INVARIANCE OF TAUTOLOGICAL EQUATIONS II: GROMOV WITTEN THEORY

INVARIANCE OF TAUTOLOGICAL EQUATIONS II: GROMOV WITTEN THEORY Y.-P. LEE Abstract. The main focus of Part II is to explore the technique of invariance of tautological equations in the realm of Gromov Witten theory. The main result is a proof of Invariance Conjecture [6], which establishes some general inductive relations of the tautological rings. This has been used in [,, 3] to obtain results in tautological relations. 0. Introduction This work is a continuation of Part I of ITE [6]. The purpose of this paper is to explore the technique of invariance of tautological equations in the realm of the Gromov Witten theory. Section gives a quick summary of geometric Gromov Witten theory. As alluded in Part I [6], the set of conjectures proposed there are motivated by study of Givental s axiomatic Gromov Witten theory. In Sections and 3, we summarize Givental s theory. Some geometric results on the tautological classes, definitely known to experts, are given in Section 4 for readers convenience. In Section 5, it is proved that each tautological class are invariant under the quantized lower triangular loop groups. Thus the invariance under lower triangular loop groups gives no constraints. In Section 6, we study the invariance of tautological equations under the quantized upper triangular loop groups. It turns out that this last invariance gives very strong constraints on the possible forms of the tautological equations. As a matter of fact, the constraints are so tight as to, conjecturally, uniquely determine the tautological equations. That is the motivation of the three Invariance Conjectures advanced in [6]. Using techniques in Gromov Witten theory, The main result proved here is Invariance Conjecture, via Gromov Witten theoretic techniques. At the end of the paper, we indicate how this vindicates a uniform derivation of all known tautological equations and a simple derivation of Faber type statement for Research partially supported by NSF and AMS Centennial Fellowship.

Y.-P. LEE the tautological rings. An appendix (jointly with Y. Iwao demonstrates some properties of the upper triangular loop groups associated to P. Acknowledgments. I wish to thank, D. Arcara, E. Getzler, A. Givental, R. Pandharipande and R. Vakil for many useful discussions.. Geometric Gromov Witten theory.. Preliminaries of Gromov Witten theory. Gromov Witten theory studies the tautological intersection theory on M g,n (X, β, the moduli spaces of stable maps from curves C of genus g with n marked points to a smooth projective variety X. The intersection numbers, or Gromov Witten invariants, are integrals of tautological classes over the virtual fundamental classes of M g,n (X, β n ev i(γ i ψ k i i. [M g,n(x,β] vir i= Here γ i H (X and ψ i are the cotangent classes (gravitational descendents. For the sake of the later reference, let us fix some notations. (i H := H (X, Q is a Q-vector space, assumed of rank N. Let {φ µ } N µ= be a basis of H. (ii H carries a symmetric bilinear form, Poincaré pairing, Define, : H H Q. g µν := φ µ, φ ν and g µν to be the inverse matrix. (iii Let H t := k=0h be the infinite dimensional complex vector space with basis {φ µ ψ k }. H t has a natural Q-algebra structure: φ µ ψ k φ ν ψ k (φ µ φ ν ψ k +k, where φ µ φ nu is the cup product in H. (iv Let {t µ k }, µ =,..., N, k = 0,...,, be the dual coordinates of the basis {φ µ ψ k }. We note that at each marked point, the insertion is H t -valued. Let t := k,µ t µ k φ µψ k denote a general element in the vector space H t. To simplify the notations, t k will stand for the vector (t k,...,tn k and tµ for (t µ 0, t µ,....

(v Define INVARIANCE OF TAUTOLOGICAL EQUATIONS II 3 µ k... µn k n g,n,β := and define by multi-linearity. (vi Let n [M g,n(x,β] vir i= t n g,n,β = t...t g,n,β F X g (t := n,β n! tn g,n,β ev i (φ µ i ψ k i i be the generating function of all genus g Gromov Witten invariants. The τ-function of X is the formal expression ( τ X GW := e P g=0 ~g F X g... Gravitational ancestors and the (3g -jet properties. Let ( st : M g,m+l (X, β M g,m+l be the stabilization morphism defined by forgetting the map and ft : M g,m+l M g,m be the forgetful morphism defined by forgetting the last l points. The gravitational ancestors are defined to be (3 ψi := (ft st ψ i and genus g ancestor potential is defined by F X g (t, s := Q β m t µ k m!l! ( ψ i k ev i (φ µ m,l,β [M g,m+l (X,β] vir i= k m+l i=m+ The following property is called the (3g -jet property [9] (4 m t µ k +... F X g (t, s t 0 =0 = 0 for k i 3g. tµm k m+ The ancestors and descendents are different, but easy to compare: The difference is a virtual boundary divisor. See (9 and [5] for details. s µ ev i (φ µ. In Gromov Witten theory, one usually has to deal with the coefficients in the Novikov ring, due to some convergence issues. We shall not touch upon this subtleties here but refer the readers to [9]. µ

4 Y.-P. LEE. Genus zero axiomatic Gromov Witten theory Let H be a Q-vector space of dimension N with a distinguished element. Let {φ µ } be a basis of H and φ =. Assume that H is endowed with a nondegenerate symmetric Q-bilinear form, or metric,,. Let H denote the infinite dimensional vector space H[z, z ] consisting of Laurent polynomials with coefficients in H. Introduce a symplectic form Ω on H: Ω(f(z, g(z := Res z=0 f( z, g(z, where the symbol Res z=0 means to take the residue at z = 0. There is a natural polarization H = H q H p by the Lagrangian subspaces H q := H[z] and H p := z H[z ] which provides a symplectic identification of (H, Ω with the cotangent bundle T H q with the natural symplectic structure. H q has a basis {φ µ z k }, µ N, 0 k with dual coordinates {q k µ}. The corresponding basis for H p is {φ µ z k }, µ N, 0 k with dual coordinates {p k µ}. For example, let {φ i } be an orthonormal basis of H. An H-valued Laurent formal series can be written in this basis as... + (p,...,p N ( z + (p 0,...,p N 0 ( z + (q0,...,qn 0 + (q,...,qn z +.... In fact, {p i k, qi k } for k = 0,,,... and i =,...,N are the Darboux coordinates compatible with this polarization in the sense that Ω = i,k dp i k dqi k. The parallel between H q and H t is evident, and is in fact given by the following affine coordinate transformation, called the dilaton shift, t µ k = qµ k + δµ δ k. Definition. Let G 0 (t be a (formal function on H t. The pair T := (H, G 0 is called a g = 0 axiomatic theory if G 0 satisfies three sets of genus zero tautological equations: the Dilaton Equation (5, the String Equation (6 and the Topological Recursion Relations (TRR (7. Different completions of H are used in different places. This will be not be discussed details in the present article. See [9] for the details.

INVARIANCE OF TAUTOLOGICAL EQUATIONS II 5 (5 (6 (7 G 0 (t (t = t G 0 (t t 0 3 G 0 (t t α k+ tβ l tγ m t µ G 0 (t k t µ G 0 (t, k=0 µ k = t 0, t 0 + = µν k=0 ν G 0 (t t α k tµ 0 t ν k+ g µν G 0 (t, t ν k 3 G 0 (t, α, β, γ, k, l, m. t ν 0 tβ l tγ m In the case of geometric theory, G 0 = F0 X It is well known that F0 X satisfies the above three sets of equations (5 (6 (7. The main advantage of viewing the genus zero theory through this formulation, seems to us, is to replace H t by H where a symplectic structure is available. Therefore many properties can be reformulated in terms of the symplectic structure Ω and hence independent of the choice of the polarization. This suggests that the space of genus zero axiomatic Gromov Witten theories, i.e. the space of functions G 0 satisfying the string equation, dilaton equation and TRRs, has a huge symmetry group. Definition. Let L ( GL(H denote the twisted loop group which consists of End(H-valued formal Laurent series M(z in the indeterminate z satisfying M ( zm(z = I. Here denotes the adjoint with respect to (,. The condition M ( zm(z = I means that M(z is a symplectic transformation on H. Theorem. [3] The twisted loop group acts on the space of axiomatic genus zero theories. Furthermore, the action is transitive on the semisimple theories of a fixed rank N. Remarks. (i In the geometric theory, F0 X (t is usually a formal function in t. Therefore, the corresponding function in q would be formal at q = z. Furthermore, the Novikov rings are usually needed to ensure the well-definedness of F0 X (t. (cf. Footnote. (ii It can be shown that the axiomatic genus zero theory over complex numbers is equivalent to the definition of abstract (formal Frobenius manifolds, not necessarily conformal. The coordinates on the corresponding Frobenius manifold is given by the following map [5] (8 s µ := t µ 0 t 0 G 0 (t.

6 Y.-P. LEE From now on, the term genus zero axiomatic theory is identified with Frobenius manifold. (iii The above formulation (or the Frobenius manifold formulation does not include the divisor axiom, which is true for any geometric theory. (iv Coates and Givental [4] (see also [3] give a beautiful geometric reformation of the genus zero axiomatic theory in terms of Lagrangian cones in H. When viewed in the Lagrangian cone formulation, Theorem becomes transparent and a proof is almost immediate. 3. Quantization and higher genus axiomatic theory 3.. Preliminaries on quantization. To quantize an infinitesimal symplectic transformation, or its corresponding quadratic hamiltonians, we recall the standard Weyl quantization. A polarization H = T H q on the symplectic vector space H (the phase space defines a configuration space H q. The quantum Fock space will be a certain class of functions f(~, q on H q (containing at least polynomial functions, with additional formal variable ~ ( Planck s constant. The classical observables are certain functions of p, q. The quantization process is to find for the classical mechanical system on H a quantum mechanical system on the Fock space such that the classical observables, like the hamiltonians h(q, p on H, are quantized to become operators ĥ(q, q on the Fock space. Let A(z be an End(H-valued Laurent formal series in z satisfying (A( zf( z, g(z + (f( z, A(zg(z = 0, then A(z defines an infinitesimal symplectic transformation Ω(Af, g + Ω(f, Ag = 0. An infinitesimal symplectic transformation A of H corresponds to a quadratic polynomial P(A in p, q P(A(f := Ω(Af, f.

INVARIANCE OF TAUTOLOGICAL EQUATIONS II 7 Choose a Darboux coordinate system {qk i, pi k }. The quantization P P assigns (9 =, pi k = ~, qi qk i k = qk i / ~, p i k pj l = p i kp j l = ~, p i k qj l = q j l q i k q i k qj l = q i k qj l /~,, qk i q j l In summary, the quantization is the process A P(A [ P(A inf. sympl. transf. quadr. hamilt. operator on Fock sp.. It can be readily checked that the first map is a Lie algebra isomorphism: The Lie bracket on the left is defined by [A, A ] = A A A A and the Lie bracket in the middle is defined by Poisson bracket {P (p, q, P (p, q} = k,i P P P p i k qk i p i k P. qk i The second map is not a Lie algebra homomorphism, but is very close to being one. Lemma. [ P, P ] = \ {P, P } + C(P, P, where the cocycle C, in orthonormal coordinates, vanishes except C(p i k pj l, qi k qj l = C(qi k qj l, pi k pj l = + δij δ kl. Example. Let dim H = and A(z be multiplication by z. It is easy to see that A(z is infinitesimally symplectic. (0 P(z = q 0 m=0 \ P(z = q 0 m=0 q m+ p m q m+ q m. Note that one often has to quantize the symplectic instead of the infinitesimal symplectic transformations. Following the common practice in physics, define ( ea(z := e [ A(z,

8 Y.-P. LEE for e A(z an element in the twisted loop group. 3.. τ-function for the axiomatic theory. Let X be the space of N points and H Npt := H (X. Let φ i be the delta-function at the i-th point. Then {φ i } N i= form an orthonormal basis and are the idempotents of the quantum product φ i φ j = δ ij φ i. The genus zero potential for N points is nothing but a sum of genus zero potentials of a point F Npt 0 (t,...,t N = F pt 0 (t +... + F pt 0 (t N. In particular, the genus zero theory of N points is semisimple. By Theorem, any semisimple genus zero axiomatic theory T of rank N can be obtained from H Npt by action of an an element O T in the twisted loop group. By Birkhoff factorization, O T = S T (z R T (z, where S(z (resp. R(z is an matrix-valued functions in z (resp. z. In order to define the axiomatic higher genus potentials G T g for the semisimple theory T, one first introduces the τ-function of T. Definition 3. [] Define the axiomatic τ-function as ( τ T G := ŜT ( R T τ Npt GW, where τ Npt GW is defined in (. Define the axiomatic genus g potential GT g via the formula (cf. ( (3 τ T G =: e P g=0 ~g G T g. Remark. (i It is not obvious that the above definitions make sense. The function ŜT ( R T τ Npt is well-defined, due to the (3g -jet properties (4, proved in [9] for geometric Gromov Witten theory and in [] in the axiomatic framework. The fact log τg T can be written as g=0 ~g (formal function in t is also nontrivial. The interested readers are referred to the original article [] or [9] for details. An immediate question regarding Definition 3: When the axiomatic semisimple theory actually comes from a projective variety X, is the τ-function defined in the axiomatic theory the same as the τ-function defined in the geometric Gromov Witten theory? This is known as Givental s conjecture. Givental himself establishes the special cases when X is toric Fano. Recently, C. Teleman [0] has announced a complete proof based on his classification of semisimple D cohomological field theories.

INVARIANCE OF TAUTOLOGICAL EQUATIONS II 9 Theorem. ([] [0] Let X be a projective variety whose quantum cohomology is semisimple, then (4 τ X G = τ X GW. 4. Tautological equations in Gromov Witten theory Due to the stabilization morphism (, any tautological equation in M g,n can be pulled back and become an equation on tautological classes on M g,n (X, β. The ψ-classes can be either transformed to ancestor classes ψ or to ψ-classes on the moduli spaces of stable maps. Due to the functorial properties of the virtual fundamental classes, the pullbacks of the tautological equations hold for the Gromov Witten theory of any target space. The term tautological equations will also be used for the corresponding equations in Gromov Witten theory and in the theory of spin curves. Equations in Gromov Witten theory which are valid for all target spaces are called the universal equations. Tautological equations are universal equations. However, these induced equations will produce relations among generalized Gromov Witten invariants, which involves not only ψ-classes but also κ-classes and boundary classes. Since the integration over boundary classes can be written in terms of ordinary Gromov Witten invariants by the splitting axiom, what is really in question is the κ- classes. We will start with some results in tautological classes on moduli of curves. Let ft l : M g,n+l M g,n be the forgetful morphism, forgetting the last l marked points. Lemma. (ft l ( n (ψ i k i i= n+l i+n+ (ψ i k i+ = K kn+...k n+l n i= ψ k i i, where K kn+...k n+l is defined as follows. Let σ S l be an element in the symmetric group permuting the set {n +,... n + l}. σ can be written as a product of disjoint cycles σ = c c..., where c i := (c i...ci a i with c i j {n +,...n + l}. Define K kn+...k n+l := σ S l a i i(σ= Q c i j= κ c i j.

(6 ft (κ l = κ l ψ l n+. 0 Y.-P. LEE For example, when l =, the formula becomes n (ft (( (ψ i k i (ψ n+ kn++ (ψ n+ k n++ i= =(κ kn+ κ kn+ + κ kn+ +k n+ n (ψ i k i. Proof. The proof follows from induction on l and the following three geometric ingredients, which are well-known in the theory of moduli of curves. Let ft : M g,n+ M g,n be the forgetful morphism, forgetting the last marked point and D i,n+ be the boundary divisor in M g,n+ defined as the image of the section of ft, considered as the universal curve, by the i-th marked point. Then i= (5 ft (ψ i = ψ i D i,n+. ψ i for i n on M g,n+ vanishes when restricted to D i,n+. The following result follows by combining the above ingredients and induction on power of κ-classes. Corollary. A tautological equations on M g,n involving κ-classes of highest power less than l (e.g. κ k...κ kl can be written as a pushforward, via forgetful morphism ft l, of a tautological equation on M g,n+l, involving only boundary strata and ψ-classes. By pulling-back to moduli of stable maps, one has the following corollary. Corollary. (i The system of generalized Gromov Witten invariants involving κ-classes and λ-classes is the same as the system of usual Gromov Witten invariants. (ii Any induced equation of generalized Gromov Witten invariants can be written as an equation of ordinary Gromov Witten invariants. Proof. The fact that the system of generalized GW invariants invovling λ-classes can be reduced to ordinary GW invariants follows from [6]. The part involving κ-classes follows from Lemma. With this Corollary, one can talk about the induced equations of (ordinary Gromov Witten invariants from any tautological equations.

INVARIANCE OF TAUTOLOGICAL EQUATIONS II Remark. It is not difficult to see, from the above discussions (in particular equations (5, (6 and Lemma, that the three graphical operations introduced in [6] (cutting edges, genus reduction, and splitting vertices are compatible with the pull-back operations. 5. Invariance under lower triangular subgroups The twisted loop group is generated by lower triangular subgroup and the upper triangular subgroup. The lower triangular subgroup consists of End(H-valued formal series S(z = e s(z in z satisfying S ( zs(z = or equivalently l= n=0 i,j s ( z + s(z = 0. 5.. Quantization of lower triangular subgroups. The quadratic hamiltonian of s(z = l= s lz l is (s l ij q j l+n pi n + ( n (s l ij qnq i j l n. The fact that s(z is a series in z implies that the quadratic hamiltonian P(s of s is of the form q -term + qp-term where q in qp-term does not contain q 0. The quantization of the P(s ŝ = (s l ij q j l+n qn i + ( n (s l ij qn i ~ qj l n. Here i, j are the indices of the orthonormal basis. (The indices µ, ν will be reserved for the gluing indices at the nodes. For simplicity of the notation, we adopt the summation convention to sum over all repeated indices. Let dτ G dǫ s := ŝ(zτ G. Then dg 0 (ǫ s dǫ s = dg g (ǫ s dǫ s = l= l= (s l ij q j l+n qn i G 0 + ( n (s l+n+ ij qn i qj l. n=0 i,j (s l ij q j l+n qn i G g, for g. n=0 i,j Define i k i k... in k n g := n G g t i k t i, k... t in k n

Y.-P. LEE and denote... :=... 0. These functions... g will be called axiomatic Gromov Witten invariants. Then d i dǫ k i k... s (7 = (s l ij q j l+n i n i k... + (s l iia k i a l i k... l= i,a ˆ k a... + δ ( ( k (sk +k + i i + ( k where δ = 0 when there are more than insertions and δ = when there are two insertions. The notation ˆ k i means that i k is omitted ia (sk +k + i i, from the summation. We assume that there are at least two insertions, as this is the case in our application. For g (8 d dǫ s i k i k... g = (s l ij q j l+n i n i k... g + a 5.. S-Invariance. (s l iia k i a l i k... ˆ k a... g Theorem 3. (S-invariance theorem All tautological equations are invariant under action of lower triangular subgroups of the twisted loop groups. Proof. Let E = 0 be a tautological equation of axiomatic Gromov Witten invariants. Suppose that this equation holds for a given semisimple Frobenius manifold, e.g. H Npt = C N. We will show that ŝe = 0. This will prove the theorem. ŝe = 0 follows from the following facts: (a The combined effect of the first term in (7 (for genus zero invariants and in (8 (for g invariants vanishes. (b The combined effect of the remaining terms in (7 and in (8 also vanishes. (a is due to the fact that the sum of the contributions from the first term is a derivative of the original equation E = 0 with respect to q variables. Therefore it vanishes. It takes a little more work to show (b. Recall that all tautological equations are induced from moduli spaces of curves. Therefore, any relations of tautological classes on M g,n contain no genus zero components of two or less marked points. However, when one writes down the induced equation for (axiomatic Gromov Witten invariants, the genus ia

INVARIANCE OF TAUTOLOGICAL EQUATIONS II 3 zero invariants with two insertions will appear. This is due to the difference between the cotangent classes on M g,n+m (X, β and the pull-back classes from M g,n. Therefore the only contribution from the third term of (7 comes from these terms. More precisely, let ψ j (descendents denote the j-th cotangent class on M g,n+m (X, β and ψ j (ancestors the pull-backs of cotangent classes from M g,n by the combination of the stabilization and forgetful morphisms (forgetting the maps and extra marked points, and stabilizing if necessary. Let D j be the divisor on M g,n+m (X, β defined by the image of the gluing morphism β +β =β m +m =m M (j 0,+m (X, β X M g,n+m (X, β M g,n+m (X, β, where M g,n+m (X, β carries all first n marked points except the j-th one, which is carried by M (j 0,+m (X, β. It is easy to see geometrically that (9 ψ j ψ j = D j. (See e.g. [5]. Let us denote µ,... the generalized (axiomatic k, l Gromov Witten invariants with ψ k ψ ev l (φ µ at the first marked point. The above relation can be rephrased in terms of invariants as i k, l... g = i k+,l... g i k µ µ l... g. Repeat this process of reducing l, one can show by induction that i k, l... g = i... k+r,l r g k+r i µ µ... l r g... r k i µ ( p+ µ k µ... µ p k p µp µp... k g. p,l r p= k +...+k p=r p Now suppose that one has an equation of tautological classes of M g,n. Use the above equation (for r = l one can translate the equation of tautological classes on M g,n into an equation of the (axiomatic Gromov Witten invariants. The term-wise cancellation of the contributions from the second and the third terms of (7 and (8 can be seen easily by straightforward computation. If the above description is a bit abstract, the reader might want to try the following simple example. ψ on M g, is translated into invariants: x g x µ µ g x µ µ g + x µ µ ν ν g. The above translation from tautological classes to Gromov Witten invariants are worked out explicitly in some examples in Sections 6 and 7 of [8].

4 Y.-P. LEE Remark. (i The S-invariance theorem actually hold at the level of (Chow or cohomology classes, rather than just the numerical invariants. The geometric content is (9. This should be clear from the proof. (ii Note that one may apply this to the tautological rings on moduli of curves due to (i, where X = pt. δ = 0 in this case as there does not exist M 0,n for n < 3. 5.3. Reduction to q 0 = 0. The arguments in this section are mostly taken from [4]. Let E = 0 be a tautological equation of (axiomatic Gromov Witten invariants. Since we have already proved ŝ(e = 0, our next goal would be to show ˆr(E = 0. In this section, we will show that it suffices to check ˆr(E = 0 on the subspace q 0 = 0. Lemma 3. It suffices to show ˆrE = 0 on each level set of the map q s in (8. Proof. The union of the level sets is equal to H +. Lemma 4. It suffices to check the relation for all ˆr(zE = 0 along zh + (i.e. q 0 = 0. Proof. Theorem 5. of [] states that a particular lower triangular matrix S s, which is called calibration of the Frobenius manifold, transforms the level set at s to zh +. S-invariance Theorem then concludes the proof. Remark 3. In fact, S s can be taken as a fundamental solution of the horizontal sections of the Dubrovin (flat connection, in z formal series. It was discovered in [], following the works in [5] and [9], that A := Ŝsτ X is the corresponding generating function for ancestors. Therefore the transformed equation ŜsEŜ s = 0 is really an equation of ancestors. 6. Invariance under upper triangular subgroups 6.. Quantization of upper triangular subgroups. The upper triangular subgroup consists of the regular part of the twisted loop groups R(z = e r(z satisfying (0 R ( zr(z = or equivalently ( r ( z + r(z = 0.

INVARIANCE OF TAUTOLOGICAL EQUATIONS II 5 The quantization of r(z is ˆr(z = (r l ij qn j qn+l i ( Therefore = + + + + ~ l= d i k i k... g dǫ r l= n=0 i,j l= n=0 i,j l m=0( m+ ij (r l ij q j n i n+l i k... g (r l iia k i a+l i k... l= l= i,a l m=0( m+ ij l l= m=0( m+ ij ˆ k a... g ia (r l ij q i l m q j m. (r l ij i l m j m i k i k... g g g =0 (r l ij i k i k...( l m i g m j g g. Here, if g = 0, the third term on the right... = 0 by definition. Also, it is understood that the formula for ˆr l extends to products of Gromov Witten invariants by Leibniz rule. 6.. Relations to invariance of tautological equations. Let E = 0 be a tautological equation on moduli of curves. As explained in Part I, it can be written in terms of a formal sum of decorated graphs. Denote E = 0 also the induced equation of Gromov Witten invariants. d Consider dǫ r E. It is clear that the first term of ( vanish as E = 0 implies (r l ij qn j E = 0. Similarly, the contribution to the second term from ˆ k i at an external marked point (i.e. not at a node cancels. d Therefore, dǫ r E consists of three parts, from the second term (with ˆ k i at a node, third term and fourth term. It follows from the usual correspondence between tautological classes and Gromov Witten invariants that these three parts corresponds to three graphical operations defined in [6]: The second term, when i a are indices at a node, corresponds to cutting the edges.

6 Y.-P. LEE The third term corresponds to genus reduction. The last term corresponds to splitting the vertices. Equation ( implies (r l ij is symmetric in i, j for l even and antisymmetric for l odd. The corresponding operation for a fixed l on the decorated graphs are denoted r l in Part I. In fact, this is the original motivation of the Invariance Conjectures. 6.3. R-invariance. Theorem 4. (R-invariance theorem All tautological equations of Gromov Witten invariants are invariant under action of upper triangular subgroups of the twisted loop groups. Proof. Since we have already established S-invariance theorem, this theorem follows from Theorem. Indeed, (4 implies that there is a loop group element (or rather its quantization taking a tautological equation on moduli of curves to that of any semisimple theory and vice versa. Therefore the set of induced tautological equations of one semisimple theory has a one-one correspondence with the set of tautological equations of another semisimple theory. 6.4. Proof of Invariance Conjecture. A set of three Invariance Conjectures are advanced in [6] to give the tautological relations an inductive structure. The main purpose of this section is to establish Conjecture there, which corresponds to an infinitesimal form of Theorem 4 on moduli of curves. Theorem 5. Invariance Conjecture (stated in [6] is true. Proof. The two ingredients in the proof are ( Teleman s classification theorem of semisimple cohomological field theories [0], applied to Givental s framework on X = P. ( An explicit calculation of r l -matrix of P in the Appendix. Note that ( is stronger than Theorem : It implies a cycle form of Givental s formula. Let E = i c i Γ i = 0 be a tautological relation. (Notations as in Part I [6]. The induced tautological equation on the Gromov Witten theory of two points is denoted Ē. ( implies that this tautological equation on the (two copies of moduli space of curves (X = pt is transformed to the

INVARIANCE OF TAUTOLOGICAL EQUATIONS II 7 corresponding tautological equation on moduli space of stable maps to P. By Theorem 3 and Remark and ( above, d Ē = (r l ij r ij l Ē = 0, dǫ r l= i,j= where r ij l is the operation r l, with the two new half-edges called i and j (for the i-th and j-th points. It remains to prove that (r l ij is non-degenerate in the sense that (r l ij 0 unless r ij l Ē = 0 due to (. This is the content of Proposition, which will be proved in the Appendix. Therefore, r ij l Ē = 0 which implies r l E = 0. 6.5. Three applications of Theorem 5. In a series of joint work with D. Arcara [,, 3] and with A. Givental [4], Theorem 5 is shown to implies A uniform derivation of all known g =, tautological equations. Derivation of a tautological equation in M 3, of codimension 3. All monimials of κ-classes and ψ-classes are independent in R k (M g,n /R k ( M g,n for all k [g/3]. The first two investigate the existence of tautological equations, while the last one deals with the non-existence of tautological equations. Theorem 5 provides a new and sysmtematic way of finding (lack of tautological equations inductively. Appendix A. The R matrix for P by Y. IWAO and Y.-P. LEE The notations here follows those in [9]. All invariants and functions are for X = P. R(z = R n z n n=0 is defined by R 0 = I (the identity matrix and the following recursive relation (.4.5 in [9], ( (3 (u u (Rn (R n (R n (R n (du du + ( (drn (dr n (dr n (dr n = ( 0 (Rn (R n 0 (du du

8 Y.-P. LEE for n, where (u, u are the canonical coordinates of QH (P. Proposition. ( (4 R n = n( n n ( n cn v n, where n k= c n := ( + 4k, v := u n u. n! and it is understood that c = /4. Proof. Equation (3 gives a recursive relation which determines all R n from R 0 = I. It is easy to check that (4 satisfies (3. Recall that r(z is defined as log R(z. Proposition. ( r l = ( 0 c l c l 0 a l b l b l al if l is even, if l is odd, such that a l, b l, c l are all nonzero rational numbers. The rest of the Appendix is used to prove this proposition. First, it is not very difficult to see the matrices r l should be of the above forms and all a l, b l, c l are rational numbers. These assertions follow from the following formula for r l (5 r l z l ( R(z n+ = log R(z = n + l= = l z l n=0 l ( m m= m i + +i m=l i j >0 R i R im, Equation (, and induction on l.

INVARIANCE OF TAUTOLOGICAL EQUATIONS II 9 Here are a few examples of r l matrices obtained from (5 and Proposition : r = ( 4 r = 3 ( 0 3 3 0 r 3 = 5 ( 4 3 3 4 r 4 = 4 ( 0 33 33 0 r 5 = ( 5 9 3 0839 0839 3 etc. The non-vanishing of a l, b l, c l requires some very elementary algebra. The idea is to get some rough estimates of the absolute values of entries of r l, which are enough to guarantee the non-vanishing. We will start with some preparations. Define a function of n, C(n, by Then for n, C(n = { n k= ( 4k if n if n = (R n = C(n (n! ( v n. Lemma 5. C(n is strictly decreasing and lim n C(n > 0.6. Proof. C(n is obviously strictly decreasing. Since C(n is strictly decreasing and positive for all n, lim n C(n exists. So does lim n [ln C(n] = ln[lim n C(n]. Now, since ln( x 4(ln 3 4 x for 0 < x 4,,

0 Y.-P. LEE [ n lim [ln C(n] = lim n n Therefore So =(ln 3 4 lim n ] ln( 4k k= [ n ] k= k lim n = π 6 ln 3 4. π lim [ln C(n] n 6 ln 3 4. [ n lim C(n e π 6 ln 3 4 = 0.699. n k= ] 4(ln 3 4 4k By abuse of notation, we let (R n j i denote the absolute value of the coefficient of v n in (R n j i. Since (R n = n (R n, we get the following corollary: Corollary 3. For n, (n! 0.6 (n! 0.6 4n For a matrix R i R il, (6 (R i R il = (7 (R i R il = a j =, j=,,l a j =, j=,,l Due to the unitary condition (0, < (R n (n! < (R n (n!. 4n (R i a (R i a a (R i3 a a 3 (R il a l a l (R il a l (R i a (R i a a (R i3 a a 3 (R il a l a l (R il a l. (R ij = ±(R ij, (R ij = ±(R ij. So each term in (6 and (7 can be written in the form (8 ± (R ij (R ij (R ijk (R ijk+ (R ijl }{{}}{{} k l k

INVARIANCE OF TAUTOLOGICAL EQUATIONS II Lemma 6. The number of terms of the form (8 is given by ( l k if k is odd ν (l, k = 0 if k is even in (6, and ( l k if k is even ν (l, k = 0 if k is odd. in (7, Lemma 7. For each l, (9 α(x, y, l := k:odd k l ( l x k y l k = (x + yl ( x + y l. k (30 β(x, y, l := k:even k l ( l x k y l k = (x + yl + ( x + y l. k These two lemmas are elementary and proofs are obvious. Now define σ l n by Lemma 8. σ l n := i + +i l =n i j >0 i!i! i l!, σn 8 n! n = 0,,, 3,. 3 Proof. By induction on n. It is easy to check the statement holds for n 7. Now, assume 8 3 n! σ n. It suffices to show 8 3 (n +! σ n+. Case. n is odd. We may assume n 3. 8 3 n! σ n = 0!n! +!(n! + + (n!( n +! + + n!0!.

Y.-P. LEE Therefore, 8 3 (n +! (n + σ n = 0!(n +! +![(n + (n!] +![(n + (n!] + + ( n 3![(n + ( n + 3!] + ( n![(n + ( n +!] + [(n + ( n +!]( n! } {{ } (a + [(n + ( n + 3!]( n 3! + (n +!0! 0!(n +! +!n! +!(n! + + ( n 3 + ( n!( n + 3! + ( n +!( n +!( n + 5!! + ( n + 3!( n! } {{ } (b + ( n + 5!( n 3! + + (n +!0! = σn+. Since n for n 3, (a (b. n + Case. n is even. We may assume n 4. 8 3 n! σ n = 0!n! +!(n! +!(n! + + ( n 4 + ( n!( n +! + ( n!(n +! + (n!( n! + ( n + 4!( n + 4!!( n 4! + n!0!.

So INVARIANCE OF TAUTOLOGICAL EQUATIONS II 3 8 3 (n +! (n + σ n = 0!(n +! +![(n + (n!] +![(n + (n!] + + ( n 4![(n + ( n + 4!] +( n![(n + ( n +!] + (n + ( n!(n +!] + [(n + (n!]( n! }{{ } (c + [(n + ( n + 4!]( n 4! + (n +!0! 0!(n +! +!n! +!(n! + + ( n 4!( n + 6! + ( n!( n + 4! + ( n!(n +! + ( n +!( n + 4! + (n!( n! } {{ } (d Since 4n + n 8 n + n Corollary 4. + ( n + 6!( n 4! + + (n +!0! = σn+. for n 4, (c (d. σ l n (8 3 l n! for n = 0,,, 3,, l =,, 3,. Proof. By induction on l. This is true for n arbitrary, l =. Assume σn l (8 3 l n! for all n. It suffices to show σn l+ all n. ( 8 3 l n! for σ l+ n = 0!σ l n +!σl n +!σl n + + (n!σl + n!σl 0 ( 8 3 l [0!n! +!(n! +!(n! + + (n!! + n!0!] = ( 8 3 l σ n (8 3 l n!. Lemma 9. Let S(m := m k=3 k 3 (m k! k for m 5.

4 Y.-P. LEE Then Proof. T(m := S(m (m 3! is a decreasing function of m for m 5. = S(m + = (m! + 3 m+ k=3 k 3 (m k +! k (m 3! 4 + (m 4! + + m 4! +m 3! +m 0! 5 m m m +. Clearly, ith term of T(m ith term of T(m+ for i =,, m 3, with equality only for i =. Now m 3 m(m! + m (m + (m! < m 3 m(m 3! So T(m + < T(m for m 5. if m 5. Corollary 5. ( T(m < for m 5. ( T(m < 0.477 for m 9. (3 S(m < (m 3! for m 5. (4 S(m < 0.477(m 3! for m 9. Lemma 0. (R i R j + (R j R i = (R i (R j if i + j is odd. Proof. Without loss of generality, we may assume ( ( a b c d R i =, R b a j =, i is even, j is odd. d c Therefore (R i R j + (R jr i = ac = (R i (R j. Corollary 6. m (R k R m k 5 (m 3! for m odd, m 7. 56 k= Proof. Since m is odd,

INVARIANCE OF TAUTOLOGICAL EQUATIONS II 5 4 m (R k R m k m k= k= (Rk Rm k m 3 3(m! 6(m + 3! 3(m 3! 3(! 6 6(m + + 6( m 3 (m! 64 m + 9! (m 3! ( 56 (m + + 3 9 (m 3! + 64 56 = 3 64 [ (m 3! + (m 3! m 3 m 3! ( m 5 [0!(m 4! +!(m 5! + + (m ] = 3 64 ( + (m 3! m 3 (m 3(! (m + 6( (m (! (m + (!( m 3! 3 64 5 5 (m 3! = 4 56 (m 3! for m 7. Lemma. (Rm 0.3(m! for m 7. Proof. By Corollary 3 (Rm (m! > 0.6 4m = 4 0.6 ( (m! m 4 0.6 6 (m! 7 > 0.3(m!. Now we are ready to prove Proposition. We start with off-diagonal entries.

6 Y.-P. LEE l (r l = (R l + ( n n= n i + +i n=l i j >0 (R i R in. } {{ } =:(R l By the triangular inequality, we have (r l (Rl (R l. So it suffices to show (Rl > (R l Now,

INVARIANCE OF TAUTOLOGICAL EQUATIONS II 7 = (R l l n= n= n i + +i n=l i j >0 (Ri R in l (Rij n (Rijk (R ijk+ (Rijn l (ij! (i j k! n n= l n= l n= n k: odd k n ( n k ( k ( 4 n k σ n l n (by Lemma 5 n 3n ( n 4 n ( 8 3 n (l n! (i j k+! 4i jk+ (i j n! 4i jn (by Corollary 3 (by Lemma 7 with x =, y = 4 = 3 l [ n ( 3 ] 6 n n (l n! < 3 6 n= l n= = 3 8 (l! + 3 n n (l n! l n=3 n n (l n! and Corollary 4 < 3 8 (l! + 3 (l 3! for l (by Corollary 5. By Corollary 3, 0.3(l! < (Rl < 0.5(l! for l. So for l 5, (Rl 3 > 0.3(l! > 8 (l! + 3 (l 3! > (R l, i.e., (r l 0. Since we know (r l 0 for l =,, 3, 4, (r l 0 for all l. Similarly for diagonal terms for l = odd.

8 Y.-P. LEE (R l i +i =l i j >0 l (Ri R i + 5 56 (l 3! + 3 6 < 5 56 (l 3! + 3 6 l n=3 l n=3 n n=3 n i + +i n=l i j >0 (Ri R in [ n + ( 3 n ] (l n! n n (l n! = 5 56 (l 3! + 3 S(l < 5 (l 3! + 0.7(l 3! for l 9, m = odd (by Corollary 5. 56 By Lemma, (Rl 5 > 0.3(l! > ( 56 + 0.7(l 3! > (R l for l 9. (r l 0 for l 9, l = odd. We know (r l 0 for l =, 3, 5. It is now left to check l = 7, which is checked by hand. (R 7 5 7 56 4! + 3 m + ( m σ m 4 m 7 m = 5 4 56 + m=3 [ 3 7 08 + 64 4 8 + 56 5 + 5 43 04 0 + 6 79 + 4096 6 + 7 87 6384 On the other hand, (R7 3 5 35 63 99 43 = = 7.04. 4 7! So (r 7 0. The proof of Proposition is now complete. References ] =.370. [] D. Arcara, Y.-P. Lee, Tautological equations in genus two via invariance constraints, math.ag/050488. [] D. Arcara, Y.-P. Lee, Tautological equation in M 3, via invariance constraints, math.ag/050384. [3] D. Arcara, Y.-P. Lee, On independence of generators of the tautological rings, math.ag/0605488.

INVARIANCE OF TAUTOLOGICAL EQUATIONS II 9 [4] T. Coates, A. Givental, Quantum Riemann - Roch, Lefschetz and Serre, math.ag/004. [5] R. Dijkgraaf, E. Witten, Mean field theory, topological field theory, and multimatrix models, Nuclear Phys. B 34 (990, no. 3, 486 5. [6] C. Faber, R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 39 (000, no., 73 99. [7] C. Faber, R. Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS 7 (005, no., 3 49. [8] E. Getzler, Topological recursion relations in genus, Integrable systems and algebraic geometry (Kobe/Kyoto, 997, 73 06, World Sci. Publishing, River Edge, NJ, 998. [9] E. Getzler, The jet-space of a Frobenius manifold and higher-genus Gromov- Witten invariants, math.ag/0338. [0] A. Givental, Semisimple Frobenius structures at higher genus, Internat. Math. Res. Notices 00, no. 3, 65 86. [] A. Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians, Dedicated to the memory of I. G. Petrovskii on the occasion of his 00th anniversary. Mosc. Math. J. (00, no. 4, 55 568, 645. [] A. Givental, A n singularities and nkdv hierarchies, math.ag/00905. [3] A. Givental, Symplectic geometry of Frobenius structures, math.ag/0305409 [4] A. Givental, Y.-P. Lee, In preparation. [5] M. Kontsevich, Yu. Manin, Relations between the correlators of the topological sigma-model coupled to gravity, Comm. Math. Phys. 96 (998, no., 385 398. [6] Y.-P. Lee, Invariance of tautological equations I: conjectures and applications, math.ag/060438. [7] Y.-P. Lee, Witten s conjecture, Virasoro conjecture, and invariance of tautological equations, math.ag/0300. [8] Y.-P. Lee, Witten s conjecture and Virasoro conjecture up to genus two, math.ag/03044. To appear in the proceedings of the conference Gromov- Witten Theory of Spin Curves and Orbifolds, Contemp. Math., AMS. [9] Y.-P. Lee, R. Pandharipande, Frobenius manifolds, Gromov Witten theory, and Virasoro constraints, in preparation. Material needed for this paper are available from http://www.math.princeton.edu/ rahulp/ [0] C. Teleman, in preparation, lecture notes available from http://online.kitp.ucsb.edu/online/strings05/teleman/. Department of Mathematics, University of Utah, Salt Lake City, Utah 84-0090, U.S.A. E-mail address: yplee@math.utah.edu