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Teichmüller deformations of super Riemann surfaces
Volume 190, number 1,2
PHYSICS LETTERSB
21 May 1987
T E I C H M ~ L L E R D E F O R M A T I O N S OF SUPER R I E M A N N SURFACES Jeffrey M. RABIN Department of Mathematics and Enrico Fermi Institute, Universityof Chicago, Chicago, IL 60637, USA Received 20 February 1987
Exact solutions are obtained for the linearized super Beltrami equations which describe infinitesimal deformations of super Riemann surfaces. The tangent and cotangent spaces to super TeichmiJllerspace, as well as the super Weil-Petersson metric, are described in terms of Beltrami differentials and superconformaltensors of weight 3/2. Martinec's approach to super Teichmiiller deformations is thereby derived from that of Crane and Rabin.
Tools from the theory of Riemann surfaces and algebraic geometry have proven extremely powerful for attacking problems in string theory [ 1-3 ]. The generalization of these tools to super Riemann surfaces (SRSs) is well under way and should prove equally important for superstrings [ 2-6 ]. One such tool is the description of (super) Riemann surfaces by (super) Beltrami differentials, with the associated analytic machinery of the Beltrami equation and schwarzian derivatives. This machinery can be used to prove the uniformization theorem, to derive the Schottky group representation of Riemann surfaces, and to prove the Bers embedding theorem which describes the global complex structure of Teichmoiler space [ 7 ]. In string theory it has been used to extract the analytic information contained in the stress tensor and the ghost system, and to express the Polyakov measure on moduli space. All these results have been extended to the supersymmetric case. The main purpose of the present work is to use the super Beltrami equations to obtain the SRS analogues of some classical results of Ahlfors on infinitesimal deformations of Riemann surfaces [8,9]. These results give the parametrization of the tangent space to Teichmiiller space in terms of Beltrami differentials, the parametrization of the cotangent space in terms of quadratic differentials, and the Weil-Petersson metric which expresses the duality ResearchpartiaUysupportedby the DepartmentofEnergy(DEAC02-82-ER-40073). 40
between these descriptions. The treatment given here is in terms of SRSs as supermanifolds with a complex structure. Since a SRS represents a superconformal equivalence class of two-dimensional supergravity geometries, the same results can be derived by studying infinitesimal deformations of supergravity geometries. Using this approach, D'Hoker and Phong have recently obtained results similar to those presented here [ 5 ]. Another purpose of this paper is to give a rigorous derivation of Martinec's treatment of super Beltrami differentials [ 3 ] from that of Crane and Rabin [ 4]. Martinec's formulation is seen to be valid for infinitesimal deformations of a "split" SRS, that is, one whose odd supermoduli all vanish. It can be derived by linearizing the general formalism of Crane and Rabin to describe infinitesimal deformations, and making a special choice of gauge for the Beltrami differentials. In ref. [4 ] we conjectured that this special gauge choice was always possible. I show here that this is true for all infinitesimal and some finite deformations of split SRSs, but almost certainly false for arbitrary deformations. I begin with a review of the super Beltrami equations as derived in ref. [ 4 ]. A super Riemann surface M is a complex supermanifold with coordinates (z, 0) and transition functions of the form 2=f(z) +Oqz(z)~,
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w h e r e f a n d ~ are analytic, and subscripts denote differentiation. This is the most general analytic solution of the superconformal constraint D 2 = 0D/~,
D=0o+00~.
(2)
M always has a body Mo, an associated Riemann surface with transition functions 2=fo(z), where fo is f with any Grassmann parameters on which it may depend set to zero. Mo will be assumed compact, with fixed genus g > 1. The choice of signs for the square roots in (1) defines a spin structure on M0, which will also be fixed throughout this discussion. There is only one SRS SU whose body is U, the upper half plane, and M can be represented as M = SU/G, where G is a group of superconformal transformations of the form
az+b ~=--+0 cz+d
yz+~ (cz+d) z ' -
~= 7z+• 0 cz+------~+ c ~
(1 + ½~y),
ad-bc=l,
(3)
with a, b, c, d real and y = i?, 3 = i& The group G has 2g generators and is isomorphic to n l (M) =rCl (Mo). The generators depend on 6 g - 6 even and 4 g - 4 odd parameters, which can be used as moduli and supermoduli characterizing the SRS. SRSs whose odd supermoduli vanish were called canonical in ref. [4]; here the description split [ I 0] will be used. Mo is represented as U/Go, where Go is G with the odd parameters set to zero. Since a SRS M represents a superconformal equivalence class of supergravity geometries, it can be characterized by the frame field E " = dZUE~vof any geometry in its class. Given the supergravity torsion constraints, and the fact that the phase of E A can be changed at will by a local Lorentz rotation, it is sufficient to specify IEZl alone. We fix some standard SRS M * = S U / G * and a super diffeomorphism W: M * ~ M , regarding M as a deformation of M*. I f coordinates on SU are chosen so that IE~(M *) I lifts to the simple form 2 l d z + 0d0 l, then the pullback to M* of IE~(M) I lifts to a super Beltrami differential of the form ]EZl =pldz+l~(z, ~, O, 0)d~
+v(z, ~ O, O)dO+a(z, ~ O, 0 ) d 0 1 .
(4)
21 May 1987
Note that every super Beltrami differential is G*invariant, since it is obtained by lifting something from M* to SU. The standard SRS M* is assumed to be split, for convenience. The diffeomorphism W lifts to SU as a "quasisuperconformal" map (z, 0)--, (w, 0) which satisfies the super Beltrami equations
we+ OO~=U(w~ + 0 0 ~ ) ,
(5a)
-Wo +0¢o =v(wz +OOz) ,
(5b)
- w o +00o = a ( w z + O 0 z ) •
(5c)
Super moduli space SMg is the space of all SRSs having compact bodies of genus g;, super Teichmiiller space STg is a covering space for which the covering group is the modular, or mapping class group. A characterization of STg (Bers embedding theorem) is obtained by the following construction. Given a Beltrami differential describing a SRS M, extend the Beltrami coefficients/z, v, ~r from SU into the entire complex super plane SC by defining/t= a = 0, v = 0 for z in the lower half plane. Let the map (z, 0 ) ~ (w, 0) satisfy the super Beltrami equations in SC and the boundary conditions that (z, 0) --. (w, 0) is continuous across the real axis and sends (0, 0 ) - , (0, 0), (1, 0 ) --, ( 1, 0), and ( oo, 0 ) --, ( oo, 0 ). This last condition means, precisely, that as z--, ~ with 0 = 0, 0--,0 and w--,~ with w/z2---,O. For z in the lower half plane, where this map is superconformal, compute the super schwarzian derivative
S(z, 0)- D4j
2 (D30)(D20)
DO
=S~ (z) + OS°( z) .
(D0) 2
(6)
Then S is an odd G*-invariant superconformal tensor of weight 3/2, S o is an even G*-invariant quadratic differential, and S ~is an odd G*-invariant 3/2differential. This construction associates with each Beltrami differential a pair of points (S °, + S 1) in the space of such invariant differentials, which is a complex supermanifold of dimension ( 3 g - 3, 2 g - 2 ) . The sign ambiguity occurs because solutions of the super Beltrami equations are not quite unique: if (w, 0) is a solution, so is (w, - 0 ) - In principle, S might depend on the particular Beltrami differential chosen to represent the SRS M. Ifit were indepen41
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dent of this choice, each S would be intrinsically associated to the SRS and would label a point of super moduli space. In fact, S depends weakly on the choice of Beltrami differential: it depends on the homotopy class of the map W which produces the differential. Thus, each S labels a SRS plus an element of the mapping class group; in other words, a point of SZg. Martinec has proposed a notion of super Beltrami differential which seems different from that above [3]. For him a quasisuperconformal map has the form (1), except that f a n d ~t need not be analytic. Infinitesimally it is described by a vector field 8Z=Sz(z, & 0) +080(z, ~., O).
(7)
There is no 0-dependence, and there is only one Beltrami coefficient, given by
i~(z, 5, O)=O,Sz.
(8)
The first step in deriving this formulation from that above is to realize that the representation of SRSs by Beltrami coefficients is highly nonunique. Any diffeomorphism of M will change the frame field, hence also the Beltrami coefficients, but not the SRS structure they represent. It may be possible to use this gauge freedom to choose particularly simple Beltrami coefficients representing M. We will see that Martinec's formulation corresponds to the gauge choice v = 0, a = 0. In ref. [4] we conjectured that a gauge with a = 0 and/z and v independent of 0 could always be chosen. If this is true, w and q}may be assumed independent of 0 as well, and eq. (5c) will be satisfied identically. To describe infinitesimal deformations of M* we linearize the remaining equations by putting
21 May 1987
#=#°+0,,,
o ,
(11) these equations become fro = g o ,
(12a) (12b)
_~1 ..~1 =01 ,
(12c)
-~'°+20° =~°.
(12d)
Now the Bers embedding construction can be carded out explicitly in this linearized context. Eq. (12a) can be solved using the generalized Cauchy integral formula [ 11 ] 1 f fa f(z, 7.) = - -~ j u-z dZu+F(z)
d 2 u = d ( R e u ) d ( I m u) ,
(13)
with F ( z ) analytic. This analytic term is uniquely determined by the boundary conditions for (12a) and can be absorbed in a change of the integral kernel:
¢v°(z, 5) = _ 1 f P(u, z)it°(u, a) dZu, d U
1
z-1
P(u, z) = - + -u-z u
z
u-1 "
(14)
and treating if, ~,/t, 0 as infinitesimals. The linearized super Beltrami equations are simply
Any additional analytic term would have to vanish at z = 0 and 1, so would have at least a double pole at infinity, which is inconsistent with the boundary conditions. The integral in (14) extends only over the upper half plane because the infinitesimal Beltrami coefficients used in the Bers construction vanish in the lower half plane. The rest of the solution of eqs. (12) is
~e+ 0 6 , =U,
(10a)
6'(z,e)=- ~
-D~+6+
(10b)
w=z+~,
0=0+6,
v=O+f,,
(9)
,f e(u,z) U
0~o = O.
Observe that with the notational change ff~Sz, ~->80, eq. (10a) is precisely Martinec's eq. (8). Introducing the expansions
x [u'(u, a)+G(u, a)]d2u, 26°(z, Z) = 0°(z, 5)
-l
~ p~(u,z)u°(u,a)d2u, U
42
(15)
(16)
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PHYSICS LETTERSB
~,'(z, e)= -~'(z, e)
-2---~fP(u,z)[/t'(u,~)+Ola(u,~)ld2u.
(17)
U
The super schwarzian derivative in the linearized approximation is simply S(z,
Therefore W fixes the attractive fixed point of every element of the group G*. Since these points are dense on the real axis, W fixes the real axis pointwise as claimed. Therefore, changing the Beltrami differential by Wwill not affect the solution of the super Beltrami equations on the real axis and will not alter S. Given a Beltrami coefficient/t representing a SRS in the gauge a = 0 , eq. (19) is an explicit formula for the corresponding superconformal tensor S. Conversely, given S, one choice of a corresponding Beltrami coefficient is
v=O,
O)~-O4()=f~lzz-'bO~° z
( u _ z ) 3 [/t'(u, a)+0~(u, a)Jd2u
=-
21 May 1987
/t°(z, g) = - 4 ( I m z)ZS°(g) ,
f3
+0 (u_z)4/t°(u,
t~)d2u
}
•
(18)
z)S' (g).
/t ' (z, Z) = 4 i ( I m
(22a) (22b)
U
If r^e~= 0, which is the case for example in the v= 0 gauge, then this formula can be written as the superspace integral
S(z, 0 ) = -
__1 f d Z u d Z /t(u, t/,X )
n
(u-z-zO)
By substituting eqs. (12) into ( 18 ) and integrating by parts, one obtains
U
_(g=77
--21ifdu{~~'(u)+O
+
3
}
(u3~_z)4~°(u)}
R
(20) which shows that S is completely determined by the effect of the map (z, 0) ~ (w, 0) on the real axis. This provides an important check on eq. (18) by showing that it has the correct gauge invariance properties: changing the Beltrami differential by a diffeomorphism of M* does not affect S. This is any diffeomorphism of M* lifts to SU as a diffeomorphism fixing the real axis pointwise. To see this, let W be the diffeomorphism, x any point of SU, and q an element of G* with attractive fixed point x*. Because the lifted diffeomorphism commutes with q, we have
because
W ( x * ) = lim
12 f
1
V ( z ) = - ~ J ~ F ( u ) ( I m u)2d2u,
(23)
U
(19)
3"
SU
S(z,o)=lfd2u{2
This is verified explicitly by substituting (22) into (18 ) and using the identity [ 8 ]
for Fanalytic in U, which can be proven directly using Stokes' theorem and residues. By exhibiting explicitly a Beltrami differential in the v = 0, a = 0 gauge corresponding to an arbitrary S, eqs. (22) constitute a proof that it is always possible to choose this gauge for infinitesimal deformations of a split SRS. Eq. (22a) differs from the classical result for ordinary Riemann surfaces by a factor of 2. This can be traced to the normalization of S in eq. (6), which is such that S Ofor a superconformal map (1) with ¥ = 0 reduces to the usual schwarzian derivative of There are also slight differences from ref. [8] due to the fact that was defined so as to be analytic in the rather than the upper, half plane. The super Weil-Petersson metric will be a metric on tangent vectors to super Teichmiiller space, that is, on infinitesimal deformations of a fixed SRS. Representing two such deformations by supertensors &, $2, the supersymmetric generalization of the ordinary Weil-Petersson scalar product is
f(z).
(S,,$2)=
twice
S(z, O) lower,
f
dZzdOdOS'(Z'O)Sz(f'O)
SU/G*
W[qn(x)]
n~oD
= lim q~[ W(x)] = x * .
X (Im z + ½00) 2 . (2I)
(24)
Performing the Grassmann integrals leads to 43
Volume 190, number 1,2
(S,,S2)= f dZz[S°(e)S°(e)(Imz)
PHYSICS LETTERS B
2
U/G*
-SI (z)S~(z)Im z],
(25)
where the first term is the ordinary Weil-Petersson scalar product. By introducing the Beltrami coefficients using eqs. (22), the super Weil-Petersson scalar product can be written in the mixed form
(s,, & ) = - ~
d2z ba°(z, ~)S°(~)
(26)
The ordinary Weil-Petersson metric in the form of the first term in (26) can be viewed as establishing a duality between the spaces of quadratic differentials and Beltrami coefficients: This interpretation does not quite work for the fermionic partners of these quantities, however. Because both/z I and S~ are odd Grassmann-valued functions, the second term of (26) is always nilpotent, though even. Considering the second term as a linear functional of S 1, not all functionals can be obtained by varying p I, but only those taking nilpotent values. Therefore the Beltrami coefficients do not span the entire dual space of superconformal tensors S. This purely technical point reflects a subtlety in the identification of infinitesimal deformations with tangent vectors to STg. If ~ is an odd coordinate on ST,,, then 0/0~ is a tangent vector, but it does not represent an infinitesimal deformation until it is multiplied by an infinitesimal odd parameter. The scalar product of two such deformations, unlike that of tangent vectors, is then necessarily nilpotent. Having established that Martinec's gauge choice g = 0, a = 0 is always possible for infinitesimal deformations of a split SRS, we wish to investigate whether this is so for finite deformations as well. Given S(z, 0) representing any deformation of M*, we try to find a map (w, O) having super schwarzian derivative S, and a Beltrami coefficient ~t in this gauge such that eqs. (5) are satisfied. Eq. (5c) is satisfied identically if w and 0 are independent of 0, and the remaining equations in this gauge become
44
wz + 0¢~ =u(wz + ¢¢z),
(27a)
Dw= 0De.
(27b)
Eq. (27b) is nothing but the superconformal condition (2). This completes the justification of Martinec's claim that quasisuperconformal maps in this gauge have the form (1) with the functions f = w ° and = ¢~ not necessarily analytic. A superconformal map in the lower half plane having super schwarzian derivative S can be constructed from solutions of the auxiliary equation D3F= -SF.
U/G*
+i/z[ (z, z ) S ~ ( ~ ) ] .
21 May 1987
(28)
It is possible to choose two even solutions F(z, 0), G(z, O) and one odd solution q(z, 0) of this equation such that
w=F/G,
q)=rl/G
(29)
is the desired map in the lower half plane [ 12 ]. For the solution in the upper half plane we try, generalizing [ 13 ]
wO= F°(~) + ( z - Z ) F °'(Z) Go(f)+(z_Z)GO,(z) ' 0 ~ = q~(Z) + (z-z)r/~ '(~)
G°(f) + (z-z)G°'(z)
(30)
These match continuously onto the solution (29) in the lower half plane, and the remaining components w ~ and 0o are determined by (27b). Direct calculation proves that this is indeed a solution of the super Beltrami equations (27) with the Beltrami coefficient # still given by (22), provided that the odd 3/2differential S ~is of the form ~h, with e an odd Grassmann number and h an ordinary commuting 3/2-differential. The solution (30) does not work for more general S ~of the form Eiht. This restriction means that in effect the deformations in odd Grassmann directions are still linearized, since it guarantees that products among S ~and any of its derivatives vanish. As for ordinary Riemann surfaces, the solution (30) is only valid if S Ois not too large, because of eq. (22a) and the fact that Beltrami differentials must obey j/~o[ < 1. Nevertheless, deformations with S O arbitrarily large can be represented in the g = 0, a = 0 gauge. First make the deformation described by S O,
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with S ~=0. This is just a quasiconformal deformation of the body of the SRS, and can be represented by a quasisuperconformal map of the form (1) with ~,=0. After this deformation, the SRS is still split, and the linear deformation described by S t can be made. The composition of these two maps is still in the desired gauge. This result shows that at least some finite deformations of split SRSs can be described by Beltrami coefficients in the u = 0, a = 0 gauge. Can all deformations be so described? After all, there may be a generalization of the solution (30) and/or a generalization of the formulas (22) which works in all cases. The following argument indicates that this is not so, and that the class of deformations identified above is the largest that can be described in this gauge. In other words, u = 0, a = 0 cannot always be achieved by a gauge transformation (diffeomorphism) and is therefore too strong to be imposed as a gauge condition in general. Certainly deformations of a nonsplit SRS M* cannot be described in this gauge. This is because the Beltrami differential must be G*-invariant, and in the nonsplit case the group G* mixes the z and 0 coordinates nontrivially. Since the Beltrami differential has Z dependence and is G*-invariant, it must depend on 0. However, examination of the components of the super Beltrami equations in this gauge shows that they are inconsistent if/t depends on 0. Now consider a deformation of a split SRS having S ~= eihi. Imagine making this deformation in successive steps corresponding to the terms in the sum over i. After the first step, the deformed SRS is no longer split, so that further deformations must introduce 0 dependence. Conceivably the 0 dependence introduced in successive deformations could cancel in special cases, but presumably not in general. This conclusion would also agree with D'Hoker and Phong [5], who found that the description of deformations of 2D supergravities becomes much more complicated at second order in the gravitino field, due to the difficulty of identifying the independent degrees of freedom when the torsion constraints are imposed. Identifying the independent degrees of freedom parametrizing deformations of a supergravity geometry is the same problem as fixing a gauge for the Beltrami differential IEZl. For applications to superstrings, the deformations of split SRSs are of most importance. The Polyakov
21 May 1987
path integral is an integral over supermoduli space. Integration over a supermanifold means a Grassmann integral followed by ordinary integration over the body of the supermanifold. The body of supermoduli space is the space of split SRSs, so the final integral is over this space only. The Polyakov measure is therefore conveniently expressed in terms of Beltrami coefficients describing deformations of split SRSs. However, in order to do the Grassmann part of the integration, one must know the highest term in the expansion of the integrand in supermoduli. Therefore the explicit description of the deformations beyond linear order is important and deserves more study. In summary, I have derived formulas relating the Beltrami coefficients and super 3/2-tensors describing infinitesimal deformations of split SRSs. The gauge choice leading to Martinec's formulation was shown to be possible for deformations linear in the supermoduli but arbitrary in the moduli. Finally, the super Weil-Petersson metric on the tangent space to STg at a split point was obtained.
References [ 1J D. Friedan, S. Shenker and E. Martinec, Nucl. Phys. B 271 (1986) 93; D. Friedan and S. Shenker, Phys. Len. B 175 (1986) 287; A.A. Belavin and V.G. Knizhnik, Phys. Len. B 168 (1986) 201; G. Moore, J. Harris, P. Nelson and I. Singer, Phys. Lett. B 178 (1986) 167; A.A. Beilinson and Yu.l. Martin, Commun. Math. Phys. 107 (1986) 359; Yu.I. Manin, Phys. Lett. B 172 (1986) 184; P. Nelson, Lectures on strings and moduli space, to be published in Phys. Rep.; J.J. Atick and A. Sen, Spin field correlators on an arbitray genus Riemann surface and nonrenormalization theorems in string theories, SLAC-PUB-4131 (1986); S.B. Giddings and E. Martinec, Conformal geometry and string field theory, Princeton preprint (April 1986); S.B. Giddings and S.A. Wolpert, A triangulation of moduli space from light cone string theory, Princeton preprint PUPT-1025 (October 1986). [2] D. Friedan, Notes on string theory and two-dimensional conformal field theory, in: Proc. Workshop on Unified string theories, eds. D. Gross and M. Green (World Scientific, Singapore, 1986). [3] E. Martinec, Nucl. Phys. B 281 (1987) 157.
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[4] L. Crane and J.M. Rabin, Super Riemann surfaces: uniformization and Teichmiiller theory, University of Chicago preprint EFI 86-25 (September 1986), Commun. Math. Phys., submitted for publication. [ 5 ] E. D'Hoker and D.H. Phong, Superholomorphic anomalies and supermoduli space, Princeton preprint PUPT-1029 (October 1986). [ 6 ] Yu.I. Manin, Quantum strings and algebraic curves, talk at the Intern. Congress of Mathematicians (Berkeley, 1986); G. Moore, P. Nelson and J. Polchinski, Phys. Lett. B 169 (1986) 47; J.M. Rabin, Super Riemann surfaces, University of Chicago preprint EF186-63 (October 1986 ), to be published in: Conf, on Mathematical aspects of string theory (U.C. San Diego, 1986); J.D. Cohn, N=2 super Riemann surfaces, University of Chicago preprint EFI-86-32 (August 1986).
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[7] L. Bers, Bull. London Math. Soc. 4 (1972) 257; C.J. Earle, Teichmiiller theory, in: Discrete groups and automorphic functions, ed. W.J. Harvey (Academic Press, London, 1977). [8] L.V. Ahlfors, J. Analyse Math. 9 (1961) 161. [ 9 ] L.V. Ahlfors, Ann. Math. 74 (1961 ) 171. [ 10] Yu.I. Martin, Russ. Math. Sure. 39, No. 6 (1984) 51; M. Rothstein, Proc. AMS 95 (1985) 255. [ 11 ] H.M. Farkas and I. Kra, Riemann surfaces (Springer, Berlin, 1980). [ 12] J.F. Arvis, Nucl. Phys. B 212 (1983) 151. [ 13] L.V. Ahlfors, Acta Math. 109 (1963) 291.
PHYSICS LETTERSB
21 May 1987
T E I C H M ~ L L E R D E F O R M A T I O N S OF SUPER R I E M A N N SURFACES Jeffrey M. RABIN Department of Mathematics and Enrico Fermi Institute, Universityof Chicago, Chicago, IL 60637, USA Received 20 February 1987
Exact solutions are obtained for the linearized super Beltrami equations which describe infinitesimal deformations of super Riemann surfaces. The tangent and cotangent spaces to super TeichmiJllerspace, as well as the super Weil-Petersson metric, are described in terms of Beltrami differentials and superconformaltensors of weight 3/2. Martinec's approach to super Teichmiiller deformations is thereby derived from that of Crane and Rabin.
Tools from the theory of Riemann surfaces and algebraic geometry have proven extremely powerful for attacking problems in string theory [ 1-3 ]. The generalization of these tools to super Riemann surfaces (SRSs) is well under way and should prove equally important for superstrings [ 2-6 ]. One such tool is the description of (super) Riemann surfaces by (super) Beltrami differentials, with the associated analytic machinery of the Beltrami equation and schwarzian derivatives. This machinery can be used to prove the uniformization theorem, to derive the Schottky group representation of Riemann surfaces, and to prove the Bers embedding theorem which describes the global complex structure of Teichmoiler space [ 7 ]. In string theory it has been used to extract the analytic information contained in the stress tensor and the ghost system, and to express the Polyakov measure on moduli space. All these results have been extended to the supersymmetric case. The main purpose of the present work is to use the super Beltrami equations to obtain the SRS analogues of some classical results of Ahlfors on infinitesimal deformations of Riemann surfaces [8,9]. These results give the parametrization of the tangent space to Teichmiiller space in terms of Beltrami differentials, the parametrization of the cotangent space in terms of quadratic differentials, and the Weil-Petersson metric which expresses the duality ResearchpartiaUysupportedby the DepartmentofEnergy(DEAC02-82-ER-40073). 40
between these descriptions. The treatment given here is in terms of SRSs as supermanifolds with a complex structure. Since a SRS represents a superconformal equivalence class of two-dimensional supergravity geometries, the same results can be derived by studying infinitesimal deformations of supergravity geometries. Using this approach, D'Hoker and Phong have recently obtained results similar to those presented here [ 5 ]. Another purpose of this paper is to give a rigorous derivation of Martinec's treatment of super Beltrami differentials [ 3 ] from that of Crane and Rabin [ 4]. Martinec's formulation is seen to be valid for infinitesimal deformations of a "split" SRS, that is, one whose odd supermoduli all vanish. It can be derived by linearizing the general formalism of Crane and Rabin to describe infinitesimal deformations, and making a special choice of gauge for the Beltrami differentials. In ref. [4 ] we conjectured that this special gauge choice was always possible. I show here that this is true for all infinitesimal and some finite deformations of split SRSs, but almost certainly false for arbitrary deformations. I begin with a review of the super Beltrami equations as derived in ref. [ 4 ]. A super Riemann surface M is a complex supermanifold with coordinates (z, 0) and transition functions of the form 2=f(z) +Oqz(z)~,
(l) 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 190, number 1,2
PHYSICS LETTERS B
w h e r e f a n d ~ are analytic, and subscripts denote differentiation. This is the most general analytic solution of the superconformal constraint D 2 = 0D/~,
D=0o+00~.
(2)
M always has a body Mo, an associated Riemann surface with transition functions 2=fo(z), where fo is f with any Grassmann parameters on which it may depend set to zero. Mo will be assumed compact, with fixed genus g > 1. The choice of signs for the square roots in (1) defines a spin structure on M0, which will also be fixed throughout this discussion. There is only one SRS SU whose body is U, the upper half plane, and M can be represented as M = SU/G, where G is a group of superconformal transformations of the form
az+b ~=--+0 cz+d
yz+~ (cz+d) z ' -
~= 7z+• 0 cz+------~+ c ~
(1 + ½~y),
ad-bc=l,
(3)
with a, b, c, d real and y = i?, 3 = i& The group G has 2g generators and is isomorphic to n l (M) =rCl (Mo). The generators depend on 6 g - 6 even and 4 g - 4 odd parameters, which can be used as moduli and supermoduli characterizing the SRS. SRSs whose odd supermoduli vanish were called canonical in ref. [4]; here the description split [ I 0] will be used. Mo is represented as U/Go, where Go is G with the odd parameters set to zero. Since a SRS M represents a superconformal equivalence class of supergravity geometries, it can be characterized by the frame field E " = dZUE~vof any geometry in its class. Given the supergravity torsion constraints, and the fact that the phase of E A can be changed at will by a local Lorentz rotation, it is sufficient to specify IEZl alone. We fix some standard SRS M * = S U / G * and a super diffeomorphism W: M * ~ M , regarding M as a deformation of M*. I f coordinates on SU are chosen so that IE~(M *) I lifts to the simple form 2 l d z + 0d0 l, then the pullback to M* of IE~(M) I lifts to a super Beltrami differential of the form ]EZl =pldz+l~(z, ~, O, 0)d~
+v(z, ~ O, O)dO+a(z, ~ O, 0 ) d 0 1 .
(4)
21 May 1987
Note that every super Beltrami differential is G*invariant, since it is obtained by lifting something from M* to SU. The standard SRS M* is assumed to be split, for convenience. The diffeomorphism W lifts to SU as a "quasisuperconformal" map (z, 0)--, (w, 0) which satisfies the super Beltrami equations
we+ OO~=U(w~ + 0 0 ~ ) ,
(5a)
-Wo +0¢o =v(wz +OOz) ,
(5b)
- w o +00o = a ( w z + O 0 z ) •
(5c)
Super moduli space SMg is the space of all SRSs having compact bodies of genus g;, super Teichmiiller space STg is a covering space for which the covering group is the modular, or mapping class group. A characterization of STg (Bers embedding theorem) is obtained by the following construction. Given a Beltrami differential describing a SRS M, extend the Beltrami coefficients/z, v, ~r from SU into the entire complex super plane SC by defining/t= a = 0, v = 0 for z in the lower half plane. Let the map (z, 0 ) ~ (w, 0) satisfy the super Beltrami equations in SC and the boundary conditions that (z, 0) --. (w, 0) is continuous across the real axis and sends (0, 0 ) - , (0, 0), (1, 0 ) --, ( 1, 0), and ( oo, 0 ) --, ( oo, 0 ). This last condition means, precisely, that as z--, ~ with 0 = 0, 0--,0 and w--,~ with w/z2---,O. For z in the lower half plane, where this map is superconformal, compute the super schwarzian derivative
S(z, 0)- D4j
2 (D30)(D20)
DO
=S~ (z) + OS°( z) .
(D0) 2
(6)
Then S is an odd G*-invariant superconformal tensor of weight 3/2, S o is an even G*-invariant quadratic differential, and S ~is an odd G*-invariant 3/2differential. This construction associates with each Beltrami differential a pair of points (S °, + S 1) in the space of such invariant differentials, which is a complex supermanifold of dimension ( 3 g - 3, 2 g - 2 ) . The sign ambiguity occurs because solutions of the super Beltrami equations are not quite unique: if (w, 0) is a solution, so is (w, - 0 ) - In principle, S might depend on the particular Beltrami differential chosen to represent the SRS M. Ifit were indepen41
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dent of this choice, each S would be intrinsically associated to the SRS and would label a point of super moduli space. In fact, S depends weakly on the choice of Beltrami differential: it depends on the homotopy class of the map W which produces the differential. Thus, each S labels a SRS plus an element of the mapping class group; in other words, a point of SZg. Martinec has proposed a notion of super Beltrami differential which seems different from that above [3]. For him a quasisuperconformal map has the form (1), except that f a n d ~t need not be analytic. Infinitesimally it is described by a vector field 8Z=Sz(z, & 0) +080(z, ~., O).
(7)
There is no 0-dependence, and there is only one Beltrami coefficient, given by
i~(z, 5, O)=O,Sz.
(8)
The first step in deriving this formulation from that above is to realize that the representation of SRSs by Beltrami coefficients is highly nonunique. Any diffeomorphism of M will change the frame field, hence also the Beltrami coefficients, but not the SRS structure they represent. It may be possible to use this gauge freedom to choose particularly simple Beltrami coefficients representing M. We will see that Martinec's formulation corresponds to the gauge choice v = 0, a = 0. In ref. [4] we conjectured that a gauge with a = 0 and/z and v independent of 0 could always be chosen. If this is true, w and q}may be assumed independent of 0 as well, and eq. (5c) will be satisfied identically. To describe infinitesimal deformations of M* we linearize the remaining equations by putting
21 May 1987
#=#°+0,,,
o ,
(11) these equations become fro = g o ,
(12a) (12b)
_~1 ..~1 =01 ,
(12c)
-~'°+20° =~°.
(12d)
Now the Bers embedding construction can be carded out explicitly in this linearized context. Eq. (12a) can be solved using the generalized Cauchy integral formula [ 11 ] 1 f fa f(z, 7.) = - -~ j u-z dZu+F(z)
d 2 u = d ( R e u ) d ( I m u) ,
(13)
with F ( z ) analytic. This analytic term is uniquely determined by the boundary conditions for (12a) and can be absorbed in a change of the integral kernel:
¢v°(z, 5) = _ 1 f P(u, z)it°(u, a) dZu, d U
1
z-1
P(u, z) = - + -u-z u
z
u-1 "
(14)
and treating if, ~,/t, 0 as infinitesimals. The linearized super Beltrami equations are simply
Any additional analytic term would have to vanish at z = 0 and 1, so would have at least a double pole at infinity, which is inconsistent with the boundary conditions. The integral in (14) extends only over the upper half plane because the infinitesimal Beltrami coefficients used in the Bers construction vanish in the lower half plane. The rest of the solution of eqs. (12) is
~e+ 0 6 , =U,
(10a)
6'(z,e)=- ~
-D~+6+
(10b)
w=z+~,
0=0+6,
v=O+f,,
(9)
,f e(u,z) U
0~o = O.
Observe that with the notational change ff~Sz, ~->80, eq. (10a) is precisely Martinec's eq. (8). Introducing the expansions
x [u'(u, a)+G(u, a)]d2u, 26°(z, Z) = 0°(z, 5)
-l
~ p~(u,z)u°(u,a)d2u, U
42
(15)
(16)
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PHYSICS LETTERSB
~,'(z, e)= -~'(z, e)
-2---~fP(u,z)[/t'(u,~)+Ola(u,~)ld2u.
(17)
U
The super schwarzian derivative in the linearized approximation is simply S(z,
Therefore W fixes the attractive fixed point of every element of the group G*. Since these points are dense on the real axis, W fixes the real axis pointwise as claimed. Therefore, changing the Beltrami differential by Wwill not affect the solution of the super Beltrami equations on the real axis and will not alter S. Given a Beltrami coefficient/t representing a SRS in the gauge a = 0 , eq. (19) is an explicit formula for the corresponding superconformal tensor S. Conversely, given S, one choice of a corresponding Beltrami coefficient is
v=O,
O)~-O4()=f~lzz-'bO~° z
( u _ z ) 3 [/t'(u, a)+0~(u, a)Jd2u
=-
21 May 1987
/t°(z, g) = - 4 ( I m z)ZS°(g) ,
f3
+0 (u_z)4/t°(u,
t~)d2u
}
•
(18)
z)S' (g).
/t ' (z, Z) = 4 i ( I m
(22a) (22b)
U
If r^e~= 0, which is the case for example in the v= 0 gauge, then this formula can be written as the superspace integral
S(z, 0 ) = -
__1 f d Z u d Z /t(u, t/,X )
n
(u-z-zO)
By substituting eqs. (12) into ( 18 ) and integrating by parts, one obtains
U
_(g=77
--21ifdu{~~'(u)+O
+
3
}
(u3~_z)4~°(u)}
R
(20) which shows that S is completely determined by the effect of the map (z, 0) ~ (w, 0) on the real axis. This provides an important check on eq. (18) by showing that it has the correct gauge invariance properties: changing the Beltrami differential by a diffeomorphism of M* does not affect S. This is any diffeomorphism of M* lifts to SU as a diffeomorphism fixing the real axis pointwise. To see this, let W be the diffeomorphism, x any point of SU, and q an element of G* with attractive fixed point x*. Because the lifted diffeomorphism commutes with q, we have
because
W ( x * ) = lim
12 f
1
V ( z ) = - ~ J ~ F ( u ) ( I m u)2d2u,
(23)
U
(19)
3"
SU
S(z,o)=lfd2u{2
This is verified explicitly by substituting (22) into (18 ) and using the identity [ 8 ]
for Fanalytic in U, which can be proven directly using Stokes' theorem and residues. By exhibiting explicitly a Beltrami differential in the v = 0, a = 0 gauge corresponding to an arbitrary S, eqs. (22) constitute a proof that it is always possible to choose this gauge for infinitesimal deformations of a split SRS. Eq. (22a) differs from the classical result for ordinary Riemann surfaces by a factor of 2. This can be traced to the normalization of S in eq. (6), which is such that S Ofor a superconformal map (1) with ¥ = 0 reduces to the usual schwarzian derivative of There are also slight differences from ref. [8] due to the fact that was defined so as to be analytic in the rather than the upper, half plane. The super Weil-Petersson metric will be a metric on tangent vectors to super Teichmiiller space, that is, on infinitesimal deformations of a fixed SRS. Representing two such deformations by supertensors &, $2, the supersymmetric generalization of the ordinary Weil-Petersson scalar product is
f(z).
(S,,$2)=
twice
S(z, O) lower,
f
dZzdOdOS'(Z'O)Sz(f'O)
SU/G*
W[qn(x)]
n~oD
= lim q~[ W(x)] = x * .
X (Im z + ½00) 2 . (2I)
(24)
Performing the Grassmann integrals leads to 43
Volume 190, number 1,2
(S,,S2)= f dZz[S°(e)S°(e)(Imz)
PHYSICS LETTERS B
2
U/G*
-SI (z)S~(z)Im z],
(25)
where the first term is the ordinary Weil-Petersson scalar product. By introducing the Beltrami coefficients using eqs. (22), the super Weil-Petersson scalar product can be written in the mixed form
(s,, & ) = - ~
d2z ba°(z, ~)S°(~)
(26)
The ordinary Weil-Petersson metric in the form of the first term in (26) can be viewed as establishing a duality between the spaces of quadratic differentials and Beltrami coefficients: This interpretation does not quite work for the fermionic partners of these quantities, however. Because both/z I and S~ are odd Grassmann-valued functions, the second term of (26) is always nilpotent, though even. Considering the second term as a linear functional of S 1, not all functionals can be obtained by varying p I, but only those taking nilpotent values. Therefore the Beltrami coefficients do not span the entire dual space of superconformal tensors S. This purely technical point reflects a subtlety in the identification of infinitesimal deformations with tangent vectors to STg. If ~ is an odd coordinate on ST,,, then 0/0~ is a tangent vector, but it does not represent an infinitesimal deformation until it is multiplied by an infinitesimal odd parameter. The scalar product of two such deformations, unlike that of tangent vectors, is then necessarily nilpotent. Having established that Martinec's gauge choice g = 0, a = 0 is always possible for infinitesimal deformations of a split SRS, we wish to investigate whether this is so for finite deformations as well. Given S(z, 0) representing any deformation of M*, we try to find a map (w, O) having super schwarzian derivative S, and a Beltrami coefficient ~t in this gauge such that eqs. (5) are satisfied. Eq. (5c) is satisfied identically if w and 0 are independent of 0, and the remaining equations in this gauge become
44
wz + 0¢~ =u(wz + ¢¢z),
(27a)
Dw= 0De.
(27b)
Eq. (27b) is nothing but the superconformal condition (2). This completes the justification of Martinec's claim that quasisuperconformal maps in this gauge have the form (1) with the functions f = w ° and = ¢~ not necessarily analytic. A superconformal map in the lower half plane having super schwarzian derivative S can be constructed from solutions of the auxiliary equation D3F= -SF.
U/G*
+i/z[ (z, z ) S ~ ( ~ ) ] .
21 May 1987
(28)
It is possible to choose two even solutions F(z, 0), G(z, O) and one odd solution q(z, 0) of this equation such that
w=F/G,
q)=rl/G
(29)
is the desired map in the lower half plane [ 12 ]. For the solution in the upper half plane we try, generalizing [ 13 ]
wO= F°(~) + ( z - Z ) F °'(Z) Go(f)+(z_Z)GO,(z) ' 0 ~ = q~(Z) + (z-z)r/~ '(~)
G°(f) + (z-z)G°'(z)
(30)
These match continuously onto the solution (29) in the lower half plane, and the remaining components w ~ and 0o are determined by (27b). Direct calculation proves that this is indeed a solution of the super Beltrami equations (27) with the Beltrami coefficient # still given by (22), provided that the odd 3/2differential S ~is of the form ~h, with e an odd Grassmann number and h an ordinary commuting 3/2-differential. The solution (30) does not work for more general S ~of the form Eiht. This restriction means that in effect the deformations in odd Grassmann directions are still linearized, since it guarantees that products among S ~and any of its derivatives vanish. As for ordinary Riemann surfaces, the solution (30) is only valid if S Ois not too large, because of eq. (22a) and the fact that Beltrami differentials must obey j/~o[ < 1. Nevertheless, deformations with S O arbitrarily large can be represented in the g = 0, a = 0 gauge. First make the deformation described by S O,
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with S ~=0. This is just a quasiconformal deformation of the body of the SRS, and can be represented by a quasisuperconformal map of the form (1) with ~,=0. After this deformation, the SRS is still split, and the linear deformation described by S t can be made. The composition of these two maps is still in the desired gauge. This result shows that at least some finite deformations of split SRSs can be described by Beltrami coefficients in the u = 0, a = 0 gauge. Can all deformations be so described? After all, there may be a generalization of the solution (30) and/or a generalization of the formulas (22) which works in all cases. The following argument indicates that this is not so, and that the class of deformations identified above is the largest that can be described in this gauge. In other words, u = 0, a = 0 cannot always be achieved by a gauge transformation (diffeomorphism) and is therefore too strong to be imposed as a gauge condition in general. Certainly deformations of a nonsplit SRS M* cannot be described in this gauge. This is because the Beltrami differential must be G*-invariant, and in the nonsplit case the group G* mixes the z and 0 coordinates nontrivially. Since the Beltrami differential has Z dependence and is G*-invariant, it must depend on 0. However, examination of the components of the super Beltrami equations in this gauge shows that they are inconsistent if/t depends on 0. Now consider a deformation of a split SRS having S ~= eihi. Imagine making this deformation in successive steps corresponding to the terms in the sum over i. After the first step, the deformed SRS is no longer split, so that further deformations must introduce 0 dependence. Conceivably the 0 dependence introduced in successive deformations could cancel in special cases, but presumably not in general. This conclusion would also agree with D'Hoker and Phong [5], who found that the description of deformations of 2D supergravities becomes much more complicated at second order in the gravitino field, due to the difficulty of identifying the independent degrees of freedom when the torsion constraints are imposed. Identifying the independent degrees of freedom parametrizing deformations of a supergravity geometry is the same problem as fixing a gauge for the Beltrami differential IEZl. For applications to superstrings, the deformations of split SRSs are of most importance. The Polyakov
21 May 1987
path integral is an integral over supermoduli space. Integration over a supermanifold means a Grassmann integral followed by ordinary integration over the body of the supermanifold. The body of supermoduli space is the space of split SRSs, so the final integral is over this space only. The Polyakov measure is therefore conveniently expressed in terms of Beltrami coefficients describing deformations of split SRSs. However, in order to do the Grassmann part of the integration, one must know the highest term in the expansion of the integrand in supermoduli. Therefore the explicit description of the deformations beyond linear order is important and deserves more study. In summary, I have derived formulas relating the Beltrami coefficients and super 3/2-tensors describing infinitesimal deformations of split SRSs. The gauge choice leading to Martinec's formulation was shown to be possible for deformations linear in the supermoduli but arbitrary in the moduli. Finally, the super Weil-Petersson metric on the tangent space to STg at a split point was obtained.
References [ 1J D. Friedan, S. Shenker and E. Martinec, Nucl. Phys. B 271 (1986) 93; D. Friedan and S. Shenker, Phys. Len. B 175 (1986) 287; A.A. Belavin and V.G. Knizhnik, Phys. Len. B 168 (1986) 201; G. Moore, J. Harris, P. Nelson and I. Singer, Phys. Lett. B 178 (1986) 167; A.A. Beilinson and Yu.l. Martin, Commun. Math. Phys. 107 (1986) 359; Yu.I. Manin, Phys. Lett. B 172 (1986) 184; P. Nelson, Lectures on strings and moduli space, to be published in Phys. Rep.; J.J. Atick and A. Sen, Spin field correlators on an arbitray genus Riemann surface and nonrenormalization theorems in string theories, SLAC-PUB-4131 (1986); S.B. Giddings and E. Martinec, Conformal geometry and string field theory, Princeton preprint (April 1986); S.B. Giddings and S.A. Wolpert, A triangulation of moduli space from light cone string theory, Princeton preprint PUPT-1025 (October 1986). [2] D. Friedan, Notes on string theory and two-dimensional conformal field theory, in: Proc. Workshop on Unified string theories, eds. D. Gross and M. Green (World Scientific, Singapore, 1986). [3] E. Martinec, Nucl. Phys. B 281 (1987) 157.
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[4] L. Crane and J.M. Rabin, Super Riemann surfaces: uniformization and Teichmiiller theory, University of Chicago preprint EFI 86-25 (September 1986), Commun. Math. Phys., submitted for publication. [ 5 ] E. D'Hoker and D.H. Phong, Superholomorphic anomalies and supermoduli space, Princeton preprint PUPT-1029 (October 1986). [ 6 ] Yu.I. Manin, Quantum strings and algebraic curves, talk at the Intern. Congress of Mathematicians (Berkeley, 1986); G. Moore, P. Nelson and J. Polchinski, Phys. Lett. B 169 (1986) 47; J.M. Rabin, Super Riemann surfaces, University of Chicago preprint EF186-63 (October 1986 ), to be published in: Conf, on Mathematical aspects of string theory (U.C. San Diego, 1986); J.D. Cohn, N=2 super Riemann surfaces, University of Chicago preprint EFI-86-32 (August 1986).
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[7] L. Bers, Bull. London Math. Soc. 4 (1972) 257; C.J. Earle, Teichmiiller theory, in: Discrete groups and automorphic functions, ed. W.J. Harvey (Academic Press, London, 1977). [8] L.V. Ahlfors, J. Analyse Math. 9 (1961) 161. [ 9 ] L.V. Ahlfors, Ann. Math. 74 (1961 ) 171. [ 10] Yu.I. Martin, Russ. Math. Sure. 39, No. 6 (1984) 51; M. Rothstein, Proc. AMS 95 (1985) 255. [ 11 ] H.M. Farkas and I. Kra, Riemann surfaces (Springer, Berlin, 1980). [ 12] J.F. Arvis, Nucl. Phys. B 212 (1983) 151. [ 13] L.V. Ahlfors, Acta Math. 109 (1963) 291.