SISSA/ISAS 67/94/EP hepth@xxx/9405174 IFUM 469/FT May, 1994 1 CONSTRAINED TOPOLOGICAL FIELD THEORY arXiv:hep-th/9405174v2 22 Jun 1994 Damiano Anselmi, Pietro Fré International School for Advanced Studies (ISAS), via Beirut 2-4, I-34100 Trieste, Italia and Istituto Nazionale di Fisica Nucleare (INFN) - Sezione di Trieste, Trieste, Italia Luciano Girardello and Paolo Soriani Dipartimento di Fisica, Università di Milano, via Celoria 6, I-20133 Milano, Italia and Istituto Nazionale di Fisica Nucleare (INFN) - Sezione di Milano, Milano, Italia Abstract We derive a model of constrained topological gravity, a theory recently introduced by us through the twist of N=2 Liouville theory, starting from the general BRST algebra and imposing the moduli space constraint as a gauge fixing. To do this, it is necessary to introduce a formalism that allows a careful treatment of the global and the local degrees of freedom of the fields. Surprisingly, the moduli space constraint arises from the simplest and most natural gauge-fermion (antighost × Lagrange multiplier), confirming the previous results. The simplified technical set-up provides a deeper understanding for constrained topological gravity and a convenient framework for future investigations, like the matter coupling and the analysis of the effects of the constraint on the holomorphic anomaly. 1 Partially supported by EEC, Science Project SC1∗ -CT92-0789. 1  In a recent paper [1], we analysed the topological twist of N=2 supergravity in two dimensions and revealed some new features with respect to the known models of topo- logical gravity. The key point (that, to our knowledge, was not previously noticed in the literature) is that the gravitini must be U(1) charged with respect to the graviphoton in order to close off-shell the supersymmetry transformations2 . This fact has crucial effects: the graviphoton A appears, after the twist, in the BRST variation of some antighost and can be interpreted as a BRST Lagrange multiplier; moreover, the U(1) current is non- trivial and its vanishing projects the moduli space Mg of the genus g Riemann surfaces onto a homology cycle Vg ∈ H2g−3 (Mg ) of codimension g. Any topological field theory projects the functional integral onto the moduli space of some instantons. Consequently, the Riemann surfaces lying in Vg are to be named gravitational instantons in two dimensions. In the present paper, we construct a model of constrained topological gravity inde- pendently of any topological twist, that however captures the main suggestion springing from the twist of the N=2 theory. For the details of the twisted model as well as for many other technical points just alluded to in the present letter, the reader is referred to ref. [1]. Here, in any case, we also trace back the match with the model of ref. [1]. We get a theory whose formal structure is essentially the same as in the Verlinde and Verlinde model [2], but such that the correlation functions are calculated on Vg instead of Mg . As discussed in detail in ref. [1], constrained topological gravity is described by the gauge-free BRST algebra of SL(2, R), the same as in the Verlinde and Verlinde model [2], se± = ψ ± − ∇c± ∓ c0 e± , sψ ± = −∇γ ± ∓ c0 ψ ± ± γ0 e± ± ψ0 c± , sγ ± = ∓c0 γ ± ± γ0 c± , sc± = γ ± ∓ c0 c± , sω = ψ0 − dc0 , sψ0 = −dγ0 , sc0 = γ0 , sγ0 = 0, (1) but the gauge-fixing BRST algebra Bgauge−f ixing is enriched with an antighost one-form ψ̄, a Lagrange multiplier one form A and the respective gauge ghosts, γ and c, sψ̄ = A + dγ, sA = dc, sγ = c, sc = 0. (2) The possibility of introducing a BRST constraint on the moduli space is due to the fact that Bgauge−f ixing contains fields (ψ̄ and A) that possess global degrees of freedom: instead of enlarging the moduli space, they reduce it. Due to this, it is necessary to develop a formalism that permits to deal conveniently with the global degrees of freedom, together with the local ones3 . Let us introduce fiducial fields ê± , ψ̂ ± , ψ̄ˆ and  satisfying the “purely topological” BRST algebræ sê± = ψ̂ ± , sψ̂ ± = 0, sψ̄ˆ = Â, s = 0. (3) 2 The corresponding supersymmetry algebra was called by us charged Poincarè superalgebra. 3 For this problem see also [3]. 2 The hatted fields will represent the purely global degrees of freedom of the corresponding unhatted fields. For example, ê± represents the conformal class of the metric represented by e± . To clarify this point, it is useful to write down the gauge-fixings of diffeomorphisms and Lorentz rotations together with their fermionic counterparts, e+ ∧ ê+ = 0, e− ∧ ê− = 0, e+ ∧ ê− + e− ∧ ê+ = 0, ψ + ê+ = ψ̂ + e+ , ψ − ê− = ψ̂ − e− , ψ + ê− + ψ − ê+ = ψ̂ + e− + ψ̂ − e+ . (4) We can solve these gauge-fixing conditions by writing e± = eϕ ê± , ψ ± = eϕ (ηê± + ψ̂ ± ), (5) ϕ being the Liouville field and η its superpartner. We see that ê± represent the differ- ± + + entials dz and dz̄. Moreover, writing ψ̂ ± = ψ̂+ ± − ê + ψ̂− ê , we are lead to identify ψ̂− and ψ̂+ − with m̂i µiz ˆ iz̄ ˆ z̄ and m̄i µ̄z , respectively. Here m̂i and m̄i , i = 1, . . . , 3g − 3 are the iz iz̄ supermoduli, while µz̄ and µ̄z are the Beltrami differentials. On the other hand, writing ψ̄ˆ = ψ̄ˆ+ ê+ + ψ̄ˆ− ê− and  = Â+ ê+ + Â− ê− , we can identify ψ̄ˆ+ and ψ̄ˆ− with ν̂j ωzj and ν̄ˆj ω̄z̄j , while Â+ and Â− can be identified with νj ωzj and ν̄j ω̄z̄j . Here ωzj is a basis of g holomorphic differentials, g being the genus of the Riemann surface, and ω̄z̄j are their complex conjugates. Finally, νj and ν̄j are the global degrees of freedom of the U(1) gauge field A, while ν̂j and ν̄ˆ j are those of ψ̄. Formula (3) should be compared with formula (9.1) of [1]. The functional measure dµ contains the integration dµ̂ over the hatted fields, dµ̂ = dê+ dê− dψ̂ + dψ̂ − dÂdψ̄ˆ, (6) where dê+ dê− is the integration over the moduli space Mg of genus g Riemann surfaces Σg , while d is the integration over the moduli space of U(1) flat connections, which is the Jacobian variety Cg /(Zg + ΩZg ), Ω being the period matrix of Σg . dψ̂ + dψ̂ − and dψ̄ˆ are the integrations over the supermoduli. The above identifications between hatted fields and moduli will be made only in the final expressions: in the intermediate steps it is convenient to retain the hatted fields, in order to avoid concepts like the “field dependent points” z and z̄, that are unusual in quantum field theory. (i) (i) Let Ω(i) = Ω+ ê+ + Ω− ê− , i = 1, 2, be two one-forms such that sΩ(1) = Ω(2) . Then (1) (1) the formulæ for sΩ+ and sΩ− read (1) +(2) − (1) (1) sΩ+ = Ω+ − ψ̂+ Ω+ − ψ̂+ Ω− , (1) +(2) − (1) (1) sΩ− = Ω− − ψ̂− Ω+ − ψ̂− Ω− . (7) These expressions will be crucial when constraining the moduli space. Notice that even (1) when Ω(2) = 0, sΩ± are nonzero. 3  Another important point concerns the gauge-fixing of the local U(1) gauge symme- tries. In section VII of ref. [1] we invented a suitable trick in order to reach a complete chiral factorization between left and right moving sectors, at least in the limit when the cosmological constant tends to zero. Here, we need to use the same trick twice, once for ψ̄ and once for A. So, let us introduce two trivial BRST systems {Γ̄, ζ} and {ξ, c′ }, with sΓ̄ = ζ, sζ = 0, sξ = c′ , sc′ = 0. (8) The gauge-fixings of the γ and c symmetries, as well as the above two trivial symmetries are ψ̄± = ψ̄ˆ± ∓ ∇± Γ̄, A± = ± ∓ ∇± ξ, (9) where ∇± are such that the exterior derivative d is ê+ ∇+ + ê− ∇− . Finally, we need to fix the topological symmetries and this can be achieved by setting T ± = 0, R = a2 e+ e− , dψ̄ = 0, (10) T ± = de± ± ω ∧ e± being the torsions. The condition dψ̄ = 0 and its BRST variation dA = 0 do not depend on the global degrees of freedom ν and ν̂. The gauge-fixing for them (moduli space constraint) will be treated in detail later on. With obvious notation, the gauge fermions for the gauge-fixings that we have so far introduced are Ψ1 = π̄± T ± + π̄(R − a2 e+ e− ), Ψ2 = χdψ̄, Ψ3 = b++ e+ ê+ + b−− e− ê− + b+− (e+ ê− + e− ê+ ) + β++ (ψ + ê+ − ψ̂ + e+ ) +β−− (ψ − ê− − ψ̂ − e− ) + β+− (ψ + ê− + ψ − ê+ − ψ̂ + e− − ψ̂ − e+ ), Ψ = β (ψ̄ − ψ̄ˆ − ∇ Γ̄)ê+ ê− + b (A −  − ∇ ξ)ê+ ê− 4 + − − − + − − − +β− (ψ̄+ − ψ̄ˆ+ + ∇+ Γ̄)ê+ ê− + b− (A+ − Â+ + ∇+ ξ)ê+ ê− . (11) Let π = sπ̄ and λ = sχ. Focusing on the local degrees of freedom, the Lagrangian L = −s 4i=1 Ψi turns out to be, after some simple field redefinitions similar to those P discussed in section VII of ref. [1] and in the limit a2 → 0, L = π∂z ∂z̄ ϕ + χ∂z ∂z̄ ξ − π̄∂z ∂z̄ η + λ∂z ∂z̄ Γ̄ −bzz ∂z̄ cz − βzz ∂z̄ γ z − βz ∂z̄ γ − bz ∂z̄ c +bz̄ z̄ ∂z cz̄ + βz̄z̄ ∂z γ z̄ − βz̄ ∂z γ̄ − bz̄ ∂z c̄. (12) Notice that there are second order fermions, differently from the model of ref. [1]. The BRST charge Q can be easily found with the method of ref. [1], i.e. by means of a local BRST variation of the Lagrangian. One can then write Q = Qs + Qv , (13) 4 where Q2s = Q2v = {Qs , Qv } = 0. Explicitly, I Qs = −∂z πη − λ∂z ξ + bzz γ z − βz c, (14) while Qv is the same as in the Verlinde and Verlinde model plus −∂z χc−∂z λγ. To recover the correct energy momentum tensor, one has to perform the following redefinitions (left moving part) bzz → bzz − bz ∂z γ + ∂z χ∂z Γ̄, βz → βz + ∂z (bz cz ), ξ → ξ − ∂z Γ̄cz , λ → λ + ∂z χcz , c → c + ∂z γcz . (15) The operator product expansions among the fields are left unchanged (and so the form of the Lagrangian). The final BRST charge Q is Qs + Qv with Qs as before and I 1 1 1 Qv = −cz Tzz − γ z Gzz − cJz′ + γGz′ . (16) 2 2 2 One has4 grav 1 gh 1 Tzz = Tzz + Tzz , Gzz = Ggrav zz + Ggh , 2 2 zz grav grav grav Tzz = T(1)zz + T(2)zz , Ggrav zz = Ggrav grav (1)zz + G(2)zz , gh gh gh gh gh Tzz = T(1)zz + T(2)zz , Ggh zz = G(1)zz + G(2)zz , (17) and grav 1 grav T(1)zz = ∂z π∂z ϕ − ∂z2 π − ∂z π̄∂z η, T(2)zz = ∂z χ∂z ξ + ∂z λ∂z Γ̄, 2 Ggrav 2 (1)zz = ∂z π̄ − 2∂z π̄∂z ϕ, Ggrav (2)zz = −2∂z χ∂z Γ̄, gh gh T(1)zz = 2bzz ∂z cz + ∂z bzz cz + 2βzz ∂z γ z + ∂z βzz γ z , T(2)zz = βz ∂z γ + bz ∂z c, Ggh z z (1)zz = −4βzz ∂z c − 2∂z βzz c , Ggh (2)zz = 2bz ∂z γ. (18) On the other hand, the U(1) current Jz′ = 2{Qv , bz } and the associated supercurrent G′z = −2[Qv , βz ] are trivial: Jz′ = 2∂z χ + 2∂z (bz cz ), Jz′ = 2∂z χ + ∂z (bz cz ), G′z = −2∂z λ + 2∂z (βz cz ) + 2∂z (bz γ z ), Gz′ = −2∂z λ + ∂z (βz cz ) + ∂z (bz γ z ), (19) in the sense that ω̄z̄j Jz′ = 0 and ω̄z̄j G′z = 0, ∀j. Recall that in ref. [1] it was precisely R R the nontriviality of ω̄z̄j Jz that was responsible of the constraint on the moduli space. R This was due to the outlined fact that the gravitini need to be U(1) charged in order to 4 We have normalized the currents in such a way that {Qs , Gzz } = 2Tzz . 5 close the N=2 supersymmetry transformations off-shell. In the present model, however, we have not dealt so far with the moduli space constraint and no field is U(1) charged. Indeed, the present approach is “constructive”, in the sense that we are not getting the topological theory from an already formulated independent model (like N=2 Liouville theory). This means that the moduli space constraint has to be introduced “by hand”. As a matter of fact, the topological Halgebra is not closed by G′z and Jz′ , but by the topological current Gz , such that Qs = Gz , and by the ghost current Jz , Gz = Ggrav grav gh gh (1)z + G(2)z + G(1)z + G(2)z , grav Jz = J(1)z grav + J(2)z gh + J(1)z gh + J(2)z , Ggrav (1)z = −∂z πη, Ggrav (2)z = −λ∂z ξ, Ggh z (1)z = bzz γ , Ggh (2)z = −βz c, grav 1 grav gh gh J(1)z = − ∂z π + η∂z π̄, J(2)z = λ∂z Γ̄, J(1)z = bzz cz + 2βzz γ z , J(2)z = bz c. 2 (20) This is more similar to what happens in the Verlinde and Verlinde model. Indeed, the topological algebra Tzz -Gzz -Gz -Jz is the tensor product of the Verlinde and Verlinde one (denoted by the subscript 1), and the constraining topological algebra, corresponding to equation (2) and denoted by the subscript 2. The Verlinde and Verlinde topological algebra can be untwisted according to Tzz → Tzz − 12 ∂z Jz , to give an N=2 superconformal algebra with central charge c1 = cgrav 1 + cgh grav 1 , c1 = 3, cgh 1 = −9, while the constraining topological algebra can be untwisted to give an N=2 superconformal algebra with central charge c2 = cgrav 2 + cgh grav 2 , c2 = 3, cgh 2 = 3. Thus the total topological algebra has central grav gh grav charge c = c +c , c = 6, cgh = −6, as in ref. [1]. Nevertheless, the above representation of the topological algebra is different from the one of ref. [1]. A map between the present conformal theory and the one of ref. [1] is easily derived as follows5 . Following a procedure similar to the one of section VIII of [1], we can write (see also [4]) I Qv = {Qs , [Qv , S]}, S= bz γ − βzz cz , S 2 = 0, U1 = e[Qv ,S] , U1 QU1−1 = Qs . (21) Now, the topological charge Qs is the same as in the theory of ref. [1] and there exists an operator U2 (see section VIII of [1]) such that U2 Q′ U2−1 = Qs , Q′ = Qs + Q′v denoting the total BRST charge of the topological model of ref. [1]. Then the operator U = U2−1 U1 maps between the conformal field theories corresponding to the two models of constrained topological gravity: Q′ = UQU −1 . (22) The “singular” character of U2 (the field redefinitions contain negative powers of 1 − γ) is thus explained by the fact that the U(1) currents are different in the two cases and the moduli space constraint is imposed in a different way. 5 For this pourpose, it is convenient to introduce first order fermions π̄z = ∂z π̄ and Γ̄z = ∂z Γ̄ (and their complex conjugates). 6  We now discuss the constraint on the moduli space. It is easy to see that the La- grangian L = 4i=1 −sΨi is independent of the global degrees of freedom ν and ν̂. In- P deed, from (7) and (11) it follows that L4 = −sΨ4 only depends on the differences A± − ± = ∓∇± ξ and ψ̄± − ψ̄ˆ± = ∓∇± Γ̄ and never on A± , ψ̄± , ± , ψ̄ˆ± , separately. On the other hand, L2 = −sΨ2 contains dψ̄ and dA, which are the same as d(ψ̄ − ψ̄ˆ) and d(A − Â), if we take into account that dψ̄ˆ = 0 and d = 0. Thus the ν-ν̂ dependence is completely confined to a fifth gauge fermion Ψ5 , by means of which we now impose the moduli space constraint. We have two possibilities that are related to two different descriptions given in ref. [1]. The first possibility is represented by a rather natural gauge-fermion: antighost × Lagrange multiplier; precisely 1 Ψ5 = (ψ̄ˆ+ Â− + ψ̄ˆ− Â+ )ê+ ê− . (23) 2 Then, we have, using (3) and (7), − ˆ + ˆ L5 = sΨ5 = (Â+ Â− − ψ̂+ ψ̄ − Â− − ψ̂− ψ̄ + Â+ )ê+ ê− Z Z Z = νj ν̄ k ωzj ω̄z̄k d2 z − m̄ ˆ i ν̄ˆ j ν̄ k µ̄iz̄ j k 2 z ω̄z̄ ω̄z̄ d z − m̂i ν̂j ν k µiz j k 2 z̄ ωz ωz d z. (24) Σg Σg Σg Now we want to perform the ν-ν̂ integration. First of all, we notice that the ν integra- tion can be performed over all Cg instead of Cg /(Zg + ΩZg ). Indeed, the restriction to Cg /(Zg + ΩZg ) is due to the invariance with respect to the U(1) gauge transformations that are not continuously deformable to the identity. However, such invariance is explic- itly broken by Ψ5 , since the Â+ Â− term is a kind of mass term for the U(1) connection. When there is a gauge-invariance, one can break it either by solving a certain gauge- fixing condition or by introducing a corresponding gauge-fermion in the action. The first possibility is not practicable, in general, since solving a gauge-fixing condition usually requires to invert derivative operators. In the present case, however, the two possibilities are equally practicable, but the second one is more convenient. The BRST equivalence of the two possibilities can be proved by using a stretching argument like the one of ref. [1] and convert the integration over Cg /(Zg + ΩZg ) to the integration over the full Cg . Using the properties (see [5] for example) Z Z ∂Ωjk Z ∂ Ω̄jk ωzj ω̄z̄k d2 z = (Ω̄ − Ω)jk , µiz j k 2 z̄ ωz ωz d z = i , µ̄iz̄ j k 2 z ω̄z̄ ω̄z̄ d z = −i , Σg Σg ∂mi Σg ∂ m̄i (25) and using the well known formulæ for a superdeterminant, we get g 1 ¯ 1 Z Y R   L5 dνj dν̄j dν̂j dν̄ˆj e Σg = det ∂ Ω̄ ∂Ω , (26) j=1 Ω − Ω̄ Ω − Ω̄ ∂ where ∂ = m̂i ∂m i and a suitable normalization factor has been introduced. This is pre- cisely the top Chern class cg (Ehol ) of the Hodge vector bundle Ehol → Mg whose sections 7 are the holomorphic differentials. This representation of cg (Ehol ) is easily obtained (see section IX of ref. [1]) by choosing the imaginary part of the period matrix as fiber metric. It is amazing to notice that this result follows naturally from the simplest gauge-fermion that comes to one’s mind, i.e. (23). In some sense, we can still say that the constraint comes automatically and is not imposed “by hand”, since at first sight there is no gauge- fixing condition in (23). Now, let us discuss a second possibility, which better “simulates” what one gets automatically by twisting the N=2 Liouville theory. Again, the form of the gauge-fermion is quite typical, namely antighost × gauge-condition. This requires, however, that we know a priori what condition to impose. Let S = Sz dz be a section of Ehol . As discussed in [1],R we can project onto the Poincarè dual of cg (Ehol ) by requiring the vanishing of aj ≡ Σg ω̄z̄j Sz d2 z ∀j [6]. This is achieved by choosing Ψ5 = ψ̄ˆ− ê− S + ψ̄ˆ+ ê+ S̄. (27) Then, we get + ˆ − ˆ + L5 = sΨ5 = (Â− − ψ̂− ψ̄ + )ê− S + ψ̂+ ψ̄ − ê S + ψ̄ˆ− ê− sS + h.c. = (νj āk − ν̄j ak + ν̂j sāk − ν̄ˆ j sak )(Ω̄ − Ω)jk + R, (28) where R is an addend made of terms proportional to aj or āj and independent of ν-ν̄. The integration over ν-ν̄ gives delta functions that permit to neglect R. At the end, noticing that saj can be replaced by daj , d being the exterior derivative on the moduli space, we get 2 g Z Y R g L5 dνj dν̄j dν̂j dν̄ˆ j e Y Σg = δ(aj ) daj , (29) j=1 j=1 which is the representation of cg (Ehol ) in the de Rham current cohomology [1]. Both choices (23) and (27) of Ψ5 do not depend, by construction, on the local degrees of freedom of the fields. Moreover, the observables σn = γ0n are independent of sector of Bgauge−f ixing that implements the constraint on the moduli space. After integrating over the global degrees of freedom ν-ν̂ of ψ̄ and A, one can also integrate over the local degrees of freedom of the constraining sector (λ, Γ̄, χ, ξ, bz , c, βz , γ). When zero modes are suitably taken into account, such integrations give a net unit factor, since fermionic and bosonic determinants compensate, as it is common in topological field theory. The surviving fields are precisely those of the Verlinde and Verlinde model and the only remnant of the constraining procedure is the insertion of cg (Ehol ). Thus, the physical amplitudes are s Z s [c1 (Li )]di , Y Y < σdi (xi ) >= cg (Ehol ) (30) i=1 Mg,s i=1 as claimed in [1]. Mg,s is the moduli space of Riemann surfaces Σg of genus g and s marked points, while c1 (Li ) are the Mumford-Morita classes [7]. The selection rule is 8 clearly s X di = 2g − 3 + s. (31) i=1 Notice that, after the above integrations, the problem of the difference between true and formal dimensions [1] is bypassed. Neverthelsss, the fact that the right hand side of (31) contains 2g − 3 instead of something proportional to g − 1 (and so to the curvature) seems intriguing, since it is not straightforward to repeat the Verlinde and Verlinde analysis of contact terms. This is, to our opinion, a challenging feature of constrained topological gravity. In the case of the sphere the correlation functions are the same as in ordinary topo- logical gravity. For g = 1, on the other hand, Ω = τ and c1 (Ehol ) = (τdτ−τ̄ ∧dτ̄ )2 , which is the Poincaré metric. Its Poincaré dual is a point, that can be chosen at infinity. This corresponds to a torus with a pinched cycle or, equivalently, a sphere with two identified points. For s = 1, (31) is zero, so that the moduli space is a point and we can write 1 < σ0 >= , (32) 2 the one half being a symmetry factor due to the identification of the two points. This correlation function is the analogue of < σ0 σ0 σ0 > in genus zero. In view of the above remarks, it is natural to expect that the correlation functions in genus one are a half of the corresponding correlation functions in genus zero, namely when ki=1 ni = k, then P k k Y 1 Y 1 k! < σ0 σni >= < σ0 σ0 σ0 σni >= Qk . (33) i=1 2 i=1 2 i=1 ni ! One can conceive several variants of (30), in which correlation functions are products of Mumford-Morita classes times a fixed moduli space cocycle. For example, one could replace cg (Ehol ) with cg−k (Ehol ), 0 < k < g. However, the case that we have considered is the one that deserves particular attention, firstly because it is suggested by physics, secondly because only cg (Ehol ) is expressible as a determinant and can be easily inserted in a field theoretical model, thirdly because cg−k (Ehol ) is not meaningful for all genera, but only for g ≥ k. (Anyway, the fact that for k = 1, the right hand side of (31) is 2g − 2 + s perhaps deserves attention). To conclude, many open questions still remain to be answered and lots of possible applications should be investigated in the future. The first question is whether the correlation functions of constrained topological gravity satisfy any integrable hierarchy. In other words, one would like to know if one can generalize the Kontsevich contruction [8], by identifying the set of fat graphs that describe the gravitational instantons in two dimensions and by finding the corresponding matrix model. A very promising chapter, still to be open, concerns the possible couplings of con- trained topological gravity to topological matter. One should generalize to this case the analysis done for standard topological gravity. In particular, one should investigate the 9 meaning of the equivariance condition [2, 4] on the physical states and what are the possible matter representatives for the gravitational observables [9]. Moreover it would be very interesting to know what are the effects of the moduli space constraint on the holomorphic anomaly [10]. Finally, one can also think of a generalization of the contraining mechanism proposed in this letter, by studying different gauge-fixing terms for the global degrees of freedom. In particular, one can wonder whether in the standard theory of topological gravity coupled to matter the moduli space contraint possesses a representation in some sort of “matter picture” similar to the ones of [4, 9]. References [1] D. Anselmi, P. Fré, L. Girardello and P. Soriani, “Constrained Topological Grav- ity from Twisted N=2 Liouville Theory”, preprint SISSA/ISAS 49/94/EP, IFUM 468/FT, hepth/9404109, April 1994. [2] E. Verlinde and H. Verlinde, Nucl. Phys. B348 (1991) 457. [3] C.M. Becchi, R. Collina and C. Imbimbo, A Functional and Lagrangian Formulation of Two Dimensional Topological Gravity, hepth 9406096, June 94; talks given at the Fubinifest, Torino, February 1994 (to appear in the Proceedings, World Scientific Publishing) and at the Trieste Workshop on String Theory, April 1994. [4] T. Eguchi, H. Kanno, Y. Yamada and S.K. Yang, Phys. Lett. B 305 (1993) 235. [5] M. Schiffer and D.C. Spencer, “Functionals of finite Riemann surfaces”, Princeton University Press, Princeton 1954; E. D’ Hoker and D. H. Phong, Rev. Mod. Phys. 60 (1988) 917. [6] P. Griffiths and J. Harris, Principles of Algebraic Geometry, a Wiley-Interscience publication, 1978, pp. 409-414. [7] S. Morita, Invent. Math. 90 (1987) 551; D. Mumford, “Towards an enumerative geometry of the moduli space of curves. Arithmetic and Geometry”, Progr. Math. 36 (1983) 271. [8] M. Kontsevich, Comm. Math. Phys. 147 (1992) 1. [9] A. Losev, Descendants constructed from matter field in topological Landau-Ginzburg model coupled to topological gravity, preprint TPI-MINN-92/40-T, 1992. [10] M. Bershadsky, S. Cecotti, S. Ooguri and C. Vafa, Nucl. Phys. B405 (1993) 279. 10