Transcript Cubic selfinteraction for higher spin gauge fields

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, [arXiv: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) 410415, [arXiv:1009.1054 [hep-th]].
3. K. Mkrtchyan, “On generating functions of Higher
Spin cubic interactions,” to apear in Physics of
Atomic Nuclei, arXiv:1101.5643 [hep-th].
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Free Higher Spin Fields
F   A    (A)
s=1

  
F   h   (h)   (h)    h
s=2
…
Equation of motion for Higher Spin gauge fields

F1...s  h1...s  [1 (h)2 ...s  ...]  [12 h3 ...s  ...]
Free Lagrangian for Higher Spin gauge fields
1 1 ... s
s( s  1) 1 ... s 2

L2 ( h )   h
F 1 ... s 
h
F 1 ... s 2
2
8
(s)
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Formalism
The most elegant and convenient way of handling symmetric tensors
i
a
is by contracting them with the s’th tensorial power of a vector
s
h ( z a)   ( a )h12  s ( z )
(s)
i
i
i 1
Tr  h ( z a )  Trh
(s)
(s)
( s  2)
1 s i a
a 

2   a  i
( s) i 1
1
(s)
( z a ) 
h
( z a )
a
s( s  1)
Grad  h( s ) ( z a)  Gradh( s1) ( z a)  (a)h( s ) ( z a)
Div  h ( z a )  Divh
(s)
( s 1)
1
( z a )  ( a )h( s ) ( z a )
s
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Fronsdal fields , Equation and Lagrangian
2
a
( s 1)

( z a)  0
a
h ( z a)  0,
(s)
Fronsdal constraints
Fronsdal Equation
de Donder operator
( s1)
D
( z a)  Dah ( z a)  0
( s)
 h ( z a)  s(a)
(s)
( s 1)
( z a)
de Donder gauge
Gauge transformation
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Cubic interactions of Higher Spin fields
Power Expansion of Lagrangian
and Gauge transformation
Gauge Symmetry
      ...
(2)
(3)
L  L  L  ...
(0)
(
(1)
 L = 0,
Noether Equation
(0)
i

(1)
i
 ...)(L  L  ...) = 0,
(2)
(3)
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Noether equation order by order
Free Lagrangian
 L  0,
(0)
(2)
Fronsdal, 1980
First nontrivial interaction – cubic Lagrangian
 L  L  0
(1)
(2)
(0)
(3)
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Noether equation in first nontrivial order
3

(1)
i
(2)
i
L (h
( si )
3
(ai ))   LI (h
(0)
i
i =1
where
( s1 )
(a1 ), h
( s2 )
(a2 ), h
( s3 )
(a3 )) = 0,
i =1
Li0 (h
( si )
1 (s )
(a)) =  h i (ai )*a F
i
2
( si )
(ai ) 
 h (ai ) = si (ai i )
(0) ( si )
i
1
(s )
Wa h i (ai )*a Wa F
i
i
8si ( si  1) i
( si 1)
(s)
(ai )
( zi ; ai )
The Noether equation in this order is equivalent to
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
(0)
i
LI (h
( s1 )
(a1 ), h
( s2 )
(a2 ), h
( s3 )
(a3 )) = O( F
( si )
(ai ))
i =1
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Gauge invariance
Unique Cubic Interaction
for arbitrary HS fields!!!
“Symmetry dictates the form of interaction.”
C. N. Yang
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Cubic Interaction Lagrangian leading term
where
With the number
of derivatives
ij  i  j
( 23 a )  ( 23 )   a
 
( c a ) 
c a 
Metsaev, 2006
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Generating Function for totally symmetric HS fields
With following gauge
transformations
Generating function for gauge parameters
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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!
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Off-shelling the On-shell expressions
Anticommuting
variables!
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Off-Shell Generating Function
Where
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Simple example: cubic selfinteraction of the
graviton in deDonder gauge
dD
1
L
1 



(h )  h    h h  h  h   h
2
(2)
Minimal selfinteractions for higher spin gauge fields is
a closed subset of all interactions in flat space.
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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 AdS cubic vertexes.
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Thank you for
your attention
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