# The AdS/CFT Correspondence for Euclidean Quantum Fields

Hanno Gottschalk (Bergische Universität Wuppertal, Germany)

The AdS/CFT correspondence conjectured by Juan Maldacena stems from string theory, where a specific string theory on AdS_{5} is identified with a Yang Mills theory on its conformal boundary. Here we study a very much simplified version of the original conjecture that applies to constructive Euclidean scalar quantum field theory on AdS_{2}.

We study two possible prescriptions for AdS/CFT correspondence by means of functional integrals. The considerations are non-perturbative and reveal certain divergencies which turn out to be harmless, in the sense that reflection-positivity and conformal invariance are not destroyed. Using decoupling inequalities for Neumann boundary conditions on a tessellation of AdS_{2}, we are however able to show that the infra-red limit for the generating functional of the conformal boundary field becomes trivial. Possible renormalization strategies, like trading IR for UV singularities, are being discussed.

The talk is based on joint work with Horst Thaler.

Gottschalk, H., Thaler, H.: A triviality result in the AdS/CFT correspondence for Euclidean quantum fields with exponential interaction, arXiv:1210.0215.

Gottschalk, H., Thaler, H.: A comment on the infra-red problem in the AdS/CFT correspondence, Proc. Int. Conf. “Recent Developments in QFT”, Leipzig 2007, arXiv:0709.4486.

Gottschalk, H., Thaler, H.: AdS/CFT correspondence in the Euclidean context, arXiv:math-ph/0611006.