On Interactions in Higher Spin Gauge Field Theory

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 1. R. Manvelyan, K. Mkrtchyan and W. Rühl, “General trilinear interaction for arbitrary even higher spin gauge fields, ” Nucl. Phys. B 836 (2010) 204, [ar. Xiv: 1003. 2877 [hep-th]]. 2. R. Manvelyan, K. Mkrtchyan and W. Rühl, “A generating function for the cubic interactions of higher spin fields, ” Phys. Lett. B 696 (2011) 410 -415, [ar. Xiv: 1009. 1054 [hep-th]]. 3. K. Mkrtchyan, “On generating functions of Higher Spin cubic interactions, ” to apear in Physics of Atomic Nuclei, ar. Xiv: 1101. 5643 [hep-th]. 2

Free Higher Spin Fields s=1 s=2 … Equation of motion for Higher Spin gauge fields Free Lagrangian for Higher Spin gauge fields 3

Formalism The most elegant and convenient way of handling symmetric tensors is by contracting them with the s’th tensorial power of a vector 4

Fronsdal fields , Equation and Lagrangian Fronsdal constraints Fronsdal Equation de Donder operator de Donder gauge Gauge transformation 5

Cubic interactions of Higher Spin fields Power Expansion of Lagrangian and Gauge transformation Gauge Symmetry Noether Equation 6

Noether equation order by order Free Lagrangian Fronsdal, 1980 First nontrivial interaction – cubic Lagrangian 7

Noether equation in first nontrivial order where The Noether equation in this order is equivalent to 8

Gauge invariance Unique Cubic Interaction for arbitrary HS fields!!! “Symmetry dictates the form of interaction. ” C. N. Yang 9

Cubic Interaction Lagrangian leading term where With the number of derivatives Metsaev, 2006 10

Generating Function for totally symmetric HS fields With following gauge transformations Generating function for gauge parameters 11

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! 12

Off-shelling the On-shell expressions Anticommuting variables! 13

Off-Shell Generating Function Where 14

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Simple example: cubic selfinteraction of the graviton in de. Donder gauge Minimal selfinteractions for higher spin gauge fields is a closed subset of all interactions in flat space. 16

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 cases of interactions in flat spacetime which are well known for many years and coincide with the flat limits of known Ad. S cubic vertexes. 17

Thank you for your attention 18