Beltrami parametrization for superstrings

Volume 198. number 3 PHYSICS LETTERS B 26 November 1987

BELTRAMI PARAMETRIZATION FOR SUPERSTRINGS

L. BAULIEU a, M. BELLON a and R. GRIMM b ” LPTHE, UniversitP Pierre et Marie Curie, Tour 16 - ler etage, 4, place Jussieu, F-75252 Paris Cede,r 05, France ’ Instltut ftir Theoretische Physik, Universitiit Karlsruhe, D-7500 Karlsruhe, Fed. Rep. Germany

Received 1 September 1987

We introduce a parametrization of the metric and supersymmetry structure of the superstring world sheet which permits the

derivation of the superstring action for general backgrounds. Factorization is explicit at each step of the construction.

In ref. [ l] a particular parametrization of the bidimensional metric and diffeomorphism ghosts has been presented. This parametrization, called Beltrami parametrization separates in a covariant way independent left- and right-moving modes of the diffeomorphism ghosts. Thereby it permits a most transparent gauge fixing of the bosonic string action. If one discards the Weyl sector, the corresponding BRS symmetry is made of two independent parts. Each of them describes a one-dimensional Yang-Mills theory with fields taking their values in the algebra of one-dimensional diffeomorphisms.

In this work, we extend these results to the case of superstring theories. On the world-sheet, the relevant symmetries are the local supersymmetry-diffeomorphism-superWey1 transformations. A first step is to build the corresponding BRS symmetry. This can be done e.g. using the methods introduced in refs. [2,3]. Then, remarkably enough, one can extract from the whole set of fields on which this BRS algebra acts a superWey1 inert system of fields which exhibits the same factorization properties as those of the purely bosonic case: from the two-dimensional gravitino components one can construct a fermionic field cy which turns out to be the supersymmetric partner of the Beltrami differential ,K Moreover the classical superstring action depends on the world-sheet variables only through the variables cr and ,u. The factorization in two independent sectors of the BRS algebra for the variables cx and p permits the gauge fixing of the superstring action while imposing at all the stages the so-called factorization property [ 41. Since we consider the case of general backgrounds for the superstring world-sheet, the same tool as in ref. [ 1 ] can now be used to explore the current algebra of the stress-energy tensor and of the supersymmetry current as well as the corresponding superconformal properties

[41. We parametrize the world-sheet 2 by local complex coordinates to make it into a Riemann surface. We use

the notation + (respectively -) to denote the index z (respectively 2). The indices 0 and 0 denote the two possible chiralities of a spinor. We have the relations r,Q=i,@x”- y’Ox”, r+Q’x= -v”xo, v/y~~=~~x~, which hold true for any spinors v and x. y + and y - stand for both two-dimensional gamma matrices ( y +z = 0, y-‘=O, y+y- + y-y+ = 1). The superstring multiplet is (xJTw). x is the string field, f the Neveu-Ramond -Schwarz field and w the corresponding auxiliary field. Solving the Bianchi identities in superspace leads one to the conclusion that the two-dimensional supergravity multiplet is made from the zweibein e’, the two-dimensional gravitino w with four independent components and an auxiliary scalar a. We introduce a spin con- nection one-form w(e,w) which satisfies the condition of vanishing torsion:

T+ =de+ +me+ +iy/@vuo =O, Tp =de- -we- -iyetyo =O . (1)

0370-2693/87/S 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Volume 198. number 3 PHYSICS LETTERS B 26 November 1987

The relevant ghosts are the anticommuting ghost vector field < and the commuting supersymmetric ghosts ~0. In the superWey1 sector one has a ghost scalar multiplet s2,= (Q, A@$) and in the Lorentz sector a scalar

ghost &. In order to construct conveniently the action of the BRS symmetry on all fields, one may use the formalism

of refs. [ 2,3]. Let s be the generator of the BRS symmetry. We define

a=d.x+ a/ax+ +tiPaldxP +s )

p+ Fe+ +i,e+ , $ Ee- +i,e- , @@ ~y@+x@ , QQ ~:‘y~+x~, Cijeo+Q,. (2)

One finds that the action of s on the relevant field is consistently defined by the following equations, the derivation of which will be displayed in a separate publication [ 51:

F+ sag+ $&3p+ _tip@p@ =QP+ , FT- sag- _$jzp _iQQpQ =Qp- ,

pO~a~~+t~a~=p+P~p+_~__iP+~Oa+/lOP++lSZ~~)

p”O~apO_t~~.O=P+P-p+_O_fP-~~a+/iOe”-+~Q~Q)

and

&~&+i~@fe-iQ@fe=Xa~“,

9”E(a+4s)f0+~“w=~~ea-~Qx__-tr;2f”,

~QE(a-~~)fQ+~Qw=~~pa_~Qx+_t~fQ,

*‘Ea;iw= -W,p”+i@a~~ -icz/Qy? -&J. (3)

Expanding eqs. (3) in ghost number determines a consistent action of s on all classical and ghost fields (i.e. with s*=O). The resulting expression is such that the fields which describe the world-sheet structure and the Neveu-Ramond-Schwarz fields have BRS variations depending on the superWey1 ghosts. Moreover, no factorization is explicit.

We now introduce the following field redefinitions which generalize to the supersymmetric case the Beltrami parametrization of ref. [ 11:

e+=exp(p+)E+ , e=exp((o-)E- , E+ =dx++p+dx- , E- =dx-+p-dx+ ,

t,@ =exp( $q+)(l@E+ Soled_-) , !+vQ =exp( fp-)(A”E- +aQd_x+) ,

Z+ E&E’ =tt +p+r- , 3- =i,E- =r- +p-r’ ,

x@ =exp($a,+)(K@ +AQE+ ) , x0 =exp(fp-)(K@ +laZ-),

x=X, w=exp(-t@)W, fQ=exp(-_4-)FQ, f@=exp(-fy,+)FO.

It is useful to define

(4)

As in ref. [ l] we assume that ,D + is restricted to the inside of the unit disk so that the metrics is nowhere singular. By inserting the field redefinitions (4) in the BRS equations (3), one finds that the BRS transformations of the fields X, PO, W, ,u’, a@, 3’, K@ do not depend on the ghosts (.Q,n@J?) while CD,= (@J@,u) transforms as a scalar multiplet with a non-homogeneous term containing the Weyl superghost Q,. Moreover the BRS transformation laws of p’, a 0, E’ , K@ do not depend on 0,. The action of s on these

fields is indeed given by the following equations:

sP+ r&3+ -,u+d+Z+ +E+d+p+ +2iKQa0 ,

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Volume 198, number 3 PHYSICS LETTERS B 26 November 1987

~~~=a_~~-~+a+~~+f~~a+~+++B+a+~~-~ff~a+~+,

-+ ss =E+a +Z++iK@K@, sKe=Z+d,K@+~Kadd.Z:“t, (5a)

sp- =a,zp -p-a_5- +z-a_p- -2iKeaa,

saO=a+~O-~~a~~O+~~Oa_~-+$~a_crO-faQa_~-,

-_ I_ -- sfi =; a -3 -iKeKe, sK@=~-&K~-/-~K’O&~:“. (5b)

Eqs. (5a), (5b) show the announced Weyl invariance and factorization properties of the fields ,u ‘, c@, .Z’, K@. We call these fields superBeltrami variables. We can define, as in ref. [ 11, unified objects:

b+ =fi+&- +z+ , p- =p-&+ +5-, &@=a@&- +KQ, &riB=a!B&+ +KQ,

a_ =dx-a/ax- +s, Cl+ =dx+aiax+ +s. (6)

Then, eqs. (5) can be expressed under the following compact form:

d_g+-~+a+fi+-i&@~@=O, a~a~_~+a+a~_I~~a+~+=o,

a+~P_p-a_p-+iaQ&QOOo ;i+&Q-p~a_~Q-q~Qa~p-=o. (7)

The BRS equations can be written in superfield notations if we introduce spinorial variables 8@ and eO. We

define

A+ =b+ +2i6@&@, AT- =p- +2ieQde.

and the covariant derivatives

(8)

D, =a/aeQ +ie@a+ , De =a/ae@ -ieea_ . (9)

One has D& =id+ ,D& = -id_. We can now write the BRS equations (5) under a superfield form:

;i_A+-~+a+A++aiD~~+D~~+=O, a+~--_~a_~--aiD,~-D,~-=O. (10)

The transformation laws of the components of the @‘, are lengthy to write. Since we shall see that these components do not appear in the classical lagrangian, we postpone their expression to ref. [ 51, where the derivation of eqs. (3) through the resolution of Bianchi identities in superspace will be detailed also. The superstring classical action can be expressed solely in terms of the variables (X,F, w) and of cy and ,LL. One gets the following action, invariant under the full BRS symmetry defined in eq. (3).

qa, -p-a_)x.(a_ -~+d+)x+2i@P.(d_ -p+a+)x-2it_PFQ.(a+ -p-a_)x+2@0PF@~FQ]

- d2x[iFya+ -p-a_)F@-iFya_ -p+a+)F~+ww]. I (11) r

This action could be deduced from the one of Brink, di Vecchia and Howe [ 61 by the change of variables of

eq. (5). The components of the classical stress-energy tensor T,lj and supersymmetry current SC9 are obtained by

differentiating YCI with respect to p”’ and a@. One gets

T; =69,,/6,u+ =-D,+X.D,+X-iFQ.a+FQ , T, =~JJ~,/~,L- = -D,_X.D,[email protected]_F@ ,

Sz =6c%a,,li5a” = -2iFQ.D,+X, 63: =64,,/6@ =2iF@*D,_X, (12)

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D,_X= l_jipP [(a_ -p+a+)X-ia”Po -ifi+a@F@] . (13)

To quantize the superstring action, i.e. to gauge fix consistently the world-sheet variables, we simply mimic our solution for the purely bosonic case. We introduce the antighosts 2’ and K@ associated to Z’ and

K@, the antighosts (O,;i@ $) associated to (Q,n@,jI), and the corresponding Sttickelberg type auxiliary fields b i-, b@ and b, da, d. The action of s on these fields is

&’ =b’ , &@ =@, &=b, s;i@=d@, sjj=d,

sb’=O, sb@=O, sb=O, sd@=O, sd=O. (14)

A general background for the superstring is defined by a set of BRS invariant functions

& > 4% @cl, 47, a,. We have implicitly used Lorentz invariance and imposed VI+ =IJ,. Knowing the BRS equations, we can gauge fix in a BRS invariant way all two-dimensional supergravity dynamical fields which describe the world-sheet by adding to the classical action the following s-exact gauge fixing action:

+a(~-~,)+nQ(nQ-ag)+;iQ(aQ-aOO)+p(a-a,)] . >

(15)

Since the fields ( @J@,a) transform non-homogeneously, one finds after expansion of the action (15) that the equations of motion of Q, /1@, p are algebraic. Therefore all ghost fields from the superWey1 sector can be integrated out from the action 9C, + 9gf. The variations of all Sttickelberg type fields b yield the constraints ~~=y$,c@=c@,@=@O,J.@=@,a=aO.Theresultingactionis

,a,,,=4,,(x,I;,w,y,,~r,)- [d*xP(8_E’-p:a+z+ +E+a+p; -2ia,OK@)

2

- d2xK”(a~K~-~u,+a+K0+tK~a+~~+z+a+a,”-ta,oa+z+) s z

(16)

This action, where pO and a+, stand now for any given consistent background for the world-sheet metric and supersymmetric structure, is still invariant under the new nilpotent Slavnov operator

~p,+=a_z+-&a+E++z+a+j~$ -2icuFK@,

(17)

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ScYgO=a+Ke -~~a~KOffKQa_~Lg+~-a_a)oO-faoOa_~-,

-_ SG =~:“-a_~-_iK@K~, sK~=Z~a_K~+$K~&~-,

SY e+=o, & =o, sP=O, sP=O. (17 cont’d)

We have therefore found the possibility of expressing the superstring action in a general background while imposing a nilpotent Slavnov type symmetry. Other types of gauges could be constructed as well, for instance a superharmonic gauge. The technics developed in ref. [ l]for exploring the factorization properties could

therefore be repeated for the field theory induced by the action (16). One must of course take into account the fact that the equations of motion identify the gravitino antighost K@ as a differential of rank 312. A further gauge fixing procedure must therefore be introduced in order to cure the degeneracy introduced by the existence of 2g- 2 zero modes for the gravitino antighost in addition to that of 3g- 3 zero modes for the diffeomorphism antighost when C is a compact surface with genus g> 1. This additional gauge fixing procedure which must preserve the BRS invariance will be developed elsewhere. Besides, one has also an extra invariance of Y,,, due to the possibility of changing the backgrounds pLo and (Ye which generalizes in the supersymmetric case the one found in ref. [ 11. The form of this supersymmetry will be displayed in ref. [ 51, as well as the local version of

the BRS equations (17). The consistent anomaly can be expressed in function of the superBeltrami variables. Indeed, starting from

the BRS equations under the form (7), one easily gets the following identities:

&(a+p+a:p+ +4ia,&@a+&@)=a+(g+a+p+a:b+ -2in@a+&@a+@+ +4i,ii+d+Z@d+&@),

a+(a_p-alp- -4ia_~ia__~)=a_(p~a_p~aZp- +2i&QaPGaaP,iP -4iF-d_5°dPao). (18)

The consistent anomaly di is immediately identified from the part with ghost number two of eq. (18):

A;=A;’ +A;’ ,

A:‘= (a+s+ )a:p+ +4i(a+K@)a+a@, A;'=(a_Z-)alpp -4i(dPKa)a_aQ . (19)

One has indeed that sJrAl ‘d*x=O from eq. (18) and thus sJzA:d2x=0. One gets therefore a factorized form for the anomaly, as in the purely bosonic case. We must however precise that this form of the anomaly is not properly defined. If one inserts the anomaly JzA:d2x on the right-hand side of the Ward identity associated to the s-invariance, one interprets the anomaly phenomenon as the non-conservation of the stress-energy tensor S4,,,,lS,uu,’ and of the supersymmetry current 6Ya,,,/6cu@. The single overall anomaly coefficient can be com- puted either from af 6 *r/S,& Sp$ or from d + 6 *r/6 0$6 I$ , and is of course proportional to D - 10.

The Schwinger type term (i.e. the second co-cycle) which represents the anomaly in a hamiltonian version of the theory is determined by the term with ghost number two in the object a+,L’ a$p + +4id+ &@ d + d @. It

can be expressed as

P P A:= {dx+[(a+E+)a:E+ +4i(a+K@)a+KQ] +dx-[(a_zp)a5ap -4i(a_P)aPKe]}. (20)

The diffeomorphism and supersymmetric invariant but non superWey1 invariant counterterm action which should be added to Ylol for restoring superWey1 invariance at the renormalized level when one uses a non- invariant regulator is the following one:

(21)

Due to the gauge fixing terms in Y,,, a must be set equal to its background value a,. The counterterm (2 1) is the supersymmetric generalization of the cosmological term which occurs in bosonic string theory.

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To summarize the results contained in this note, we have generalized for the superstring theory the so-called Beltrami parametrization. This yields the factorization in two independent sectors of the superWey1 inert sector of the bidimensional supersymmetry, diffeomorphism, superWey1 gauge symmetry. As a consequence, the technics which have been introduced in ref.

References

[ 1 ] L. Baulieu and M. Bellon, Phys. Lett. B 196 (I 987) 142.

[2] L. Baulieu and M. Bellon, Nucl. Phys. B 266 (1985) 75.

[3] L. Baulieu, M. Bellon and R. Grimm, Nucl. Phys. B 294 (1987) 279.

[ 41 P. Friedan, E. Martinet and S. Shenker, Phys. Lett. B 160 (1985) 55; Nucl. Phys. B 271 (1986) 93;

A. Belavin and V. Knizhnik, Phys. Lett. B 168 (1986) 201;

A. Belavin, A.M. Polyakov and A. Zamolodchikov, Nucl. Phys. B 241 (1984) 333.

[ 51 L. Baulieu, M. Bellon and R. Grimm, in preparation.

[ 61 L. Brink, P. di Vecchia and P. Howe, Phys. Lett. B 65 (I 976) 47 I;

A.M. Polyakov, Phys. Lett. B 103 (1981) 211.

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Beltrami parametrization for superstrings