Quantum complexity in the AdS/CFT correspondence

• Applicants: Professorin Dr. Johanna Erdmenger; Dr. René Meyer
• Subject Area: Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 438623008

The AdS/CFT correspondence, often referred to as holography, conjectures a remarkable duality between gravity theories and quantum field theories (QFT). Recently, following the holographic entanglement entropy proposal of Ryu and Takayanagi (2006), new relations between gravity theories and quantum information have been established. In particular, with the aim of gaining new insight into the quantum nature of black holes, gravity realizations of quantum information concepts such as computational complexity were proposed by Susskind and collaborators. So far, however, the explicit relation between holographic complexity proposals and the complexity of quantum information remains an open question. Further progress requires a generalization of the information-theoretic definition to QFTs, i.e. to infinite-dimensional Hilbert spaces. Recently, progress in this direction was made for free QFTs. Here, as a promising path to understanding complexity for interacting QFTs, we will propose and analyze complexity definitions in two-dimensional conformal field theories (CFTs), both for CFTs with and without AdS dual. In particular, our project consists of the following three parts and their interrelation: 1) Within AdS/CFT, we will determine the field theory duals of non-minimal geodesics in three-dimensional asymptotically AdS spaces by means of Wilson lines. These duals are expected to be CFT two-point functions for excited states. Using kinematic space, a mathematical concept from integral geometry, we will provide afield-theory interpretation of the contribution of non-minimal geodesics to proposals for holographic complexity. 2) In the context of CFT, we plan to extend a recent complexity proposal of Caputa et al that involves a single representation of the Virasoro algebra to the more general case of Kac-Moody symmetry algebras. We will explicitly calculate complexity as given by this proposal for different reference and target states. Moreover, we plan to relate a central element of this proposal, the 1+1-dimensional Polyakov action, to AdS_3 gravity. 3) Starting from examples for CFTs with known lattice regularizations such as the Ising model, we will investigate which gate transformations, reference states and complexity measures are compatible with the continuum limit and thus with conformal symmetry. We will use these results to compute complexity for states in different representations of the Virasoro algebra, in generalization of part 2). Finally, we will combine the results of the three parts of the proposal in view of contributing to clarifying the relation between complexity proposals within CFT and holography. We expect our results to indicate further new avenues for studying aspects of the quantum nature of black holes.

• DFG Programme: Research Grants