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116 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
where is given by (6). Therefore, another intermediate Lagrangian
emerges, which otherways would result from adding the degrees
of freedom
Now we integrate in the variable and after some manipulations,
the relevant part of the above Lagrangian necessary for the integration
in is
Inserting (18) into (15) and integrating out with respect to we finally
find
Therefore, the complete dual Lagrangian is
Where Of course, the condition of factorization
for the above Lagrangian still holds.
3. S–DUALITY IN
MACDOWELL–MANSOURI GAUGE
THEORY OF GRAVITY
Let us briefly review the MacDowell–Mansouri (MM) proposal [6].
The starting point for the construction of this theory is to consider
an SO(3,2) gauge theory with a Lie algebra–valued gauge potential
where the indices are space–time indices and the
indices A, B = 0,1, 2, 3,4. From the gauge potential we may introduce
the corresponding field strength
where are the structure constants of SO(3,2).
MM choose and as an action