Formal pseudodifferential operators and Witten’s r-spin numbers

Kefeng Liu; Ravi Vakil; Hao Xu

doi:10.1515/crelle-2014-0102

Published by De Gruyter December 23, 2014

Formal pseudodifferential operators and Witten’s r-spin numbers

Kefeng Liu , Ravi Vakil and Hao Xu

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2014-0102

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Abstract

We derive an effective recursion for Witten’s r-spin intersection numbers, using Witten’s conjecture relating r-spin numbers to the Gel’fand–Dikii hierarchy. Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions). We use these closed-form descriptions to prove Harer–Zagier’s formula for the Euler characteristic of ℳg,1. Finally, we extend Witten’s series expansion formula for the Landau–Ginzburg potential to study r-spin numbers in the small phase space in genus zero. Our key tool is the calculus of formal pseudodifferential operators, and is partially motivated by work of Brézin and Hikami.

A Combinatorial identities

We prove Proposition 3.6, using the Lagrange inversion formula. The following form of Lagrange inversion formula can be found in [29, p. 38].

Lemma A.1 (Lagrange inversion formula).

Let F⁢(x)=a1⁢x+a2⁢x2+⋯∈C⁢[[x]] be a power series with a1≠0 and F-1⁢(x)∈C⁢[[x]] be its inverse (defined by F-1⁢(F⁢(x))=x). For k,n∈Z we have

(A.1)1n⁢[xn-k]⁢(xF⁢(x))n=1k⁢[xn]⁢F-1⁢(x)k.

We now prove Proposition 3.6.

Proof of Proposition 3.6.

Let f⁢(x)=1+∑j=2∞aj⁢xj∈ℂ⁢[[x]] be as in Proposition 3.6. Then F⁢(x)=x/f⁢(x) is a power series with a1≠0, so we can apply Lemma A.1. Taking k=1 in equation (A.1), we see that [x2]⁢F-1⁢(x)=12⁢[x]⁢f⁢(x)2=0, so we have

(A.2)1F-1⁢(x)=1x+c3⁢x3+c4⁢x4+⋯=1x-c3⁢x-c4⁢x2-⋯.

Taking k=1 and k=2 in equation (A.1) respectively, we get

[xn+1]⁢f⁢(x)nn=-[xn]⁢1F-1,

[xn+2]⁢f⁢(x)nn=-12⁢[xn]⁢1F-1.

Substituting the above two identities into equation (3.9) and then applying equation (A.2) to the summation term on the right-hand side, equation (3.9) becomes

-[xn+1]⁢1F-1=12⁢∑j=1n-1[xj]⁢1F-1⁢[xn-j]⁢1F-1-12⁢[xn]⁢1(F-1(x)2

=(12⁢[xn]⁢1F-1⁢(x)2-[xn+1]⁢1F-1)-12⁢[xn]⁢1(F-1(x)2

=-[xn+1]⁢1F-1.

So we have proved Proposition 3.6. ∎

We now present equivalent formulations of Proposition 3.6 that may be useful elsewhere. We use the notation introduced in (5.22).

Proposition A.2.

Let 𝐚,𝐛∈N∞, 𝐜∈N∞ and ∥𝐜∥≥2. Then the following identity holds:

(|𝒄|+∥𝒄∥-3)!(|𝒄|-1)!⋅(∥𝒄∥-1)=12⁢∑𝒄=𝒂+𝒃𝒂,𝒃≠0(𝒄𝒂,𝒃)⁢(|𝒂|+∥𝒂∥-2)!⋅(|𝒃|+∥𝒃∥-2)!(|𝒂|-1)!⋅(|𝒃|-1)!,

where (𝐜𝐚,𝐛) is defined as

(𝒄𝒂,𝒃)=∏i≥1𝒄!𝒂!⁢𝒃!=∏i≥1(ciai,bi)

(cf. (5.22)).

Proof.

Take any c=(c1,c2,…)∈ℕ∞, compare the coefficient ∏j≥2ajcj-1 in both sides of equation (3.9). We have

(|𝒄|+∥𝒄∥-2)!(|𝒄|-1)!⁢𝒄!=12⁢∑𝒄=𝒂+𝒃𝒂,𝒃≠0(|𝒂|+∥𝒂∥-2)!(|𝒂|-1)!⁢𝒂!⁢(|𝒃|+∥𝒃∥-2)!(|𝒃|-1)!⁢𝒃!+(|𝒄|+∥𝒄∥-3)!(|𝒄|-2)!⁢𝒄!.

By moving the last term on the right-hand side to the left, we get the desired identity. ∎

A partition is a sequence of integers μ1≥μ2≥…≥μk>0. We write

|μ|=μ1+⋯+μk,ℓ⁢(μ)=k.

Define mj⁢(μ) to be the number of indices j among μ1,…,μk,

zμ=∏jmj⁢(μ)!⁢jmj⁢(μ) and pμ=∏jpjmj⁢(μ).

Proposition A.3.

We have

∑ℓ⁢(μ)≥2(|μ|+ℓ⁢(μ)-3)!⁢(ℓ⁢(μ)-1)⁢pμ(|μ|-1)!⁢zμ=12⁢(∑μ≠0(|μ|+ℓ⁢(μ)-2)!⁢pμ(|μ|-1)!⁢zμ)2.

Proof.

Take 𝒄=(m1⁢(μ),m2⁢(μ),…)∈ℕ∞. Then the identity in the proposition is just a reformulation of Proposition A.2. ∎

B The differential polynomial Wr⁢(z)

From Section 3, γ-1r+1 can be expressed as a differential polynomial in γ-11,…,γ-1r-1. If 2≤i≤r, denote by pi⁢(r) the coefficient of Di⁢γ-1r+1-i in the resulting differential polynomial S⁢(γ-1r+1). From the proof of Proposition 3.8, it is straightforward to obtain the following recursive formula for pi⁢(r):

(B.1)pi⁢(r)=1r+1-i⁢[Di⁢w-(r+1-i)]⁢(γ-1r+1-r+1r⁢γ-2r-r+12⁢r⁢D⁢γ-1r)

-1r+1-i⁢∑j=2i-1(r+1-ji+1-j)⁢pj⁢(r)

=(r+1i)⁢1r⁢i-12⁢(i+1)-1r+1-i⁢∑j=2i-1(r+1-ji+1-j)⁢pj⁢(r).

We have proved p2⁢(r)=r+112 in Proposition 3.8. The relation of pi⁢(r) to the coefficients of Wr⁢(z) is given by

(B.2)[zr-i(i)]⁢Wr⁢(z)=--1i⁢(r-i+1)ri2-1⁢(r+1)⁢pi⁢(r).

For i≥2, define quantities Ci by

(B.3)pi⁢(r)=1r⁢(r+1i)⁢Ci.

We will see shortly that Ci are in fact constants independent of r.

Substituting (B.3) into the recursion formula (B.1), we get

1r⁢(r+1i)⁢Ci=(r+1i)⁢1r⁢i-12⁢(i+1)-1r+1-i⁢∑j=2i-1(r+1-ji+1-j)⁢1r⁢(r+1j)⁢Cj,

i.e.

(B.4)Ci=i-12⁢(i+1)-∑j=2i-1(i+1j)⁢Cji+1.

Hence by induction (starting from C0=-12, C1=0), we see that the Ci are constants. Using the values of C0 and C1, we may simplify (B.4) as

(B.5)∑j=0i(i+1j)⁢Cj=i-22,i≥1.

Proposition B.1.

Let i≥2. Then Ci=Bi the Bernoulli numbers. In particular, we have C2⁢k+1=0.

Proof.

We define a new sequence Cj′ by C0′=1, C1′=-12 and Cj′=Cj for j≥2. From (B.5), we have

∑j=0i(i+1j)⁢Cj′=0,i≥1,

which is the usual recursion for Bernoulli numbers. Since C0′=B0 and C1′=B1, we must have Cj=Cj′=Bj for all j≥2. ∎

From (B.2) and (B.3), we thus proved Proposition 3.10.

C An identity of Bernoulli numbers

Let fn⁢(k), n≥1, be given by the recursion

(C.1)fn+1⁢(k)=-(k+1)⁢fn⁢(k+2)+(2⁢k+1)⁢fn⁢(k+1)-k⁢fn⁢(k)

starting with f1⁢(k)=-1(k+1)⁢(k+2).

Proposition C.1.

Let n≥2. We have fn⁢(0)=-Bnn.

The rest of the appendix is devoted to proving the above proposition. First we record the following combinatorial identities:

(C.2)∑k=0n(nk)⁢(-1)n-k⁢1n+i-k=n!⁢(i-1)!(n+i)!,

(C.3)∑i=wn(i-1)!(i-w)!=n!(n-w)!⁢w,

(C.4)(et-1t)w=∑j=w∞tj-wj!⁢∑s=0w(-1)s⁢(ws)⁢(w-s)j.

Consider the generating function Fn⁢(x)=∑k=0∞fn⁢(k)⁢xk. Then we have

F1⁢(x)=-1x+x-1x2⁢ln⁡(1-x)

and (C.1) implies

Fn⁢(x)=-∂∂⁡x⁢(Fn-1⁢(x)x)+2⁢∂∂⁡x⁢Fn-1⁢(x)-1x⁢Fn-1⁢(x)-x⁢∂∂⁡x⁢Fn-1⁢(x)

=-(x-1)2x⁢Fn-1⁢(x)+1-xx2⁢Fn-1⁢(x).

More precisely, Fn⁢(x) from the above recursion differs from the true generating function by a finite sum of negative powers of x. Thus Proposition C.1 is equivalent to prove that the constant term of -n⁢Fn⁢(x) equals Bn.

It is not difficult to see that Fn⁢(x) decomposes as

(C.5)Fn⁢(x)=Gn⁢(x)+Rn⁢(x)⁢ln⁡(1-x),n≥1,

where Gn⁢(x) and Rn⁢(x) are rational functions in x satisfying the recursions

(C.6)Gn⁢(x)=-(x-1)2x⁢∂∂⁡x⁢Gn-1⁢(x)+1-xx2⁢Gn-1⁢(x)+1-xx⁢Rn-1⁢(x),

(C.7)Rn⁢(x)=-(x-1)2x⁢∂∂⁡x⁢Rn-1⁢(x)+1-xx2⁢Rn-1⁢(x).

We may solve (C.7) to get

(C.8)Rn⁢(x)=(x-1)n⁢∑i=1n(-1)i+1⁢a⁢(n,i)⁢x-n-i,

where a⁢(n,i) is given by

a⁢(n,i)=∑w=0i(-1)i-w⁢(n+ii-w)⁢∑s=0w(-1)s⁢(w-s)n+ws!⁢(w-s)!.

From (C.6), we may prove that [xk]⁢Gn⁢(x)=0 for all k≥0. By using (C.2), (C.3), (C.4), the constant term of -n⁢Fn⁢(x) equals

[x0]⁢(-n)⁢Rn⁢(x)⁢ln⁡(1-x)

=(-n)⁢∑i=1n(-1)i⁢a⁢(n,i)⁢∑k=0n(nk)⁢(-1)n-k⁢1n+i-k

=(-n)⁢∑i=1n(-1)i⁢a⁢(n,i)⁢n!⁢(i-1)!(n+i)!

=(-n)⁢∑w=1n∑s=0w(-1)s⁢(ws)⁢(w-s)w+nw!⁢(-1)w⁢n!(w+n)!⁢∑i=wn(i-1)!(i-w)!

=(-n)⁢∑w=1n∑s=0w(-1)s⁢(ws)⁢(w-s)w+nw!⁢(-1)w⁢n!(w+n)!⁢n!(n-w)!⁢w

=(-n)⁢∑w=1n(-1)w⁢(nw)⁢n!w⁢[tn]⁢(et-1t)w

=-∑w=1n(-1)w⁢(nw)⁢n!⁢[tn-1]⁢((et-1t)w-1⁢dd⁢t⁢(et-1t))

=-n!⁢[tn-1]⁢(∑w=0n(-1)w⁢(nw)⁢(et-1t)w⁢t⁢et-et+1t⁢(et-1))+n!⁢[tn-1]⁢t⁢et-et+1t⁢(et-1)

=-n!⁢[tn-1]⁢((1-et-1t)n⁢t⁢et-et+1t⁢(et-1))+n!⁢[tn]⁢t⁢et-et+1et-1

=n!⁢[tn]⁢t⁢et-et+1et-1,

where the last equation follows by noting

1-et-1t=-12⁢t-16⁢t2+⋯,

t⁢et-et+1t⁢(et-1)=12+112⁢t+⋯.

Finally, Proposition C.1 follows from

(C.9)1+t⁢et-et+1et-1=t1-e-t=1+t2+∑n=2∞Bn⁢tnn!.

Remark C.2.

Another way of proving Proposition C.1 is by studying the function

hj⁢(k)=∏i=12⁢j(k+i)⁢fj⁢(k).

Then (C.1) becomes

(C.10)hj+1⁢(k)=-(k+1)2⁢(k+2)⁢hj⁢(k+2)

+(k+1)⁢(2⁢k+1)⁢(k+2⁢j+2)⁢hj⁢(k+1)

-k⁢(k+2⁢j+1)⁢(k+2⁢j+2)⁢hj⁢(k)

starting with h1⁢(k)=-1, h2⁢(k)=4⁢k-2, h3⁢(k)=-36⁢k2+84⁢k.

We may prove from the recursion (C.10) (although more difficult) that hj⁢(k) is a degree j-1 polynomial whose leading term equals (-1)j⁢(j!)2⁢kj-1 and the constant term equals -(2⁢j)!⁢Bj/j when j≥2, as claimed in Proposition C.1.

Acknowledgements

We thank H. Chang, J. Li, W. Luo, M. Mulase, Y. B. Ruan, and J. Zhou for helpful conversations. The third author thanks Professor D. Zeilberger for answering a question on computer proof of combinatorial identities. We thank the referees for helpful comments.

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Received: 2012-1-28

Revised: 2014-6-18

Published Online: 2014-12-23

Published in Print: 2017-7-1

Formal pseudodifferential operators and Witten’s r-spin numbers

Abstract

A Combinatorial identities

Lemma A.1 (Lagrange inversion formula).

Proof of Proposition 3.6.

Proposition A.2.

Proof.

Proposition A.3.

Proof.

B The differential polynomial Wr⁢(z)

Proposition B.1.

Proof.

C An identity of Bernoulli numbers

Proposition C.1.

Remark C.2.

Acknowledgements

References

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