Abstract
We present a systematic procedure to extract the dynamics of the low energy soft mode in SYK type models with a single energy scale J and emergent reparametrization symmetry in the IR. This is given in the framework of the perturbative scheme of arXiv:1608.07567 based on a specific (off-shell) breaking of conformal invariance in the UV, adjusted to yield the exact large-N saddle point. While this breaking term formally vanishes on-shell, it has a non-trivial effect on correlation functions and the effective action. In particular, it leads to the Schwarzian action with a specific coupling to bi-local matter. The method is applied to the evaluation of O(1) corrections to the correlation function of bi-locals. As a byproduct we confirm precise agreement with the explicit, symmetry breaking procedure. We provide a verification in the large q limit (Liouville theory), where the correlators can be calculated exactly at all length scales. In this case, our scheme illuminates how the enhanced O(J) and the subleading O(1) contributions originate from the Schwarzian dynamics of the soft mode and its interaction with h = 2 (bi-local) matter.
Highlights
We present a systematic procedure to extract the dynamics of the low energy soft mode in SYK type models with a single energy scale J and emergent reparametrization symmetry in the IR
The low energy dynamics of the soft modes arising from the spontaneous breaking of an approximate symmetry in a quantum field theory is an important problem which appears in many areas of physics
We concentrated on the development of a complete understanding of systematic near conformal perturbation expansion in SYK type models
Summary
In the large N dynamics of models of SYK type [1,2,3,4,5,6,7,8,9,10,11,12,13,14]1 (including tensor models [35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54]), a central role is played by the emergent Schwarzian mode which is dual to the gravitational mode in the dual theory. While the origin of this mode can be traced to (time) reparametrization symmetry of the critical theory, its dynamics and its couplings to the matter degrees of freedom emerges in the near-critical region. In [9] a framework was given for both evaluating the leading Schwarzian action, and developing a systematic perturbation expansion near criticality It relied on understanding the mechanism for (spontaneously) broken conformal symmetry in these nonlinear theories: which was introduced in [8] through an off-shell (source) mechanism. We prove that once the regularized source is chosen such that one gets the correct large N saddle, this diagrammatic procedure yields the correct 1/J expansion of the exact two point correlation function These results receive a verification as follows. For correlation functions we describe techniques for evaluation of contributing Feynman diagrams using various Schwarzian identities We apply this to the O(1) correction to the enhanced propagator and show agreement with a perturbative expansion around the exact saddle point solution of the theory.
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