The document discusses a model that combines dual graviton with a topological BF model. It introduces the dual graviton and topological BF model, then discusses their coupling using the Batalin-Vilkovisky anti-field BRST formalism. The coupled model satisfies the master equation for consistent interactions up to order two. The coupling provides consistent interactions when the dimension is D=k+3, where dual linearized gravity is equivalent to Pauli-Fierz theory.

1. School of Mathematics and Physics1
Ashkbiz Danehkar
School of Mathematics & Physics, Queen’s University Belfast,
Belfast BT7 1NN, United Kingdom
Faculty of Physics, University of Craiova, 200585 Craiova, Romania
Dual Graviton coupled with
a Topological BF Model
Talk at Max-Planck-Institut für Quantenoptik, München, Germany, September 23, 2009
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2. School of Mathematics and Physics
Layout
1. Introduction:
Dual Graviton, Topological Background Field (BF) Model
1. Anti-field-BRST Formalism
2. Master Equation and Gauge Coupling
3. Dual Graviton + Topological BF Model
–Consistent Interaction
1. Conclusion
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3. School of Mathematics and Physics3
1. Introduction
( )5 |1 1
|12 4tS d x F F F Fµνρ α µν
µνρ α µν= − +∫
Dual Graviton
Topological BF Model
Dual Formulation of Linearized Gravity
Topological Background Field of Linearized Gravity
Bekaert, Boulanger, & Henneaux 2003, PRD 67, 044010
Cioroianu & Sarau 2005, JHEP 0507, 056

4. School of Mathematics and Physics
{ } { }0
0
* * *
, , ,s
s
A
AC Cα α
α αφ φΦ = Φ =
* *
( , ) .l lr r
A A
A A
Y YX X
X Y
∂ ∂∂ ∂
= −
∂Φ ∂Φ ∂Φ ∂Φ
0α
φ
• BRST Symmetry appears as replacement for original gauge
transformations
Becchi, Rouet & Stora, 1974, PLB 52, 344
Tyutin 1975 hep-th/0812.0580
Anti-field Formalism:
Anti-bracket Formalism:
field, anti-field,s
Cα
0
*
αφ *
s
Cα
ghosts, anti-ghosts
2. Anti-field-BRST Formalism
4
Gomis, Paris, Smuel 1995, PR, 259, 1
Batalin & Vilkovisky 1983 PRD 28, 2567

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,s δ γ= +
( , ) 0,S S =
. (., )s S=
• BRST Couplings:
BRST differential:
s: BRST differential, δ: Koszul-Tate differential, γ: exterior longitudinal derivative
Master Equation:
Gauge Coupling: 2
1 2 ,S S gS g S= + + +L
1
2
2 1 1
3
3 1 2
( , ) 0,
: 2( , ) 0,
: 2( , ) ( , ) 0,
: ( , ) ( , ) 0,
S S
g S S
g S S S S
g S S S S
=
=
+ =
+ =
M M
g: coupling constant
2
1
2 1 1
3 1 2
0,
2 0,
2 ( , ) 0,
( , ) 0,
s
sS
sS S S
sS S S
=
=
+ =
+ =
M
(nilpotent)
3. Master Equation and Gauge Coupling
Henneaux 1998, CM, 219, 93

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4. Dual Graviton + Topological BF Model
( , ) 0S S =
( )5 |1 1
|12 4tS d x F F F Fµνρ α µν
µνρ α µν= − +∫
• Dual Linearized Gravity:
• BRST Couplings:
• Background Field (BF):
2
1 2S S gS g S= + + +L
L
0 t BFS S S= +
L
0 ,ghostS S S= +
Master Eq.: Gauge Couplings:
6
Bekaert, Boulanger, & Henneaux 2003, PRD 67, 044010
Cioroianu & Sarau 2005, JHEP 0507, 056

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{ } { }0
0
* * *
, , ,s
s
A
AC Cα α
α αφ φΦ = Φ =
• Consistent Interaction (BRST Couplings)
Anti-field Formalism:
0
|( , , , , , , )A H B K tα µ µ µν µνρ
µν µν ρφ ϕ φ= 0
* * * * * * * * |
( , , , , , , )A H B K tµν µν ρ
α µ µ µν µνρφ ϕ φ=
1
( , , , , , , )C C G C S Aα µν µνρ µνρλ
µ µν µνη η= 1
* * * * * * * *
( , , , , , , )C C G C S Aµ µν µν
α µν µνρ µνρλη η=
0, 2 , ,H Aµ µν µ µ
ε ε ν εδ ϕ δ ε δ ε= = ∂ = ∂ [ ]3 , ,Bµν µνρ
ε ε µν µ νδ ε δ φ ξ= − = ∂
| [ ] [ ]4 , 2 ,K tµνρ µνρλ
ε λ ε µν α µ ν α µ ν α α µνδ ε δ θ χ χ= ∂ = ∂ + ∂ − ∂
2
( , , , , )C C G C Sα µνρ µνρλ µνρλσ
µη= 2
* * * * * *
( , , , , )C C G C S µ
α µνρ µνρλ µνρλση=
3
( , )C Cα µνρλ µνρλσ
η= 3
* * *
( , )C Cα µνρλ µνρλση=
4
( )C Cα µνρλσ
= 4
* *
( )C Cα µνρλσ=
4. Dual Graviton + Topological BF Model
Bizdadea, Cioroianu, Danehkar, et al, 2009, EPJC 63, 491

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,s δ γ= +( , ) 0,S S = . (., )s S=
• BRST Couplings:
Master Equation:
1
2
2 1 1
3
3 1 2
( , ) 0,
: 2( , ) 0,
: 2( , ) ( , ) 0,
: ( , ) ( , ) 0,
S S
g S S
g S S S S
g S S S S
=
=
+ =
+ =
M M
2
1
2 1 1
3 1 2
0,
2 0,
2 ( , ) 0,
( , ) 0,
s
sS
sS S S
sS S S
=
=
+ =
+ =
M
L
0 t BFS S S= +L
0 ,ghostS S S= +
( )5 |1 1
|12 4tS d x F F F Fµνρ α µν
µνρ α µν= − +∫ Dual Graviton
Topological BF Model
4. Dual Graviton + Topological BF Model
8
Bizdadea, Cioroianu, Danehkar, et al, 2009, EPJC 63, 491

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5. Summary
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• Gauge structure of interacting theory:
deformed gauge algebra & higher-order structure
• Combined Dual graviton with topological BF model in D=5
• Lagrangian action includes only interaction of order one and two
2
1 2 ,S S gS g S= + +
• Coupling between dual graviton and topological BF provides
consistent interactions in dimension D=k+3 where dual formulation
of linearized gravity is dual to the Pauli-Fierz theory:
4
, det .S d gR g gµν= − =∫
0, 3kS k= ≥

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Special thanks to:
• Dr. Eugen M. Cioroianu (Craiova U, Romania)
• Prof. Constantin Bizdadea & Prof. Solange O. Saliu (Craiova U, Romania)
• EU Marie Curie Actions FP6 (MRTN-CT-2004-005104) via Craiova U.
Acknowledgments:

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Thank You!