An Integrable Deformation of the AdS5 x S5 Superstring (1409.1538v2)

Abstract: The S-matrix on the world-sheet theory of the string in AdS5 x S5 has previously been shown to admit a deformation where the symmetry algebra is replaced by the associated quantum group. The case where q is real has been identified as a particular deformation of the Green-Schwarz sigma model. An interpretation of the case with q a root of unity has, until now, been lacking. We show that the Green-Schwarz sigma model admits a discrete deformation which can be viewed as a rather simple deformation of the F/F_V gauged WZW model, where F=PSU(2,2|4). The deformation parameter q is then a k-th root of unity where k is the level. The deformed theory has the same equations-of-motion as the Green-Schwarz sigma model but has a different symplectic structure. We show that the resulting theory is integrable and has just the right amount of kappa-symmetries that appear as a remnant of the fermionic part of the original gauge symmetry. This points to the existence of a fully consistent deformed string background.

• The paper demonstrates a discrete integrable deformation of the Green-Schwarz sigma model using a q-deformation of the symmetry algebra.
• The introduced q-parameter, particularly at roots of unity, transforms the model into a gauged WZW framework with a modified symplectic structure.
• The work lays groundwork for exploring exact S-matrices and extending the gauge/gravity correspondence in deformed string theories.

An Integrable Deformation of the AdS$_5 \times S^5$ Superstring

The paper explores deformations of the world-sheet S-matrix for string theory in $AdS_5 \times S^5$, with an emphasis on maintaining integrability. The underlying mathematical framework revolves around a deformation where the elementary symmetry of the theory is altered via quantum group $U_q(\mathfrak{g})$ for the algebra $\mathfrak{g}=psu(2,2|4)$. This deformation introduces key modifications to the Green-Schwarz sigma model underpinning the superstring theory in this spacetime. The authors expand on introducing discrete transformations parameterized by $q$, which, under specific conditions, equate to a root of unity.

In contrast to the real $q$ case, where the model constitutes an $\eta$-deformation yielding a well-defined target space deformed background that maintains the original equations of motion yet alters its symplectic structure, the transformation to $q$ as a root of unity requires distinct handling and remains less explored.

Key Findings

• The Green-Schwarz model is demonstrated to accommodate a discrete deformation transforming it into a gauged Wess-Zumino-Witten (WZW) model with the deformation parameter $q$ interpreted as a $k$-th root of unity.
• Though achieving the same equations of motion as the conventional sigma model, the resultant theory reports a different symplectic structure.
• The proposed deformation maintains integrability while retaining sufficient kappa-symmetries, signalling a consistent string background.

Technical Implications

These deformations suggest a significant avenue to explore consistent string backgrounds that lie outside the conventional paradigms associated with a symmetric space. The resultant theory deviates in terms of its symplectic properties yet remains integrable, suggesting an appealing symmetry structure essential for the consistency of the string theory at stake. These transformations imply not only adjustments in the algebraic structure but also initiate a broader class of integrable theories with potential implications for the paper of solvable models within string theory.

Future Directions

The deformation approach offers fertile ground for exploring exact S-matrices of underlying field theories with deformed symmetry algebras. The intent to correlate these observations with broader aspects of the gauge/gravity correspondence will likely require scrutinizing quantum corrections and extending classical analyses to uncover potential exactness of these deformed theories as world-sheet theories. Furthermore, evaluating the potential cancellation of one-loop beta contributions could reinforce the integrability and symmetry attributes suggested by this paper.

Conclusion

This paper forges a critical path in extending the integrability of the AdS$_5 \times S^5$ superstring beyond typical constructs. By reconsidering deformation parameters and symplectic structures, it lays the groundwork for novel exploration within string theory frameworks, offering hints at immense potential for new system symmetries and consistency properties, thereby yielding intriguing questions and challenges for future investigation within theoretical physics ecosystems.