Aspects of Higher Spin Theories Conformal Field Theories and Holography
Abstract
This dissertation consist of three parts. The first part of the thesis is devoted to the study of gravity and higher spin gauge theories in 2+1 dimensions. We construct cosmological solutions of higher spin gravity in 2+1 dimensional de Sitter space. We show that a consistent thermodynamics can be obtained for their horizons by demanding appropriate holonomy conditions. This is equivalent to demanding the integrability of the Euclidean boundary CFT partition function, and reduces to GibbonsHawking thermodynamics in the spin2 case. By using a prescription of Maldacena, we relate the thermodynamics of these solutions to those of higher spin black holes in AdS3. For the case of negative cosmological constant we show that interpreting the inverse AdS3 radius 1=l as a Grassmann variable results in a formal map from gravity in AdS3 to gravity in flat space. The underlying reason for this is the fact that ISO(2,1) is the InonuWigner contraction of SO(2,2). We show how this works for the ChernSimons actions, demonstrate how the general (Banados) solution in AdS3 maps to the general flat space solution, and how the Killing vectors, charges and the Virasoro algebra in the BrownHenneaux case map to the corresponding quantities in the BMS3 case. Our results straightforwardly generalize to the higher spin case: the flat space higher spin theories emerge automatically in this approach from their AdS counterparts. We also demonstrate the power of our approach by doing singularity resolution in the BMS gauge as an application. Finally, we construct a candidate for the most general chiral higher spin theory with AdS3 boundary conditions. In the ChernSimons language, the leftmoving solution has DrinfeldSokolov reduced form, but on the rightmoving solution all charges and chemical potentials are turned on. Altogether (for the spin3 case) these are 19 functions. Despite this, we show that the resulting metric has the form of the “most general” AdS3 boundary conditions discussed by Grumiller and Riegler. The asymptotic symmetry algebra is a product of a W3 algebra on the left and an affine sl(3)k current algebra on the right, as desired. The metric and higher spin fields depend on all the 19 functions.
The second part is devoted to the problem of Neumann boundary condition in Einstein’s gravity. The GibbonsHawkingYork (GHY) boundary term makes the Dirichlet problem for gravity well defined, but no such general term seems to be known for Neumann boundary conditions. In our work, we view Neumann boundary condition not as fixing the normal derivative of the metric (“velocity”) at the boundary, but as fixing the functional derivative of the action with respect to the boundary metric (“momentum”). This leads directly to a new boundary term for gravity: the trace of the extrinsic curvature with a specific dimensiondependent coefficient. In three dimensions this boundary term reduces to a “onehalf” GHY term noted in the literature previously, and we observe that our action translates precisely to the ChernSimons action with no extra boundary terms. In four dimensions the boundary term vanishes, giving a natural Neumann interpretation to the standard EinsteinHilbert action without boundary terms. We also argue that a natural boundary condition for gravity in asymptotically AdS spaces is to hold the renormalized boundary stress tensor density fixed, instead of the boundary metric. This leads to a welldefined variational problem, as well as new counterterms and a finite onshell action. We elaborate this in various (even and odd) dimensions in the language of holographic renormalization. Even though the form of the new renormalized action is distinct from the standard one, once the cutoff is taken to infinity, their values on classical solutions coincide when the trace anomaly vanishes. For AdS4, we compute the ADM form of this renormalized action and show in detail how the correct thermodynamics of KerrAdS black holes emerge. We comment on the possibility of a consistent quantization with our boundary conditions when the boundary is dynamical, and make a connection to the results of Compere and Marolf. The difference between our approach and microcanonicallike ensembles in standard AdS/CFT is emphasized.
In the third part of the dissertation, we use the recently developed CFT techniques of Rychkov and Tan to compute anomalous dimensions in the O(N) GrossNeveu model in d = 2 + dimensions. To do this, we extend the “cowpie contraction” algorithm of Basu and Krishnan to theories with fermions. Our results match perfectly with Feynman diagram computations.
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