Chaotic string dynamics in deformed $T^{1,1}$

Recently, Arutyunov, Bassi and Lacroix have shown that 2D non-linear sigma model with a deformed $T^{1,1}$ background is classically integrable [arXiv:2010.05573 [hep-th]]. This background includes a Kalb-Ramond two-form with a critical value. Then the sigma model has been conjectured to be non-integrable when the two-form is off critical. We confirm this conjecure by explicitly presenting classical chaos. With a winding string ansatz, the system is reduced to a dynamical system described by a set of ordinary differential equations. Then we find classical chaos, which indicates non-integrability, by numerically computing Poincar\'{e} sections and Lyapunov spectra for some initial conditions.


Introduction
The AdS/CFT correspondence [1][2][3] is one of the most significant topics in String Theory. A typical example is the duality between type IIB string theory on AdS 5 ×S 5 and the N = 4 SU (N ) super Yang-Mills theory in large N limit. A possible generalization, preserving the conformal symmetry of the boundary theory, is to replace the internal manifold S 5 by an Einstein manifold T 1,1 [4][5][6]. The metric of T 1,1 is given by (dψ + cos θ 1 dφ 1 + cos θ 2 dφ 2 ) 2 (1.1) (in units that the dimension of the curvature scale is unity). The AdS 5 × T 1,1 geometry has been elaborated in [7]. In particular, the classical dynamics of a string moving on T 1,1 is chaotic and hence non-integrable [8]. This non-integrability has been shown also in an analytic manner [9]. Furthermore, the chaotic dynamics has been shown in a near Penrose limit [10]. The coset construction of the T 1,1 metric based on the coset has been discussed in [11]. Yang-Baxter deformations [12][13][14][15][16][17] of T 1,1 are discussed in [11,18,19] (For a short summary, see [20]).
It should be remarked that when k = λ 2 , some interesting things happen. The classical string dynamics on R × T (λ 1 ,λ 2 ,λ) λ 2 , where R describes the time direction in target space, is integrable [21] in the sense of Lax pair. The special case with λ 2 1 = λ 2 2 = 3 2 λ 2 gives rise to the original T 1,1 metric (1.1) . Note that even for this original metric, the integrability is still preserved due to the existence of the B-field (1.3b) . Also, the case with λ 1 = λ 2 = λ has already been known as the Guadagnini-Martellini-Mintchev (GMM) model [22] and used to construct an NS-NS supergravity background with the flux k = λ 2 [23]. For recent progress on the RG flow in this model, see [24].
On the other hand, it is conjectured that the system would be non-integrable when k = λ 2 [21] 1 . Hence it should be significant to show this non-integrability by following some standard manners. In this letter, we will consider cases with k = λ 2 and show that the string sigma model on R × T (λ 1 ,λ 2 ,λ) k exhibits classical chaos in string motion. To reduce the sigma model to a dynamical system, we employ a winding string ansatz in which a string is wrapped around all of the isometry directions, i.e. φ 1 , φ 2 and ψ . Then the chaotic behaviors are explicitly shown by numerically computing Poincaré sections and Lyapunov spectra for some initial values.
This letter is organized as follows. In section 2, we introduce the classical string sigma model action on R × T (λ 1 ,λ 2 ,λ) k . Then the system is reduced by using a winding string ansatz. In section 3, Poincaré sections and Lyapunov spectra are computed numerically for some initial values and classical chaos appears when k = λ 2 . Section 4 is devoted to conclusion and discussion.
2 Reducing string sigma model on R × T (λ 1 ,λ 2 ,λ) k We shall consider a classical string on the background R × T (λ 1 ,λ 2 ,λ) k . As discussed in [21], the original isometries as T 1,1 are retained in the presence of B-field. In particular, one can easily see in (1.3) that constant shifts in (φ 1 , φ 2 , ψ)-directions are still symmetries.
Note that we suppose that the background R × T (λ 1 ,λ 2 ,λ) k should be obtained from a supergravity embedding of the background (1.3) , though it has not been done other than for special values of the parameters. So this is just analogy of R × S 3 in the case of AdS 5 ×S 5 , where R comes from AdS time.

String sigma model on
Let G µν and B µν (µ, ν = t, θ 1 , φ 1 , θ 2 , φ 2 , ψ) denote the background metric and B-field, h ab (a, b = τ, σ) the world-sheet metric, and X µ the target-space coordinates. The classical action of the string σ-model is given by where h ≡ det h ab and the antisymmetric symbol is defined as τ σ = −1 . The equation of motion for X µ is given by 3) The equation of motion for h ab imposes that the stress-energy tensor T ab on the world-sheet should vanish: Note here that the variation of h ab is irrelevant to the antisymmetric symbol. The vanishing stress-energy tensor leads to the Virasoro constraints which restrict the string embedding specified by X µ .
Thanks to the world-sheet diffeomorphism, one can take the conformal gauge h ab = e 2ω(τ,σ) η ab , where η ab = diag(−1, 1) . In this gauge, the action takes the form: The conjugate momentum is given by Then the Hamiltonian is given by While the B-field appears in (2.6), it cancels with the B-field from L in the above expression. As a result, the canonical stress-energy tensor made of the conjugate momentum gives the same result as (2.4) as it can be checked easily.
Once the gauge has been fixed, the Virasoro constraints can be expressed as follows. The (τ, τ )-component of the stress-energy tensor is precisely the Hamiltonian, i.e. T τ τ = H and hence the Virasoro constraints indicate that H = 0 (the Hamiltonian constraint) . We also find that T τ τ = T σσ because T a a = 0 . Substituting (2.6) into (2.4) in the conformal gauge, the other independent component T τ σ can be rewritten as (2.8) When k = λ 2 , this string sigma model has a classical Lax pair and is classically integrable [21]. Our purpose here is to see the anticipated non-integrability when k = λ 2 .

Reduction of the system with a winding string ansatz
In order to show non-integrability, it is sufficient to find out chaotic motions in a subsector. Hence it is convenient to consider the simplest subsector which contains chaotic motions. To reduce the system to a set of ordinary differential equations, let us suppose the following winding string ansatz: This ansatz describes a string wrapped on (φ 1 , φ 2 , ψ)-directions. Since the periodicity of the compact coordinates are φ i φ i + 2π (i = 1, 2) and ψ ψ + 4π , and σ σ + 2π , the winding numbers α i (i = 1, 2) and α should satisfy α i , α/2 ∈ Z .
By substituting the ansatz (2.9) into (2.6), the momentum components associated with the isometries (and hence conserved) are given by Under the ansatz (2.9) , the constraint (2.8) can be rewritten as From this expression, one finds that this constraint can be satisfied by simply taking p φ 1 = p φ 2 = p ψ = 0 . It is also helpful to introduce the energy E defined as E ≡ −p t . Then, from (2.10), we obtaiṅ (2.12d) The other components are given by The string motion is described by coupled ordinary differential equations for θ i . Substituting (2.9) into (2.2) and using (2.12), we obtain the evolution equations for θ i as Note that the two equations are decoupled when k = λ 2 and no chaos appears obviously.

The Hamiltonian becomes
where the potential V (θ 1 , θ 2 ) is defined as Recall that the Hamiltonian has to be zero, i.e. H = 0, due to the Virasoro constraints. Since the Hamiltonian is conserved, this constraint is always satisfied along time evolution once the initial data is prepared so as to satisfy H = 0 .
The presence of the background B-field restricts the domain of the string motion. When k = 0 , the string cannot reach the poles of S 2 (θ i = 0 and π), because V (θ 1 , θ 2 ) diverges there. Therefore, the string can move in the range 0 < θ i < π . When k = 0 , the divergent terms are absent in the potential, and the string can shrink and pass through the poles.

Chaotic string dynamics
In this section, we present classical chaos in the reduced system (2.14) by computing Poincaré sections and Lyapunov spectra for some values when k = λ 2 .
For numerical computations, we set the parameters as for simplicity. We have examined various values of k, but here we will show results for the cases that k = 0 , 0.5 and 10 . For other values of k as far as we have examined, classical chaos has been observed (except the special case with k = λ 2 ) . Poincaré sections are computed for θ 2 = π (k = 0) or θ 2 = π/2 (k = 0.5, 10) and p θ 2 ≥ 0 .

(3.2)
In each figure, the energy E is fixed and the initial conditions for the differential equations are changed. Different colors correspond to different initial conditions.
In addition, to measure the strength of chaos, we have examined the Lyapunov spectrum as well (For the detail of the computation, for example, see [25]) . In the system (2.14) , the Lyapunov spectrum contains four exponents and the largest one is the most significant for the growth of chaos.
As a remark on α , for α = 0 and λ 1 = λ 2 = λ, we could not find any chaos in string motion even for k = λ 2 . This would be simply because this case corresponds to an integrable subsector.
In the following, let us present Poincaré sections and Lyapunov spectra for k = 0 , 0.5 and 10 , respectively.
(iii) k = 10 case Figure 3 shows Poincaré sections for k = 10 . The potential minimum is (θ 1 , θ 2 ) ∼ (π/2, π/2) . When E = 5 , there are only KAM tori and no chaos is found [ Fig. 3 (a)] . When E = 40 , some of KAM tori collide with each other and chaos appears [ Fig. 3 (b)] . As the energy becomes much higher (E = 300) , there are only KAM tori and chaos disappears again [ Fig. 3 (c)] . Notice here that higher energy is necessary to see chaos in this case in comparison to the previous two cases. This is simply because the large value of k means a strong magnetic flux. The magnetic flux acts to force the particles to move in a circle and hence the KAM tori can survive even at relatively high energy. Figure 3 (d) is a Lyapunov spectrum with E = 40 . The initial values of θ 1 (t), θ 2 (t) , p θ 1 (t) , and p θ 2 (t) are taken as θ 1 (0) = θ 2 (0) = π/2 , p θ 1 (0) = 15 , and p θ 2 (0) = 37 . Then the maximum Lyapunov exponent is 2.0341 ± 0.0182 .

Conclusion and Discussion
In this letter, we have studied a string sigma model on R × T (λ 1 ,λ 2 ,λ) k by using a winding string ansatz. We have shown the presence of classical chaos for k = λ 2 by looking at Poincaré sections. We have also calculated the Lyapunov spectra and showed that the largest Lyapunov exponent is positive for chaotic string motions. Thus we conclude that the string sigma model is not classically integrable when k = λ 2 as conjectured in [21].
In our analysis, we have supposed a supergravity embedding of the background (1.3) . It is a significant future problem to find out this supergravity embedding for general values of the parameters. Notably, in [21], the background (1.3) has been derived as a certain limit of a more general background. It is a challenging problem to try to find out a supergravity embedding of it. It is also a nice practice to study non-integrable parameter region of it by following the standard manner as performed in this letter.