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Critical dimensions for strungs on group-manifolds from path-integral methods
Volume 189, number 1,2
PHYSICSLETTERSB
30 April 1987
CRITICAL D I M E N S I O N S F o R STRINGS O N G R O U P - M A N I F O L D S FROM PATH-INTEGRAL M E T H O D S A. CERESOLE l, A. LERDA 1, p. P I Z Z O C H E R O 1 and P. VAN N I E U W E N H U I Z E N Institutefor TheoreticalPhysics, State Universityof New York at Stony Brook, Stony Brook, NY 11794, USA
Received 17 December 1986
We derive the critical dimensionsof bosonic string on SO(N) group-manifoldsfor arbitrary levelk, using path-integralmethods and non-abelian bosonization techniques. We obtain a recursive equation for the critical dimensions, whose solution is in agreement with the operator approach results. We also discuss the application of this method to the linear sigma models.
There are (at least) three ways to compute critical dimensions of strings: by operator methods [ 1,2], by requiring nilpotency of the BRST charge Q [ 3 ], and by using Fujikawa's method for path integrals [ 4-6 ]. In this letter we shall consider strings propagating in a curve background space M d × G , where Md is a ddimensional Minkowski spacetime and G is a compact group. The action on the group-manifold G is a non-linear sigma model with parallelizing torsion [ 7 ], with overall normalization constant k and we shall derive the critical dimensions d(crit) Of Md for arbitrary k from the path-integral method. The operator method starts from the Kac-Moody algebra of currents [ 8 ], computes the resulting central charge in the Virasoro algebra and equates it to the central charge that one would need in the Virasoro algebra in order to have closure of the Lorentz algebra in the light-cone gauge [ 1,2 ]. In this way one finds a critical dimension lower than 26 or 10 in the bosonic or fermionic case, respectively d(crit) = 2 6 - d i m G(1 + ½C2(G)k - t ) -~ , d(crit) = 1 0 - ~dim G -- ~dim G(1 + ½ C 2 ( G ) k - ' ) - 1 ,
(1)
where C2 (G) is the quadratic Casimir operator in the
A. Della Riccia fellow. 34
adjoint representation. For the normalization of the generators in this formula, see ref. [ 2 ]. In the pathintegral method, one works in the (super)conformal gauge and rescales all matter fields in little steps in such a way that (super)gravity decouples and the effective action for the (super)gravity fields, due to integrating the corresponding infinitesimal jacobians, vanishes in the critical dimension. We shall apply the path-integral method to bosonic non-linear sigma models with G = S O ( N ) , and generalize previous results for k = 1 and k = 2 [9,10,6] to all k. Our methods are general and can be applied to other choices for G and to supersymmetric non linear sigma models as well (for some examples see ref. [ 6 ]), but we shall restrict our attention here to G = SO (N). We shall use as input certain proposed non-abelian bosonization formulas [ 11,12 ], which convert complicated non-linear bosonic path-integrals into linear fermionic path-integrals containing NMajorana fermions minimally coupled to an SO(k) gauge field A, [ 13 ]. In particular we shall follow ref. [9] and assume that these path-integral equivalences remain valid when gravity is present. Our main result is that the QCD2 determinant obtained by rescaling the fermions in such a way that they decouple from A~, contributes to the critical dimension. (In the process we will extend flat space results for QCD2 determinants [14,15] to curved space.) Phrased differently: although one can can use the local phase and local chiral symmetries to classically gauge A, to zero [ 16,17 ], at the quantum level 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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the chiral anomaly does couple to gravity and the integrated chiral anomaly (i.e. the effective action) contributes to the trace anomaly. This seems to contradict claims in the literature concerning linear sigma models, which state that one must put A u = 0 in order to get the correct critical dimensions [16]. I f we would apply the methods we have developed for nonlinear sigma models to the vector fields A u in the N = 2 and N = 4 linear sigma models, we would indeed find different critical dimensions than d = 2 [ 18] and d = - 2 [ 19]. We trace this ambiguity for the linear sigma models to a well-known ambiguity in the operator approach with respect to normal ordering [20]. Namely, normal ordering with respect to fields yields d = 2 and d - = 2 , whereas normal ordering with respect to currents (which is necessary in non-linear sigma models, in order that the non-abelian bosonization formulas are valid [ 10]) yields different critical dimensions. Having related the ambiguity in our results for linear sigma models to an ambiguity in the operator approach, we then return to path-integrals. The chiral anomaly yields a contribution to the effective action given for k = 2 by [ 14 ]
F=exp(~nfd2xA,,(~-O~O~/O~2DA,
) . (2)
In this form, F apparently does not depend on gravity. However, Physically the non-local operator (OZOgz)-1 is expected to feel the effects of curved spacetime, and changing the integration variables from A u to the two scalar fields q and ~ into which a d--2 vector can be decomposed, one finds a new jacobian, which may depend on gravity, according to the choice for the integration measure of r/and ~. This ambiguity in the measure corresponds, thus, to the ambiguity in the operator method mentioned above. We shall argue that the vector fields in the N = 2 and N = 4 linear sigma models should not be treated like the vector fields in our non-linear N-- 1 models, but rather like the graviton. In that case the integration measure of r/and ~o is unity and we obtain unambiguously the correct results both for the linear and the non-linear models. Let us now give details. The path-integral for the non-linear bosonic sigma model reads
30 April 1987
Z= fdh exp[kW(h)] ,
+r(h)
(3)
,
where h e S O ( N ) , F(h) is the Wess-Zumino action, and k must be an integer [1 1 ]. We shall assume [ 9,13 ] that this path-integral is equivalent to the following path-integral with A u in the fundamental represent_at!on of SO(k) ( i = 1, ..., N; a,b= 1..... k)
Z = ~ dAudx exp( f d 2 x x / ~ \
X~,~ t~, r~7~a? m e It m ( 6 ab Olt+Ap a b ) X i b ] | i a,b
J
(4)
(since Z ia are Majorana fermions one may omit the spin connection). For our purpose, however, we must be more careful in specifying the integration measure. We shall use as integration variables, ~ = h (det e)t/z, ~u=A u and )~= x(det e) 1/2 [ 21 ], since with this choice gravitational anomalies are absent in both formulas (3) and (4). As usual, we choose the conformal gauge e" u =e~'O'u. From the d spacetime scalars and the coordinate ghosts and antighosts, one finds the following contribution to the effective action [ 4 ] ( ½ d - 13)ao, 1 ~ IZT~d
Oo= - = - : - - t d
2
x a(x)O~O~a(x) .
(5)
By rescaling the fermions in (4) such that they decouple from the gravitational field a (x), one finds another contribution to the effective action equal to + ~ao per Majorana fermion, and for d = d ( c r i t ) the whole effective action vanishes [ 5 ]. We will now compute this contribution, which is due to the scalars in (3) when expressed into fermions and A u as in (4). We begin by writing A u in terms of SO (k) Lie-algebra valued hermitian scalars as A = (hi) -~(dh - 1), where h = e x p ( i q + y W ) [ 14]. Hence, in the conformal gauge, the action in (4) can be written as 35
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× (dh -1 e-"/2~ ') .
(6)
We will rescale ;~ such that the matrix h - 1 e - a / 2 is removed in little steps. To regulate the corresponding jacobians we use as regulator the square of the action [ 5,6 ]. Reverting temporarily to the covariant notation, we derive the following lemma for the evaluation of the jacobians f
d2 k
(__~__~) 2{e-igx exp[(l/M2)e-2~,~
= - (1/12rt)[ - 3M 2 e (4ot--2fl)o'(x)
-- ~y,U~VF/at
+ 2(Ot+fl)OxO~tT(X)] + O ( M -2) , where a
and # are constants, D = d + A ,
(7)
and
Fu~= OuA,-O,,Au+ [Au, A,]. This lemma extends the formula for A u = 0 as obtained in refs. [4,5 ]. The term with M2 cancels for supersymmetric models, but here we ignore it, since it does not influence the critical dimension but only leads to a renormalization of the cosmological constant. Under an infinitesimal rescaling of 2 in (6), we must compute the trace of dt(tr/2+i~/+ys~o) regulated as in (7) with at = ~ and fl = - ½, but using in (7), t r , = ( l - t ) t r instead of a, and A,=(ht*) -] ×(dht) -] instead of A, where h t = e x p [ ( l - t ) × (it/+ys~0)]. The result for the infinitesimal jacobian reads N l n J / = - l-~n dt f d2xTr[(½a+iq+ys~) t × ( - - ~ 1 /zipFu~,--½OaOaat)] ,
(8)
where the trace Tr is over spinor and group indices. This result splits into a purely gravitational term and a flat space Yang-Mills term, because the generators in the fundamental representation of SO(k) are traceless. This latter term is rewritten as in ref. [ 14 ] leading, aftert integrating over t, to the following expression for the total j acobian J = exp ( - f~ In J,) (where the minus sign is due to fermionic statistics) 36
30 April 1987
+2i f d t tr(AtUOurl)
-fdtTr(~,sA,~oAt))],
(9)
and where tr is the trace only over group indices. As argued in the introduction, if one were to express r/and ~o into Au, one would find non local operators which sense the gravitational field. Hence we expect that also the three last terms in (9) contain terms proportional to 0o. At this point our treatment of the case k = 2 and the case k> 2 bifurcates. For k=2, the last term in (9) vanishes, and Af = (1 - t ) A u, while Au= -iOj, q+i~u,O~to. The jacobian for the change from A u to (q, ~0) is proportional to det(0a0~), and exponentiating it by means of an anticommuting scalar ghost and antighost, this jacobian gives after introducing twiddled variables
J(A~q, ~0) = -Do.
(10)
The A-dependent terms in the jacobian in (9) reduce to (2) so that r/cancels, reflecting the original SO (k) invariance, and one is left with a term proportional to ~00a0~0. Using as integration variable O=~o(det e) 1/2, the removal of this gravity dependence yields a term + ½noand the jacobian in (9) becomes J = exp( ¼kN~o+ ½0o + a term with 00x0aO).
(11 )
We still have to the local SO(k) invariance and add corresponding ghosts. We choose the gauge 7/=0, which leads to algebraic SO (k) ghosts and antighosts which consequently do not contribute. Adding the jacobians for the rescaling of)~ and in (11 ) and the change of basis fromA u to t/and ~0in (10) to the result in (5), finally yields for the coefficient of 0o
(½d-13)+(~4kN+½)-l,
for k = 2 .
(12)
From (1) we find d(crit) = 2 6 - I N ( N - 1) × [I+¼C2(G)] -1 and since C z ( G ) = 2 ( N - 2 ) , we easily see that the critical dimensions obtained from
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(12) and from (1) agree. Our results agree with those of refs. [ 9,10 ]; even the separate contributions agree in 1-1 fashion, although derived in a different manner. In a previous publication [ 6 ] the gauge O~Au = 0 was used; in that gauge the ghosts, needed to exponentiate the d'alembertian above (10), were replaced by standard Faddeev-Popov ghosts and gave the same contribution as in (10). However, the extra contribution ½~ocoming from (2) (i.e., from replacing ~ by ~) was not taken into account. Let us now turn to the case k> 2. We pick up the discussion at (9), but since it would be very difficult to express At~ in terms of r/and ~ and to perform the t-integral explicitly, we instead choose a convenient gauge. We fix the gauge A + = 0, in which case the Adependent terms in (9) become proportional to W(h) in (3), but now with heSO(k), rather than heSO(N). Following for example section 4 of ref. [ 14], we find for the jacobian J i n (9) J = e x p [ ~kNOo - N W ( h ) ] ,
(13)
where the minus sign agrees with ref. [22]. We next pass from the integration variable A_ to h. According to Polyakov and Wiegmann [22], the jacobian for this basis change is again proportional to W(h), but now scaled by a factor C2(G = SO(k)) = 2 ( k - 2). Thus J(A _ ~ h ) =exp[ - 2 ( k - 2 )
W(h)].
(14)
Again we must consider the ghosts corresponding to the gauge A + = 0. The ghost action becomes C*O + C. Scaling C = ~ ( d e t e) -~/2 one obtains the action 6-'0 + ( ~(det e) - ~/2). To remove the gravitational dependence we scale ~ ( d e t e) 1/2 and using as regulator the iterated action exp( - crt)O_0+ exp( - a t ) M -2, with 0_O+ =Oaa, we find J(ghosts) =exp[ - ½ k ( k - I )0o] .
(15)
Adding the results in ( 13 ) - (15 ) we find an iterative equation for the coefficient of 0o. Namely, denoting the total coefficient of 0ocoming from (3), or equivalently (4), by C(N, k), we arrive at the equation C(N, k) = ~ k N - C( Ikl , - N - 2 ( k - 2 ) ) -½k(k-l).
(16)
For k = 1, C(N, k) = ~Nbecause in this case, according to Witten [ 11 ], (3) is equivalent to Nfee Major-
30 April 1987
ana spinors. With this boundary value we can solve (16 ) and find 2C(N,k)=½N(N-1)[I+N-2)/k]
-~
(17)
Comparing the critical dimensions obtained from (17) with (1), and using C2(SO(N)) = 2 ( N - 2 ) , we evidently have obtained agreement. The last issue to be discussed concerns the critical dimensions of linear sigma models containing vector fields. Consider for instance the N = 2 model [23]. From the complex scalars, Dirac spinors and real coordinate and complex supersymmetry (anti) ghosts, one finds [ 5,6 ] ( d + ½d- 13 + 11)0o.
(18)
In previous analyses, the contribution of At, was ignored [ 6 ], while the Maxwell ghosts gave a contribution -0o. (In ref. [ 5] the Maxwell ghost and antighost each gave -½, while in ref. [6] a simpler computation gave twice - ½for the ghost alone.) This leads with (18) to d ( c r i t ) = 2 . However, let us now look more closely at the role ofA~. Rescaling the fermions as in (9), but with k = 2 (because the N = 2 model has a group SO(2)) and N = d (because there are d Dirac fermions, which form d Majorana SO (2) doublets), we find the following total contribution due to A~. From the transition from Aa to (r/, q0 we find, as in (10), -0o; 'this is the same contribution as in refs. [5,6] due to the Faddeev-Popov ghosts for the gauge A~,= 0. Extra now, however, would be the effect due to the replacing (a by #, which yields + ½0o as indicated in (11 ). Not replacing ~ by # would lead one back to d(crit) = 2, but with this effect we instead find d(crit) = ]. It is interesting that we can obtain these two different results also in the operator approach with suitable choices of normal orderings. Using the ideas of refs. [ 13,20], one can write the Virasoro generators of the fermionic theory either ordering the interaction part with respect to the fundamental oscillators of the fermions or with respect to the gauge current modes. According to ref. [20] the first choice corresponds to "switch off" the interaction Aa and indeed yields a central charge in the Viraso algebra, which gives d ( c r i t ) = 2 upon requesting the closure of the Lorentz algebra in the light-cone gauge. On the other hand, the second normal ordering prescription yields a central charge consistent with d(crit) = ~. While in 37
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the N = 2 linear sigma model the argument of ref. [ 16 ] suggests the use of the n o r m a l ordering with respect to the Fermi fields, in our original bosonic non-linear sigma model (3) one must use the other prescription, i.e. normal ordering with respect to the current modes, because, as pointed out in refs. [ 13,10 ], only in this way the fermionized model in (4) is equivalent to the bosonic one in (3). However, for the linear sigma models there is really no ambiguity. The scalars ¢ which are the left-overs from the vectors A~, belong before rescaling to the following rigid N = 2 supersymmetry multiplet, containing also the conformal parts of the graviton and gravitino
L = ( ½etrOa Oxtr+ ~u'TdT"g/ + ~0~ 0 ~ ) •
(19)
It is the overall coefficient of this expression which determines the critical dimension. Indeed, in the critical d i m e n s i o n all extra modes from supergravity should vanish, and if one were to first rescale ~ a n d then require that the resulting modified coefficient of aO~O~a were to vanish, one would still find extra modes in the theory. Thus, for the linear sigma models one should not rescale ~, and one may indeed ignore the contribution from A~. In the non-linear N = 1 model instead, the C-fields are matter, not part of the supergravity multiplet, a n d in that case one should rescale them as we have done. It is a pleasure to thank K. Fujikawa, L. Mczincescu a n d H.J. Schnitzer for useful discussions.
References [ 1] D. Nemeschanskyand S. Yankielovicz,Phys. Rev. Lett. 54 (1985) 620. [2] E. Bergshoeff, S. Randjbar-Daemi, A. Salam, H. Sarmadi and E. Sezgin,Nucl. Phys. B 269 (1986) 77.
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[3] M.K. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443. [4] K. Fuijkawa, Phys. Rev. D 25 (1982) 2584. [5] P. Bouwknegt and P. van Nieuwenhuizen, Class. Quant. Gray. 3 (1986) 207. [6] A. Eastaugh, L. Mezincescu,E. Sezginand P. van Nieuwenhuizen, Phys. Rev. Lett. 57 (1986) 29. [7] T. Curtrightand C. Zachos,Phys. Rev. Lett. 53 (1984) 1799. [8] V.G. Knizhnik and A.B. Zamolodchikov,Nucl. Phys. B 247 (1984) 83. [9] A.N. Redlich and H.J. Schnitzer, Phys. Lett. B 167 (1986) 315. [10] D. Chang, A. Kumar and R.N. Mohapatra, Z. Phys. C 32 (1986) 417. [ 11 ] E. Witten, Commun. Math. Phys. 92 (1984) 455. [ 12] S. Coleman, Phys. Rev. D 11 (1975) 2088; S. Mandelstam, Phys. Rev. D 11 (1975) 3026; A.M. Polyakov and P.B. Wiegrnann, Phys. Lett. B 131 (1983) 121;B 141 (1984)223; P. Di Vecchia, B. Durhuus and J.L. Petersen, Phys. Lett. B 144 (1984) 245; D. Gonzales and A.N. Redlich, Phys. Lett. B 147 (1984) 150; C.M. Naon, Phys. Rev. D 31 (1985) 2035. [ 13] I. Antoniadis and C. Bachas, Nucl. Phys. B 278 (1986) 343. [ 14] K.D. Rothe, Nucl. Phys. B 269 (1986) 269; O. Alvarez,Nucl. Phys. B 238 (1984) 61; R.I. Nepomechie, Ann. Phys. 158 (1984) 67. [15] R.E. Gamboa Saravi, F.A. Schaposnik and J.E. Solomin, Nucl. Phys. B 185 (1981) 239; R. Roskies and F.A. Schaposnik, Phys. Rev. D 23 (1981) 558. [ 16] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. B 106 (1981) 63. [ 17] P. van Nieuwenhuizen,Int. J. Mod. Phys. A 1 (1986) 155. [ 18] M. Ademollo, L. Brink, A. D'Adda, R. D'Auria, N. Napolitano, S. Seiuto, E. Del Giudice, P. Di Vecchia, S. Ferrara, F. Gliozzi, R. Pettorino and J.H. Sehwarz, Nucl. Phys. B 111 (1976) 77. [19] M. Pernici and P. van Nieuwenhuizen, Phys. Lett. B 169 (1986) 381. [20] P. Doddard and D. Olive,Nucl, Phys. B 257 (FS14) (1985) 226. [21 ] K. Fujikawa, Nucl. Phys. B 226 (1983) 437; K. Fujikawa and O. Yasuda, Nucl. Phys. B 245 (1984) 436. [22] A.M. Polyakov and P.B. Wiegmann, Phys. Lett. B 131 (1983) 121;B 141 (1984) 223. [23] L. Brink and J.H. Schwarz, Nucl. Phys. B 121 (1977) 285.
PHYSICSLETTERSB
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CRITICAL D I M E N S I O N S F o R STRINGS O N G R O U P - M A N I F O L D S FROM PATH-INTEGRAL M E T H O D S A. CERESOLE l, A. LERDA 1, p. P I Z Z O C H E R O 1 and P. VAN N I E U W E N H U I Z E N Institutefor TheoreticalPhysics, State Universityof New York at Stony Brook, Stony Brook, NY 11794, USA
Received 17 December 1986
We derive the critical dimensionsof bosonic string on SO(N) group-manifoldsfor arbitrary levelk, using path-integralmethods and non-abelian bosonization techniques. We obtain a recursive equation for the critical dimensions, whose solution is in agreement with the operator approach results. We also discuss the application of this method to the linear sigma models.
There are (at least) three ways to compute critical dimensions of strings: by operator methods [ 1,2], by requiring nilpotency of the BRST charge Q [ 3 ], and by using Fujikawa's method for path integrals [ 4-6 ]. In this letter we shall consider strings propagating in a curve background space M d × G , where Md is a ddimensional Minkowski spacetime and G is a compact group. The action on the group-manifold G is a non-linear sigma model with parallelizing torsion [ 7 ], with overall normalization constant k and we shall derive the critical dimensions d(crit) Of Md for arbitrary k from the path-integral method. The operator method starts from the Kac-Moody algebra of currents [ 8 ], computes the resulting central charge in the Virasoro algebra and equates it to the central charge that one would need in the Virasoro algebra in order to have closure of the Lorentz algebra in the light-cone gauge [ 1,2 ]. In this way one finds a critical dimension lower than 26 or 10 in the bosonic or fermionic case, respectively d(crit) = 2 6 - d i m G(1 + ½C2(G)k - t ) -~ , d(crit) = 1 0 - ~dim G -- ~dim G(1 + ½ C 2 ( G ) k - ' ) - 1 ,
(1)
where C2 (G) is the quadratic Casimir operator in the
A. Della Riccia fellow. 34
adjoint representation. For the normalization of the generators in this formula, see ref. [ 2 ]. In the pathintegral method, one works in the (super)conformal gauge and rescales all matter fields in little steps in such a way that (super)gravity decouples and the effective action for the (super)gravity fields, due to integrating the corresponding infinitesimal jacobians, vanishes in the critical dimension. We shall apply the path-integral method to bosonic non-linear sigma models with G = S O ( N ) , and generalize previous results for k = 1 and k = 2 [9,10,6] to all k. Our methods are general and can be applied to other choices for G and to supersymmetric non linear sigma models as well (for some examples see ref. [ 6 ]), but we shall restrict our attention here to G = SO (N). We shall use as input certain proposed non-abelian bosonization formulas [ 11,12 ], which convert complicated non-linear bosonic path-integrals into linear fermionic path-integrals containing NMajorana fermions minimally coupled to an SO(k) gauge field A, [ 13 ]. In particular we shall follow ref. [9] and assume that these path-integral equivalences remain valid when gravity is present. Our main result is that the QCD2 determinant obtained by rescaling the fermions in such a way that they decouple from A~, contributes to the critical dimension. (In the process we will extend flat space results for QCD2 determinants [14,15] to curved space.) Phrased differently: although one can can use the local phase and local chiral symmetries to classically gauge A, to zero [ 16,17 ], at the quantum level 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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the chiral anomaly does couple to gravity and the integrated chiral anomaly (i.e. the effective action) contributes to the trace anomaly. This seems to contradict claims in the literature concerning linear sigma models, which state that one must put A u = 0 in order to get the correct critical dimensions [16]. I f we would apply the methods we have developed for nonlinear sigma models to the vector fields A u in the N = 2 and N = 4 linear sigma models, we would indeed find different critical dimensions than d = 2 [ 18] and d = - 2 [ 19]. We trace this ambiguity for the linear sigma models to a well-known ambiguity in the operator approach with respect to normal ordering [20]. Namely, normal ordering with respect to fields yields d = 2 and d - = 2 , whereas normal ordering with respect to currents (which is necessary in non-linear sigma models, in order that the non-abelian bosonization formulas are valid [ 10]) yields different critical dimensions. Having related the ambiguity in our results for linear sigma models to an ambiguity in the operator approach, we then return to path-integrals. The chiral anomaly yields a contribution to the effective action given for k = 2 by [ 14 ]
F=exp(~nfd2xA,,(~-O~O~/O~2DA,
) . (2)
In this form, F apparently does not depend on gravity. However, Physically the non-local operator (OZOgz)-1 is expected to feel the effects of curved spacetime, and changing the integration variables from A u to the two scalar fields q and ~ into which a d--2 vector can be decomposed, one finds a new jacobian, which may depend on gravity, according to the choice for the integration measure of r/and ~. This ambiguity in the measure corresponds, thus, to the ambiguity in the operator method mentioned above. We shall argue that the vector fields in the N = 2 and N = 4 linear sigma models should not be treated like the vector fields in our non-linear N-- 1 models, but rather like the graviton. In that case the integration measure of r/and ~o is unity and we obtain unambiguously the correct results both for the linear and the non-linear models. Let us now give details. The path-integral for the non-linear bosonic sigma model reads
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Z= fdh exp[kW(h)] ,
+r(h)
(3)
,
where h e S O ( N ) , F(h) is the Wess-Zumino action, and k must be an integer [1 1 ]. We shall assume [ 9,13 ] that this path-integral is equivalent to the following path-integral with A u in the fundamental represent_at!on of SO(k) ( i = 1, ..., N; a,b= 1..... k)
Z = ~ dAudx exp( f d 2 x x / ~ \
X~,~ t~, r~7~a? m e It m ( 6 ab Olt+Ap a b ) X i b ] | i a,b
J
(4)
(since Z ia are Majorana fermions one may omit the spin connection). For our purpose, however, we must be more careful in specifying the integration measure. We shall use as integration variables, ~ = h (det e)t/z, ~u=A u and )~= x(det e) 1/2 [ 21 ], since with this choice gravitational anomalies are absent in both formulas (3) and (4). As usual, we choose the conformal gauge e" u =e~'O'u. From the d spacetime scalars and the coordinate ghosts and antighosts, one finds the following contribution to the effective action [ 4 ] ( ½ d - 13)ao, 1 ~ IZT~d
Oo= - = - : - - t d
2
x a(x)O~O~a(x) .
(5)
By rescaling the fermions in (4) such that they decouple from the gravitational field a (x), one finds another contribution to the effective action equal to + ~ao per Majorana fermion, and for d = d ( c r i t ) the whole effective action vanishes [ 5 ]. We will now compute this contribution, which is due to the scalars in (3) when expressed into fermions and A u as in (4). We begin by writing A u in terms of SO (k) Lie-algebra valued hermitian scalars as A = (hi) -~(dh - 1), where h = e x p ( i q + y W ) [ 14]. Hence, in the conformal gauge, the action in (4) can be written as 35
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× (dh -1 e-"/2~ ') .
(6)
We will rescale ;~ such that the matrix h - 1 e - a / 2 is removed in little steps. To regulate the corresponding jacobians we use as regulator the square of the action [ 5,6 ]. Reverting temporarily to the covariant notation, we derive the following lemma for the evaluation of the jacobians f
d2 k
(__~__~) 2{e-igx exp[(l/M2)e-2~,~
= - (1/12rt)[ - 3M 2 e (4ot--2fl)o'(x)
-- ~y,U~VF/at
+ 2(Ot+fl)OxO~tT(X)] + O ( M -2) , where a
and # are constants, D = d + A ,
(7)
and
Fu~= OuA,-O,,Au+ [Au, A,]. This lemma extends the formula for A u = 0 as obtained in refs. [4,5 ]. The term with M2 cancels for supersymmetric models, but here we ignore it, since it does not influence the critical dimension but only leads to a renormalization of the cosmological constant. Under an infinitesimal rescaling of 2 in (6), we must compute the trace of dt(tr/2+i~/+ys~o) regulated as in (7) with at = ~ and fl = - ½, but using in (7), t r , = ( l - t ) t r instead of a, and A,=(ht*) -] ×(dht) -] instead of A, where h t = e x p [ ( l - t ) × (it/+ys~0)]. The result for the infinitesimal jacobian reads N l n J / = - l-~n dt f d2xTr[(½a+iq+ys~) t × ( - - ~ 1 /zipFu~,--½OaOaat)] ,
(8)
where the trace Tr is over spinor and group indices. This result splits into a purely gravitational term and a flat space Yang-Mills term, because the generators in the fundamental representation of SO(k) are traceless. This latter term is rewritten as in ref. [ 14 ] leading, aftert integrating over t, to the following expression for the total j acobian J = exp ( - f~ In J,) (where the minus sign is due to fermionic statistics) 36
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+2i f d t tr(AtUOurl)
-fdtTr(~,sA,~oAt))],
(9)
and where tr is the trace only over group indices. As argued in the introduction, if one were to express r/and ~o into Au, one would find non local operators which sense the gravitational field. Hence we expect that also the three last terms in (9) contain terms proportional to 0o. At this point our treatment of the case k = 2 and the case k> 2 bifurcates. For k=2, the last term in (9) vanishes, and Af = (1 - t ) A u, while Au= -iOj, q+i~u,O~to. The jacobian for the change from A u to (q, ~0) is proportional to det(0a0~), and exponentiating it by means of an anticommuting scalar ghost and antighost, this jacobian gives after introducing twiddled variables
J(A~q, ~0) = -Do.
(10)
The A-dependent terms in the jacobian in (9) reduce to (2) so that r/cancels, reflecting the original SO (k) invariance, and one is left with a term proportional to ~00a0~0. Using as integration variable O=~o(det e) 1/2, the removal of this gravity dependence yields a term + ½noand the jacobian in (9) becomes J = exp( ¼kN~o+ ½0o + a term with 00x0aO).
(11 )
We still have to the local SO(k) invariance and add corresponding ghosts. We choose the gauge 7/=0, which leads to algebraic SO (k) ghosts and antighosts which consequently do not contribute. Adding the jacobians for the rescaling of)~ and in (11 ) and the change of basis fromA u to t/and ~0in (10) to the result in (5), finally yields for the coefficient of 0o
(½d-13)+(~4kN+½)-l,
for k = 2 .
(12)
From (1) we find d(crit) = 2 6 - I N ( N - 1) × [I+¼C2(G)] -1 and since C z ( G ) = 2 ( N - 2 ) , we easily see that the critical dimensions obtained from
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(12) and from (1) agree. Our results agree with those of refs. [ 9,10 ]; even the separate contributions agree in 1-1 fashion, although derived in a different manner. In a previous publication [ 6 ] the gauge O~Au = 0 was used; in that gauge the ghosts, needed to exponentiate the d'alembertian above (10), were replaced by standard Faddeev-Popov ghosts and gave the same contribution as in (10). However, the extra contribution ½~ocoming from (2) (i.e., from replacing ~ by ~) was not taken into account. Let us now turn to the case k> 2. We pick up the discussion at (9), but since it would be very difficult to express At~ in terms of r/and ~ and to perform the t-integral explicitly, we instead choose a convenient gauge. We fix the gauge A + = 0, in which case the Adependent terms in (9) become proportional to W(h) in (3), but now with heSO(k), rather than heSO(N). Following for example section 4 of ref. [ 14], we find for the jacobian J i n (9) J = e x p [ ~kNOo - N W ( h ) ] ,
(13)
where the minus sign agrees with ref. [22]. We next pass from the integration variable A_ to h. According to Polyakov and Wiegmann [22], the jacobian for this basis change is again proportional to W(h), but now scaled by a factor C2(G = SO(k)) = 2 ( k - 2). Thus J(A _ ~ h ) =exp[ - 2 ( k - 2 )
W(h)].
(14)
Again we must consider the ghosts corresponding to the gauge A + = 0. The ghost action becomes C*O + C. Scaling C = ~ ( d e t e) -~/2 one obtains the action 6-'0 + ( ~(det e) - ~/2). To remove the gravitational dependence we scale ~ ( d e t e) 1/2 and using as regulator the iterated action exp( - crt)O_0+ exp( - a t ) M -2, with 0_O+ =Oaa, we find J(ghosts) =exp[ - ½ k ( k - I )0o] .
(15)
Adding the results in ( 13 ) - (15 ) we find an iterative equation for the coefficient of 0o. Namely, denoting the total coefficient of 0ocoming from (3), or equivalently (4), by C(N, k), we arrive at the equation C(N, k) = ~ k N - C( Ikl , - N - 2 ( k - 2 ) ) -½k(k-l).
(16)
For k = 1, C(N, k) = ~Nbecause in this case, according to Witten [ 11 ], (3) is equivalent to Nfee Major-
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ana spinors. With this boundary value we can solve (16 ) and find 2C(N,k)=½N(N-1)[I+N-2)/k]
-~
(17)
Comparing the critical dimensions obtained from (17) with (1), and using C2(SO(N)) = 2 ( N - 2 ) , we evidently have obtained agreement. The last issue to be discussed concerns the critical dimensions of linear sigma models containing vector fields. Consider for instance the N = 2 model [23]. From the complex scalars, Dirac spinors and real coordinate and complex supersymmetry (anti) ghosts, one finds [ 5,6 ] ( d + ½d- 13 + 11)0o.
(18)
In previous analyses, the contribution of At, was ignored [ 6 ], while the Maxwell ghosts gave a contribution -0o. (In ref. [ 5] the Maxwell ghost and antighost each gave -½, while in ref. [6] a simpler computation gave twice - ½for the ghost alone.) This leads with (18) to d ( c r i t ) = 2 . However, let us now look more closely at the role ofA~. Rescaling the fermions as in (9), but with k = 2 (because the N = 2 model has a group SO(2)) and N = d (because there are d Dirac fermions, which form d Majorana SO (2) doublets), we find the following total contribution due to A~. From the transition from Aa to (r/, q0 we find, as in (10), -0o; 'this is the same contribution as in refs. [5,6] due to the Faddeev-Popov ghosts for the gauge A~,= 0. Extra now, however, would be the effect due to the replacing (a by #, which yields + ½0o as indicated in (11 ). Not replacing ~ by # would lead one back to d(crit) = 2, but with this effect we instead find d(crit) = ]. It is interesting that we can obtain these two different results also in the operator approach with suitable choices of normal orderings. Using the ideas of refs. [ 13,20], one can write the Virasoro generators of the fermionic theory either ordering the interaction part with respect to the fundamental oscillators of the fermions or with respect to the gauge current modes. According to ref. [20] the first choice corresponds to "switch off" the interaction Aa and indeed yields a central charge in the Viraso algebra, which gives d ( c r i t ) = 2 upon requesting the closure of the Lorentz algebra in the light-cone gauge. On the other hand, the second normal ordering prescription yields a central charge consistent with d(crit) = ~. While in 37
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the N = 2 linear sigma model the argument of ref. [ 16 ] suggests the use of the n o r m a l ordering with respect to the Fermi fields, in our original bosonic non-linear sigma model (3) one must use the other prescription, i.e. normal ordering with respect to the current modes, because, as pointed out in refs. [ 13,10 ], only in this way the fermionized model in (4) is equivalent to the bosonic one in (3). However, for the linear sigma models there is really no ambiguity. The scalars ¢ which are the left-overs from the vectors A~, belong before rescaling to the following rigid N = 2 supersymmetry multiplet, containing also the conformal parts of the graviton and gravitino
L = ( ½etrOa Oxtr+ ~u'TdT"g/ + ~0~ 0 ~ ) •
(19)
It is the overall coefficient of this expression which determines the critical dimension. Indeed, in the critical d i m e n s i o n all extra modes from supergravity should vanish, and if one were to first rescale ~ a n d then require that the resulting modified coefficient of aO~O~a were to vanish, one would still find extra modes in the theory. Thus, for the linear sigma models one should not rescale ~, and one may indeed ignore the contribution from A~. In the non-linear N = 1 model instead, the C-fields are matter, not part of the supergravity multiplet, a n d in that case one should rescale them as we have done. It is a pleasure to thank K. Fujikawa, L. Mczincescu a n d H.J. Schnitzer for useful discussions.
References [ 1] D. Nemeschanskyand S. Yankielovicz,Phys. Rev. Lett. 54 (1985) 620. [2] E. Bergshoeff, S. Randjbar-Daemi, A. Salam, H. Sarmadi and E. Sezgin,Nucl. Phys. B 269 (1986) 77.
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[3] M.K. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443. [4] K. Fuijkawa, Phys. Rev. D 25 (1982) 2584. [5] P. Bouwknegt and P. van Nieuwenhuizen, Class. Quant. Gray. 3 (1986) 207. [6] A. Eastaugh, L. Mezincescu,E. Sezginand P. van Nieuwenhuizen, Phys. Rev. Lett. 57 (1986) 29. [7] T. Curtrightand C. Zachos,Phys. Rev. Lett. 53 (1984) 1799. [8] V.G. Knizhnik and A.B. Zamolodchikov,Nucl. Phys. B 247 (1984) 83. [9] A.N. Redlich and H.J. Schnitzer, Phys. Lett. B 167 (1986) 315. [10] D. Chang, A. Kumar and R.N. Mohapatra, Z. Phys. C 32 (1986) 417. [ 11 ] E. Witten, Commun. Math. Phys. 92 (1984) 455. [ 12] S. Coleman, Phys. Rev. D 11 (1975) 2088; S. Mandelstam, Phys. Rev. D 11 (1975) 3026; A.M. Polyakov and P.B. Wiegrnann, Phys. Lett. B 131 (1983) 121;B 141 (1984)223; P. Di Vecchia, B. Durhuus and J.L. Petersen, Phys. Lett. B 144 (1984) 245; D. Gonzales and A.N. Redlich, Phys. Lett. B 147 (1984) 150; C.M. Naon, Phys. Rev. D 31 (1985) 2035. [ 13] I. Antoniadis and C. Bachas, Nucl. Phys. B 278 (1986) 343. [ 14] K.D. Rothe, Nucl. Phys. B 269 (1986) 269; O. Alvarez,Nucl. Phys. B 238 (1984) 61; R.I. Nepomechie, Ann. Phys. 158 (1984) 67. [15] R.E. Gamboa Saravi, F.A. Schaposnik and J.E. Solomin, Nucl. Phys. B 185 (1981) 239; R. Roskies and F.A. Schaposnik, Phys. Rev. D 23 (1981) 558. [ 16] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. B 106 (1981) 63. [ 17] P. van Nieuwenhuizen,Int. J. Mod. Phys. A 1 (1986) 155. [ 18] M. Ademollo, L. Brink, A. D'Adda, R. D'Auria, N. Napolitano, S. Seiuto, E. Del Giudice, P. Di Vecchia, S. Ferrara, F. Gliozzi, R. Pettorino and J.H. Sehwarz, Nucl. Phys. B 111 (1976) 77. [19] M. Pernici and P. van Nieuwenhuizen, Phys. Lett. B 169 (1986) 381. [20] P. Doddard and D. Olive,Nucl, Phys. B 257 (FS14) (1985) 226. [21 ] K. Fujikawa, Nucl. Phys. B 226 (1983) 437; K. Fujikawa and O. Yasuda, Nucl. Phys. B 245 (1984) 436. [22] A.M. Polyakov and P.B. Wiegmann, Phys. Lett. B 131 (1983) 121;B 141 (1984) 223. [23] L. Brink and J.H. Schwarz, Nucl. Phys. B 121 (1977) 285.