Introduction to holography

Abstract

These are course notes for the 'Introduction to holography' Master level course at University of Cologne. The goal of the course is to give a pedogogical introduction to holography. Holography is a popular approach to quantum gravity, in which a theory of gravity can be described by a lower-dimensional boundary theory that itself has no gravity. The most concrete known example of a holographic model is the AdS/CFT correspondence, where the gravitational theory has a negative cosmological constant (the universe is asymptotically Anti-de Sitter) and the boundary theory is a conformal field theory. Symmetry plays a very important role in this duality. We therefore start the course with a review of Poincaré symmetry in quantum field theory, before moving on in the second chapter to conformal symmetry in conformally invariant quantum field theories or CFT's. Then we move to the basics of AdS physics in chapters 3 and 4, which will already reveal hints to the existence of a duality with CFT. After gathering the basic ingredients (CFT and AdS), in the second half of the course we are ready to formulate the AdS/CFT correspondence (chapter 5), including finite temperature AdS/CFT (chapter 6), which involves black holes and their thermodynamics in the gravitational theory (chapter 7). We end the course with an introduction to entanglement in AdS/CFT and the origin of statements that 'gravity emerges from entanglement' in holography.

Figures (25)

• Figure 1: Integration volume.
• Figure 2: Conformal transformation from plane to cylinder.
• Figure 3: Radial quantization on the $z$-plane: evolution in $\tau$ from one state $|\psi\rangle$ to another $|\psi'\rangle$ is generated by the scaling generator $D$. We can equivalently think of this as usual time evolution generated by $H_\tau = P^\tau$ in cylinder coordinates.
• Figure 4: Asymptotic state $|\psi_0 \rangle = \mathcal{O}(0) |0\rangle$ at $\tau = -\infty$.
• Figure 5: Topological charge conservation.