Phd. defense by Lorenzo Menculini

Non-relativistic strings, Spin Matrix Theory and Holography

Even after the advent of holographic dualities, the interplay of gravity with quantum mechanics keeps many obscure corners. Non-relativistic limits of holography and strings can offer different viewpoints and open new windows on quantum gravity. 

This thesis work deals with Spin Matrix Theory (SMT) and non-relativistic strings, with a special focus on the connection between the two. Spin Matrix Theories are quantum-mechanical theories obtained as decoupling limits of the AdS/CFT holographic correspondence, zooming in on near-BPS sectors of N = 4 SYM while also taking the ’t Hooft coupling λ to zero. SMT limits were devised with the aim of simplifying the study of the non-planar regime, and they can also be understood as non-relativistic limits of the duality. On the other hand, on the bulk side of the correspondence, SMT is connected to non-relativistic strings. These make up a sector of string theory of their own, with intrinsically interesting properties.

Non-relativistic spacetimes are covariantly described by non-Riemannian geometries. Starting from the Polyakov action of relativistic string theory in NS-NS background fields, we discuss a null-reduction procedure to obtain actions for strings moving in a so-called Torsional Newton-Cartan (TNC) spacetime. This procedure is reminiscent of the ordinary T-duality in string theory, but differs in some important aspects. Particularly interesting is the Polyakov-type action for TNC strings, allowing the study of world-sheet symmetries.
Different kinds of non-relativistic string theories exist, obtained from different procedures. We can link TNC strings to other proposals for non-relativistic string theory, and in particular to one where strings move in a string Newton-Cartan (SNC) geometry, showing they are equivalent under one assumption. This relationship also helps clarifying some issues related to a field redefinition freedom found in SNC strings.

By taking a scaling limit of the TNC string action, involving both background and world-sheet fields as well as the string tension, one can make the world-sheet also non-relativistic, thereby finding a Galilean Conformal Algebra (GCA) of generators. The dilaton coupling also drops out in the limit, constraining allowed world-sheet geometries. This scaling limit is shown to implement SMT limits on strings in AdS5 × S5, hence the non-relativistic world-sheet string theories obtained in this way represent string duals of SMT quantum-mechanical theories. We analyze specific examples of these sigma-models, and find an important and general connection with spin chains.