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### On Interactions in Higher Spin Gauge Field Theory

Karapet Mkrtchyan

Supersymmetries and Quantum Symmetries July 18-23, 2011 Dubna

Based on work in collaboration with Ruben Manvelyan and Werner Rühl

Based On

- R. Manvelyan, K. Mkrtchyan and W. Rühl, “General trilinear interaction for arbitrary even higher spin gauge fields,” Nucl. Phys. B 836 (2010) 204, [arXiv:1003.2877 [hep-th]].
- R. Manvelyan, K. Mkrtchyan and W. Rühl, “A generating function for the cubic interactions of higher spin fields,” Phys.Lett.B696 (2011) 410-415,[arXiv:1009.1054 [hep-th]].
- K.Mkrtchyan, “On generating functions of Higher Spin cubic interactions,” to apear in Physics of Atomic Nuclei, arXiv:1101.5643 [hep-th].

Free Higher Spin Fields

s=1

s=2

…

Equation of motion for Higher Spin gauge fields

Free Lagrangian for Higher Spin gauge fields

Formalism

The most elegant and convenient way of handling symmetric tensors is by contracting them with the s’th tensorial power of a vector

Fronsdal fields , Equation and Lagrangian

Fronsdal constraints

Fronsdal Equation

de Donder operator

de Donder gauge

Gauge transformation

Cubic interactions of Higher Spin fields

Power Expansion of Lagrangian

and Gauge transformation

Gauge Symmetry

Noether Equation

Noether equation order by order

Free Lagrangian

Fronsdal, 1980

First nontrivial interaction – cubic Lagrangian

Gauge invariance

Unique Cubic Interaction for arbitrary HS fields!!!

“Symmetry dictates the form of interaction.” C. N. Yang

Generating Function for totally symmetric HS fields

With following gauge transformations

Generating function for gauge parameters

Generating Function for HS cubic interactions

Sagnotti-Taronna GF (On-Shell)

Where

With vertex operator

This result is derived from String Theory side and in complete agreement with results presented here, derived by pure field theory approach!

Off-shelling the On-shell expressions

Anticommuting variables!

Simple example: cubic selfinteraction of the graviton in deDonder gauge

Minimal selfinteractions for higher spin gauge fields is a closed subset of all interactions in flat space.

Conclusions

- Local, higher derivative cubic interactions for HS gauge fields in flat space-time are completely classified and explicitly derived in covariant form.
- All possible cases of cubic interactions (including selfinteractions) between different HS gauge fields in any dimensions are presented in one compact formula.
- These interactions between HS gauge fields are unique and include all lower spin casesof interactions in flat spacetime which are well known for many years and coincide with the flat limits of known AdS cubic vertexes.

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