The effective action for 2d-gravity in Weyl-invariant regularization is
extended to supersymmetric case. The super area-preserving invariance and
cocyclic properties under general supergravitational transformations of the
last action is shown.Comment: latex file, no figures, 5 pages. Some references are adde

Karakhanyan, D. R.

English

arXiv

The effective action for 2d-gravity in Weyl-invariant regularization is extended to supersymmetric case. The super area-preserving invariance and cocyclic properties under general supergravitational transformations of the last action is shown

Karakhanian, D R

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AREA-PRESERVING ALGEBRA STRUCTURE OF
TWO-DIMENSIONAL SUPERGRAVITY
D.R. Karakhanyan
Theoretical Physics Department,
Yerevan Physics Institute
Alikhanyan Br. st.2, Yerevan, 375036 Armenia
Abstract
The eective action for 2d-gravity in Weyl-invariant regularization is
extended to supersymmetric case. The super area-preserving invariance
and cocyclic properties under general supergravitational transformations
of the last action is shown.
E-MAIL:karakhian@vxc. erphy.am
1 Introduction.
The Weyl symmetry in the string theory is a fundamental principle which governs
the string dynamics [1].
The super Weyl transformations, which rstly had been considered by Howe
[2] are useful for investigation of the superspace constraints in the supergravity
theories [3].
Even being spoiled by quantum corrections the Weyl symmetry of classical
theory implies the cocyclic property of the eective action. It is so restrictive that
xes action up to numerical constants [4]. This form of the eective action causes
an independent interest as the theory of matter eld in (super) Weyl invariant
(super) gravitational background [5].
In previous work [6] we have studied transformation properties of the eective
action of 2d-gravity under Weyl rescaling of metric. The well known non-local
functional of eective action of 2d-gravity is
W [g] =
Z
R
1
2
R (1.1)
The Weyl-coboundary of 1
W [eg]−W [g]  SL(; g) (1.2)
coincides with the famous Liouville action
SL(; g) =
Z
d2x
p
g(
1
2
g@@ +R) (1.3)
The action 3 satises to cocyclic condition
SL(1 + 2; g)− SL(1; e
2g)− SL(2; g) = 0 (1.4)
which follows from 2.The relation 4 allow us to consider 3 as the anomaly arising
from mesure’s transformation in the bosonic string theory:
DegX
 = e−DSL(;g)DgX
 (1.5)
whereD is dimension of target space. The eective action of 2d-gravity, computed
in the Weyl invariant regularization is simply functional1, depending on the Weyl
invariant combination g=
p
g
~W [g] = W
"
g
p
g
#
=
Z
(R(g) − 2 log
p
g)
p
g
1
p
g2
(R(g)− 2 log
p
g)
p
g: (1.6)
and dier from 1 by the local counterterm:
~W [g] = W [g] + SL(; g)j=− logpg (1.7)
1
However the Weyl invariant action 7 is not scalar under general coordinate trans-
formation Diff2 now, it is invariant only under area- preserving SDiff2 sub-
group of Diff2. So under shifting 7 we trade Diff2 for Weyl  SDiff2 and
corresponding functional mesure is not dieomorphism invariant.
~DfgX
 = e−dS(f;g) ~DgX
: (1.8)
Here we used the notations
S(f; g)  SL(; g)j=log Fx ; 
F
x = det k
@F 
@x
k
and F  is inverse to f: F(f+; f−) = x
2 The super Liouville action
The extension of procedure, described above, to case of the 2d supergravity is
transparent. The super Liouville action consist from two parts: kinetic and
topological.
SS−L(;  ; e
a
; ) = S0 + S1 (2.1)
The kinetic part of 9 is well known Neveu-Schwarz string action:
S0 =
Z
d2xe(g@@ − i  γ
γ@ + 2γ
γ @ +
1
2
  γ
γ) (2.2)
and topological part is
S1 =
Z
d2x(!(e) +
1
2
γ3γ
)(@ + γγ
 ) (2.3)
The S0 and S1 are separately invariant under local supersymmetry transforma-
tions. Their sum satises to the cocyclic condition:
SS−L(1 + 2;  1; e
a
; ) = SS−L(2;  2; e
a
; ) +
+SS−L(1; e
−
2
4  1; e
2
2 ea; e
2
4 ( + γ 2) (2.4)
In agreement with general analisis of the cocycles of the Weyl group [4] we note
that expression 9 is polynomial on parameters of the super Weyl group ( and
 ) of degree no more that two (dimension of the space-time). The maximal order
terms on  and  i.e. S0 are invariant under super Weyl transformations:
 ! e−

4 
ea ! e

2 ea (2.5)
 ! e

2+ γ
2
while the low-order part S1 is not in complete agreement with general analisis of
the cocycles of the Weyl group [4].
The supereld formulation is rather simple. In terms of supereld ’ =  +
 +   +B expression 9 takes the form
SS−L =
Z
d2xd2E(
1
2
D’ D’+ <’) (2.6)
where < is supercurvature.
The expression 14 is super Weyl coboundary of the non-local eective action
W [E] =
Z
<
1
D D
<: (2.7)
It has general form
(Anomaly)−1(Anomaly) (2.8)
where  is (super)Weyl invariant dierential operator of order d (space-time
dimension). To obtain super Weyl invariant expression one needs to replace
zweibein and gravitino by super Weyl invariant combinations
ea !
eap
g
;  ! e
−1
4γγ (2.9)
To dene, which part of two-dimensional superdieomorphysm group is compat-
ible with super Weyl symmetry, it should to be nd local supersymmetric and
general coordinate transformations whose preserve gauge conditions:
det keak = 1; γ
 = 0 (2.10)
The answer is following: it is dieomorphysms with divergentless vector eld and
local supersymmetry with the parameter, which satises constraint
γ(@ −
1
2
!(e))(x) = 0 (2.11)
Let us notice that the commutator of such transformations gives rise to area-
preserving dieomorphysm. This is quite natural, because that transformations
form maximal subalgebra of two-dimensional superdieomorphisms. This sub-
group is dened by the symplectic form
Ω = pdq − AdA (2.12)
in phase superplane. The corresponding Poisson braket is
fA;Bg = @A@B + 
AB @
LA
@A
@RB
@B
(2.13)
If we choose super Weyl invariant regularization for functional mesure, then under
general superdieomorphysms it is transformed according to rule
~DFE = e
S(F;E) ~DE (2.14)
Here F is superdieomorphysm, acting on supervielbein E, constrained by unit
determinant and  is matter supereld.
3
3 Conclusion and outlook
As we seen, all regularities of ordinary 2d-gravity can be transformed to the case
of supergravity in natural way.
It seems real to do this in four and higher dimensions.
This approach can be applied to w-gravity. Because the 2d supergravity in
many aspects seems like to w-gravity [7] it is reasonable to try nd nonchiral
expression for the eective action of w3-gravity in flat and general ordinary- and
w3- gravitational background.
4 Acknowledgments
I would like to thank F. David, P. West, I. Kostov, R. Manvelyan and R.
Mkrtchyan for helpful discussions. Finally, I would like to thank the CEA Saclay
for hospitality while the part of this work was done.
This work is supported in part by the NATO linkage grant LG 9303057,
the grant 211-5291 YPI of the German Bundesministerium fu¨r Forshung und
Technologie, Federal Republic of Germany and the grant INTAS { 2858
References
[1] A.M.Polyakov, Phys.Lett.B 103 (1981) 893
A.M.Polyakov, Mod.Phys.Lett.A 2 (1987) 893
[2] P.S. Howe J.Phys. A: Math.Gen. 12 (1979) 393
[3] S. James Gates Jr., K.S. Stelle, P.C. West, Nucl. Phys. B 169 (1980) 347
[4] D.R. Karakhanyan, R.P. Manvelyan, R. L. Mkrtchyan, hep-th/9411068 (sub-
mitted to Nucl. Phys.)
[5] R. Jackiw, Another View on Massless Matter-Gravity Fields in Two-
Dimension, preprint MIT-CTP-2377
D. Candemi, R. Jakiv, B. Zwiebach, Physical States in Matter-Coupled Dila-
ton Gravity, preprint UCLA/95/TEP/16, MIT-CTP-2436
[6] D.R. Karakhanyan, R.P. Manvelyan and R.L. Mkrtchyan, Phys. Lett. B 329
(1994)185
[7] H. Ooguri, K. Schoutens, A. Sevrin, P. van Nieuwenhuizen Comm. Math.
Phys. 145 (1992) 515.
4


Area-preserving algebra structure of two-dimensional supergravity

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Area-Preserving Algebra Structure of Two-Dimensional Supergravity

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