brst symmetry in cohomological hamiltonian mechanics - Institute for ...

These ghost sectors define scalar distributions, with commuting coefficients
ρ 0 and ρ 12 .
The equations for the ”vector” distribution, ⃗ρ ≡ (ρ 1 , ρ 2 ), can be rewritten
in the following form:
or, taking β > 0,
∂ 1 ρ 2 − ∂ 2 ρ 1 = 0, 2β(h 1 ρ 2 − h 2 ρ 1 ) = 0,
h 2
h 1
∂ 1 ρ 1 − ∂ 2 ρ 1 = −ρ 1 ∂ 1
h 2
h 1
, ρ 2 = h 2
h 1
ρ 1 .
The characteristic equations for the non-homogeneous first-order partial differential
equation above are
da 1
ds = h 2
h 1
,
da 2
ds = −1, dρ 1
ds = −ρ 1∂ 1
h 2
h 1
,
from which one can easily find the first and the second integrals
∫
U 1 =
(h 1 da 1 + h 2 da 2 ), U 2 = h 2
h 1
ρ 1 .
The general solution then is of the form Φ(U 1 , U 2 ) = 0, that is, we can write
ρ 1 = h 1
h 2
f(U 1 ),
and, accordingly, ρ 2 = f(U 1 ), where f is an arbitrary function. We see that
the solutions for the one-ghost (odd-ghost) sector, ρ 1 and ρ 2 , characterizing
the vector distribution ⃗ρ, do not depend on the parameter β > 0. This
remarkable property might go beyond the two-dimensional case.
References
[1] E.Witten, Comm. Math. Phys. 117,353 (1988) 353.
[2] E.Witten, Comm. Math. Phys. 118,411 (1988) 411.
[3] D.Birmingham, M.Blau, M.Rakowski, Phys. Rep. 209,129 (1991) 129.
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