VI International Conference “Models in Quantum Field Theory” (MQFT-2018) 27-31 August 2018

To the memory of Joseph Polchinski and Lev Lipatov Qualitative foundations of AdS/CFT correspondence

Sergey Afonin Saint Petersburg State University The Greatest Equation Ever? Physics World magazine:

(relates the five most important numbers, via the three basic operations, ) In XXI century?

Many theorists believe

Anti de-Sitter Conformal Field Theory

To be precise, it is a particular case of much more general concept of gauge/gravity dualities

Joseph Polchinski:

Motivation: Strong coupling problem

Possible solution: Dual theory

Strings: Closed and Open

Brane

At low energies: gravitons Quantum fields SCHEMATIC ORIGIN OF THE GAUGE/GRAVITY CORRESPONDENCE

OR

D-brane interaction via closed strings via open strings (at low energies) (= gravitation) (= gauge theory)

A remote analogy with dual description of hadron scattering via resonance exchange 1 3 1 3 1 3

OR 1+3 1+2 2 4 2 4 2 4 hadron interaction annihilation (= direct) channel (at low energies) exchange (= cross) channel Formulation

AdS/CFT correspondence (gauge/gravity duality = holographic duality) is a conjectured equivalence between a quantum gravity (in terms of string theory or M-theory) compactified on anti-de Sitter space (AdS) and a Conformal Field Theory (CFT) on AdS boundary

The most promoted example (Maldacena, 1997 - the most cited work in theoretical physics!):

5 SYM theory with SU(N) gauge Type IIB string theory on AdS5  S in the low-energy (i.e. supergravity)  group on AdS5 boundary (= 4D Minkowski) approximation in the limit

Here one has a one-to-one mapping of the global symmetries Isometries of S5  SO(6) R-symmetry of Super Yang-Mills theory

Isometries of AdS5  Conformal group SO(2,4) in 4D space 5 S : AdS5: - AdS - strong coupling How to understand the necessity of Essential ingredients: - supersymmetry all these ingredients on a qualitative - large number of fields (intuitive) level? IIB string:

with

Both theories share IIB string the symmetry:

contains Why AdS?

! A hand-waving motivation of gauge/gravity duality (without strings)

Is it possible to make the spin-2 graviton as a bound state of two spin-1 gauge bosons?

The Weinberg-Witten no-go theorem: It is impossible. Because if there is a massless spin-2 particle in the spectrum, then the matrix element of the energy-momentum tensor has impossible properties.

However, no-go theorems usually make some unstated assumptions (as they look too obvious) which may turn out to be a weak point (a classic example: the Coleman-Mandula theorem).

The t’Hooft holographic principle: Quantum gravity in any volume is naturally formulated in terms of degrees of freedom on its surface, one per Planck area.

The hidden assumption: The graviton bound state moves in the same spacetime as its gauge boson constituents. But it could move in one additional dimension! In some sense, similar things were already anticipated in QCD phenomenology – in the situations where the magnitude z of the separation of hadron constituents behaves like a spacetime coordinate because interactions are approximately local in z and the pair wavefunction satisfies a five-dimensional wave equation. The BFKL analysis of Regge scattering:

The color transparency:

So when we look at the gluon pair we may picture it as a graviton four of whose coordinates are the center of mass of the pair, and the fifth is the separation. Why strong coupling?

The gauge boson pair must behave like a graviton and not like a pair of gauge bosons.

Why supersymmetry? QFTs tend to become unstable at strong coupling, through the production of pairs whose negative potential energy exceeds their kinetic energy. In continuum theories this can happen at all scales, and the theory ceases to exist. SUSY protects against this: schematically the Hamiltonian is the sum of the squares of Hermitian supercharges, so the energy is bounded below.

Why a large number of fields?

We want the AdS scale R to be large compared to the Planck length Lp, so that we can use Einstein gravity. This means that we can fit a large black hole into the space, one with many Planckian pixels and so a large entropy, and so the field theory had better have a correspondingly large number of degrees of freedom. In exact AdS/CFT correspondence, the number of fields is a power of R/Lp. Strings vs. QCD-like theories*

The problem: soft behavior of amplitudes in high-energy scattering vs. hard behavior in QCD.

In the AdS5 space, the amplitudes of string scattering has the same behavior as in QCD! IN CONCLUSION

Source for major inspiration! (uncountable number of related models in the last 20 years)

But still to be proven… THE END