hepth/9801179 ITEP-TH-31/97 YCTP-P5-97 ”Hodge strings” and elements of K.Saito’s theory of the Primitive form 8 9 9 A.Losev ∗ 1 n a J To appear in the proceedings of the Taniguchi Symposium 7 2 Topological field theories, Primitive forms and related topics, Kyoto, December 1996 1 v 9 7 1 Abstract 1 0 The ”Hodge strings” construction of solutions to associativity equa- 8 9 tions is proposed. From the topological string theory point of view this / h construction formalizes the ”integration over the position of the marked t - point” procedure for computation of amplitudes. From the mathematical p point of view the ”Hodge strings” construction is just a composition of e h elements of harmonic theory (known among physicists as a t-part of t t∗ : − v equations) and the K.Saito construction of flat coordinates (starting from i X flat connection with a spectral parameter). r We also show how elements of K.Saito theory of primitive form ap- a pear naturally in the ”Landau-Ginzburg” version of harmonic theory if we consider the holomorphic pieces of germs of harmonic forms at the singularity. 1. Introduction and summary The first aim of this article is to explain how and why some version of the Hodge (harmonic) theory leads to the specific map from tensor powers of the vector space to the cohomologies of the Deligne-Mumford compactification of the moduli space of rational curves with n marked points. In physics such a map is called ”generalized amplitudes in topo- ∗This work was supported by RFFI grant 96-01-01101, PYI grant PHY9058501 , DOE grant DE-FG02-92ER40704 and by grant 96-15-96455 for support of scientific schools logical strings”; in mathematics, a particular case of this map is called ”Gromov-Witten” invariants. The second aim is to explain how elements of the K.Saito theory of Primitive form (that ,among other things, provides solutions to associativ- ity ( WDVV) equation [BV]) naturally arise from the ”Landau-Ginzburg” version of Hodge ( harmonic) theory. In section 2, we remind that ”generalized amplitudes” in genus zero (in a ”topological string without gravitational descendents”) are in a one- to-one correspondence with the solutions to the WDVV equation(4), thus our construction (called ”Hodge strings”) should end with the solution to this equation. Sections 3 and 4 contain motivations coming from the topological stringtheoryfortheconstruction(theypartlyanswerthequestion”why?”) and could be omitted by a reader who is a mathematician. In section 3, we review the concept and structures of ”conformal topological strings”, and in section 4 we describe the ”integration over the position of the marked point” procedure of computation of the ”amplitudes” in genus zero. Along these lines we explain the origin of the QG -system, that − contains Z graded vector space H, odd operators Q and G , even com- 2 − muting Q-closed operators Φ (all operators are acting on H) and the i bilinear pairing <> on H. The ”integration over the marked point” pro- cedure shows what kind of structure should we expect to see. Section 5 is axiomatic: here we introduce the notion and general properties of an abstract QG system. Then we show that starting − with the QG -system (H,Q,G ,Φ ,<>) having Hodge property, Pair- − − i ing of the cohomology property and the Primitive element property one can canonically construct a solution to the WDVV equation. The con- struction is done in two steps. First, by comparing two flat connections (the”Hodge” connection and the ”Gauss-Manin” connection) on the bun- dle of Q(t)+zG cohomologies, we show the existence of a flat connection − with the spectral parameter ( H z−1C), which is known in physics as ∇ − the t-part of t t∗ equations [CV]. Then, using the Primitive element − property, we construct a solution to the WDVV equations, like K.Saito did in the theory of Primitive form. This section answers the question ”how” and formally is independent of the previous sections. Nevertheless, we try to comment ”why” the construction goes this way by referring to section 4. In section 6, we review the so called ”Landau-Ginzburg” realization of the ”Hodge strings” input (in physics such a system is known as N = 2 supersymmetric Landau-Ginzburg quantum mechanics). Then,in section 7, we start out by briefly reviewing (in subsection 7.1) elements of K.Saito’s theory of primitive form in the form of ”good section” and in terms of QG systems. ( In the Appendix we relate − it to the original formulation). To reach K.Saito’s theory of Primitive form from the ”Landau-Ginzburg” system (for quasihomogeneous case) we first pass from the smooth quickly vanishing forms of the ”Landau- Ginzburg” system to the non-holomorphic germs of forms at a singularity, and then take holomorphic pieces of germs. We find that holomorphic pieces of germs coming from the development of harmonic forms of the ”Landau-Ginzburg” theory satisfy two of K.Saito’s conditions for a ”good section” and, with a quasihomogeneous ”antiholomorphic superpotential U¯ ”, satisfy the third condition (this third condition is not necessary for construction of the solution to the WDVV equations) . We explain the ambiguity of the solutions for K.Saito’s conditions for a ”good section” as coming from the ”antiholomorphic superpotential U¯” that disappears in the ”taking holomorphic pieces” procedure (this phe- nomena in topological strings is called the ”holomorphic anomaly”). We expect that methods developed here could be useful in the under- standing of non-quasihomogeneous systems. Conventions. The sum over repeating indexes (the physicist’s con- vention) is adopted in the text . 2. ”Compact” topological strings and the associativity equation ”Topological string theory”[Wi, DW, VV, DVV, Wi2, KM, BCOV] studies genus q ”generalized amplitudes” GA , taking values in cohomolo- q ¯ gies of the Deligne-Mumford compactification M of the moduli space q,n of complex structures of genus q Riemann surfaces with n marked points. Pairing between GA and the cycle C M¯ is given by the functional q q,n ∈ integral [Wi, DW, KM] (GA ,C)(V ,...,V ) = φV (φ(z ))...V (φ(z ))exp(S (φ)), q 1 n ZC∈M¯q,nZ D 1 1 n n TS (1) fields V (φ(z)) are called ”vertex operators” and ordinary ”amplitudes” i ¯ A (V ,...,V ) correspond to C = M . q 1 n q,n Deligne-Mumford compactification M¯ is a union of M (a set of n 0,n 0,n noncoincident points on CP moduli SL(2,C) action) and the compacti- 1 fication divisor Comp.The divisor Comp is a union of components C(S), where S partitions n marked points into two groups consisting of n (S) 1 and n (S) points, n > 1. A surface corresponding to a general point 2 i in C(S) is a union of two spheres having one common point with n (S) 1 marked points on the first sphere and n (S) on the second. The set of 2 general points in C(S) form the space M M . 0,n1+1 ⊗ 0,n2+1 In this paper, we will consider a class of ”compact” topological string theories that have no gravitational descendents [Wi] among its ”vertex operators” and have a nondegenerate pairing on the space of ”vertex op- erators”. This class of theories includes, for example, topological sigma models of type A on compact Kahler manifolds and twisted unitary su- perconformal theories. It is believed that these theories play the same role among all theories as smooth compact manifolds among all manifolds. It expected that, in ”compact” topological theories, the functional integral for surfaces corresponding to points in C(S) factorizes and [Wi] (GA ,C(S))(V ,...,V ) = 0 i1 in ηjkA (V ,...,V ,V )A (V ,...,V ,V ) (2) 0 i1 in1 j 0 in1+1 in2+n1 k where η is a matrix of symmetric bilinear nondegenerate products on ver- tex operators. Keel [Ke]found that the homologue ringH of M¯ is generated by cy- ∗ 0,k cles C(S).He described relations between these cycles in homologies lead- ing to constraints on GA because of (2). 0 An elegant way of formulating these constraints uses the generating function for ”amplitudes”. Introducing formal parameters T , we define i the germ F(T): ∞ 1 F(T) = A (T V ,...,T V ) (3) X k! 0 i1 i1 ik ik k=3 Then, Keel’s relations lead to: ∂3F(T) ∂3F(T) ∂3F(T) ∂3F(T) ηkl = ηkl (4) ∂T ∂T ∂T ∂T ∂T ∂T ∂T ∂T ∂T ∂T ∂T ∂T i j k l p q i p k l j q Using the factorization property and Keel’s description of homologies of moduli space, we can reconstruct GA from A [KM], see also [DVV]. 0 0 3. Amplitudes in conformal topological strings theory The ”Hodge string” construction generalizes the ”integration over the positionofthemarkedpoint”procedure[VV,DVV,Lo1,LP,BCOV,Lo2] of computation of amplitudes in ”conformal topological theory coupled to topological gravity” also known as ”conformal topological string theory”. The general covariant action S of topological field theory is a sum m of a ”topological”(metric independent) Q-closed term S and a Q-exact top term for a fermionic scalar symmetry Q: S = S (φ)+Q(R(φ),g), m top whereg denotesthemetricontheRiemannsurface. Theenergy-momentum tensor T is Q-exact: δR T = Q( ) = Q(G) (5) δg We call topological field theory conformal, if R is conformally invariant, i.e. G is traceless. We introduce fermionic two-tensor fields ψ, such that functions of g,ψ are forms on the space of metrics. An external differential on these forms could be written as follows: Q = ψ δ . g δg The action for a topological theory coupled to topological gravity is S = S +ψG = S +(Q+Q )(R). TS m top g The functional integral Z(g,ψ) over the set of fields φ with the action S is a closed form on the space of metrics. Since G is traceless, Z is TS a horizontal [DVV, Di] † form with respect to the action of conformal transformations of metrics and diffeomorphisms of the Riemann surface; thus, it defines a closed form on the moduli space of conformal(=complex) structures on the genus q Riemann surface. To construct generalized amplitudes we insert fields (zero-observables =”vertex operators”) V at marked points on Riemann surface. They i should satisfy Q(V ) = 0,G (V ) = 0. (6) i 0,− i Here G is the superpartner of the component of the energy-momentum 0,− tensor T that corresponds to the rotation with the constant phase 0,− z eiθz of the local coordinate at the marked point. The first condi- → tion in (6) is needed to construct a closed form on the space of metrics, while the second provides horizontality of the corresponding form with respect to diffeomorphisms that leave marked points fixed but rotate local coordinates [Al, DN, DVV, Eg, Di]. 4. Integration over positions of marked points The ”integration over marked points” procedure reduces all genus zero amplitudes to the three point amplitude: F = A (V ,V ,V ), ijk 0 i j k †Differentialformonthe principalbundle is calledhorizontalifits contractionwith the vertical(tangent to fiber) vector is zero.Closedhorizontalforms on the total space correspond to closed forms on the base of the bundle which can be computed from topological matter theory. (2) In conformal topological theory, we associate a two-observable V = i G G V to a zero observable V . Thus, we deform topological theory L,−1 R,−1 i i to a family of theories parametrized by t, with the action S (t) = S + m m (2) t V ; thus, zero-observables V form a tangent bundle to this space of i i theories [DVV]. If, in the functional integral that computes the n-point amplitude, we first pick up one of the marked points (we will call it a ”moving point”), integrate over the position of the moving point, and only then take the functional integral, the n-point amplitude becomes the derivative in t of the n 1 point amplitude. − In the process of integration, we should take special care about the region where the moving point tends to hit a fixed point because the ge- ometry there is not a naive one. The contribution fromthis region(contact terms [VV,Lo1,Lo2,LP,Di,BCOV]) leads to a specific contact term con- nection on the bundle of zero-observables over the space of theories and thus on the tangent space to the space of theories. Repeating this procedure again and again, we can recover amplitudes from F (t). The amplitudes should be symmetric and independent of the ijk order of integration over positions of marked points. In other terms, generating parameters T from (3) should become the so-called special coordinates on the space of theories, the derivatives with respect to the special coordinates should become covariantly constant sec- tions of the contact term connection, and symmetric tensor F (in the ijk special coordinate frame) should be a third derivative of F(T). Moreover, F(T) has to solve the WDVV equations (4). All this implies that the contact term connection is quite a special one! To gain better understanding of this connection, we will study the space of states in 2d theory associated with the circle (considered as a component of the boundary of the Riemann surface). Moreover, we will restrictourselves tothesubspaceH ofthesestatesthatareinvariantunder constant rotation of the circle. Fermionic symmetry Q of the theory and G reduce to odd anticom- 0,− muting operators Q and G on H. − Zero-observables V (being inserted at the middle of the punctured i disc) generate states h that are Q and G closed: i − Qh = G h = 0, (7) i − i the zero observable 1 generates the distinguished state h . The operation 0 ofsewing two discs together corresponds to thebilinear pairing <,>. Inte- grals of zero observables along the boundary give operators Φ = V dσ. i S1 i R One can show from the functional integral that the objects defined above have the following properties: Q2 = G2 = QG +G Q = 0,[Q,Φ ] = 0,[Φ ,Φ ] = 0, (8) − − − i i j QT = EQ,GT = EG,ΦT = Φ (9) − Here transposition ”T” is taken with respect to the pairing <,>, and operator E commutes with Φ and anticommutes with Q and G . − In the deformed theory, Q(t) = Q+ [G ,t Φ ] in the first order in t. − i i To ensure it globally we will take for simplicity‡ [[G ,Φ ],Φ ] = 0. (10) − i j The contribution from the region near the place where the ”mov- ing” i-th point hits the marked j-th one gives the ”cancelled propaga- tor argument”(CPA) connection on states h over the space of theories j [VV, Di, LP, Lo2]: ∞ (CPA) δ h = G dτG exp( τT )Φ h , (11) i j −Z 0,+ − 0,+ i j 0 thus δ(CPA)h is G -exact. Here T is the Hamiltonian acting on the − 0,+ space H, and G is its superpartner: T = Q(G ). 0,+ 0,+ 0,+ Covariantly constant sections§ of the CPA connection will be denoted as h (t). This connection induces the connection on the space of zero- i observables: covariantly constant sections of contact term connection V (t) = uj(t)V i i j are such that, being inserted in the middle of the disc in the t-deformed theory, they produce covariantly constant sections h (t): i h (t) = lim uj(t)rT0,+Φ h (t). (12) i r→0 i j 0 Let us denote as C (t) the linear operator representing the action of Φ in i i Q(t)-cohomologies. Then, the relation (12) reads: [h (t)] = uj(t)C (t)[h (t)] (13) i Q(t) i j 0 Q(t) here and below [h] stands for a class of a Q-closed element h in Q- Q cohomologies. ‡in general case one has to go in for Kodaira-Spencer type arguments,see [BCOV] §Flatness of CPA connection is necessary for the consistency of the procedure From the functional integral we get: F (t) =< h (t),Φ h (t) > ul(t). (14) ijk i l k j While the string origin of the described procedure is quite natural, its consistency is far from being obvious. 5. The ”Hodge string” QG -system − 5.1 General facts about QG systems − Definition. The QG system (Q,G ,Φ,H) is a collection of Z -graded − − 2 vector space H, odd operators Q and G , and a set of even operators Φ , − i i = 1,...,µ, acting on this space, that have the properties (8,10). Given a QG system, one can construct a family Q(t) of nilpotent odd − operators in H: Q(t) = Q+t [G ,Φ ] (15) i − i over a deformation space with coordinates t . i Definition. Cohomologies of QG -systems. − Let H be the space of Q(t) cohomologies in H. Let Q(t) Q(t,z) = Q(t)+zG (16) − Let H be the space of cohomologies of Q(t) in H Q(t) • Let Hˆ be the space of cohomologies of Q(t,z) in H C[[z]] Q(t,z) • ⊗ Let H be the space of cohomologies of Q(t,z) in the space H Q(t,z) • ⊗ C << z >>, where C << z >> is the space of Laurent expansions in z Let Hl Hˆ be the space of ”little” cohomologies, defined • Q(t,z) ⊂ Q(t,z) as those classes in Hˆ that have representatives in H: Q(t,z) Hl = [ω] Hˆ ω H,Qω = G ω = 0 (17) Q(t,z) { Q(t,z) ∈ Q(t,z)| ∈ − } Remark.The space Hˆ has a natural decreasing filtration by powers Q(t,z) of z: Hˆ = Hˆ(0) Hˆ(1) ... (18) Q(t,z) ⊃ ⊃ a class is in Hˆ(k) if it contains element zkω. The inclusion Hl Hˆ Q(t,z) ⊂ Q(t,z) induces the decreasing filtration on ”little” cohomologies. Remark from string theory. In string theory of the general type the space of ”little” cohomologies corresponds to the space of states created by ”vertex operators”. One can show [VV, Lo1, Eg, Lo2] that states l,(k) from H are created by ”vertex operators” that are k-th gravitational Q(t,z) descendents. Definition. Let C (t) : H H be a linear operator representing i Q(t) Q(t) → the action of Φ in Q(t) cohomologies: i C (t)[ω] = [Φ ω] (19) i Q(t) i Q(t) Remark. Operators C (t) should be considered as components of the i one-form on the deformation space with values in EndH . From the Q(t) definition it follows that these operators commute with each other: [C (t),C (t)] = 0 (20) i j Definition. A morphism of QG -systems − (Q1,G1,Φ1,H1) (Q2,G2,Φ2,H2) (21) − i → − i isamorphismH1 H2commutingwiththeactionofoperatorsQβ,Gβ,Φβ → − i in Hβ, β = 1,2. It is clear that the morphism of QG systems induces the morphism − of cohomologies of QG systems. − Definition. A morphism of QG -systems will be called a quasiisomor- − phism of QG systems if it induces an isomorphism in all cohomologies of − QG systems. − Definition. By the ”Gauss-Manin” connection in a QG system, we call − a canonical flat connection GM in H over C[[t]], whose horizontal Q(t,z) ∇ sections [ωGM(t,z)] satisfy the following: Q(t,z) [ωGM(t,z)] = [exp( t Φ /z)ωGM(0,z)] (22) Q(t,z) i i Q(t,z), − i.e. their representatives solve the following differential equation: ∂ ωGM(t)+z−1Φ ωGM(t) Im(Q(t,z)) (23) i ∂t ∈ i Remark. We call this canonical connection ”Gauss-Manin” following K.Saito (see the Appendix). Remark. It is clear that morphisms of QG systems induce morphisms − of the ”Gauss-Manin” connections, and quasiisomorphisms induce the iso- morphism of these connections. Below we will list some additional properties that the QG system − could have and that would be important for the ”Hodge string” system . The Hodge property. ImQ KerG = ImG KerQ = Im(QG ) (24) − − − ∩ ∩ Statement 5.1ItfollowsfromtheHodgepropertythatdimH = dimH Q G− and there exists a set h of Q and G closed elements of H (these ele- a − { } ments are unique up to ImQG ), such that classes [h ] and [h ] form − a Q a G− bases in Q and G cohomologies. − Remark. In harmonic theory such elements are just harmonic forms; that is why below we will call these elements ”harmonic” . Statement 5.2 If the QG system has a Hodge property, then − H = Hˆ C[z−1] = H C << z >> (25) Q(0,z) ∼ Q(0,z) ∼ Q(0) ⊗ ⊗ and Hl = H . (26) Q(0,z) ∼ Q(0) Proof. Classes in the first line are identified by considering hP(z) and hP(z,z−1) for ”harmonic” h as representatives of classes in Hˆ and Q(0,z) H respectively. (Here, P are polynomials. ) The statement on the Q(0,z) second line becomes clear as a generalization of the following reasoning: [ω ] = z[ω ], if ω = Qω′ and ω = G ω′. But from the Hodge property 1 2 1 2 − it follows that ω ImQG , and th−us ω Im(Q+zG ). 2 i − i − ∈ ∈ Remark. The difference between Hl and H is one of the criteria Q(0,z) Q(0) showing the failure of the Hodge property. From the string theory point of view this difference means that corresponding string theory has gravi- tational descendents among its vertex operators, and is ”noncompact” in the sense of section 2. The QG system with pairing is a QG system with the bilinear pair- − − ing <,> satisfying property (9). Remark.From (9) it follows that the pairing <,> descends to both Q(t) and G cohomologies, i.e. for Q(t)-closed ω H − 2 ∈ < Q(t)ω ,ω >=< ω ,Q(t)ω >= 0, (27) 1 2 2 1 and for G closed ω − 2 < G ω ,ω >=< ω ,G ω >= 0, (28) − 1 2 2 − 1 Pairing of the cohomology property. The pairing <,> is non-degenerate when restricted to Q-cohomologies.