Formal pseudodifferential operators and Witten's r-spin numbers
aa r X i v : . [ m a t h . AG ] D ec FORMAL PSEUDODIFFERENTIAL OPERATORS ANDWITTEN’S r -SPIN NUMBERS KEFENG LIU, RAVI VAKIL, AND HAO XU
Abstract.
We derive an effective recursion for Witten’s r -spin intersectionnumbers, using Witten’s conjecture relating r -spin numbers to the Gel’fand-Dikii hierarchy (Theorem 4.1). Consequences include closed-form descriptionsof the intersection numbers (for example, in terms of gamma functions: Propo-sitions 5.2 and 5.4, Corollary 5.5). We use these closed-form descriptions toprove Harer-Zagier’s formula for the Euler characteristic of M g, . Finally in §
6, we extend Witten’s series expansion formula for the Landau-Ginzburg po-tential to study r -spin numbers in the small phase space in genus zero. Ourkey tool is the calculus of formal pseudodifferential operators, and is partiallymotivated by work of Br´ezin and Hikami. Contents
1. Introduction 12. Review: Witten’s r -spin intersection numbers 33. Formal pseudodifferential operators 54. An algorithm for computing Witten’s r -spin numbers 105. The Euler characteristic of M g, W r ( z ) 26Appendix C. An identity of Bernoulli numbers 27References 291. Introduction
Motivated by two dimensional gravity, E. Witten proposed two influential con-jectures relating integrable hierarchies to the intersection theory of moduli spacesof curves, see [30, 31].We begin by recalling Witten’s definition of r -spin intersection numbers. Wit-ten’s original papers [31, 32] remain the best introduction to the mathematical andphysical background of this subject. Other excellent expositions can be found in[13, 26]. For an introduction to relevant facts about the moduli spaces of curves,see [29].Let Σ be a Riemann surface of genus g with marked points x , x , . . . , x s . Fixan integer r ≥
2. Label each marked point x i by an integer m i , 0 ≤ m i ≤ r − S = K ⊗ O ( − P si =1 m i x i ) over Σ, where K as usualdenotes the canonical line bundle. If 2 g − − P si =1 m i is divisible by r , then there are r g isomorphism classes of line bundles T such that T ⊗ r ∼ = S . The choice of anisomorphism class of T determines a finite ´etale cover M /rg,s of M g,s , the modulispace of r -spin curves, which comes with a universal curve π : C /rg,n → M /rg,s , onwhich lives a universal bundle, which we also sloppily denote T . A compactificationof M /rg,s , denoted by M /rg,s , was constructed in [1, 12].Let V be a vector bundle over M /rg,s whose fiber is the dual space to H (Σ , T ).More precisely, V := R π ∗ T . The top Chern class c top ( V ) of this bundle has degree( g − r − /r + P si =1 m i /r . The algebro-geometric constructions of c top ( V ) canbe found in [4, 24].We associate with each marked point x i an integer n i ≥
0. Witten’s r -spinintersection numbers are defined by(1) h τ n ,m . . . τ n s ,m s i g = 1 r g Z M /rg,s s Y i =1 ψ ( x i ) n i · c top ( V ) , which is non-zero only if(2) ( r + 1)(2 g −
2) + rs = r s X j =1 n j + s X j =1 m j . Fix an integer r ≥
2. Consider the pseudodifferential operator(3) Q = D r + r − X i =0 γ i ( x ) D i , where D = √− √ r ∂∂x . It is easy to see that there is a unique pseudodifferential operator L such that L r = Q (see Lemma 3.1),which we denote Q /r = D + X i> w − i D − i , where the coefficients { w − i } are universal differential polynomials in the { γ i } .The Gel’fand–Dikii equations read i ∂Q∂t n,m = [ Q n +( m +1) /r + , Q ] · c n,m √ r , where the constants c n,m are given by c n,m = ( − n r n +1 ( m + 1)( r + m + 1) · · · ( nr + m + 1) . Consider the formal series F in variables t n,m , n ≥ ≤ m ≤ r − F ( t , , t , , . . . ) = X d n,m h Y n,m τ d n,m n,m i Y n,m t d n,m n,m d n,m ! . Witten conjectured in [31] that the above F is the string solution of the r -Gel’fand–Dikii hierarchy, namely that F satisfies(4) ∂ F∂t , ∂t n,m = − c n,m Res( Q n + m +1 r ) , DO AND WITTEN’S r -SPIN NUMBERS 3 where Q satisfies the Gel’fand–Dikii equations and t , is identified with x . Inaddition, F satisfies the string equation(5) ∂F∂t , = 12 r − X i,j =0 δ i + j,r − t ,i t ,j + ∞ X n =0 r − X m =0 t n +1 ,m ∂F∂t n,m . This should be regarded as a boundary condition for F .When r = 2, the above assertion is the celebrated Witten-Kontsevich theorem[15], to which there are a number of enlightening proofs. Witten’s conjecture forany r ≥ r -spin theory corresponds to A r − singularity in the Landau-Ginzburg theory. Fan, Javis and Ruan [9] have developeda Gromov-Witten type quantum theory for all non-degenerate quasi-homogeneoussingularities and proved the ADE-integrable hierarchy conjecture of Witten. Changand Li [6] have initiated a program to give an algebro-geometric construction ofLandau-Ginzburg theory.Witten’s constraints (4) (the r -Gel’fand–Dikii equation) and (5) (the stringequation) uniquely determine F . There is much interest in understanding thestructure of r -spin intersection numbers both in mathematics and physics (cf.[2, 3, 5, 14, 21, 27]).The paper is organized as follows. In §
2, we recall useful identities of r -spinnumbers. In §
3, we prove a structure theorem of formal pseudodifferential operatorsand use it to derive/define “universal differential polynomials” W r ( z ), which willplay a central role in the rest of the paper. In §
4, we present a recursive algorithm forcomputing Witten’s r -spin numbers for all genera. Consequences include closed-form descriptions of the one-point r -spin numbers, which we use in § M g, . In §
6, we study r -spinnumbers on small phase spaces in genus zero. Acknowledgements.
We thank J. Li, W. Luo, M. Mulase, Y.B. Ruan, and J.Zhou for helpful conversations. The third author thanks Professor D. Zeilbergerfor answering a question on computer proof of combinatorial identities.2.
Review: Witten’s r -spin intersection numbers In this section, we collect fundamental properties of r -spin intersection numbersthat we will use in this paper. The proof of the these identities can be found in[31, 13]. The r -spin numbers satisfy the following:i) If m i = r −
1, for some 1 ≤ i ≤ s , then h τ n ,m · · · τ n s ,m s i g = 0 . ii) (string equation) (6) h τ , s Y i =1 τ n i ,m i i g = s X j =1 h τ n j − ,m j s Y i =1 i = j τ n i ,m i i g . This, along with h τ , τ ,i τ ,j i = δ i + j,r − , is equivalent to (5). KEFENG LIU, RAVI VAKIL, AND HAO XU iii) (dilaton equation) (7) h τ , s Y i =1 τ n i ,m i i g = (2 g − s ) h s Y i =1 τ n i ,m s i g . iv) (genus zero topological recursion relation) (8) h τ n +1 ,m τ n ,m τ n ,m s Y i =4 τ n i ,m i i = X { ··· s } = I ` J r − X m ′ ,m ′′ =0 h τ n ,m Y i ∈ I τ n i ,m i τ ,m ′ i · η m ′ ,m ′′ h τ ,m ′′ τ n ,m τ n ,m Y i ∈ J τ n i ,m i i , where η m ′ ,m ′′ = δ m ′ + m ′′ ,r − .v) (WDVV equation in genus zero) (9) r − X m ′ ,m ′′ =0 Y { ··· s } = I ` J h τ n ,m τ n ,m Y i ∈ I τ n i ,m i τ ,m ′ i η m ′ ,m ′′ h τ ,m ′′ τ n ,m τ n ,m Y i ∈ J τ n i ,m i i = r − X m ′ ,m ′′ =0 Y { ··· s } = I ` J h τ n ,m τ n ,m Y i ∈ I τ n i ,m i τ ,m ′ i η m ′ ,m ′′ h τ ,m ′′ τ n ,m τ n ,m Y i ∈ J τ n i ,m i i Witten gives a detailed study of r -spin numbers in genus zero in [31]. As hepoints out, the genus zero topological recursion relation can be used to eliminateall descendent indices (those τ i,j with i > h τ ,m , · · · τ ,m s i on the small phase space. Witten provesthat the WDVV equation uniquely determines primary r -spin intersection numbersin genus zero. For the reader’s convenience, we record Witten’s work below in amore explicit form. We will denote h τ ,a , · · · , τ ,a s i by either h τ a , · · · , τ a s i or h a , · · · , a s i . Witten proves that h τ a τ a τ a i = δ a + a + a ,r − , h τ a τ a τ a τ a i = 1 r · min( a i , r − − a i ) . Theorem 2.1 (Witten, [31]) . Let s ≥ , a ≥ · · · ≥ a s and P sj =1 a j = r ( s − − .Define z = a , y = a , x = a and m = x + z − ( r − , m = r − − z, m = y, m = z. Then Witten’s formula can be written as (10) h a , · · · , a s i = (cid:28) x + y + z − ( r − , r − − z, z, s Y i =4 a i (cid:29) + X I ` J = { ,...,s } I,J = ∅ r − X j =0 (cid:28) j, m , m , Y i ∈ I a i (cid:29)(cid:28) r − − j, m , m , Y i ∈ J a i (cid:29) − (cid:28) j, m , m , Y i ∈ I a i (cid:29)(cid:28) r − − j, m , m , Y i ∈ J a i (cid:29)! . DO AND WITTEN’S r -SPIN NUMBERS 5 This formula recursively computes all primary r -spin numbers.Proof. The argument is due to Witten. From s ≥
5, and 0 ≤ a i ≤ r −
2, it is notdifficult to check that 0 ≤ m i ≤ r −
2. By the WDVV equation (9), we have X I ` J = { ,...,s } r − X j =0 (cid:28) j, m , m , Y i ∈ I a i (cid:29)(cid:28) r − − j, m , m , Y i ∈ J a i (cid:29) = X I Q J = { ,...,s } r − X j =0 (cid:28) j, m , m , Y i ∈ I a i (cid:29)(cid:28) r − − j, m , m , Y i ∈ J a i (cid:29) . Then Witten’s formula follows from the inequalities m + m > r − m + m >r − z ′ ≥ y ′ ≥ x ′ are the three largest numbers in the index set { x + y + z − ( r − , r − − z, z, a , . . . , a s } , then r − − z is not one of x ′ , y ′ , z ′ as long as s ≥
5. On theother hand, each bracket in the quadratic terms in the right hand side of (10) hasstrictly less than s points. (cid:3) Formal pseudodifferential operators
A formal pseudodifferential operator is an expression of the form L = N X i = −∞ u i ( x ) ∂ i , where ∂ = ∂∂x . Its positive and negative parts are defined to be L + = N X i =0 u i ( x ) ∂ i , L − = − X i = −∞ u i ( x ) ∂ i . For k ∈ Z , we define ∂ k · f = X j ≥ (cid:18) kj (cid:19) f ( j ) ∂ k − j , where f ( j ) = ∂ j f∂x j . We follow the usual convention that (cid:18) − a − b (cid:19) = (cid:18) a + bb (cid:19) ( − b , a, b ≥ . In particular, ∂ · f = f ′ + f ∂ . Note that we reserve the notation ∂f for the derivativeof f . It is straightforward to check that the set of all formal pseudodifferentialoperators forms an associative algebra, denoted by ΨDO.The idea of fractional powers appeared in the work of Gel’fand and Dikii [10].It plays an important role in integrable systems (cf. [25]). The following lemma iswell-known. Lemma 3.1.
Recall the pseudodifferential operator Q defined in (3) Q = D r + r − X i =0 γ i ( x ) D i . KEFENG LIU, RAVI VAKIL, AND HAO XU
There exists a unique pseudodifferential operator of the form Q /r = D + X i ≥ w − i D − i , whose r -th power is Q ; and w = 0 .Proof. Let Q r = D + w + w − D − + · · · . Then ( Q r ) r = D r + rw D r − + · · · .Since there is no D r − term on Q , we have w = 0. Thus we may write( Q r ) r = D r + rw − D r − + ( rw − + r ( r − Dw − D r − + · · · . In general, we have rw − i + p i ( w − , · · · w − i +1 ) = γ r − − i , where p i is a differential polynomial of its argument. So w − i can be uniquelydetermined recursively as differential polynomials of γ i . (cid:3) Fix k ≥
1. Write Q k/r = D k + k − X i =0 γ ki D i + ∞ X i =1 γ k − i D − i . Here we emphasize that throughout this paper, the superscript k in γ ki never denotesa power. In particular, we have γ ri = γ i .Since Q ( k +1) /r = Q /r · Q k/r , for ℓ ≤ k − γ k +1 ℓ = w ℓ − k + Dγ kℓ + γ kℓ − + k − − ℓ X j =1 w − j k − X i = j + ℓ (cid:18) − ji − j − ℓ (cid:19) D i − j − ℓ γ ki . This identity can be used to determine γ k +1 ℓ recursively as differential polynomialsof { w − i } . Lemma 3.2.
With the notation above, if we assign w ( j ) − i = D j w − i the weight i + j + 1 , then γ kℓ is homogeneous of weight k − ℓ .Proof. Since γ ℓ = w ℓ is of weight 1 − ℓ , the general statement follows from theequation (11). (cid:3) Lemma 3.3.
Let [ w ( j ) − i ] γ kℓ denote the coefficient of w ( j ) − i in γ kℓ . If k ≥ , ℓ ≤ k − and ≤ i ≤ k − ℓ − , then we have (12) [ D k − ℓ − i − w − i ] γ kℓ = (cid:18) kk − ℓ − i (cid:19) . In particular, [ w ℓ − k +1 ] γ kℓ = k and [ Dw ℓ − k +2 ] γ kℓ = k ( k − .Proof. When k = 1, by definition, γ ℓ = w ℓ for ℓ <
0. The identity (12) obviouslyholds in this case. So we apply the recursive equation (11) and use induction on k .When i = k − ℓ −
1, we have[ w ℓ − k +1 ] γ kℓ = 1 + [ w ℓ − k +1 ] γ k − ℓ − = 1 + k − k DO AND WITTEN’S r -SPIN NUMBERS 7 and similarly when i < k − ℓ −
1, we have[ D k − ℓ − i − w − i ] γ kℓ = [ D k − ℓ − i − w − i ] γ k − ℓ + [ D k − ℓ − i − w − i ] γ k − ℓ − = (cid:18) k − k − ℓ − i − (cid:19) + (cid:18) k − k − ℓ − i (cid:19) = (cid:18) kk − ℓ − i (cid:19) as desired. (cid:3) Lemma 3.4 (Witten, [31]) . With the above notation, γ i +1 − = Res( Q ( i +1) /r ) , wecan express coefficients γ i of Q as differential polynomials in γ i +1 − , ≤ i ≤ r − .Proof. By Lemmas 3.2 and 3.3, we have γ i +1 − = ( i + 1) w − − i + p i ( w − , · · · w − i )(13) = ( i + 1) γ r − − i r + p ′ i ( γ r − , · · · γ r − − i ) , where p i and p ′ i are differential polynomials of their arguments. Thus we canrecursively express γ i as differential polynomials in γ i +1 − , ≤ i ≤ r − (cid:3) Denote by P ( γ kℓ ) the sum of monomials in γ kℓ that does not contain derivativesof w − i . Then we have P ( γ kℓ ) = [ p k − ℓ ] X i> w − i p i +1 ! k (14) = Res p =0 (1 + P i> w − i p i +1 ) k p k − ℓ +1 . Fix an integer r ≥
2. From the Gel’fand-Dikii equation (4), we have γ m +1 − = Res( Q m +1 r ) = − m + 1 r hh τ , τ ,m ii , for 0 ≤ m ≤ r −
2= ( m + 1) w − m − + · · · and γ r +1 − = Res( Q r ) = r + 1 r hh τ , τ , ii (15) = ( r + 1) w − r − + r ( r + 1)2 Dw − r + · · · . For the first time, we use the fact that Q is a differential operator (i.e. Q − = 0),which implies that(16) 0 = γ r − = r · w − r + · · · and(17) 0 = γ r − = r · w − r − + · · · . The leading coefficients of the above equations come from Lemma 3.3.We first substitute (17) and then (16) into (15) to eliminate w − r − and w − r re-spectively. Next we substitute γ r − − , γ r − − , . . . , γ − consecutively into (15) to elim-inate w − r +1 , w − r +2 , . . . , w − successively. Then it is easy to see that γ r +1 − is nowexpressed in terms of differential polynomials of γ m +1 − , ≤ m ≤ r −
2. From now weon will use S ( γ r +1 − ) to denote this differential polynomial in γ m +1 − , ≤ m ≤ r − KEFENG LIU, RAVI VAKIL, AND HAO XU resulting from substitutions in γ r +1 − . We will keep the notation γ r +1 − for the differ-ential polynomial (15) in w − i .If we use the notation(18) z ( j ) m = − rm + 1 · ∂ j γ m +1 − ∂x j = hh τ j +10 , τ ,m ii , then we have the following structure theorem of formal pseudodifferential operators. Theorem 3.5. (As discussed above, we may regard S ( γ r +1 − ) as a differential poly-nomial in z m .) We have r r + 1 S ( γ r +1 − ) = 12 r − X j =0 z j z r − − j + W r ( z ) , where W r ( z ) represents the terms containing derivatives of some z m .Proof. Since (16) is used to eliminate w − r − in γ r +1 − , it is not difficult to see thatthe identity of Theorem 3.5 is equivalent to r r + 1 P ( γ r +1 − ) − rP ( γ r − ) = 12 r − X j =0 − rj + 1 P ( γ j +1 − ) − rr − − j P ( γ r − − j − ) . From equation (14), this is precisely the combinatorial identity shown in the nextproposition. (cid:3)
Proposition 3.6.
Let a j be formal variables and f ( x ) = 1 + ∞ X j =2 a j x j ∈ C [[ x ]] be a formal series satisfying f (0) = 1 and f ′ (0) = 0 . Then for any n ≥ , (19) [ x n +2 ] f n +1 n + 1 = 12 n − X j =1 [ x j +1 ] f j j · [ x n − j +1 ] f n − j n − j + [ x n +2 ] f n n , where [ x n ] f k denotes the coefficient of x n in the series expansion of f k . The proof of Proposition 3.6 along with other interesting equivalent formulationscan be found in Appendix A.
Example 3.7.
We illustrate the above procedure explicitly for r = 4. Let Q / = D + P i> w − i D − i . Then − hh τ , τ , ii = Res( Q / ) = w − , − hh τ , τ , ii = Res( Q / ) = 2 w − + Dw − , − hh τ , τ , ii = Res( Q / ) = 3 w − + D w − + 3 Dw − + 3 w − . We also have0 = Res( Q ) =4 w − + D w − + 4 D w − + 6 Dw − + 6 w − Dw − + 12 w − w − , γ − =4 w − + 6 Dw − + 4 D w − + D w − + 6 w − Dw − − ( Dw − ) + 12 w − w − + 6 w − + 4 w − + 2 w − D w − . DO AND WITTEN’S r -SPIN NUMBERS 9 Substituting the above two groups of identities into γ − = 516 hh τ , τ , ii = Res( Q / )= 5 w − + D w − + 5 D w − + 10 D w − + 10 Dw − + 5( Dw − ) + 10 w − D w − + 10 w − + 10 w − + 20 w − Dw − + 10 w − Dw − + 20 w − w − and using D = √− ∂∂x , we get165 γ − = z z + 12 z + 14 z (2)2 + 148 z z (2)0 + 132 z ′ z ′ + 1480 z (4)0 . If we substitute the z m using equation (18), we get exactly the recursion formula(25).The universal differential polynomial W r ( z ) in z , . . . , z r − is particularly inter-esting in view of Theorem 3.5. We present W r ( z ) for 2 ≤ r ≤ W ( z ) = 112 z (2)0 , W ( z ) = 16 z (2)1 ,W ( z ) = 14 z (2)2 + 148 z z (2)0 + 132 z ′ z ′ + 1480 z (4)0 ,W ( z ) = 110 z ′ z ′ + 130 z z (2)1 + 130 z (2)0 z + 13 z (2)3 + 1150 z (4)1 ,W ( z ) = 5864 z (3)0 z ′ + 1144 z ( z ′ ) + 18 z ′ z ′ + 124 z z (2)2 + 1432 z z (2)0 + 124 z z (2)0 + 172 z (4)2 + 19072 z (6)0 + 112592 ( z (2)0 ) + 112 ( z ′ ) + 118 z z (2)1 + 1720 z z (4)0 + 512 z (2)4 . We now study their coefficients.
Proposition 3.8.
We have [ z (2) r − ] W r ( z ) = r − .Proof. From equation (18) and D = √− √ r ∂∂x , we have ∂ z r − ∂x = − rr − · ∂ γ r − − ∂x = r r − D γ r − − . So from Theorem 3.5, we get(20) [ z (2) r − ] W r ( z ) = r − r + 1 [ D γ r − − ] S ( γ r +1 − ) . Recall that in γ r +1 − = ( r +1) w − r − + r ( r +1)2 Dw − r + · · · , we first substitute w − r − using γ r − and then substitute w − r using γ r − , see equations (16), (17). Then γ r +1 − becomes a differential polynomial in w − , . . . , w − r +1 . We need to take care thatwhen substituting w − r − by γ r − , a new term of Dw − r will appear. With the abovesubstitutions in mind and note that γ r − − = ( r − w − r +1 + · · · , we may applyLemma 3.3 to get r − r + 1 [ D γ r − − ] S ( γ r +1 − ) = 1 r + 1 [ D w − r +1 ] (cid:18) γ r +1 − − ( r + 1) · r γ r − (cid:19) + 1 r + 1 [ D w − r +1 ] (cid:18)(cid:18) r + 1 r [ Dw − r ] γ r − − ( r + 1) r (cid:19) · r Dγ r − (cid:19) = 1 r + 1 (cid:18) ( r + 1) r ( r − − r + 1 r · r ( r − r − (cid:19) + 1 r + 1 (cid:18) r + 1 r [ Dw − r ] γ r − − ( r + 1) r (cid:19) · r [ Dw − r +1 ] γ r − = r −
13 + − r [ Dw − r +1 ] γ r − = r − . From (20), we get the desired result. (cid:3)
Corollary 3.9 (Witten, [31]) . We have the following identity for r -spin numbers: h τ , i = r − . Proof.
From the dilaton equation, h τ , τ , τ , i = 2 h τ , i . On the other hand,from Theorems 4.1 and 3.5, we have h τ , τ , τ , i = [ z (2) r − ] W r ( z ) h τ , τ ,r − τ , i = r − . In the right hand side of the first equation, all other terms vanish for dimensionalreason (see (22), (23)). Hence h τ , i = r − . (cid:3) Proposition 3.8 generalizes as follows.
Proposition 3.10.
Suppose ≤ i ≤ r . If i is odd, then [ z ( i ) r − i ] W r ( z ) = 0 . If i iseven ( k , say), then [ z (2 k ) r − k ] W r ( z ) = ( − k +1 ( r + 1 − k ) r k ( r + 1) (cid:18) r + 12 k (cid:19) B k , where B k are Bernoulli numbers. See Appendix B for a proof. By similar arguments, we have the following fact,which means that the genera in the right-hand side of (21) are integers (see (23)).We omit the details.
Proposition 3.11.
The order of derivatives in each monomial of W r ( z ) is an evennumber. An algorithm for computing Witten’s r -spin numbers Let η ij = δ i + j,r − and hh τ n ,m . . . τ n s ,m s ii = ∂∂t n ,m . . . ∂∂t n s ,m s F ( t , , t , , . . . ) . The main result of this paper is the following simple and effective recursionformula for computing all r -spin intersection numbers. Theorem 4.1.
For fixed r ≥ , we have (21) hh τ , τ , ii g = 12 hh τ , τ ,m ′ ii g ′ η m ′ m ′′ hh τ ,m ′′ τ , ii g − g ′ + Lower( r ) , where Lower( r ) is a explicit sum of products of hh . . . ii with genera strictly lowerthan g . DO AND WITTEN’S r -SPIN NUMBERS 11 Proof.
Since r r +1 γ r +1 − = hh τ , τ , ii , from Theorem 3.5, we need only prove thatthose monomials in W r ( z ) must have genera strictly less than the left hand side.Let us compare hh τ , τ , ii g and Y k z ( j k ) i k = Y k hh τ j k +10 , τ ,i k ii g k . Since the weight of γ r +1 − is r + 2 and the weight of z ( j ) m is m + j + 2, we have(22) X k ( i k + j k + 2) = r + 2 . Combining with the dimensional constraints (2), we have(23) (2 r + 2) g − X k g k ! = ( r + 1) X k j k . So g = P k g k if and only if all j k = 0. (cid:3) Remark 4.2.
Following a suggestion of Witten [31, p.248], Shadrin [26] derivedan expansion of hh τ n,m τ , ii when r = 3 and used it to compute some special r -spinnumbers. Because of a lack of an elegant structural description, Shadrin’s formula(and its generalization to higher r ) results in a much more complicated algorithmthan (21).For example, when r = 3, (21) gives(24) hh τ , τ , ii g = hh τ , τ , ii g ′ hh τ , ii g − g ′ + 16 hh τ , τ , ii g − . When r = 4, we have(25) hh τ , τ , ii g = hh τ , τ , ii g ′ hh τ , ii g − g ′ + 12 hh τ , τ , ii g ′ hh τ , τ , ii g − g ′ + 14 hh τ , τ , ii g − + 148 hh τ , ii g ′ hh τ , ii g − − g ′ + 132 hh τ , ii g ′ hh τ , ii g − − g ′ + 1480 hh τ , ii g − . Now we show how to use Theorem 1.1 to compute intersection numbers. Itconsists of three steps. (i)
When g = 0, these intersection numbers can be computed by WDVV equa-tions, using Witten’s algorithm [31], as discussed in § (ii) Assume now that g ≥
1. For an intersection number containing a punc-ture operator h τ , τ n ,m . . . τ n s ,m s i g , we have from Theorem 1.1 and the dilatonequation(26) (2 g − s − a ) h τ , τ n ,m . . . τ n s ,m s i g = 12 ∼ X s = I ` J h τ , τ ,m ′ Y i ∈ I τ n i ,m i i g ′ η m ′ m ′′ h τ ,m ′′ τ , Y i ∈ J τ n i ,m i i g − g ′ + Lower( r )where a = { i | n i = 0 } . Note that in the summation of the right-hand side, werule out the cases I = { i } and n i = 0 or J = { i } and n i = 0. Then the righthand side follows by induction on genera or numbers of marked points. (iii) For any intersection number h τ n ,m . . . τ n s ,m s i g with n ≥ n ≥ · · · ≥ n s ,we apply the string equation first: h τ n ,m . . . τ n s ,m s i g = h τ , τ n +1 ,m . . . τ n s ,m s i g − s X j =2 h τ n +1 ,m τ n j − ,m j Y i =1 ,j τ n i ,m i i g The first term in the right hand side follows from step (ii) and the second termfollows by induction on the maximum descendent index. This ends the algorithm.The results of the above algorithm agree with the table of r -spin numbers when r = 3 and 4 given in [18]. Some r -spin numbers when r = 5 are presented in Table1. Table 1.
Witten’s r -spin numbers ( r = 5) h τ , i h τ , τ , i h τ , τ , τ , i h τ , i h τ , τ , i h τ , τ , τ , i h τ , i h τ , τ , i h τ , τ , τ , i h τ , i h τ , τ , i h τ , τ , τ , i h τ , i × − h τ , τ , i h τ , τ , τ , i h τ , i × − h τ , τ , i h τ , τ , τ , i h τ , i × − h τ , τ , i h τ , τ , τ , i h τ , i
10 2660573 × − h τ , τ , i h τ , τ , τ , i h τ , i
11 21324511 × − h τ , τ , i h τ , τ , τ , i h τ , i
12 87572287 × − h τ , τ , i h τ , τ , τ , i h τ , i
14 7787064791 × − h τ , τ , i h τ , τ , τ , i h τ , i
15 538156369 × − h τ , τ , i h τ , τ , τ , i The Boussinesq hierarchy ( r = 3 ). For the remainder of this section, let r = 3. The 3-KdV hierarchy is also calledthe Boussinesq hierarchy . We see from (24), (25) that compared with the recursiveformula for 3-spin intersection numbers, the recursive formula for r -spin numbersare much more complicated for r ≥ r = 3. Thisgeneralizes the special case k = 0 obtained by Br´ezin and Hikami in [2]. Proposition 4.3.
Let k ≥ and ≤ j ≤ . Then h τ k , τ g +2 k − − j ,j i g = 112 g g ! Γ( g + k +13 )Γ( − j ) , where Γ( z ) is the gamma function.Proof. We first prove the identity in g = 0 by induction on k , namely h τ k , τ k − − j ,j i = Γ( k +13 )Γ( − j ) . DO AND WITTEN’S r -SPIN NUMBERS 13 When k = 3 , , h τ , τ , i = 13 , h τ , τ , i = 23 , h τ , τ , i = 0 . Note the last identity is consistent with the fact that Γ( z ) has a simple pole at z = 0.Thus we may assume k ≥
6. We apply the genus zero topological recursionrelation (8) to obtain: h τ k , τ k − − j ,j i = k − X i =0 (cid:18) k − i (cid:19) h τ k − − j ,j τ , τ i , i h τ , τ k − − i , i = ( k − h τ k − − j ,j τ , τ k − , i h τ , τ , i = k − · Γ( k − )Γ( − j )= Γ( k +13 )Γ( − j ) . The second equation comes from dimensional constraints. Thus we have provedProposition 4.3 when g = 0.We next assume g ≥ g . We have h τ , τ , τ k , τ g +2 k − − j ,j i g = k h τ , τ k , τ g +2 k − − j ,j i g h τ , τ , i + 16 h τ , τ k +10 , τ g +2 k − − j ,j i g − . Applying the dilaton equation (7) and the string equation (6) to the above iden-tity and combining the first term in the right hand side with the left hand side, weget h τ k , τ g +2 k − − j ,j i g = 112 g h τ k +10 , τ g +2 k − − j ,j i g − = 112 g · g − ( g − (cid:16) ( g − k +1)+13 (cid:17) Γ( − j )= 112 g g ! Γ( g + k +13 )Γ( − j )as desired. (cid:3) We now show that the 3-spin numbers in genus zero in general do not haveclean closed formulas in contrast to the case of r = 2. We will compute intersectionnumbers of the form h τ k , τ ℓ , i , which is nonzero only if k ≡ k − ℓ =8. We will use the temporary notation a m = h τ m +10 , τ m − , i , for m ≥
1. Byapplying (26) to h τ , τ , τ m +10 , τ m − , i = (2 m − m − a m and using the dilaton and string equations, it is not difficult to obtain(27) (2 m − m − m − a m = m − X i =1 (cid:18) m + 13 i + 1 (cid:19)(cid:18) m − i (cid:19) i (2 i − i − i − m − i − m − i − a i a m − i . For example, we recursively find a = h τ , i = , a = , a = , a = .To simplify the above equation, we substitute b m = (5 m − a m (3 m + 1)!(2 m − . For example, b = , b = , b = , b = . Then (27) becomes(28) (2 m − b m = m − X i =1 (5 i − m − i + 1) b i b m − i . In terms of the generating function y ( x ) = P ∞ i =1 b i x i , we can rewrite (28) as15 x (cid:18) dydx (cid:19) + (2 xy − x ) dydx − y + 2 y = 0 , from which we get a first order ODE dydx = 1 − y − p y − y x . Integrating both sides of 15 dy − y − p y − y = dxx , we get x = exp Z dy − y − p y − y + C ! = exp (cid:18) ln y + ln 36 − y − y − y − y + O ( y ) (cid:19) = 36 y − y − y − y + O ( y ) . The constant of integration C is uniquely determined by the initial value b = .Thus b i can also be computed using the Lagrange inversion formula (see LemmaA.1). b i = 1 i Res y =0 (cid:18) x ( y ) i (cid:19) , i ≥ . We note that the above derivation becomes more difficult if we instead use thegenus zero topological recursion relation (8) to compute h τ k , τ ℓ , i .5. The Euler characteristic of M g, We now give a proof for Harer and Zagier’s formula of the Euler characteristicof the moduli space of curves:
Theorem 5.1 (Harer-Zagier [11], see also [3, 15, 20, 22, 23]) . Let g ≥ . Then χ ( M g, ) = − B g g . For example, χ ( M , ) = − , χ ( M , ) = , χ ( M , ) = − . DO AND WITTEN’S r -SPIN NUMBERS 15 The early proofs of Harer-Zagier’s formula [15, 20, 22, 23] all exploit the celldecomposition of decorated moduli space in terms of Ribbon graphs. There isan intriguing fact from Witten’s construction [32] that the r → − r -spinnumbers actually gives χ ( M g, ). The main difficulty is to derive an explicit formulafor the one-point r -spin numbers. This was obtained recently by Brezin and Hikami[3] using rather complicated techniques from matrix integrals. We will give a proofusing only properties of ΨDO. The proof will conclude just after Lemma 5.3.We will use the case s = 1, and general r . Our discussion so far has assumed r ≥
2. However, for any r , there is a generalized Kontsevich (Airy) matrix model,and under the limit r → −
1, the model gives a logarithmic potential correspondingto the Penner matrix model, whose asymptotic expansion gives the generatingfunction of the Euler characteristic of M g, , [23]. (We do not understand how tomake [32, (3.55-3.57)] precise, so we instead refer the reader to [2, §
6] or [19] fora complete discussion.) Thus by taking r = − χ ( M g, ), as follows. Setting s = 1 in (1),we have(29) lim r →− h τ n,m i g (cid:12)(cid:12)(cid:12) m =0 = χ ( M g, ) , where χ ( M g, ) is the orbifold Euler characteristic of M g, . We now proceed tocompute the left side of (29), thereby computing χ ( M g, ).By the Gel’fand-Dikii equation (4), in order to compute h τ , τ n,m i g , we need tocompute the coefficient of ( Dγ r − − ) g in Res( Q n +( m +1) /r ). Note that(30) γ r − − = − r − r hh τ , τ ,r − ii . By Lemma 3.4 and (13), we know that when expressing γ i (0 ≤ i ≤ r −
2) interms of γ i +1 − (0 ≤ i ≤ r − γ contains the term γ r − − , with(31) γ = rr − γ r − − + p ( γ − , . . . , γ r − − ) , where p is a differential polynomial in its arguments.If we replace γ by x and denote by L = D r + x , it is not difficult to see from(30) and (31) that(32) h τ , τ n,m i g = ( − g c n,m r g × the constant term in Res( L n +( m +1) /r ) . There exists a pseudodifferential operator K ∈ ΨDO of the form K = 1 + ∞ X i =1 b i ( x ) D − i , such that KLK − = D r .We can determine K by comparing the coefficients at both sides of(33) KL = D r K. The first few terms are(34) K = x r D − ( r − + (1 − r ) x r D − r + x r D − (2 r − + 7(1 − r ) x r D − (2 r − + ( r − r − x r D − r + (1 − r )(10 r − r − x r D − (2 r +1) + · · · . In general, we have K = 1 + ∞ X u =1 2 u X i =1 b ur + u − i D − ( ur + u − i ) , where b ur + u − i = a u,i x i with a u,i rational functions of r . In particular, from (34),we have a , = 1 − r r , a , = 12 r . Given u ≥ ≤ i ≤ u , if we equate the coefficient of D − u + i − − ( u − r in(33), we get(35) a u − ,i − + (cid:0) i − u − ( u − r (cid:1) a u − ,i − = u − i X k =0 (cid:18) rk + 1 (cid:19) k Y j =0 ( i + j ) · a u,i + k . By a tedious but straightforward calculation, we find the recursion(36) i ! a u,i = u − X j =0 a u − ,j (cid:0) ( j + 1)! s j +2 − i + ( j − ( u − r + 1)) j ! s j +1 − i (cid:1) , where s k is the coefficient of x k in x (1 + x ) r − r + (cid:0) r (cid:1) x + (cid:0) r (cid:1) x + · · · . For convenience, let e u,i = i ! a u,i . Then (36) becomes(37) e u,i = u − X j =1 e u − ,j (cid:0) ( j + 1) s j +2 − i + ( j − ( u − r + 1)) s j +1 − i (cid:1) , with initial values e , = s = 1 − r r , e , = s = 1 r . From the recursion, we see e u,i is nonzero only when 1 ≤ i ≤ u . Proposition 5.2.
Let g ≥ . We have the following formula for one-point r -spinnumbers (38) h τ n,m i g = ( − g Γ( − g − g − r ) r g Γ(1 − m +1 r ) E g , where E u = u X i =1 (cid:18) u ( r + 1) − i (cid:19) e u,i . Proof.
Since L = K − D r K , we have L n +1+( m +1) /r = K − D ( n +1) r + m +1 K . From h τ n,m i g = h τ , τ n +1 ,m i g and ( n + 1) r + m + 1 = 2 g ( r + 1) −
1, it is not difficultto see that the constant term in Res( L n +1+( m +1) /r ) equals the constant term inRes D ( n +1) r + m +1 K , which is E g . Finally (38) follows from (32) and c n +1 ,m = ( − n +1 r n +2 ( m + 1)( r + m + 1) · · · (( n + 1) r + m + 1)= Γ( − n − − m +1 r )Γ(1 − m +1 r ) = Γ( − g − g − r )Γ(1 − m +1 r ) . DO AND WITTEN’S r -SPIN NUMBERS 17 (cid:3) Setting m = 0 and taking r → − r →− Γ (cid:18) − g − g − r (cid:19) E g , which is computed by applying L’Hˆopital’s Rule to the following Lemma. Lemma 5.3.
For any integer u ≥ , we have lim r →− Γ( − u − u − r ) E u = − B u u . Proof.
Since the residue of Γ( z ) at z = − −
1, we have(39) lim r →− ddr (cid:18) − u − u − r ) (cid:19) = 1 − u. We also have(40) ddr (cid:12)(cid:12)(cid:12) r = − (cid:18) u ( r + 1) − i (cid:19) = ( − i +1 uH i , where H i = P ≤ k ≤ i k is the i th harmonic number.Setting i = 1 in (35), we get 0 = u X k =1 (cid:18) rk (cid:19) e u,k , which, after taking derivative with respect to r , becomes(41) 0 = u X k =1 (cid:16) ( − k +1 H k e u,k ( −
1) + ( − k e ′ u,k ( − (cid:17) . From (40) and (41), we havelim r →− E u = u X k =1 (cid:16) ( − k +1 uH k e u,k ( −
1) + ( − k e ′ u,k ( − (cid:17) (42) = ( u − u X k =1 ( − k +1 uH k e u,k ( − . By (39) and (42), we see that Lemma 5.3 is equivalent to(43) u X k =1 ( − k +1 H k e u,k ( −
1) = B u u , u ≥ . Setting r = − e u,i ( −
1) = (1 − i ) e u − ,i − ( −
1) + (1 − i ) e u − ,i − ( − − ie u − ,i ( − . Here we use s ( −
1) = s ( −
1) = − s k ( −
1) = 0, k > u − X k =1 ( − k ( k + 1)( k + 2) e u − ,k ( −
1) = B u u , u ≥ . By substituting (44) successively into (45), we get(46) u − j ) X k =0 ( − k +1 f j ( k ) e u − j,k ( −
1) = B u u , ≤ j ≤ u, where f j , j ≥ f j +1 ( k ) = − ( k + 1) f j ( k + 2) + (2 k + 1) f j ( k + 1) − kf j ( k )starting with f ( k ) = − k +1)( k +2) .Let e ,i = δ i , which is compatible with the recursion (44). Then (45) (hence(43)) is equivalent to(48) f u (0) = − B u u , u ≥ . We leave the proof to the Appendix C. (cid:3)
From (29), (38) and Lemma 5.3, we recover the Harer-Zagier formula (Theo-rem 5.1).Now we make a connection to the matrix integral approach of Br´ezin and Hikami[3]. First we note that a refined argument in the proof of Lemma 5.3 will give thefollowing identity of generating functions.(49) Γ (cid:18) r (cid:19) + ∞ X u =1 E u Γ (cid:18) − u − u − r (cid:19) y u = r Z ∞ exp (cid:18) − r + 1) y (cid:20)(cid:16) x + y (cid:17) r +1 − (cid:16) x − y (cid:17) r +1 (cid:21)(cid:19) dx. The integral expression of the right-hand side appeared in [3].Consider the semigroup N ∞ of sequences d = ( d , d , . . . ) where d i are nonneg-ative integers and d i = 0 for sufficiently large i . For d ∈ N ∞ , we define(50) | d | := X i ≥ id i , || d || := X i ≥ d i , d ! := Y i ≥ d i ! . In the following exposition, we set t i = − (cid:0) r i (cid:1) / ((2 i + 1)4 i ), i ≥ Proposition 5.4.
Let g ≥ . We have the following closed formula for one-point r -spin numbers h τ n,m i g = ( − g r g Γ (cid:0) − mr (cid:1) X | d | = g Γ (cid:18) || d || − g − r (cid:19) Q i ≥ t d i i d ! . Proof.
By expanding the right-hand side of (49) − r + 1) y (cid:18)(cid:16) x + y (cid:17) r +1 − (cid:16) x − y (cid:17) r +1 (cid:19) = − x r − X i ≥ t i y i x r − i and using Γ( z ) = r Z ∞ x rz − exp( − x r ) dx, we see that Proposition 5.4 follows from (38). (cid:3) DO AND WITTEN’S r -SPIN NUMBERS 19 Corollary 5.5.
Let k ≥ and g ≥ . We have the following closed formula for r -spin numbers h τ k , τ n,m i g = ( − g r g Γ(1 − mr ) X | d | = g Γ (cid:18) || d || − g − k − r (cid:19) Q i ≥ t d i i d ! . This follows from the same inductive argument as in the proof of Proposition4.3.
Remark 5.6.
For r -spin numbers, we do not have the analogue of the divisorequation as in Gromov-Witten theory. But the identity of Proposition 5.4 suggeststhat some form of the “divisor equation” may still exist for r -spin numbers. Corollary 5.7.
Let g ≥ . Setting r = − , m = 0 in the right-hand side ofProposition 5.4, we get X | d | = g Γ( || d || + 2 g −
1) ( − || d || d ! Q i ≥ ((2 i + 1)4 i ) d i = − B g g . Proof.
We follow the method used [3, § r → − RHS = − Z ∞ (cid:18) x − y x + y (cid:19) /y dx, which is the generating function of the left-hand side of Corollary 5.7.Making the change of variables x − y x + y = e − z , i.e. x = y (cid:18) e z − e z (cid:19) . we have RHS = − Z ∞ e − z/y − ye − z (1 − e − z ) dz (51) = − Z ∞ e − z/y − e − z dz (52) = − y Z ∞ e − t dt − e − yt (53) = − ∞ X k =1 B k k y k . (54)Here (52) follows from ddz (cid:18) e − z/y − e − z (cid:19) = − e − z/y y (1 − e − z ) + − e − z e − z/y (1 − e − z ) and (54) follows from 11 − e − t = ∞ X k =0 B k t k − k ! . This completes the proof. (cid:3) Small phase space in genus zero
In this section, we extend Witten’s exposition in [31]. We first reorganize Wit-ten’s argument and highlight important relevant results of Witten for the reader’sconvenience. We then prove a full series expansion formula for the Landau-Ginzburgpotential W ( p, x ) in the small phase space ( t n,m = 0 , n >
0) of genus zero.For dimensional reasons (equation (2)), a primary intersection number h τ ,m · · · τ ,m s i g can be nonzero only when g = 0. Furthermore, for each r , there are only finitenumber of nonzero primary intersection numbers h τ ,m · · · τ ,m s i , since we have r ( s − − m + · · · + m s ≤ ( r − s (so in particular s ≤ r + 1).As observed by Witten, the genus zero Gel’fand-Dikii equation is obtained byreplacing the differential operator Q by a function W ( p, x ) = p r + r − X i =0 u i ( x ) p i and replacing commutators by Poisson brackets { A, B } = ∂A∂p ∂B∂x − ∂A∂x ∂B∂p . So in genus zero, the Gel’fand-Dikii equations reduce to ∂W∂t n,m = c n,m r { W n +( m +1) /r + , W } , where c n,m is the same constant defined in § Lemma 6.1 (Witten, [31]) . On the small phase space, we have ∂F∂t m = r ( m + 1)( r + m + 1) Res( W m +1) /r ) . Proof.
A special case of Witten’s conjecture is ∂ F∂t , ∂t ,m = r ( m + 1)( r + m + 1) Res( W m +1) /r ) . The string equation implies that on small phase space, we actually have ∂ F∂t . ∂t ,m = ∂F∂t ,m . The desired equation follows. (cid:3)
Below, all of our computations will be done entirely on the small phase space( t n,m = 0 , n >
0) and we set t m = t ,m and τ m = τ ,m . Lemma 6.2 (Witten, [31]) . For ≤ m ≤ r − , we have ∂W∂t m = − m + 1 ∂∂p W ( m +1) /r + . Proof.
A special case of Witten’s conjecture is(55) ∂ F∂t ∂t m = − rm + 1 Res( W ( m +1) /r ) . By (13) in the proof of Lemma 3.4 (the differential polynomials p, p ′ there shouldbe replaced by plain polynomials of their arguments), we can use equation (55) toexpress the coefficients u i of W as differential polynomials in ∂ F/∂t ∂t m . Hence DO AND WITTEN’S r -SPIN NUMBERS 21 W can be regarded as a function in p, t , . . . , t m . If we set all t m = 0, then the lefthand side of (55) is obviously zero for dimensional reasons, so all u i = 0 by Lemma3.4. Thus W = p r when all t m = 0. We then get the constant term of W .Differentiating (55) with respect to x = t , we get(56) δ m,r − = − rm + 1 ∂∂x Res( W ( m +1) /r ) . Since ∂W/∂x is a polynomial in p of degree at most r −
2, if this polynomial is ofdegree k , then from (13) in the proof of Lemma 3.4, the right hand side of (56) isnon-zero for m = r − − k . Thus k = 0 and ∂W∂x = ∂u ∂x = rr − ∂∂x Res( W r − r ) = − . From this and ∂u i /∂x = 0 when 1 ≤ i ≤ r −
2, we have for 0 ≤ m ≤ r − ∂∂x W ( m +1) /r + = 0 , since the coefficients of W ( m +1) /r + do not contain u . This follows from a weightcount, since by our convention (see Lemma 3.2), W ( m +1) /r + is homogeneous of weight m + 1 ≤ r −
1, while the weight of u i is r − i . (cid:3) Theorem 6.3 (Witten, [31]) . For ≤ m ≤ r − , define φ m = − ∂W∂t m . Then ∂ F∂t j ∂t m ∂t s = r · Res (cid:26) φ j φ m φ s ∂ p W (cid:27) . Now we can state our new results: the full series expansion for W in t , . . . , t m ,extending Witten’s computation up to linear terms [31]. Theorem 6.4.
We have the following series expansion for W : W = p r + r − X k =0 p k ∞ X n =1 ( − n n ! · r n − X v + ··· + v n =( n − r + k ( k + n − k ! t v · · · t v n = p r + r − X k =0 p k − t k + 12! · r X u + v = r + k ( k + 1) t u t v − · r X u + v + w =2 r + k ( k + 1)( k + 2) t u t v t w + · · · ! Proof.
We can compute the degree n term of W ( m +1) /r + from terms of W up todegree n . Then we use Lemma 6.2 to compute the degree n + 1 term of W fromthe degree n term of W ( m +1) /r + . W ( m +1) /r + = p m +1 − m + 1 r X u ≥ r − m − t u p m + u − r +1 + m + 12! · r X u + v ≥ r − m − ( m + u + v + 2 − r ) t u t v p m + u + v − r +1 + · · · = p m +1 + ( m + 1) ∞ X n =1 ( − n n ! · r n X v + ··· + v n ≥ nr − m − ( m + n P i =1 v i + n − nr )!( m + n P i =1 v i + 1 − nr )! × t v i · · · t v n p m + v + ··· + v n − nr +1 Thus the theorem can be proved inductively. (cid:3)
Corollary 6.5.
Let ≤ m ≤ r − . The series expansion for φ m is φ m = − ∂W∂t m = p m − X u ≥ r − m m + u + 1 − rr t u p m + u − r + 12! · r X u + v ≥ r − m ( u + v + m + 1 − r )( u + v + m + 2 − r ) t u t v p m + u + v − r + · · · = p m + ∞ X n =1 ( − n n ! · r n X v + ··· + v n ≥ nr − m ( m + n P i =1 v i + n − nr )!( m + n P i =1 v i − nr )! t v i · · · t v n p m + v + ··· + v n − nr Proof.
This follows from the definition of φ m and a direct computation. (cid:3) In [7], Dijkgraaf, Verlinde and Verlinde give a closed formula of φ m in termsof the determinant of matrices. Presumably their formula is equivalent to ours,although we have not checked the details.In view of Lemma 6.1, it would also be interesting to have a series expansion for W m +1 r . Here we write out terms up to degree 3. Let θ ( x ) be the Heaviside functionthat is 1 for x ≥ x < W m +1 r = p m +1 − m + 1 r X u t u p m +1+ u − r + m + 12! · r X u,v (( m + 1 − r ) + ( u + v − r + 1) θ ( u + v − r )) t u t v p m + u + v +1 − r − ( m + 1)3! · r X u,v,w (( m + 1 − r )( m + 1 − r ) + ( m + 1 − r )( u + v + 1 − r ) θ ( u + v − r )+ ( m + 1 − r )( u + w + 1 − r ) θ ( u + w − r ) + ( m + 1 − r )( v + w + 1 − r ) θ ( v + w − r )+ ( u + v + w − r + 1)( u + v + w − r + 2) θ ( u + v + w − r )) t u t v t w p m + u + v + w +1 − r + · · · By Lemma 6.1, we can use the degree 3 term of the above expansion to get aformula for 4-point correlation functions.
Corollary 6.6. h τ m τ u τ v τ w i = 1 r ( r − m − − ( u + v − r + 1) θ ( u + v − r ) − ( u + w − r + 1) θ ( u + w − r ) − ( v + w − r + 1) θ ( v + w − r )) . Proof.
Replace m by m + r in the expansion of W m +1 r and take the coefficient of p − . We get the desired result from Lemma 6.1. (cid:3) DO AND WITTEN’S r -SPIN NUMBERS 23 The formula in Corollary 6.6 is slightly different with Witten’s formula [31,(3.3.36)] h τ m τ u τ v τ w i = 1 r ( m − ( m + u − r + 1) θ ( m + u − r ) − ( m + v − r + 1) θ ( m + v − r ) − ( m + w − r + 1) θ ( m + w − r )) , but it is not difficult to prove that they are both equivalent to h τ a τ a τ a τ a i = r · min( a i , r − − a i ).Our motivation in studying r -spin numbers on the small phase space is to provethe following conjectural properties of these numbers. Conjecture 6.7.
In the small phase space, we have i) (Integrality) r s − ( s − h τ m · · · τ m s i ∈ Z . ii) (Vanishing) If m i < s − for some ≤ i ≤ s , then h τ m · · · τ m s i = 0 . iii) (Multinomial distribution) If m > m , then h τ m − τ m +1 τ m · · · τ m s i ≥h τ m τ m · · · τ m s i . We have verified this conjecture in low genus or when s is small. One caneven prove r s − h τ m · · · τ m s i ∈ Z by extending (57). However, the combinatorialdifficulty for the general case is still considerable, even though we can write downexplicit formulae for general s -point correlation functions h τ m · · · τ m s i via Theorem6.4, Corollary 6.5 and Witten’s Theorem 6.3.See [17] for more on denominators and multinomial-type properties of intersec-tion numbers.Part of our motivation comes from Gromov-Witten invariants of CP n on thesmall phase space. For fixed n ≥ d ≥
0, consider Gromov-Witten invariantsof CP n on the small phase space in genus zero h m , · · · , m s i := h τ ,m , · · · , τ ,m s i CP n ,d , which is nonzero (for dimensional reasons) only when(58) s X i =1 m i = n + ( n + 1) d + s − . The following properties are analogues of corresponding statements in Conjec-ture 6.7:i) (Integrality) h m , · · · , m s i ∈ N ≥ .ii) (Vanishing) If d > s = d + 2, then h m , · · · , m s i = 0.iii) (Multinomial distribution) If m > m , then h m − , m + 1 , m , · · · , m s i ≥ h m , m , · · · , m s i . The integrality (i) is clear, since genus zero Gromov-Witten invariants of CP n are intersections on a scheme, and hence integral. We conjecture the multinomialdistribution (iii), based on numerical evidence. We now prove (ii). Proposition 6.8.
With the notation above, we have the vanishing h m , · · · , m s i =0 of degree d genus invariants in P n , where s = d + 2 , with any m i , and d > .Proof. We show that for any choice of m i , the intersection theory problem h m , · · · , m s i ,interpreted as counting stable maps “meeting” generally chosen linear spaces ofcodimension m , . . . , m s , corresponds to the empty intersection. Let c i = n − m i be the dimension of these linear spaces for convenience; P c i = n − d + 1 from(58).As m i ≤ n , we have n ( d + 2) ≥ s X i =1 m i = n + ( n + 1) d + ( d + 2) − d ≤ ( n + 1) / < n . The image of any degree d stable map lies insidea P d . We show that there isn’t even a P d inside P n meeting the (generally chosen)linear spaces of dimension c i . The codimension of the condition (on G ( d, n )) thata P d in P n meet a P c i is max(0 , n − d − c i ). Then s X i =1 max(0 , n − d − c i ) ≥ s X i =1 ( n − d − c i )= ( n − d )( d + 2) − ( n − d + 1) > ( d + 1)( n − d ) (using d > G ( d, n )so there is no P d in P n meeting the desired linear spaces, and thus no degree d stable map meeting these linear spaces. (cid:3) Appendix A. Combinatorial identities
We prove Proposition 3.6, using the Lagrange inversion formula. The followingform of Lagrange inversion formula can be found in [28, p. 38].
Lemma A.1 (Lagrange inversion formula) . Let F ( x ) = a x + a x + · · · ∈ C [[ x ]] be a power series with a = 0 and F − ( x ) ∈ C [[ x ]] be its inverse (defined by F − ( F ( x )) = x ). For k, n ∈ Z we have (59) 1 n [ x n − k ] (cid:18) xF ( x ) (cid:19) n = 1 k [ x n ] F − ( x ) k . We now prove Proposition 3.6.Let f ( x ) = 1+ P ∞ j =2 a j x j ∈ C [[ x ]] be as in Proposition 3.6. Then F ( x ) = x/f ( x )is a power series with a = 0, so we can apply Lemma A.1. Taking k = 1 in equation(59), we see that [ x ] F − ( x ) = [ x ] f ( x ) = 0, so we have1 F − ( x ) = 1 x + c x + c x + · · · (60) = 1 x − c x − c x − · · · . Taking k = 1 and k = 2 in equation (59) respectively, we get[ x n +1 ] f ( x ) n n = − [ x n ] 1 F − , [ x n +2 ] f ( x ) n n = −
12 [ x n ] 1 F − . DO AND WITTEN’S r -SPIN NUMBERS 25 Substituting the above two identities into equation (19) and then applying equation(60) to the summation term on the right hand side, equation (19) becomes − [ x n +1 ] 1 F − = 12 n − X j =1 [ x j ] 1 F − [ x n − j ] 1 F − −
12 [ x n ] 1( F − ( x ) = (cid:18)
12 [ x n ] 1 F − ( x ) − [ x n +1 ] 1 F − (cid:19) −
12 [ x n ] 1( F − ( x ) = − [ x n +1 ] 1 F − . So we have proved Proposition 3.6. (cid:3)
We now present equivalent formulations of Proposition 3.6 that may be usefulelsewhere. We use the notation introduced in (50).
Proposition A.2.
Let a , b ∈ N ∞ , c ∈ N ∞ and || c || ≥ . Then the followingidentity holds ( | c | + || c || − | c | − · ( || c || −
1) = 12 X c = a + ba , b =0 (cid:18) ca , b (cid:19) ( | a | + || a || − · ( | b | + || b || − | a | − · ( | b | − , where (cid:0) ca , b (cid:1) is defined as Q i ≥ c ! a ! b ! = Q i ≥ (cid:0) c i a i ,b i (cid:1) (cf. (50) ).Proof. Take any c = ( c , c , . . . ) ∈ N ∞ , compare the coefficient Q j ≥ a c j − j in bothsides of equation (19). We have( | c | + || c || − | c | − c ! = 12 X c = a + ba , b =0 ( | a | + || a || − | a | − a ! ( | b | + || b || − | b | − b ! + ( | c | + || c || − | c | − c ! . By moving the last term in the right hand side to the left, we get the desiredidentity. (cid:3)
A partition is a sequence of integers µ ≥ µ ≥ · · · ≥ µ k >
0. We write | µ | = µ + · · · + µ k , ℓ ( µ ) = k. Define m j ( µ ) to be the number of j ’s among µ , . . . , µ k , z µ = Q j m j ( µ )! j m j ( µ ) ,and p µ = Q j p m j ( µ ) j . Proposition A.3. X ℓ ( µ ) ≥ ( | µ | + ℓ ( µ ) − ℓ ( µ ) − p µ ( | µ | − z µ = 12 X µ =0 ( | µ | + ℓ ( µ ) − p µ ( | µ | − z µ Proof.
Take c = ( m ( µ ) , m ( µ ) , . . . ) ∈ N ∞ . Then the identity in the proposition isjust a reformulation of Proposition A.2. (cid:3) Appendix B. The differential polynomial W r ( z )From § γ r +1 − can be expressed as a differential polynomial in γ − , . . . , γ r − − . If2 ≤ i ≤ r , denote by p i ( r ) the coefficient of D i γ r +1 − i − in the resulting differentialpolynomial S ( γ r +1 − ). From the proof of Proposition 3.8, it is straightforward toobtain the following recursive formula for p i ( r ), p i ( r ) = 1 r + 1 − i [ D i w − ( r +1 − i ) ]( γ r +1 − − r + 1 r γ r − − r + 12 r Dγ r − ) − r + 1 − i i − X j =2 (cid:18) r + 1 − ji + 1 − j (cid:19) p j ( r )(61) = (cid:18) r + 1 i (cid:19) r i − i + 1) − r + 1 − i i − X j =2 (cid:18) r + 1 − ji + 1 − j (cid:19) p j ( r ) . We have proved p ( r ) = r +112 in Proposition 3.8. The relation of p i ( r ) to thecoefficients of W r ( z ) is given by(62) [ z ( i ) r − i ] W r ( z ) = −√− i ( r − i + 1) r i − ( r + 1) p i ( r ) . For i ≥
2, define quantities C i by(63) p i ( r ) = 1 r (cid:18) r + 1 i (cid:19) C i . We will see shortly that C i are in fact constants independent of r .Substituting (63) into the recursion formula (61), we get1 r (cid:18) r + 1 i (cid:19) C i = (cid:18) r + 1 i (cid:19) r i − i + 1) − r + 1 − i i − X j =2 (cid:18) r + 1 − ji + 1 − j (cid:19) r (cid:18) r + 1 j (cid:19) C j C i = i − i + 1) − i − X j =2 (cid:18) i + 1 j (cid:19) C j i + 1 . (64)Hence by induction (starting from C = − , C = 0), we see that the C i areconstants. Using the values of C and C , we may simplify (64) as(65) i X j =0 (cid:18) i + 1 j (cid:19) C j = i − , i ≥ . Proposition B.1.
Let i ≥ . Then C i = B i the Bernoulli numbers. In particular, C k +1 = 0 .Proof. We define a new sequence C ′ j by C ′ = 1 , C ′ = − and C ′ j = C j , j ≥ i X j =0 (cid:18) i + 1 j (cid:19) C ′ j = 0 , i ≥ , which is the usual recursion for Bernoulli numbers. Since C ′ = B , C ′ = B , wemust have C j = C ′ j = B j for all j ≥ (cid:3) From (62) and (63), we thus proved Proposition 3.10.
DO AND WITTEN’S r -SPIN NUMBERS 27 Appendix C. An identity of Bernoulli numbers
Let f n ( k ) , n ≥ f n +1 ( k ) = − ( k + 1) f n ( k + 2) + (2 k + 1) f n ( k + 1) − kf n ( k )starting with f ( k ) = − k +1)( k +2) . Proposition C.1.
Let n ≥ . We have f n (0) = − B n /n . The rest of the appendix is devoted to proving the above proposition. First werecord the following combinatorial identities. n X k =0 (cid:18) nk (cid:19) ( − n − k n + i − k = n !( i − n + i )! , (67) n X i = w ( i − i − w )! = n !( n − w )! w , (68) (cid:18) e t − t (cid:19) w = ∞ X j = w t j − w j ! w X s =0 ( − s (cid:18) ws (cid:19) ( w − s ) j . (69)Consider the generating function F n ( x ) = P ∞ k =0 f n ( k ) x k , then we have F ( x ) = − x + x − x ln(1 − x ) and (66) implies F n ( x ) = − ∂∂x (cid:18) F n − ( x ) x (cid:19) + 2 ∂∂x F n − ( x ) − x F n − ( x ) − x ∂∂x F n − ( x )= − ( x − x F n − ( x ) + 1 − xx F n − ( x ) . More precisely, F n ( x ) from the above recursion differs from the true generatingfunction by a finite sum of negative powers of x . Thus Proposition C.1 is equivalentto prove that the constant term of − nF n ( x ) equals B n .It is not difficult to see that F n ( x ) decomposes as(70) F n ( x ) = G n ( x ) + R n ( x ) ln(1 − x ) , n ≥ , where G n ( x ) and R n ( x ) are rational functions in x satisfying the recursions G n ( x ) = − ( x − x ∂∂x G n − ( x ) + 1 − xx G n − ( x ) + 1 − xx R n − ( x ) , (71) R n ( x ) = − ( x − x ∂∂x R n − ( x ) + 1 − xx R n − ( x ) . (72)We may solve (72) to get(73) R n ( x ) = ( x − n n X i =1 ( − i +1 a ( n, i ) x − n − i , where a ( n, i ) is given by a ( n, i ) = i X w =0 ( − i − w (cid:18) n + ii − w (cid:19) w X s =0 ( − s ( w − s ) n + w s !( w − s )! . From (71), we may prove that [ x k ] G n ( x ) = 0 , ∀ k ≥
0. By using (67), (68), (69),the constant term of − nF n ( x ) equals (74) [ x ]( − n ) R n ( x ) ln(1 − x ) = ( − n ) n X i =1 ( − i a ( n, i ) n X k =0 (cid:18) nk (cid:19) ( − n − k n + i − k = ( − n ) n X i =1 ( − i a ( n, i ) n !( i − n + i )!= ( − n ) n X w =1 w X s =0 ( − s (cid:18) ws (cid:19) ( w − s ) w + n w ! ( − w n !( w + n )! n X i = w ( i − i − w )!= ( − n ) n X w =1 w X s =0 ( − s (cid:18) ws (cid:19) ( w − s ) w + n w ! ( − w n !( w + n )! n !( n − w )! w = ( − n ) n X w =1 ( − w (cid:18) nw (cid:19) n ! w [ t n ] (cid:18) e t − t (cid:19) w = − n X w =1 ( − w (cid:18) nw (cid:19) n ![ t n − ] (cid:18) e t − t (cid:19) w − ddt (cid:18) e t − t (cid:19)! = − n ![ t n − ] n X w =0 ( − w (cid:18) nw (cid:19) (cid:18) e t − t (cid:19) w te t − e t + 1 t ( e t − ! + n ![ t n − ] te t − e t + 1 t ( e t − − n ![ t n − ] (cid:18)(cid:18) − e t − t (cid:19) n te t − e t + 1 t ( e t − (cid:19) + n ![ t n ] te t − e t + 1 e t − n ![ t n ] te t − e t + 1 e t − , where the last equation follows by noting1 − e t − t = − t − t + · · · ,te t − e t + 1 t ( e t −
1) = 12 + 112 t + · · · . Finally, Proposition C.1 follows from(75) 1 + te t − e t + 1 e t − t − e − t = 1 + t ∞ X n =2 B n t n n ! . Remark C.2.
Another way of proving Proposition C.1 is by studying the function h j ( k ) = Q ji =1 ( k + i ) f j ( k ). Then (66) becomes(76) h j +1 ( k ) = − ( k + 1) ( k + 2) h j ( k + 2)+ ( k + 1)(2 k + 1)( k + 2 j + 2) h j ( k + 1) − k ( k + 2 j + 1)( k + 2 j + 2) h j ( k )starting with h ( k ) = − h ( k ) = 4 k − h ( k ) = − k + 84 k .We may prove from the recursion (76) (although more difficult) that h j ( k ) is adegree j − − j ( j !) k j − and the constantterm equals − (2 j )! B j /j when j ≥
2, as claimed in Proposition C.1.
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Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027,China; Department of Mathematics,University of California at Los Angeles, Los An-geles
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