Conformal fields from AdS flows - Construction of conformal field theories using renormalisation group flow equations in Anti-de-Sitter space
Final Report Abstract
The first result of the project is the system of renormalisation group flow equations for a conformal field theory (CFT) that arises as the boundary limit of a weakly coupled scalar quantum field theory (QFT) in Anti de Sitter (AdS) spacetime. In general, the flow equations describe the change of correlation functions with the energy scale, which would naively break the conformal symmetry and could lead to possible anomalies. However, in this context, they instead connect a family of CFTs, and thus allow for a novel and anomaly-free construction of the theory. Furthermore, I have shown that the perturbative expansion of this system in AdS, which corresponds to 1/N corrections in the CFT, exactly reproduces existing results at tree level and one loop order for the corresponding Witten diagrams. In the future, this system could be used to systematically compute corrections also at higher orders for CFTs of physical interest. The second result is a formula that describes the change of the coefficients in the operator product expansion (OPE) with a change of couplings in a general curved spacetime. This formula generalises existing results in flat space, and allows to recursively compute the OPE coefficients at each order in perturbation theory. The main advantage over other methods is that the formula is self-consistent, namely only OPE coefficients of lower orders are needed as input, with the free-theory coefficients determined by Wick’s theorem. I have used it to give a new very simple proof of the existence and associativity of the OPE in a general curved spacetime. In the future, it could be used to compute OPE coefficients for theories of interest, for example for the OPE of the stress tensor and the field strength tensor in quantum chromodynamics, which are important to determine properties of the quark-gluon plasma. The third result is the add-on package FIELDSX for the Mathematica computer algebra system (CAS), which allows the user to perform computations with fermions and gauge fields in generic curved spacetimes, building on and extending the well-known XACT suite for tensor algebra. A highlight is the decomposition of products of fermions into irreducible representations, which provides an alternative to Fierz identities and is not implemented in any other CAS. This is particularly useful for computations in supersymmetric QFTs and supergravity theories, where it provides a fully automatic way of checking the closure of the supersymmetry algebra or the invariance of the action. The package is already in use by other researchers, and I have used it for the fourth result: the removal of supersymmetry anomalies of supersymmetric matter QFTs with an anomalous R symmetry. Together with C. Imbimbo from Genoa and N. Risso from Padua, we verified that generally there exists a supersymmetric completion of the well-known chiral anomaly, and thus supersymmetry is potentially anomalous. However, in the case that this chiral anomaly pertains to the R symmetry of the theory, we showed that it is possible to deform the gauge and supersymmetry transformations of the theory in such a way that the anomaly is canceled. While this result was derived for theories with a single supersymmetry, our methods are quite general and can in the future also be applied to other theories. For various technical reasons, the originally planned methodology did not work and a workaround needed to be devised to at least reach some of the original goals. On the other hand, the obtained results are very reasonable.
Publications
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Deformations of supergravity and supersymmetry anomalies. Journal of High Energy Physics, 2021(12).
Fröb, Markus B.; Imbimbo, Camillo & Risso, Nicolò
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Recursive construction of the operator product expansion in curved space. Journal of High Energy Physics, 2021(2).
Fröb, Markus B.
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Constructing CFTs from AdS flows. Journal of High Energy Physics, 2022(9).
Fröb, Markus B.
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FieldsX – An extension package for the xAct tensor computer algebra suite to include fermions, gauge fields and BRST cohomology
M. B. Fröb
