Abstract
We show that the $AdS_5 \times L^{a,b,c}$ solution in type IIB theory is non-integrable. To do so, we consider a string embedding and study its fluctuations which do not admit Liouville integrable solutions. We, also, perform a numerical analysis to study the time evolution of the string and compute the largest Lyapunov exponent. This analysis indicates that the string motion is chaotic. Finally, we consider the point-like limit of the string that corresponds to BPS mesons of the quiver theory.
Highlights
The gauge/string correspondence has evolved from the archetypical duality proposal [1,2,3] suggesting the equivalence of string theory in AdS5 × S5 and the four-dimensional N 1⁄4 4 super Yang-Mills theory to more elaborate constructions with a reduced amount of symmetry in an effort to probe toy models for field theories that appear in nature and gain intuition for the latter
To that end, involves the replacement of the five-dimensional sphere of the original AdS5 × S5 background geometry by a five-dimensional Sasaki-Einstein manifold,1 which we generically denote by M5, and we obtain a duality between type IIB string theory on the AdS5 × M5 background and a quiver gauge theory that lives on the boundary [5]
We can find relations for x1; x2; α; β in terms of the quantities a, b, c, d. They have been obtained in [45]; we find it convenient and useful to repeat the analysis here
Summary
The gauge/string correspondence has evolved from the archetypical duality proposal [1,2,3] suggesting the equivalence of string theory in AdS5 × S5 and the four-dimensional N 1⁄4 4 super Yang-Mills theory to more elaborate constructions with a reduced amount of symmetry in an effort to probe toy models for field theories that appear in nature and gain intuition for the latter. In a complementary approach toward the deeper understanding of gauge theories, an important role is played by integrability as its existence uncovers an affluent structure of conserved quantities. If the result of the Kovacic method yields no Liouville integrable solutions or no solutions for the NVE, we can declare the full theory as being nonintegrable. At this point we would like to stress that even if a background is characterized as being nonintegrable in all generality, this does not preclude the existence of integrable subsectors in the theory. The study of the NVE does not yield a solution, and we declare the quiver gauge theory to be generally nonintegrable. We consider the pointlike limit of the strings such that they are related to the BPS meson states of the field theory
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