|XXXVII Workshop on Geometric Methods in Physics||1-7.07.2018|
|VII School on Geometry and Physics||25-29.06.2018|
Carlos Ignacio Perez Sanchez
The Schwinger-Dyson equations for complex tensor models
Colored tensor models form a random geometry framework—in spirit similar to random matrices, in reality substantially different from them—that has been used to model quantum gravity in arbitrary dimensions. We define the correlation functions of colored tensor models and interpret them in terms of piecewise linear bordisms. We show that they are classified not by integers, as usual, but by graphs that triangulate the cobordant spaces. We also find a Ward-Takahashi identity that is helpful in order to find the Schwinger-Dyson equations; different from the existing algebraic equations known under that very same name, ours are functional, integro-differential-like equations. While the first aim was to treat quartic theories, we obtained rather universal results (the only restriction being to have one melonic insertion in each of the interaction vertices). We also apply these techniques to a purely 3-spherical geometrical model, which, likely, could be the first solvable tensor model. We shortly elaborate a plan, in order to employ the techniques developed here to the SYK-like (Sachdev-Ye–Kitaev) Gurău-Witten model.
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