Aspects Of Higher-Spin Conformal Field Theory

Abu DardaDepartment of Appied ChemistryAligarh Muslim UNIVERSITY

Aligarh U.p. 202002

OUTLINE • Need of CFT.

• AdS/CFT correspondence .

• Unfolded dynamics

• Fronsdal equation

• Higher spin algebra

• Vasiliev theory

• 3d conformal equations

• Action of higher spin

• No-go theorem

• Conclusion

Need of CFT• In quantum field theory during renormalization we fix the

mass terms which gave rise to the problem of broken scale

symmetry, to get rid of this problem we use CFT.

• To calculate the exact correlation function in quantum field

theory.

• In the theory of world-sheet the vibrations of strings are

described by conformal field theory.

• To solve the puzzle of higher derivatives in interaction which

demand a dimensionful coupling constant

Historical Background of Higher-Spin CFT

• In 1939 Eugene Wigner gave the idea of higher-spins than one seen

in nature.

• Further interest in higher spin arose with the discovery of super

gravity in 1976 by Daniel Freedman , Sergio Ferrara .

• First step was taken in constructing the field equations and

Lagrangian by Fronsdal and by introducing the infinite tower of

higher spin.

• Complete nonlinear field equations for higher spin theories were

constructed by Vasiliev in 1992 and extended for any dimensions in

2003

• A very important achievement of the last years resulted from the

analysis of loop effects in higher spin theory.

HS AdS/CFT correspondence

• AdS/CFT or Maldecena duality is a conjectured relationship between

two theories . On One side are the anti-de sitter spaces which find their

use in models quantum gravity and on other side are the conformal field

theories which are quantum field theories that are invariant under

conformal transformations .

• Analysis of HS holography helps to uncover the origin of AdS/CFT

• AdS4 HS theory is dual to 3d vectorial conformal models

• AdS is the vacuum solution of Einstein free field equation

𝑅𝜇𝜈 = 0

AdS/CFT duality

• Any consistent quantum gravity theory have a CFT dual

Quantum gravity on 𝐴𝑑𝑆𝑑+1 ⟷ 𝐶𝐹𝑇𝑑

• Operators in the CFT are dual to fields in the AdS theory

String theory on Ad𝑆5 × 𝑆5 ≃ 𝒩 = 4 SU gauge theory in 4d

Unfolded Dynamics• Unfolded dynamics is a method for analyzing the dynamical

system.

• In this formalism we replace the time derivative by the de-

Rham derivative.

• The covariant first order unfolded differential equation :

• 𝐺Ω 𝑊 are some functions of supercoordinate 𝑊Ω

• Gauge transformation for the unfolded dynamics is

Fronsdal Equation

• This is the generalized equation for spin-s gauge field.

• This equation describes a massless bosons of arbitrary

spin

• This equation can be constructed by a generalized

christoffel symbol

HS algebra and star product• The general element 𝑃 𝑌 of infinite dimensional

associated algebra has the form:

• Higher spin algebra is the algebra of all even function f(y)

• We define the star product in the way the 𝑃1 ∗ 𝑃2 be a symbol of the operator 𝑃1 𝑃2

• The star product is associative :f∗(g∗h) = (f∗g)∗h

• [𝑦𝑎, 𝑦𝑏] = 2i𝐶𝑎𝑏

Vsiliev theory • Crucial feature of the theory: consistency with non-linear gauge

symmetries requires the infinite tower of higher spin fields.

• In the simpleset version of 4d theory ,the spectrum around the AdS

vacuum solution is

Spectrum S= 1, 2 3 ,. ……………., ∞ gauge fields

• 4d Vasiliev equations are

d𝒜 +𝒜 ∗𝒜 = 𝑒𝑖𝜃0ℬ ∗ 𝜅𝑑𝒵𝛼𝑑𝒵𝛼 + 𝑒−𝑖𝜃0ℬ ∗ 𝜅𝑑 𝒵 𝛼d 𝒵 𝛼

dℬ + 𝒜 ∗ ℬ − ℬ ∗ Π 𝒜 = 0

• Equation has vacuum solutions corresponding to AdS space time .

3d Conformal equations

• Rank one conformal massless equations

• Unfolded equation of the 3d conserve current equation

has a rank two form

• Bosons (Fermions) are even (odd) functions of 𝑦𝛼:

𝐶𝑖(−𝑦|x)= −1 𝑝𝑖𝐶𝑖(𝑦|𝑥)

Action of HS theories

• The free master action for the higher spin theories is

• This action is the extension of the free action which contain the

photon field 𝐴𝜇 and a massless half integer spin .

• This action is obtain by introducing the grassmann odd bosonic

ghost C and the anti field.

• The set of field and anti field now becomes

Colemen-Manduala no-go theorem

• This theorem states that no massless particle with spin s>2 has

been observed in nature.

• There is no string compactification that gives minkowski space

with massless particle spin greater than 2.

• Scattering occurs at almost all energies (except for some

isolated set of energies)

Conclusions• The interacting theories of massless higher spin particles can

constructed using the unfolded dynamics.

• They involve infinite tower of fields of all spins includinggraviton s=2. They are in particular the models of gravity.

• The proposed formulation is coordinate independent andapplicable to any boundaries and bulk solution.

• Vasilev theory in AdS was conjectured to be exactly dual tosimple vector model of CFT,s

Thank you