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\(\eta \)-Deformed Neumann–Rosochatius System

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Spinning Strings and Correlation Functions in the AdS/CFT Correspondence

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Abstract

In this chapter we study the dynamics of another deformation of the \(AdS_3 \times S^3\) background that does not break the integrability of the string Lagrangian. This deformation can be classified among the Yang-Baxter sigma models. These kind of models were proposed as a way to construct integrable deformations of the PCM using classical R-matrices that solve the modified classical Yang-Baxter equation. We will study the dispersion relation of spinning string in the truncation of such deformation to \(AdS_3 \times S^3\). General solutions to this truncated background can be written in term of elliptic functions. We also analyse the reduction of the elliptic solutions for some limiting values of the deformation parameter.

-What makes you think that the theory will still be integrable?

-Unlimited optimism

M. Staudacher, replying to A. A. Migdal at the Itzykson Meeting 2007 [1]

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Notes

  1. 1.

    We have chosen to follow the normalization of [15] instead of the one used in [5].

  2. 2.

    The MCYBE written in [34] and the one written in [2] differ by a sign in the right-hand side. We have chosen the sign convention of the later.

  3. 3.

    Note that we can use the constraint \(r_1^2 +r_2^2 +r_3^2 =1\) to bring the first term in (4.2.2) to the form

    $$ \frac{(r_1 r'_2 -r'_1 r_2)^2}{(r_1^2 +r_2^2)[1+\varkappa ^2 (r_1^2 +r_2^2)r_2^2]}= \frac{r_1^{\prime 2} +r_2^{\prime 2} +r_3^{\prime 2}}{1+\varkappa ^2 (r_1^2 +r_2^2)r_2^2}- \frac{r_3^{\prime 2}}{(r_1^2 +r_2^2) [1+\varkappa ^2 (r_1^2 +r_2^2) r_2^2]} \ . $$

    Furthermore the term before the Lagrange multiplier is just a total derivative,

    $$ \frac{2\varkappa \omega _3 r_3 r'_3}{1+\varkappa ^2 (r_1^2+r_2^2)}= -\frac{\omega _3}{\varkappa } \left[ \ln (1+\varkappa ^2 (r_1^2+r_2^2)) \right] ' \ . $$
  4. 4.

    However this is not the only reduction that we can perform to obtain a consistent truncation from the \(S^{5}_{\eta }\) to \(S^{3}_{\eta }\). For instance, from the equation of motion for \(r_{1}\),

    $$\begin{aligned} \frac{r''_1}{r}&=\varkappa ^2 \frac{2(r_1^2 + r_2^2) r'_1 r_2 r'_2 +r_1 r_1^{\prime 2} r_2^2 +2 r'_1 r_2^3 r'_2 -r_1 r_2^2 xr^{\prime 2}_2}{r^2} \nonumber \\&- 4 \varkappa \omega _1 \frac{r_1 r_2 r'_2}{r^2}+\Lambda r_1+\frac{r_1 (\alpha _{1}^{\prime 2} - \omega _1^2)}{r} \left( 1-\varkappa ^2 \frac{r_1^2 r_2^2}{r} \right) \ , \end{aligned}$$

    with \(r=1+\varkappa ^2 r_2^2 (r_1^2+r_2^2)\), we conclude that the choice \(r_1=0\) provides indeed another possible truncation. When we set \(r_{1}=0\) the Lagrangian becomes

    $$ L=\frac{1}{2} \left[ \frac{r_3^{\prime 2}}{r_2^2 (1+\varkappa ^2 r_2^2)} +r_2^ 2 (\alpha _2^{\prime 2} -\omega _2^2) + \frac{r_3^2 (\alpha _3^{\prime 2} -\omega _3^2)}{1+\varkappa ^2 r_2^2} \right] +\frac{\Lambda }{2} (r_2^2+r_3^2-1) \ , $$

    which can be easily seen to be equivalent to the one for the \(r_{3}=0\) truncation.

  5. 5.

    The Uhlenbeck constants for the \((AdS_5 \times S^5)_\eta \) Neumann–Rosochatius system were constructed using the Lax representation in [36]. Some immediate algebra shows that those more general constants reduce to the one we present in here along the \(r_{3}=0\) truncation.

  6. 6.

    We should note that this solution is well defined not only for real values of the parameter \(\varkappa \), but also for purely imaginary values of this parameter (although an analytical continuation of the \(r_i\) coordinates may be needed for that). If we define \(\varkappa =i\hat{\varkappa }\) we have

    $$ r_1^2 (\sigma ) =\frac{\omega _1^2 -\zeta _4}{\omega _1^2 -\omega _2^2} + \frac{\zeta _{34}}{\omega _1^2 -\omega _2^2} \frac{ \zeta _{24} \text { sc}^2\left[ \pm \frac{\hat{\varkappa } \omega _2 \sqrt{\zeta _{14} \zeta _{23}} \sigma }{\omega _1^2 -\omega _2^2} , -\frac{\zeta _{12} \zeta _{34}}{\zeta _{14} \zeta _{23}} \right] }{ \zeta _{23} - \zeta _{24} \text { sc}^2 \left[ \pm \frac{\hat{\varkappa } \omega _2 \sqrt{\zeta _{14} \zeta _{23}} \sigma }{\omega _1^2 -\omega _2^2} , -\frac{\zeta _{12} \zeta _{34}}{\zeta _{14} \zeta _{23}} \right] } \ . $$
  7. 7.

    It is immediate to see that in the limit \(\varkappa \rightarrow i\) the solution reduces to \(r_1=0\), \(r_2=1\), together with either zero total angular momentum or zero winding \(m_2\) because of the Virasoro constraint (4.2.9).

  8. 8.

    There is an additional possible expansion, depending on the choice of signs of the winding numbers.

  9. 9.

    From an algebraic point of view, the \(\varkappa =i\) limit behaves in the same way as the limit of pure NS-NS flux in the analysis of the deformation by flux of the Neumann–Rosochatius system presented in the previous chapter.

  10. 10.

    A similar result was obtained for the pulsating string ansatz in [23].

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  1. Departamento de Física Teórica I, Universidad Complutense de Madrid, Madrid, Spain

    Juan Miguel Nieto

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Nieto, J.M. (2018). \(\eta \)-Deformed Neumann–Rosochatius System. In: Spinning Strings and Correlation Functions in the AdS/CFT Correspondence. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-96020-3_4

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