TOPOLOGICAL RECURSION RELATIONS AND GROMOV-WITTEN INVARIANTS IN HIGHER GENUS

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1 TOPOLOGICAL RECURSION RELATIONS AND GROMOV-WITTEN INVARIANTS IN HIGHER GENUS ANDREAS GATHMANN Abstract. We state and rove a toological recursion relation that exresses any genus-g Gromov-Witten invariant of a rojective manifold with at least a 3g )-st ower of a cotangent line class in terms of invariants with fewer cotangent line classes. For rojective saces, we rove that these relations together with the Virasoro conditions are sufficient to calculate the full Gromov- Witten otential. This gives the first comutationally feasible way to determine the higher genus Gromov-Witten invariants of rojective saces. Consider a moduli sace M g,n X, β) of n-ointed genus-g stable mas of class β to a comlex rojective manifold X. For any i n let ψ i A M g,n X, β)) be the i-th cotangent line class, i.e. the first Chern class of the line bundle on M g,n X, β) whose fibers are the cotangent saces of the underlying curves at the i- th marked oint. Equations in the Chow grou of Mg,n X, β) that exress roducts of cotangent line classes in terms of boundary classes i.e. classes on moduli saces of reducible stable mas) are called toological recursion relations. It has been roven recently by E. Ionel that any roduct of at least g cotangent line classes on M g,n X, β) is a sum of boundary cycles [I]. Unfortunately, the corresonding toological recursion relations are not yet known exlicitly for general g. The g cases can be found in [Ge]. In theory, it should be ossible to derive the equations for other at least low) values of g from Ionel s work. As g grows however, the terms in the toological recursion relations become very comlicated, and their number seems to grow exonentially. Consequently, Ionel s result is barely useful for actual comutations, although it is of course very interesting from a theoretical oint of view. In this aer, we will rove a seemingly much weaker toological recursion relation that exresses only a roduct of at least 3g cotangent line classes at the same oint in terms of boundary cycles. The idea to obtain this relation is simle: we just ull back the obvious relation ψ 3g = 0 on M g, to M g,n X, β) along the forgetful ma, and kee track of the various ull-back correction terms in a clever way. The result is a toological recursion relation that is extremely easy to state and aly. To be recise, denote by τ m γ ) τ mn γ n ) g the genus-g Gromov- Witten correlation function i.e. the generating function for all invariants containing evγ ψ m evnγ n ψn mn ; see section for details). Let {T a } be a basis of the 99 Mathematics Subject Classification. 4N35,4N0,4J70.

2 ANDREAS GATHMANN cohomology of X, and denote by {T a } the Poincaré-dual basis. Then for any m 0 τ 3g +m γ) g = i+j=3g τ m γ)t a i τ j T a ) g, where we use the Einstein summation convention over a, and where the auxiliary correlation functions i are defined recursively by τ m γ )γ i = τ m+ γ )γ i τ m γ )T a 0 T a γ i with the initial condition 0 = 0. Unlike other toological recursion relations, our relations involve neither sums over grahs nor invariants of genus other than g and 0. Moreover, the auxiliary functions i are universal in the sense that they do not deend on g. All this makes our relations very easy and fast to aly. The alication that we have in mind in this aer is the Virasoro conditions for the Gromov-Witten invariants of rojective saces. It has been roven recently by Givental that the Gromov-Witten otential of a rojective sace P r satisfies an infinite series of differential equations, called the Virasoro conditions [Gi]. It is easily checked that these equations allow for recursion over the genus and the number of marked oints in the following sense: given g > 0, n, cohomology classes γ,..., γ n A X), and non-negative integers m,..., m n, the Virasoro conditions can exress linear combinations of invariants τ m γ)τ m γ ) τ mn γ n ) g,d where m 0, γ A X), and the degree d 0 vary) in terms of other invariants with either smaller genus, or the same genus and smaller number of marked oints. There is one such invariant for every choice of m, i.e. r + invariants for every choice of d. There is however only one non-trivial Virasoro condition for every d. Consequently, the Virasoro conditions alone are not sufficient to comute the Gromov-Witten invariants. This is where the toological recursion relations come to our rescue. By inserting them into the Virasoro conditions, we can effectively bound the value of m in the set of unknown invariants above, leaving only the invariants with 0 m < 3g. This way we arrive at infinitely many linear Virasoro conditions one for every choice of d) for only 3g invariants. It is now of course strongly exected that this system should be solvable, i.e. that the coefficient matrix of this system of linear equations has maximal rank 3g. We will show that this is indeed always the case. In fact, we will show that any choice of 3g distinct non-trivial Virasoro conditions leads to a system of linear equations that determines the invariants uniquely. We do this by comuting the determinant of the corresonding coefficient matrix: if we ick the Virasoro conditions associated to the degrees d 0,..., d 3g and reduce the cotangent line owers by our toological recursion relations, we arrive at a system

3 TOPOLOGICAL RECURSION RELATIONS IN HIGHER GENUS 3 of 3g linear equations for 3g invariants whose determinant is simly i>j d 3g i d j ) 3g i= i! i + 3g i, ) i= which is obviously always non-zero. Therefore the Virasoro recursion works, i.e. we have found a constructive and not too comlicated) way to comute the Gromov- Witten invariants of P r in any genus. One should note that the result for the determinant above is remarkably simle, given the comlicated structure of the Virasoro equations and our somewhat arbitrary choice of toological recursion relations. It would be interesting to see whether there is some deeer connection between the Virasoro conditions and the toological recursion relations that exlains this easy result. It would also be interesting to extend our result to other Fano varieties. We should also mention that our results have already been conjectured some time ago by Eguchi and Xiong [EX]. In their aer, they motivate our toological recursion relations by arguments from string theory. Assuming that these relations hold, Eguchi and Xiong use them together with the Virasoro conditions to comute a few examles of higher genus Gromov-Witten invariants of rojective saces. From this oint of view one can regard our aer as roviding a solid mathematical footing for [EX]. The aer is organized as follows. In section we will establish the toological recursion relations mentioned above. We will then describe in section how to aly these results to the Virasoro conditions to get systems of linear equations for the Gromov-Witten invariants of rojective saces. The roof that these systems of equations are always solvable i.e. the comutation of the determinant mentioned above) is given in section 3. Finally, we will list some numbers obtained with our method in section 4. A C++ rogram that imlements the algorithm of our aer and comutes the Gromov-Witten invariants of rojective saces can be obtained from the author on request.. Toological recursion relations The goal of this section is to rove the toological recursion relation stated in the introduction. To do so, we will first comare certain cycles in the moduli saces of stable and restable curves. For any g, n 0 let M g,n be the moduli sace of comlex n-ointed genus-g restable curves, i.e. the moduli sace of tules C, x,..., x n ) where C is a nodal curve of arithmetic genus g, and the x i are distinct smooth oints of C. This is a roer but not searated) smooth Artin stack of dimension 3g 3 + n. The oen substack corresonding to irreducible curves is denoted M g,n. Recall that a restable curve C, x,..., x n ) is called stable if every rational comonent has at least 3 and every ellitic comonent at least secial oint, where the secial oints are the nodes and the marked oints. The oen substack of Mg,n corresonding to stable curves is denoted M g,n. It is a roer, searated,

4 4 ANDREAS GATHMANN smooth Deligne-Mumford stack. If g + n 3 there is a stabilization morhism s : M g,n M g,n that contracts every unstable comonent. On any of the above moduli saces and for any of the marked oints x i we define the cotangent line class, denoted ψ xi, to be the first Chern class of the line bundle on the moduli sace whose fiber at the oint C, x,..., x n ) is the cotangent sace TC,x i. Cotangent line classes do not remain unchanged under stabilization they receive correction terms from the locus of reducible curves where the marked oint is on an unstable comonent. Our first task is therefore to comare the cycles s ψ xi and ψ xi on M g,n for g > 0. For simlicity, we will do this here only in the case n =. As there is then only one cotangent line class, we will simly write it as ψ with no index). Let us define the correction terms that we will ick u when ulling back cotangent line classes. For fixed g > 0 and any k 0 let M k be the roduct M k = M g, M 0, M 0,. }{{} k coies Obviously, oints of M k corresond to a collection C 0),..., C k) ) of restable curves with C 0) of genus g and C i) of genus 0 for i > 0, together with marked oints x 0 on C 0) and x i, y i on C i) for i > 0. There is a natural roer gluing morhism π : M k M g, that sends any oint C 0),..., C k), x 0, x, y,..., x k, y k ) to the nodal -ointed curve C 0) C k), x k ), where C i ) is glued to C i) for all i =,..., k by identifying x i with y i. The morhism π is an isomorhism onto its image on the oen subset M g, M 0, M 0,. Obviously, a generic oint in the image of π is just a curve with k + comonents aligned in a chain, with the first comonent having genus g, and all others being rational. The marked oint is always on the last comonent. For any collection λ = λ 0,..., λ k ) of k + non-negative integers we denote by Zλ) the cycle π ψ λ0 x 0 ψ λ k x k [M k ]) A M g, ) in the Chow grou of M g, for intersection theory on Artin stacks we refer to [K]). The cycle Zλ) has ure codimension k+λ 0 + +λ k in M g,. It can be reresented grahically as genus g ψ λ ψ λ k x ψ λ 0 ψ λ k genus 0 where the cotangent line classes sit at the left oints of the nodes excet for the last one that sits on the remaining marked oint). With these cycles Zλ) we can now formulate the ull-back transformation rule for cotangent line classes.

5 TOPOLOGICAL RECURSION RELATIONS IN HIGHER GENUS 5 Lemma.. For any k 0 and λ 0,..., λ k 0 we have s ψ Zλ 0,..., λ k ) = Zλ 0 +, λ,..., λ k ) Z0, λ 0, λ,..., λ k ) in A M g, ), where s : M g, M g, is the stabilization ma. Proof. Let : M k = M g, M 0, M 0, M g, be the rojection onto the first factor. Note that s = s π. By [Ge] roosition 5 we have the transformation rule genus g s* ψ = ψ _ genus 0 on M g,. Pulling this equation back by, intersecting with ψx λ0 0 ψ λ k x k, and ushing it forward again by π then gives the equation of the lemma. It is easy to iterate this lemma to get an exression for s ψ) N : Corollary.. For all N 0 we have s ψ) N = k 0 ) k λ 0,...,λ k ) k+λ 0+ +λ k =N Zλ 0, λ,..., λ k ) in A M g, ). In articular, the right hand side is zero if N 3g. Proof. The statement is obvious for N = 0 as M g, = Z0). The equation now follows immediately by induction from lemma.. Moreover, note that M g, is a Deligne-Mumford stack of dimension 3g, so its Chow grous in codimension at least 3g vanish. Therefore s ψ) N = s ψ N ) = 0 for N 3g. Remark.3. Note that in the saces M k the marked oint is always on the last comonent. So by intersecting the equation of corollary. with the m-th ower of the cotangent line class on M g, we get ψ m s ψ) N = ) k Zλ 0, λ,..., λ k, λ k + m) k 0 λ 0,...,λ k ) k+λ 0+ +λ k =N in A M g, ) for all N, m 0. As in the corollary, the right hand side will be zero if N 3g. We will now aly this result to moduli saces of stable mas. So let X be a comlex rojective manifold, and let β be the homology class of an algebraic curve in X. As usual we denote by M g,n X, β) the moduli sace of n-ointed genus-g stable mas of class β to X see e.g. [FP]). It is a roer Deligne-Mumford stack of virtual dimension vdim M g,n X, β) = K X β + dim X 3) g) + n. The actual dimension of Mg,n X, β) might and for g > 0 usually will) be bigger than this virtual dimension. There is however always a canonically defined virtual fundamental class [ M g,n X, β)] virt A vdim Mg,nX,β) M g,n X, β))

6 6 ANDREAS GATHMANN that is used instead of the true fundamental class in intersection theory, and that therefore makes the moduli sace aear to have the correct dimension for intersection-theoretic uroses see e.g. [BF], [B]). The oints in this moduli sace can be written as C, x,..., x n, f), where C, x,..., x n ) M g,n, f : C X is a morhism of degree β, and every rational res. ellitic) comonent on which f is constant has at least 3 res. ) secial oints. The moduli saces M g,n X, β) come equied with cotangent line classes ψ xi in the same way as above. In addition, for every marked oint x i we have an evaluation morhism ev xi : Mg,n X, β) X given by ev xi C, x,..., x n, f) = fx i ). For any cohomology classes γ,..., γ n A X) and non-negative integers m,..., m n we define the Gromov-Witten invariant τ m γ ) τ mn γ n ) g,β := ψ [ M x m g,nx,β)] virt ev x γ ψ mn x n ev x n γ n Q, where the integral is understood to be zero if the integrand is not of dimension vdim M g,n X, β). If m i = 0 for some i we abbreviate τ mi γ i ) as γ i within the brackets on the left hand side. As the Gromov-Witten invariants are multilinear in the γ i, it suffices to ick the γ i from among a fixed basis. So let us choose a basis {T a } of the cohomology modulo numerical equivalence) of X and let {T a } be the Poincaré-dual basis. Remark.4. It is often convenient to encode the Gromov-Witten invariants as the coefficients of a generating function. So we introduce the so-called correlation functions τ m γ ) τ mn γ n ) g := ) τ m γ ) τ mn γ n ) ex t a mτ m T a ) q β β m where the t a m and q β are formal variables satisfying q β q β = q β+β. Here and in the following we use the summation convention for the cohomology index a, i.e. an index occurring both as a lower and uer index is summed over. The correlation functions are formal ower series in the variables t a m and q β whose coefficients describe all genus-g Gromov-Witten invariants containing at least the classes τ m γ ) τ mn γ n ). With this notation we can now rehrase corollary. in terms of correlation functions for Gromov-Witten invariants: Proosition.5. For all g > 0, N 3g, m 0, and γ A X) we have 0 = ) k k 0 λ 0,...,λ k ) k+λ 0+ +λ k =N τ λ0 T a ) g T a τ λ T a ) 0 T ak τ λk T a k ) 0 T ak τ λk +mγ) 0 as ower series in t a m and q β. g,β

7 TOPOLOGICAL RECURSION RELATIONS IN HIGHER GENUS 7 Proof. For every n and any homology class β there is a forgetful morhism q : Mg,n X, β) M g,, C, z,..., z n, f) C, z ). We denote the marked oints by z i in order not to confuse them with the x i and y i above that are used to glue the comonents of the reducible curves.) We claim that the statement of the roosition is obtained by ulling back the equation of remark.3 by q in the case N 3g and evaluating the result on the virtual fundamental class of Mg,n X, β). To show this, we obviously have to comute q Zλ) for all λ = λ 0,..., λ k ). Let B = β 0,..., β k ) be a collection of homology classes with i β i = β, and let I = I 0,..., I k ) be a collection of subsets of {,..., n} whose union is {,..., n}. Set M B,I = M g,+#i0 X, β 0 ) X M0,+#I X, β ) X X M0,+#Ik X, β k ), where the fiber roducts are taken over the evaluation mas at the first marked oint of the i )-st factor and the second marked oint of the i-th factor for i =,..., k. In other words, the moduli sace M B,I describes stable mas with the same configuration of comonents as in M k, and with the homology class β and the marked oints z,..., z n slit u onto the comonents in a rescribed way the oint z is always the first marked oint of the last factor). Note that the sace M B,I carries a natural virtual fundamental class induced from their factors. By [B] axiom III it is equal to the roduct cycle of the virtual fundamental classes of the factors, intersected with the ull-backs of the diagonal classes X X X along the evaluation mas at every air of marked oints where two comonents are glued together. Now by [B] axiom V we have a Cartesian diagram B,I M π B,I M g,n X, β) q M k π M g, with π! [ M g,n X, β)] virt = B,I [M B,I] virt. As q does not change the cotangent line classes we conclude that q Zλ) [ M g,n X, β)] virt = q π ψ λ0 x 0 ψ λ k x k ) [ M g,n X, β)] virt q = π q ψ λ0 x 0 ψ λ k x k ) [ M g,n X, β)] virt = B,I π ψ λ 0 x 0 ψ λ k x k [M B,I ] virt), ) where x i denotes the first marked oint of the i-th factor. Now choose cohomology classes γ,..., γ n and non-negative integers m,..., m n. Intersecting exression ) with evz γ ψz m evz n γ n ψz mn n and taking the degree of the resulting homology class if it is zero-dimensional), we get exactly the Gromov-Witten invariants τ λ0 T a )T 0 g,β0 T a τ λ T a )T 0,β T ak τ λk +l γ )T k 0,βk, B,I

8 8 ANDREAS GATHMANN where T i is a short-hand notation for j I i τ mj γ j ). If we choose γ i = T ai for some a i, this can obviously be rewritten as the q β n ) i= tai m i -coefficient of the function τ λ0 T a ) g T a τ λ T a ) 0 T ak τ λk T a k ) 0 T ak τ λk +mγ) 0. Inserting this into the formula of remark.3 gives the desired result. Corollary.6. Toological recursion relation) For all g > 0, N 3g, m 0, and γ A X) we have τ N+m γ) g = τ m γ)t a i τ j T a ) g, i+j=n where the auxiliary correlation functions i are defined recursively by τ m γ )γ i = τ m+ γ )γ i τ m γ )T a 0 T a γ i with the initial condition 0 = 0. Proof. Note that the k = 0 term in lemma.5 is just τ N+m γ) g. So we find that the equation of the corollary is true if we set τ m γ )γ i = ) k k 0 λ,...,λ k ) k+λ + +λ k =N T a τ λ T a ) 0 T ak τ λk T a k ) 0 T ak τ λk +mγ) 0. It is checked immediately that these correlation functions satisfy the recursive relations stated in the corollary. Remark.7. As in the case of the Gromov-Witten invariants, we will exand the correlation functions i as a ower series in q β and t a m and call the resulting coefficients i β according to the formula τ m γ )γ i = ) i τ m γ )γ ex t a mτ m T a ) q β. β m Note however that, in contrast to the Gromov-Witten numbers, the invariants i must have at least two entries, of which the second one contains no cotangent line class. For future comutations it is convenient to construct a minor generalization of corollary.6 that is mostly notational. Note that all genus-0 degree-0 invariants with fewer than 3 marked oints are trivially zero, as the moduli saces of stable mas are emty in this case. It is an imortant and interesting fact that many formulas concerning Gromov-Witten invariants get easier if we assign virtual values to these invariants in the unstable range: Convention.8. Unless stated otherwise, we will from now on allow formal negative owers of the cotangent line classes i.e. the index m in the τ m γ) can be any integer). Invariants g,β and i β are simly defined to be zero if they contain a negative ower of a cotangent line class at any oint, excet for the following cases of genus-0 degree-0 invariants with fewer than 3 marked oints: β

9 TOPOLOGICAL RECURSION RELATIONS IN HIGHER GENUS 9 i) τ t) 0,0 =, ii) τ m γ )τ m γ ) 0,0 = ) maxm,m) γ γ )δ m+m,, iii) τ i γ )γ i 0 = γ γ ) for all i 0. The correlation functions are changed accordingly so that the equations of remarks.4 and.7 remain true in articular these functions will now deend additionally on the variables t a m for m < 0). Remark.9. Note that this convention is consistent with the general formula for genus-0 degree-0 invariants ) n 3 τ m γ ) τ mn γ n ) 0,0 = γ γ n )δ m+ +m m,..., m n,n 3, n as well as with the recursion relations for the i of corollary.6. Using this convention, we can now restate our toological recursion relations as follows: Corollary.0. For all g > 0, N 3g, m Z, and γ A X) we have τ N+m γ) g = i+j=n τ m γ)t a i τ j T a ) g, where the auxiliary correlation functions i are defined recursively by the formulas given in corollary.6, together with convention.8. Proof. The equations in the corollary are the same as in corollary.6 if m 0. For m < 0 they reduce to the trivial equations τ N+m γ) g = τ N+m γ) g by convention.8.. The Virasoro conditions We now want to aly our toological recursion relation in conjunction with the Virasoro conditions to comute Gromov-Witten invariants. The Virasoro conditions are certain relations among Gromov-Witten invariants conjectured in [EHX] that have recently been roven for rojective saces by Givental [Gi]. We will therefore from now on restrict to the case X = P r. It is exected that the same methods would work for other Fano varieties as well. To state the Virasoro conditions we need some notation. We ick the obvious basis {T a } of A X) where T a denotes the class of a linear subsace of codimension a for a = 0,..., r. Let R : A X) A X) be the homomorhism of multilication with the first Chern class c X). In our basis, the -th ower R of R is then given by R ) b a = r + ) δ a+,b. For any x Q, k Z, and 0 k + denote by [x] k the z -coefficient of k j=0 z + x + j), or in other words the k + )-th elementary symmetric olynomial in k + variables evaluated at the numbers x,..., x + k.

10 0 ANDREAS GATHMANN Then the Virasoro conditions state that for any k and g we have an equation of ower series in t a m and q β see e.g. [EHX]) k+ [ 3 r 0 = =0 k+ + =0 m=0 + + k+ ] k =0 m= k k+ R ) b 0 τ k+ T b ) g A) [ a + m + r =0 m= k h=0 ] k [ ) m a + m + r R ) b a t a m τ k+m T b ) g B) ] k g [ ) m a + m + r R ) b a τ m T a )τ k+m T b ) g C) ] k R ) b a τ m T a ) h τ k+m T b ) g h, D) where convention.8 is not yet alied i.e. the genus-0 degree-0 invariants in the unstable range are defined to be zero). We should also mention that there are versions of these relations also for k and all g 0, but the equations will then get additional correction terms that we have droed here for the sake of simlicity. First of all let us aly convention.8 to these formulas. It is checked immediately that this realizes the A) and B) terms as art of the D) terms via the conventions i) and ii), resectively. So by alying our convention we can dro the A) and B) terms above if we allow arbitrary integers in the sum over m). Let us now analyze how these equations can be used to comute Gromov-Witten invariants. First of all we will comute the invariants recursively over the genus of the curves. The genus-0 invariants of P r are well-known and can be comuted by the WDVV equations see e.g. the first reconstruction theorem of Kontsevich and Manin [KM]). So let us assume that we want to comute the invariants of some genus g > 0, and that we already know all invariants of smaller genus. In the Virasoro equations above this means that we know all of C), as well as the terms of D) where h 0 and h g. Noting that the D) terms are symmetric under h g h, we can therefore rewrite the Virasoro conditions as k+ [ ) m a + m + r =0 m ] k = recursively known terms). R ) b a τ m T a ) 0 τ k+m T b ) g Next, we will comute the invariants of genus g recursively over the number of marked oints. So let us assume that we want to comute the n-oint genus-g invariants, and that we already know all invariants of genus g with fewer marked oints. In the equations above this means that we fix a degree d 0, integers m,..., m n, and n cohomology classes T a,..., T an, and comare the qd n ) i= tai m i -coefficients of the equations. By the recursion rocess we then know all the invariants in which at least one of the marked oints x,..., x n is on the

11 TOPOLOGICAL RECURSION RELATIONS IN HIGHER GENUS genus-0 invariant. So we can write k+ =0 m d +d =d [ ) m a + m + r ] k R ) ab τ m T a ) 0,d τ k+m T b )τ m T a ) τ mn T an ) g,d = recursively known terms). These are equations for the unknown invariants τ j T b )τ m T a ) τ mn T an ) g,e, where j, b, and e vary. Note that for a given j 0 there is exactly one such invariant τ j T bj )τ m T a ) τ mn T an ) g,ej : the values of b j and e j are determined uniquely by the dimension condition r + )e j + r 3) g) + n = j + b j + n m i + a i ) ) as we must have 0 b j r. Let us denote this invariant by x j. Of course it may haen that e j < 0, in which case we set x j = 0. Our equations now read k+ [ ) m+ k a + m + k + r =0 m = recursively known terms). ] k i= R b ) m a τ m +k T a ) 0,d em x m Let us now check how many non-trivial equations of this sort we get. Together with ), the dimension conditions d e m )r + ) + r 3 + = m + k + r a for the genus-0 invariant) and a + = b m from the R factor) give k = dr + ) + r 3) g) + n n m i + a i ), ) which means that the value of k is determined by d. To avoid overly comlicated notation, in what follows we will denote the number k determined by ) by kd). Moreover, let δ be the smallest value of d for which kd) is ositive. We are then getting one equation for every degree d δ. As there are r + unknown invariants x j in every degree however, it is clear that our equations alone are not sufficient to determine the x j. Let us now aly our toological recursion relations. In terms of the recursion at hand, these relations can exress every invariant x m as a linear combination of invariants of the same form with m < 3g, lus some terms that are known recursively because they contain only invariants with fewer than n marked oints. More recisely, we have x m = τ m N+ T bm )T bj i e m e j x j + recursively known terms) i+j=n i=

12 ANDREAS GATHMANN for all N 3g by corollary.0. Inserting this into the Virasoro conditions, we get kd)+ =0 m i+j=n [ ) m+ kd) a + m + kd) + r ] kd) R ) a bm τ m +kd) T a ) 0,d em τ m N+ T bm )T bj i e m e j x j = recursively known terms). Using the dimension conditions again, and noting that the sum over m is equivalent to indeendent sums over b m and e m, we can rewrite this as kd)+ =0 e i+j=n [ 3 r ) a d e)r+) τ T a ) 0,d e τ T b )T bj i e e j x j = recursively known terms), ] kd) d e)r + ) R ) ab where the dots in the τ functions denote the uniquely determined numbers so that the invariants satisfy the dimension condition. We are thus left with infinitely many equations one for every d δ) for finitely many variables x 0,..., x N. It is of course strongly exected that this system of equations should be solvable, i.e. that the matrix V N) = V N) d,j ) d δ,0 j<n with V N) d,j := kd)+ =0 e ) a d e)r+) [ 3 r ] kd) d e)r + ) R ) ab τ T a ) 0,d e τ T b )T bj N j e e j 3) has maximal rank N. This is what we will show in the next section. In fact, we will rove that every N N submatrix of V is invertible. So we have shown Theorem.. The Virasoro conditions together with the toological recursion relations of corollary.0 give a constructive way to determine all Gromov-Witten invariants of rojective saces. In contrast to other known relations that in theory determine the Gromov-Witten invariants see [GP], [LLY]), our algorithm is easily imlemented on a comuter. No comlicated sums over grahs occur anywhere in the rocedure. It should be noted however that the calculation of some genus-g degree-d invariant usually requires the recursive calculation of invariants of smaller genus with bigger degree and more marked oints. This is the main factor for slowing down the algorithm as the genus grows. Some numbers that have been comuted using this algorithm can be found in section Comutation of the determinant The goal of this section is to rove the technical result needed for theorem.:

13 TOPOLOGICAL RECURSION RELATIONS IN HIGHER GENUS 3 Proosition 3.. Fix any N, and let V N) = V N) d,j ) d δ,0 j<n be the matrix defined in equation 3). Then any N N submatrix of V N), obtained by icking N distinct values of d, has non-zero determinant. We will rove this statement in several stes. In a first ste, we will make the entries of the matrix indeendent of N and reduce the invariants i to ordinary rational Gromov-Witten invariants: Lemma 3.. Let W = W d,j ) d δ,j 0 be the matrix with entries W d,j = V j+) d,j. Then: i) For all d δ, N, and 0 j < N we have V N+) d,j = V N) d,j T bn T bj N j e N e j W d,n. ii) For any N and any N N submatrix of W obtained by taking the first N columns of any N rows, the determinant of this submatrix is the same as the corresonding submatrix of V N). Proof. i): Comaring the q e ej -terms of the recursive relations of corollary.6 we find that τ T b )T bj N j e e j = τ T b )T bj N j e e j τ T b )T b N 0,e en T bn T bj N j e N e j, from which the claim follows. ii): We rove the statement by induction on N. There is nothing to show for N =. Now assume that we know the statement for some value of N, i.e. any two corresonding N N submatrices of the matrices with columns W,0,..., W,N ) and V N),0,..., V N),N ) have the same determinant. Of course, the same is then also true for any corresonding N + ) N + ) submatrices of W,0,..., W,N, W,N ) and V N),0,..., V N),N, W,N ). But by i), the latter matrix is obtained from V N+),0,..., V N+),N, W,N ) = V N+),0,..., V N+),N ) by an elementary column oeration, so the result follows. So by the lemma, it suffices to consider the matrix W. Let us now evaluate the genus-0 Gromov-Witten invariants contained in the definition of W. Convention 3.3. For the rest of this section, we will make the usual convention that a roduct i i=i A i is defined to be i i=i + A i if i > i. Lemma 3.4. For all d δ and j 0 the matrix entry W d,j is equal to the z j - coefficient of kd) i=0 r r + i ) z ) + d e j )z) bj+ d e j i=0 + iz). r+

14 4 ANDREAS GATHMANN Proof. Recall that by equation 3) the matrix entries W d,j are given by [ ] kd) 3 r d e)r + ) R b ) a ) a d e)r+) τ T a ) 0,d e τ T b )T b j 0,e ej. e, }{{}}{{}}{{} A) B) C) The three terms in this exression can all be exressed easily in terms of generating functions. Recalling that R ) b a = r+) δ a+,b, the A) term is by definition equal to the z -coefficient of kd) δ a+,b i=0 r + )z + 3 r ) d e)r + ) + i. The B) and C) terms are rational -oint invariants of P r which have been comuted in [P] section.4: the Gromov-Witten invariant in B) without the sign) is equal to the z a -coefficient of d e i= z+i). So including the sign factor we get r+ the z a -coefficient of d e i= z i). The C) term is again by [P] equal to the r+ z r b e ej -coefficient of z+e e j) b j + i= z+i). r+ Multilying these exressions and erforming the sums over a, b, and, we find that W d,j is the z r -coefficient of e kd) i=0 r + )z + 3 r d e)r + ) + i ) z + e e j ) bj+ d e i= z i)r+ e e j i= z + i) r+, which can be rewritten as the sum of residues e res z=0 kd) i=0 r + )z + 3 r d e)r + ) + i ) z + e e j ) bj+ e e j i=e d z + i)r+ dz. Note that this fraction deends on z and e only in the combination z + e. Consequently, instead of summing the above residues at 0 over all e we can as well set e = d and sum over all oles z C of the rational function. So we see that W d,j is equal to kd) i=0 r + )z + 3 r + i ) res z=z0 z 0 C z + d e j ) bj+ d e j dz. i=0 z + i) r+ By the residue theorem this is nothing but the residue at infinity of our rational function. So we conclude that kd) r+ i=0 z + 3 r + i ) W d,j = res z=0 z + d e j) bj+ d e j i=0 z + d i)r+ z ). Finally note that by equations ) and ) we have the dimension condition kd) + = j + b j + d e j )r + ), so multilying our exression with z kd)+ in the numerator and denominator we get kd) i=0 r r + i ) z ) W d,j = res z=0 z j+ + d e j )z) bj+ d e j dz. i=0 + iz) r+ This roves the lemma.

15 TOPOLOGICAL RECURSION RELATIONS IN HIGHER GENUS 5 To avoid unnecessary factors in the determinants, let us divide row d of W by the non-zero number r + ) kd)+ and call the resulting matrix W. So we will now consider N N submatrices of W = Wd,j ), obtained by icking the first N columns of any N rows, where W d,j is the z j -coefficient of kd) i=0 + 3 r r+ + i r+ + d e j )z) bj+ d e j. 4) i=0 + iz) r+ The following technical lemma is the main ste in comuting their determinants. Lemma 3.5. Assume that we are given N, n N, M Z, q, c R, and distinct integers a 0,..., a N. Set N fz) = na k + c + i ) a k a i n z + a k z) q + iz) n k=0 i k i=m i=a k as a formal ower series in z. i) For any i 0 the z i -coefficient of fz) is a symmetric olynomial in a 0,..., a N of degree at most i N. In articular, it is zero for i < N.) ii) The z N -coefficient of fz) is equal to N! i= ) z N c + q N + n + ) + i. Proof. In the following roof, we will slightly abuse notation and vary the arguments given exlicitly for the function f. So if we e.g. want to study how fz) changes if we vary c, we will write fz) also as fz, c), and denote by fz, c + ) the function obtained from fz) when substituting c by c +. i): It is obvious by definition that fz) is symmetric in the a i. We will rove the olynomiality and degree statements by induction on N. N = 0 : In this case we have fz) = na 0 i=m + c + i n z ) + a 0 z) q + iz) n. i=a 0 We have to show that the z i -coefficient of fz) is a olynomial in a 0 of degree at most i. Note that this roerty is stable under taking roducts, so if we write fz) = n j=0 f j) z) with 0 f 0) z) = + c + i ) n z + a 0 z) q and f j) z) = i=m a 0 i=0 + c + j n ) z + iz ) for j n then it suffices to rove the statements for the f j) searately. But the statement is obvious for f 0), so let us focus on f j) for j > 0. Note that f j) z, a 0 + ) = f j) z, a 0 ) + c + j ) z. n + a 0 z

16 6 ANDREAS GATHMANN So if f i denotes the z i -coefficient of fz) we get f j) i a 0 + ) f j) i a 0 ) = c + j n i a 0 ) k f j) i k a 0). 5) The statement now follows by induction on i: it is obvious that the constant z-term of f j) z) is. For the induction ste, assume that we know that f j) i is olynomial of degree at most i in a 0 for i = 0,..., i 0. Then the right hand side of 5) is olynomial of degree at most i 0 in a 0, so f j) i 0 is olynomial of degree at most i 0. This comletes the roof of the N = 0 art of i). N N + : Note that fz, N +, a 0,..., a N+ ) = fz, N, a 0,..., a N ) fz, N, a,..., a N+ ). 6) a 0 a N+ By symmetry we have fz, N, a 0,..., a N ) = fz, N, a,..., a N+ ) if a 0 = a N+. Hence every z i -coefficient of this exression is a olynomial in the a k. Its degree is at most i N) by the induction hyothesis. This roves i). ii): By i) the z N -coefficient of fz, N) does not deend on the choice of a k, so we can set a k = a + k for all k and kee only a as a variable. It does not deend on M either, as a shift M M ± corresonds to multilication of fz) with + αz) for some α, which does not affect the leading coefficient of fz). So we can set M = without loss of generality. The recursion relation 6) now reads fz, N +, a) = k=0 fz, N, a + ) fz, N, a). 7) N + By i) the z i -coefficient of fz) has degree at most i N in a. So if we denote by f i the a i N -coefficient of the z i -coefficient of fz, a), comaring the z i -coefficients in 7) yields f i N + ) = i N N+ f in) and therefore f N N) = n n n f N0) = f N 0). In other words, instead of comuting the z N -coefficient of fz, N) we can as well set N = 0 and comute the a N -coefficient i.e. the leading coefficient in a) of the z N -coefficient of fz, N = 0). So let us set N = 0 to obtain na fz) = + c + i ) n z + az) q + iz) n, i= and denote by g N the a N -coefficient of the z N -coefficient of fz, a). Moreover, set gz) = N 0 g Nz N. Our goal is then to comute gz). We will do this by analyzing how fz) and thus gz)) varies when we vary q, c, or n. To start, it is obvious that for all α R. Next, note that i=a gz, q + α) = + z) α gz, q) 8) fz, c + ) = fz, c) + c++na n z + c+ n z.

17 TOPOLOGICAL RECURSION RELATIONS IN HIGHER GENUS 7 For gz) we can dro all terms in which the degree in a is smaller than the degree in z. So we conclude gz, c + ) = + z)gz, c). Combining this with 8) we see that gz) will deend on q and c only through their sum q + c. So in what follows we can set c = 0, and relace q by q + c in the final result. Varying n is more comlicated. We have ) n a j fz, n + ) = fz, n) + nn + ) z + i j j=0 i= n )z. }{{} =: fz) Recall that for gz) we only need the summands in fz) in which the degree in a is equal to the degree in z. So let us denote the a N -coefficient of the z N -coefficient of fz, a) by g N, and assemble the g N into a generating function gz) = N 0 g Nz N, so that gz, n + ) = gz, n) gz). To determine gz) comare the a N -coefficient of the z N -coefficient in the recursive equation fz, a) fz, a ) z = fz, a ) z n + j=0 j nn + ) z + a j n )z ). On the left hand side this coefficient is N + ) g N+. On the right hand side it is the z N -coefficient of N gz) j nn + ) + z = gz) + z. j=0 So we see that d gz) = dz + z gz). Together with the obvious initial condition g0) = we conclude that gz) = + z, and therefore gz, n + ) = gz, n) + z. Comaring this with 8) we see that gz) deends on n and q only through the sum q + n n. We can therefore set n = and then relace q by q + in the final result. But setting n to and c to 0) we are simly left with a fz) = + iz) + az) q + iz) = + az) q+. i= So it follows that gz) = + z) q+ and therefore ) q + g N = = N q + N + i). N N! Setting back in the c and n deendence, i.e. relacing q by q + c + n, we get the desired result. i=a i= We are now ready to comute our determinant.

18 8 ANDREAS GATHMANN Proosition 3.6. Let W ) d δ,j 0 be the matrix defined in equation 4). Pick N distinct integers d 0,..., d N with d i δ for all i. Then the determinant of the N N submatrix of W obtained by icking the first N columns of rows d 0,..., d N is equal to i>j d i d j ) N i= i! N i= In articular, this determinant is never zero. i + ) N i. Proof. We rove the statement by induction on N. The result is obvious for N = as every entry in the first column i.e. j = 0) of W is equal to. So let us assume that we know the statement for a given value N. We will rove it for N +. Denote by d 0,..., d N ) the determinant of the N N submatrix of W obtained by icking the first N columns of rows d 0,..., d N. Then by exansion along the last column and the induction assumtion we get N d 0,..., d N ) = ) k+n Wdk,N d 0,..., d k, d k+,..., d N ) = k=0 i>j d i d j ) N i= i! N i= i + ) N i N d k d W dk,n. i i k But by lemma 3.5, alied to the values n = r +, a k = d k e N, q = b N, M = r + )d k e N ) kd k ) = N b N, and c = 3 r M = r + N + b N, the sum in this exression is equal to N i + ). N! Inserting this into the exression for the determinant, we obtain i>j d 0,..., d N ) = d i d j ) N N i= i! i + N+ i, ) as desired. i= Remark 3.7. It should be remarked that the exression for the determinant in roosition 3.6 is surrisingly simle, given the comlicated structure of the Virasoro conditions and the toological recursion relations. It would be interesting to see if there is a deeer relation between these two sets of equations that is not yet understood and exlains the simlicity of our results. Combining the arguments of this section, we see that the systems of linear equations obtained from the Virasoro conditions and the toological recursion relations in section are always solvable. This comletes the roof of theorem.. i= 4. Some numbers In this section we will give some examles of invariants that have been comuted using the method of this aer. k=0

19 TOPOLOGICAL RECURSION RELATIONS IN HIGHER GENUS 9 Examle 4.. The Caoraso-Harris numbers, see [CH]) The following table shows some numbers of curves in P of genus g and degree d through 3d + g oints, i.e. the Gromov-Witten invariants t 3d +g g,d. They can be found either by the Caoraso-Harris method or by alying the techniques of this aer. d = d = d = 3 d = 4 d = 5 d = 6 d = 7 g = g = g = g = g = Examle 4.. We list some -oint invariants of P, i.e. invariants of the form τ m γ) g,d, where m is determined by the dimension condition 3d + g = m + deg γ. γ = t γ = H γ = d = 0 d = d = d = 0 d = d = d = 0 d = d = g = g = g = g = 3 0 g = Examle 4.3. The following table gives some Gromov-Witten invariants t d g,d of P 3, i.e. the virtual number of genus-g degree-d curves in P 3 through d oints. Note that these Gromov-Witten invariants are not enumerative for g > 0. The recise relationshi between the Gromov-Witten invariants and the enumerative numbers is not yet known. d = d = d = 3 d = 4 d = 5 d = 6 d = 7 g = g = g = g = g = References [B] K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math ), no. 3, [BF] K. Behrend, B. Fantechi, The intrinsic normal cone, Inv. Math ), no., [CH] L. Caoraso, J. Harris, Counting lane curves of any genus, Invent. Math ), no., [EHX] T. Eguchi, K. Hori, C. Xiong, Quantum cohomology and Virasoro algebra, Phys. Lett. B ), [EX] T. Eguchi, C. Xiong, Quantum cohomology at higher genus: toological recursion relations and Virasoro conditions, Adv. Theor. Math. Phys. 998), 9 9. [FP] W. Fulton, R. Pandhariande, Notes on stable mas and quantum cohomology, Proc. Sym. Pure Math ) art,

20 0 ANDREAS GATHMANN [Ge] E. Getzler, Toological recursion relations in genus, in: M.-H. Saito ed.) et al., Integrable systems and algebraic geometry, Proceedings of the 4st Taniguchi symosium, Singaore World Scientific 998), [Gi] A. Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 00), no. 4, [GP] T. Graber, R. Pandhariande, Localization of virtual classes, Invent. Math ), no., [I] E. Ionel, Toological recursive relations in H g M g,n), Invent. Math ), no. 3, [K] A. Kresch, Cycle grous for Artin stacks, Invent. Math ), no. 3, [KM] M. Kontsevich, Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys ), no. 3, [LLY] B. Lian, K. Liu, S. Yau, Mirror rincile IV, Surv. Differ. Geom. VII, International Press, Somerville, MA 000), [P] R. Pandhariande, Rational curves on hyersurfaces after A. Givental), Astérisque 5 998), ex. no. 848, 5, Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, Kaiserslautern, Germany address: gathmann@mathematik.uni-kl.de