Aspects Of Higher-Spin Conformal Field Theory
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This document provides an overview of higher-spin conformal field theory. It discusses the need for CFT in quantum field theory, the AdS/CFT correspondence, unfolded dynamics, Fronsdal equations, higher spin algebras, Vasiliev theory, 3D conformal equations, actions for higher spin theories, and Coleman-Mandula no-go theorem. The document outlines how interacting theories of massless higher spin particles can be constructed using unfolded dynamics and how Vasiliev theory in AdS space is conjectured to be dual to simple vector models in conformal field theory.
![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
• [𝑦 𝑎, 𝑦 𝑏] = 2i𝐶 𝑎𝑏](https://image.slidesharecdn.com/presentation2-171030071458/85/Aspects-Of-Higher-Spin-Conformal-Field-Theory-9-320.jpg)
Aspects Of Higher-Spin Conformal Field Theory
- 1. Aspects Of Higher-Spin Conformal Field Theory Abu Darda Department of Appied Chemistry Aligarh Muslim UNIVERSITY Aligarh U.p. 202002
- 2. 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
- 3. 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
- 4. 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.
- 5. 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 𝑅 𝜇𝜈 = 0
- 6. AdS/CFT duality • Any consistent quantum gravity theory have a CFT dual Quantum gravity on 𝐴𝑑𝑆 𝑑+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
- 7. 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
- 8. 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
- 9. 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 • [𝑦 𝑎, 𝑦 𝑏] = 2i𝐶 𝑎𝑏
- 10. 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 = 𝑒 𝑖𝜃0ℬ ∗ 𝜅𝑑𝒵 𝛼 𝑑𝒵 𝛼 + 𝑒−𝑖𝜃0ℬ ∗ 𝜅𝑑 𝒵 𝛼d 𝒵 𝛼 dℬ + 𝒜 ∗ ℬ − ℬ ∗ Π 𝒜 = 0 • Equation has vacuum solutions corresponding to AdS space time .
- 11. 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 𝑦 𝛼 : 𝐶𝑖(−𝑦|x)= −1 𝑝 𝑖 𝐶𝑖(𝑦|𝑥)
- 12. 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 𝐴 𝜇 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
- 13. 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)
- 14. Conclusions • The interacting theories of massless higher spin particles can constructed using the unfolded dynamics. • They involve infinite tower of fields of all spins including graviton s=2. They are in particular the models of gravity. • The proposed formulation is coordinate independent and applicable to any boundaries and bulk solution. • Vasilev theory in AdS was conjectured to be exactly dual to simple vector model of CFT,s
- 16. Thank you