less refined (although see Forste:1996hy Borlaf:1996na Klimcik:1995np Klimcik:1996hp |
Albertsson:2006zg Albertsson:2007it Hlavaty:2008jv and recent work in Fraser:2018atd Cordonier-Tello:2018zdw Borsato:2018idb ) one reason being that the geometries concerned are not flat making it harder to identify the appropriate boundary conditions. Here however we will have access to an elegant description of D-bra... |
Sfetsos:2015nya Klimcik:2015gba Hoare:2017ukq . By examining this analytic continuation on our D-brane configurations we will too gain information about the brane spectrum of the Poisson-Lie T-dual theory. We study this interplay of duality with D-brane configurations in the case of the [MATH] theory. Here we find that... |
The structure of the paper is as follows. We will briefly review in section the saliant features of integrable [MATH] -deformations of WZW models, in section we will explain the strategy used to derive integrable boundary conditions and then apply it directly to the [MATH] -deformed WZW. We illustrate this in section i... |
[MATH] -deformations First, we will briefly review the construction of Sfetsos:2013wia in order to set up the sigma model action from which the background fields can be read off and a Lax connection representing the equations of motion can be found. The isotropic [MATH] deformation from group manifolds is obtained by s... |
[EQUATION] and a WZW model on [MATH] for a different group element [MATH] as in eq. ( 130 ). Altogether, this action has a global [MATH] symmetry for the PCM and a [MATH] symmetry group for the WZW. One continues by gauging simultaneously the left symmetry action of the PCM and the diagonal action of the WZW acting as, |
[EQUATION] with [MATH] , using a common gauge field [MATH] transforming as, [EQUATION] The total model is made gauge invariant by replacing the derivatives in the PCM by a covariant derivative [MATH] (i.e. by minimal substitution), and by replacing the WZW model ( 130 ) with the [MATH] gauged WZW model, |
[EQUATION] Finally, we can fix the gauge to [MATH] to find, [EQUATION] where we introduced the useful operator, [EQUATION] given in terms of the adjoint action [MATH] For the isotropic [MATH] -model, which is the model we consider throughout this paper, we have, |
[EQUATION] The gauge fields are now auxiliary and can be integrated out. Varying the action [MATH] with respect to [MATH] we find the constraints, |
[EQUATION] Substituting these constraints into eq. ( ) gives, [EQUATION] in which Lie algebra indices out of position are raised and lowered with the metric [MATH] and we work in term of the Maurer-Cartan forms [MATH] and [MATH] . From the above [MATH] action it is straightforward to read off the target space data whic... |
[EQUATION] where we have used that [MATH] and in which [MATH] . Equivalently we can use the identity (and this will proof useful later), |
[EQUATION] to express the target space metric as, [EQUATION] with [MATH] the vielbein bringing us to the flat frame. In addition the Gaussian elimination of the gauge fields, when performed in a path integral, results in a non-constant dilaton profile, |
[EQUATION] in which constants are absorbed into [MATH] . Finally, one can derive the classical energy momentum tensor of the [MATH] -model to find, |
[EQUATION] There are two interesting limits at play here Sfetsos:2013wia ; for [MATH] we see from eq. ( ) that the fields [MATH] will freeze out and, hence, one will reproduce the well-understood WZW model (see appendix ), allowing consistency checks of analyses of the deformation. For small [MATH] the WZW is deformed ... |
To establish the classical integrability of the [MATH] model it is convenient to work with eq. ( ) but where we take the gauge fields on-shell eq. ( ) (although see Sfetsos:2013wia |
for the proof of integrability starting from eq. ( )). In this way any variation of the action with respect to [MATH] vanishes and calculations simplify. From eq. ( ) we then find the equation of motion for the group element [MATH] to be, |
[EQUATION] where we introduced the derivative [MATH] and the field strength [MATH] Using the on-shell expression for [MATH] , eq. ( ), this can be rewritten as, |
[EQUATION] Hence, effectively we have recast one second order equation for the field [MATH] , by the constraints, as two first order equations for [MATH] . It is now straightforward to show that the following Lax connection, |
[EQUATION] satisfying the flatness condition [MATH] is equivalent to the equations of motion ( 16 ) thereby ensuring its classical integrability Zakharov:1973pp |
(see also the next section). Integrable boundary conditions 3.1 General methodology This section closely follows Mann:2006rh Dekel:2011ja |
for obtaining open string boundary conditions that preserve integrability based on a method first introduced by Cherdnik and Sklyanin Cherednik:1985vs |
Sklyanin:1988yz in the context of two-dimensional integrable systems. We add here a slightly more general procedure applicable to integrable sigma models which has not been clearly spelled out yet in the literature (however, see Gombor:2018ppd |
for a recent usage hereof in the case of the PCM) but which can lead to distinct integrable D-brane configurations. Consider first a general [MATH] -dimensional field theory in a spacetime (or worldsheet) [MATH] parametrised by [MATH] on a periodic or infinite line with a global symmetry group [MATH] . The model is sai... |
[MATH] -valued Lax connection one-form [MATH] depending on a spectral parameter [MATH] Zakharov:1973pp [EQUATION] The Lax [MATH] is defined up to a local gauge transformation by a Lie group element [MATH] given by, |
[EQUATION] leaving the zero curvature condition ( 18 ) invariant. In this case, an infinite set of conserved charges can be obtained from the usual transport matrix [MATH] defined by, |
[EQUATION] where we included the explicit dependence on the (worldsheet) coordinates and the arrow specifies the ordering of the integral as per Babelon |
. The transport matrix satisfies the following useful properties, [EQUATION] and under the gauge transformation ( 19 ) it transforms as, |
[EQUATION] Using the flatness of the Lax [MATH] together with the above properties, one can now show that, [EQUATION] Therefore, under periodic boundary conditions [MATH] (for e.g. the closed string) we find that the trace of the monodromy matrix |
[MATH] is conserved [EQUATION] Different sets of conserved charges (local or non-local) can then be obtained from expanding the monodromy matrix or its gauge transformed form around suitable values of the spectral parameter leading typically to Yangian algebra’s or quantum groups for the non-local sets of charges, see ... |
For later convenience, we define here also a generalised transport matrix, [EQUATION] where [MATH] is a constant Lie algebra automorphism. This generalised transport matrix behaves under time derivation as, |
[EQUATION] such that [MATH] for all [MATH] and under gauge transformations as, [EQUATION] where the map [MATH] is defined as [MATH] for [MATH] small and [MATH] . Here we assumed the corresponding Lie group [MATH] to be connected to the identity (in this case [MATH] is a constant Lie group automorphism also). |
When the model is considered on a finite line, e.g. [MATH] in the context of sigma models describing open strings, the charges obtained from the above procedure are generically not conserved. Similarly to the loss of conservation of momentum along the spatial direction, integrability might be spoiled. However, with the... |
[EQUATION] where the allowed boundary conditions are encoded in conditions on the reflection matrices and the automorphism [MATH] As discussed in Mann:2006rh |
Dekel:2011ja the reflection matrices are taken to be constant in time and independent of the spectral parameter . The transport matrix [MATH] in the reflected region is constructed from the transformation [MATH] when [MATH] . We will assume in the following the reflected generalised transport matrix to have the form, |
[EQUATION] which will indeed be the case for the [MATH] model. In general this strongly depends on the specific form of the Lax connection [MATH] |
but the following procedure can be easily adapted to other cases. Similar to the bulk model, we impose integrability by requiring that the time derivative of the boundary monodromy matrix is given by a commutator, |
[EQUATION] for some matrix [MATH] , such that [MATH] is conserved for any [MATH] . Explicitly we find using the formulae ( 26 ) and ( 29 ) that, |
[EQUATION] discarding here the [MATH] -dependence and using the notation [MATH] . One can show that the integrability condition ( 33 ) sufficiently holds for [MATH] when we require the following boundary conditions on both the open string ends: |
[EQUATION] and similarly on [MATH] (but where in principal the reflection matrix [MATH] can be different allowing the open string to connect different D-branes). Substituting the Lax connection of the considered model in eq. ( 35 ) imposes integrable boundary conditions on the field variables, together with consistency... |
However this is not the end of the story: the above procedure leads to sufficient conditions for integrability of the boundary model but they are not necessary. We can cook up any exotic boundary monodromy matrix [MATH] ; as long as it satisfies [MATH] for some [MATH] an infinite set of conserved charges can be constru... |
[EQUATION] where we used eq. ( 30 ) and eq. ( 32 ) and we included in the transport matrix the possible dependence on [MATH] representing (multiple) deformation parameters. Moreover, in the reflected region there is the possibility that the spectral parameter and the deformation parameters change, meaning [MATH] and [M... |
3.2 Applied to [MATH] -deformations Having the Lax connection of the isotropic [MATH] -deformation on group manifolds at hand one can now derive the integrable boundary conditions corresponding to the boundary monodromy matrix [MATH] given in eq. ( 31 ). One can check that eq. ( 32 ) indeed holds for the Lax ( 17 ). Us... |
[EQUATION] where recall [MATH] . Plugging the above into the result ( 35 ) and requiring the reflection matrices to be [MATH] -independent leads to the following boundary conditions |
[EQUATION] where, for consistency, [MATH] should be a constant involutive Lie algebra automorphism. A further restriction comes from demanding the (classical) conformal boundary condition (see eq. ( 125 )) which requires the energy-momentum tensor to satisfy [MATH] . Using the form of the stress tensor of the [MATH] mo... |
Ishibashi:1988kg Cardy:1989ir Kato:1996nu Stanciu:1999id Alekseev:1998mc Felder:1999ka Figueroa-OFarrill:1999cmq . Summarising, [MATH] is a constant Lie algebra metric-preserving involutive automorphism: |
[EQUATION] The integrable boundary conditions thus obtained reduce in the [MATH] limit exactly to chiral-algebra preserving symmetric D-branes of the WZW model Ishibashi:1988kg |
Cardy:1989ir Kato:1996nu Stanciu:1999id Alekseev:1998mc Felder:1999ka (in the terminology of Stanciu:1999id the type D conditions ( 137 )). |
3.3 Interpretation as twisted conjugacy classes Like in the WZW case, we desire a geometrical interpretation of the (integrable) boundary conditions ( 38 ) as Dirichlet and Neumann conditions on [MATH] defining a D-brane [MATH] . In this regard it is important to realise that eq. ( 38 ) takes values in the tangent spac... |
Felder:1999ka Alekseev:1998mc First, we should split the tangent space [MATH] at [MATH] orthogonally to the D-brane [MATH] with respect to the [MATH] -deformed metric ( 12 ) (assuming the metric restricts non-degenerately to [MATH] ), |
[EQUATION] Important here is that the object [MATH] gluing left to right currents in eq. ( 38 ) is easily shown to preserve the deformed metric ( 12 ) at [MATH] provided that [MATH] preserves the inner product [MATH] . Indeed, writing the metric as |
[EQUATION] we have [MATH] Moreover, it is straightforward to show that the deformed metric at [MATH] is invariant under an adjoint action by [MATH] , i.e. [MATH] |
With these properties of the deformed metric at hand we can now exactly follow the procedure used in the WZW case of Stanciu:1999id |
Figueroa-OFarrill:1999cmq for finding the tangent space [MATH] to the D-brane when treating [MATH] as the gluing matrix. This leads to, |
[EQUATION] Using the transitivity property of left and right translations on group manifolds, together with [MATH] being an automorphism (and thus bijective), there exists for every [MATH] a Lie algebra element [MATH] such that, |
[EQUATION] Hence, [MATH] is equivalently given by, [EQUATION] Contrary to the WZW case, we have here the extra property on the Lie algebra automorphism [MATH] that it is involutive, [MATH] , and thus defines a symmetric space decomposition of the Lie algebra [MATH] |
[EQUATION] where [MATH] [MATH] and, [EQUATION] Hence, also [MATH] splits accordingly, [EQUATION] It is now possible to rescale [MATH] and [MATH] such that, |
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