Since the early ages of Physics, Math has been identified and promoted to be the language of our Universe. Geometry, in particular, has been playing a fundamental role in the construction and understanding of modern theoretical physics [231]. Two beautiful examples are given by the Principle of least action and the Theory of General Relativity. Holography represents another wonderful example of geometrization. The phase space of the dual field theory and its renormalization group (RG) flow structure are beautifully encoded in the geometric properties of the dual gravitational spacetime.
This is even a more pressing question given the fact that our Universe is definitely not Anti-de Sitter. All the cosmological observations indicate that our Universe is expanding; we live in a spacetime with small but positive cosmological constant (even if String Theory does not like it).
2.
The horizon is defined as the hypersurface where the killing vector \(\partial t\) has zero norm.
3.
Some concrete examples are (I) \(z=2\)—onset of antiferromagnetism in clean “itinerant” fermion systems [234]; (II) \(z=3\)—onset of ferromagnetism in clean itinerant fermion systems [235–237]; \(z=2.6\)—CeCu\(_{6-x}\)Au\(_x\) at critical doping [238].
4.
In the rest of this section, we will use conformal invariant and scale invariant as a synonym. Be aware that this is a highly non-trivial statement which does not hold generically. For a review about this topic, see [243].
5.
A scalar clearly would not break Lorentz invariance.
6.
After you have derived the equations of motion as suggested in Exercise 13, you will realize that the equations of motion for the function N(u) are not dynamical and once the solution in terms of \(\beta (u), w(u)\) is obtained, the profile of N(u) follows automatically.
7.
Alternatively, you can impose the boundary conditions in the UV. For \(m^2<3\), everything follows directly. In the range \(m^2>3\), the flow you obtain depends on the type of deformation you impose around the UV Lifshitz fixed point. Try!
Author information
Authors and Affiliations
Physics Department, IFT Instituto de Fisica Teorica Madrid, Madrid, Spain
Baggioli, M. (2019). The Geometrization Process and Holographic RG Flows.
In: Applied Holography. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-35184-7_5
