j. differential geometry 65 (2003) 289-340

A PROOF OF A CONJECTURE OF MARIÑO-VAFA ON HODGE INTEGRALS

CHIU-CHU MELISSA LIU, KEFENG LIU & JIAN ZHOU

Abstract

We prove a remarkable formula for Hodge integrals conjectured by Mariño and Vafa, 2002, based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable morphisms.

1. Introduction

Let \overline{M}_{g,n} denote the Deligne-Mumford moduli stack of stable curves of genus g with n marked points. Let π : \overline{M}_{g,n+1} → \overline{M}_{g,n} be the universal curve, and let ω_{π} be the relative dualizing sheaf. The Hodge bundle

𝔼 = π_{∗}ω_{π}

is a rank g vector bundle over \overline{M}_{g,n} whose fiber over [(C, x_{1}, …, x_{n})] ∈ \overline{M}_{g,n} is H^{0}(C, ω_{C}). Let s_{i} : \overline{M}_{g,n} → \overline{M}_{g,n+1} denote the section of π which corresponds to the i-th marked point, and let

𝕃_{i} = s_{i}^{∗}ω_{π}

be the line bundle over \overline{M}_{g,n} whose fiber over [(C, x_{1}, …, x_{n})] ∈ \overline{M}_{g,n} is the cotangent line T_{x_{i}}^{∗}C at the i-th marked point x_{i}. A Hodge integral is an integral of the form

∫_{\overline{M}_{g,n}} ψ_{1}^{j_{1}} ⋯ ψ_{n}^{j_{n}}λ_{1}^{k_{1}} ⋯ λ_{g}^{k_{g}}

where ψ_{i} = c_{1}(𝕃_{i}) is the first Chern class of 𝕃_{i}, and λ_{j} = c_{j}(𝔼) is the j-th Chern class of the Hodge bundle.

Received 06/30/2003.

Hodge integrals arise in the calculations of Gromov-Witten invariants by localization techniques [14, 8]. The explicit evaluation of Hodge integrals is a difficult problem. The Hodge integrals involving only ψ classes can be computed recursively by Witten’s conjecture [26] proved by Kontsevich [13]. Algorithms of computing Hodge integrals are described in [3].

In [23], M. Mariño and C. Vafa obtained a closed formula for a generating function of certain open Gromov-Witten invariants, some of which has been reduced to Hodge integrals by localization techniques which are not fully clarified mathematically. This leads to a conjectural formula of Hodge integrals by comparing with the calculations in [12]. To state this formula, we introduce some notation. Let

Λ_{g}^{∨}(u) = u^{g} − λ_{1}u^{g−1} + ⋯ + (−1)^{g}λ_{g}

be the Chern polynomial of 𝔼^{∨}, the dual of the Hodge bundle. For a partition μ given by

μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{l(μ)} > 0,

let |μ| = ∑_{i=1}^{l(μ)} μ_{i}, and define

C_{g,μ}(τ) = −\frac{\sqrt{−1}^{|μ|+l(μ)}}{| Aut(μ)|} (τ(τ + 1))^{l(μ)−1} ∏_{i=1}^{l(μ)} \frac{∏_{a=1}^{μ_{i}−1}(μ_{i}τ + a)}{(μ_{i} − 1)!}

· ∫_{\overline{M}_{g,l(μ)}} \frac{Λ_{g}^{∨}(1)Λ_{g}^{∨}(−τ − 1)Λ_{g}^{∨}(τ)}{∏_{i=1}^{l(μ)}(1 − μ_{i}ψ_{i})},

C_{μ}(λ; τ) = ∑_{g≥0} λ^{2g−2+l(μ)}C_{g,μ}(τ)

Note that

∫_{\overline{M}_{0,l(μ)}} \frac{Λ_{0}^{∨}(1)Λ_{0}^{∨}(−τ − 1)Λ_{0}^{∨}(τ)}{∏_{i=1}^{l(μ)}(1 − μ_{i}ψ_{i})} = ∫_{\overline{M}_{0,l(μ)}} \frac{1}{∏_{i=1}^{l(μ)}(1 − μ_{i}ψ_{i})} = |μ|^{l(μ)−3}

for l(μ) ≥ 3, and we use this expression to extend the definition to the case l(μ) < 3.

Introduce formal variables p = (p_{1}, p_{2}, …, p_{n}, …), and define

p_{μ} = p_{μ_{1}} ⋯ p_{μ_{l(μ)}}

for a partition μ. Define generating functions

C(λ; τ; p) = ∑_{|μ|≥1} C_{μ}(λ; τ)p_{μ},
C(λ; τ; p)^{•} = e^{C(λ;τ;p)}.

As pointed out in [23], by comparing computations in [23] with computations in [12], one obtains a conjectural formula for C_{μ}(τ). This formula is explicitly written down in [28]:

(1) C(λ; τ; p) = ∑_{n≥1} \frac{(−1)^{n−1}}{n} ∑_{μ} ( ∑_{∪_{i=1}^{n}μ^{i}=μ} ∏_{i=1}^{n} ∑_{|ν^{i}|=|μ^{i}|} \frac{χ_{ν^{i}}(C(μ^{i}))}{z_{μ^{i}}}

· e^{\sqrt{−1}(τ+ \frac{1}{2} )κ_{ν^{i}}λ/2}V_{ν^{i}}(λ))p_{μ},

C(λ; τ; p)^{•} = ∑_{|μ|≥0} ( ∑_{|ν|=|μ|} \frac{χ_{ν}(C(μ))}{z_{μ}}e^{\sqrt{−1}(τ+ \frac{1}{2} )κ_{ν}λ/2}V_{ν}(λ)) p_{μ}, (2)

where

(3)  V_{ν}(λ) = ∏_{1≤a<b≤l(ν)} \begin{matrix}sin & [(ν_{a} − ν_{b} + b − a)λ/2] \\ sin & [(b − a)λ/2] \end{matrix}

· \frac{1}{∏_{i=1}^{l(ν)} ∏_{v=1}^{ν_{i}} 2 sin [(v − i + l(ν))λ/2]}.

The right-hand side of (1) is actually some truncated version of the more general formula [23, (5.6)] given by Mariño and Vafa.

We now explain the notation on the right-hand sides of (1) and (2). For a partition μ, χ_{μ} denotes the character of the irreducible representation of S_{d} indexed by μ, where d = |μ| = ∑_{i=1}^{l(μ)} μ_{i}. The number κ_{μ} is defined by

κ_{μ} = |μ| + ∑_{i}(μ_{i}^{2} − 2iμ_{i}).

For each positive integer i,

m_{i}(μ) = |{j : μ_{j} = i}|.

Denote by C(ν) the conjugacy class of S_{d} corresponding to the partition ν, and by χ_{μ}(C(ν)) the value of the character χ_{μ} on the conjugacy class C(ν). Finally,

z_{μ} = ∏_{j} m_{j}(μ)!j^{m_{j}(μ)}.

In this paper, we will call (1) the Mariño-Vafa formula. The third author proved in [27] some special cases of the Mariño-Vafa formula and found some applications [29, 30].

We now describe our approach to the Mariño-Vafa formula (1). Denote the right-hand sides of (1) and (2) by R(λ; τ; p) and R(λ; τ; p)^{•} respectively. In [28], the third author proved the following two equivalent cut-and-join equations similar to the one satisfied by Hurwitz numbers (see [7], [15], [11, Section 15.2]):

Theorem 1.

\frac{∂R}{∂τ} = \frac{\sqrt{−1}λ}{2} ∑_{i,j≥1}(ijp_{i+j} \frac{∂^{2}R}{∂p_{i}∂p_{j}} + ijp_{i+j} \frac{∂R}{∂p_{i}} \frac{∂R}{∂p_{j}} + (i + j)p_{i}p_{j} \frac{∂R}{∂p_{i+j}} ), (4)

\frac{∂R^{•}}{∂τ} = \frac{\sqrt{−1}λ}{2} ∑_{i,j≥1} (ijp_{i+j} \frac{∂^{2}R^{•}}{∂p_{i}∂p_{j}} + (i + j)p_{i}p_{j} \frac{∂R^{•}}{∂p_{i+j}} ) . (5)

Here is a crucial observation: One can rewrite (5) as a sequence systems of ordinary equations, one for each positive integer d, hence if C(λ; τ; p)^{•} satisfies (5), then it is determined by the initial value C(λ; 0; p)^{•}. To prove (1) or (2), it suffices to prove the following two statements:

(a) Equation (4) is satisfied by C(λ; τ; p).

(b) C(λ; 0; p) = R(λ; 0; p).

Or equivalently,

(a′) Equation (5) is satisfied by C(λ; τ; p)^{•}.

(b′) C(λ; 0; p)^{•} = R(λ; 0; p)^{•}.

The generating function C(λ; 0; p) of Hodge integrals has a closed formula [4, Theorem 2]. This closed formula is shown to be equal to R(λ; 0; p) in [28]. Therefore, the Mariño-Vafa formula (1) follows from the following theorem.

Theorem 2.

(6) \frac{∂C}{∂τ} = \frac{\sqrt{−1}λ}{2} ∑_{i,j≥1} (ijp_{i+j} \frac{∂^{2}C}{∂p_{i}∂p_{j}}

+ ijp_{i+j} \frac{∂C}{∂p_{i}} \frac{∂C}{∂p_{j}} + (i + j)p_{i}p_{j} \frac{∂C}{∂p_{i+j}} ).

Our earlier paper [21] contains an essentially complete proof of the Mariño-Vafa formula based on the approach described above. The purpose of this paper is to supply various computational details, and to present some related results. See [25] for another approach to the Mariño-Vafa formula.

The rest of this paper is organized as follows. In Section 2, we give a proof of the initial condition (b). In Section 3, we give a proof of Theorem 1. The materials in these two sections are already contained in [28] and some in [21], and included here for the convenience of the readers. In Section 4, we recall the moduli spaces of relative stable morphisms and obstruction bundles which will be used in the proof of Theorem 2. In Section 5, we use the graph notation to describe the torus fixed points in the moduli spaces introduced in Section 4. In Section 6, we prove Theorem 2 by functorial virtual localization. Details of virtual localization are given in Appendix A. In Section 7, we reproduce the cutand-join equation of Hurwitz numbers (proved in [7], [15], [11, Section 15.2]) by virtual localization.

2. Initial condition

We recall some notations for partitions. For a partition ν, let ν′ denote its transpose. Define

n(η) = ∑_{i}(i − 1)η_{i} = ∑_{i} (\begin{matrix}η′_{i} \\ 2 \end{matrix}) . (7)

The hook length of ν at the square x located at the i-th row and j-th column is defined to be:

h(x) = ν_{i} + ν′_{j} − i − j + 1.

294 c.-c.m. liu, k. liu & j. zhou

Then one has the following two identities [22, pp. 10 -11]:

(8)  ∏_{x∈ν}(1 − t^{h(x)}) = \frac{∏_{i=1}^{l(ν)} ∏_{j=1}^{ν_{i}−i+l(ν)}(1 − t^{j})}{∏_{i<j}(1 − t^{ν_{i}−ν_{j}−i+j})},
(9)  ∑_{x∈ν} h(x) = n(ν) + n(ν′) + |ν|.

We first obtain a simple expression for V_{ν}(λ). Proposition 2.1.

V_{ν}(λ) = \frac{1}{2^{l} ∏_{x∈ν} sin[h(x)λ/2]}.

Proof. We rewrite the right-hand side of (8) as follows.

RHS = \frac{∏_{i=1}^{l(ν)} ∏_{j=1}^{−i+l(ν)}(1 − t^{j})}{∏_{i<j}(1 − t^{ν_{i}−ν_{j}−i+j})} · ∏_{i=1}^{l(ν)} ∏_{j=1−i+l(ν)}^{ν_{i}−i+l(ν)}(1 − t^{j})

= \frac{∏_{i<j}(1 − t^{j−i})}{∏_{i<j}(1 − t^{ν_{i}−ν_{j}−i+j})} · ∏_{i=1}^{l(ν)} ∏_{j=1}^{ν_{i}}(1 − t^{j−i+l(ν)})

= t^{(∑_{i<j}(j−i)−∑_{i<j}(ν_{i}−ν_{j}−i+j)+∑_{i=1}^{l(ν)} ∑_{j=1}^{ν_{i}}(j−i+l(ν)))/2}

· \frac{∏_{i<j}(t^{−(j−i)/2} − t^{(j−i)/2})}{∏_{i<j}(t^{−(ν_{i}−ν_{j}−i+j)/2} − t^{(ν_{i}−ν_{j}−i+j)/2})}

· ∏_{i=1}^{l(ν)} ∏_{j=1}^{ν_{i}}(t^{−(j−i+l(ν))/2} − t^{(j−i+l(ν))/2}).

Now

∑_{i<j}(j − i) − ∑_{i<j}(ν_{i} − ν_{j} − i + j) + ∑_{i=1}^{l(ν)} ∑_{j=1}^{ν_{i}}(j − i + l(ν))

= − ∑_{i<j} ν_{i} + ∑_{i<j} ν_{j} + ∑_{i=1}^{l(ν)} ∑_{j=1}^{ν_{i}} j − ∑_{i=1}^{l(ν)} ∑_{j=1}^{ν_{i}} i + ∑_{i=1}^{l(ν)} ∑_{j=1}^{ν_{i}} l(ν)

= − ∑_{i=1}^{l(ν)}(l(ν) − i)ν_{i} + ∑_{j=1}^{l(ν)}(j − 1)ν_{j} + ∑_{i=1}^{l(ν)} \frac{ν_{i}(ν_{i} + 1)}{2} − ∑_{i=1}^{l(ν)} iν_{i} + |ν|l(ν)

a proof of a conjecture of mariño-vafa 295

= −|ν|l(ν) + ∑_{i=1}^{l(ν)} iν_{i}

+ ∑_{j=1}^{l(ν)}(j − 1)ν_{j} + ∑_{i=1}^{l(ν)} \frac{ν_{i}(ν_{i} − 1)}{2} + |ν| − ∑_{i=1}^{l(ν)} iν_{i} + |ν|l(ν)

= ∑_{j=1}^{l(ν)}(j − 1)ν_{j} + ∑_{i=1}^{l(ν)} \frac{ν_{i}(ν_{i} − 1)}{2} + |ν|

= n(ν) + n(ν′) + |ν|

= ∑_{x∈ν} h(x).

Comparing with the left-hand side, one then gets:

∏_{x∈ν}(t^{−h(x)/2} − t^{h(x)/2})

= \frac{∏_{i<j}(t^{−(j−i)/2} − t^{(j−i)/2})}{∏_{i<j}(t^{−(ν_{i}−ν_{j}−i+j)/2} − t^{(ν_{i}−ν_{j}−i+j)/2})}

· ∏_{i=1}^{l(ν)} ∏_{j=1}^{ν_{i}}(t^{−(j−i+l(ν))/2} − t^{(j−i+l(ν))/2}).

The proof is completed by taking t = e^{−\sqrt{−1}λ}. q.e.d. We next compute the initial values C(λ; 0; p). Proposition 2.2.

C(λ; 0; p) = − ∑_{d>0} \frac{\sqrt{−1}^{d+1}p_{d}}{2d sin(dλ/2)}.

Proof. When l(μ) > 1, we clearly have C_{μ}(λ; 0) = 0.

296 c.-c.m. liu, k. liu & j. zhou

When μ = (d) we have

C_{(d)}(λ; 0) = − ∑_{g≥0} λ^{2g−1}\sqrt{−1}^{d+1} \frac{∏_{a=1}^{d−1}(d · 0 + a)}{(d − 1)!}

· ∫_{\overline{M}_{g,1}} \frac{Λ_{g}^{∨}(1)Λ_{g}^{∨}(0)Λ_{g}^{∨}(−1)}{1 − dψ_{1}}

= −\frac{\sqrt{−1}^{d+1}}{d} ((dλ)^{−1} + ∑_{g≥1}(dλ)^{2g−1} ∫_{\overline{M}_{g,1}} λ_{g}ψ_{1}^{2g−2})

= −\frac{\sqrt{−1}^{d+1}}{d^{2}λ} · \frac{dλ/2}{sin(dλ/2)}

= −\frac{\sqrt{−1}^{d+1}}{2d sin(dλ/2)}.

In the second equality we have used the Mumford’s relations [24, 5.4]:

Λ_{g}^{∨}(1)Λ_{g}^{∨}(−1) = (−1)^{g}.

In the third equality we have used [4, Theorem 2]. This completes the proof. q.e.d.

Proposition 2.3. We have the following identity:

log(∑_{n≥0} ∑_{|ρ|=n} ∑_{|η|=|ρ|} \frac{e^{\frac{1}{4} κ_{ρ}\sqrt{−1}λ}}{∏_{e∈ρ} 2 sin(h(e)λ/2)} \frac{χ_{ρ}(η)}{z_{η}}p_{η})= −∑_{d≥1} \frac{\sqrt{−1}^{d+1}p_{d}}{2d sin(dλ/2)}.

To prove Proposition 2.3, we need the following two lemmata.

Lemma 2.4. Introducing formal variables x_{1}, …, x_{n}, … such that

p_{i}(x_{1}, …, x_{n}, …) = x_{1}^{i} + ⋯ + x_{n}^{i} + · · · .

Then we have

∑_{n≥0} t^{n} ∑_{|ρ|=n} ∑_{|η|=|ρ|} \frac{q^{n(ρ)}}{∏_{e∈ρ}(1 − q^{h(e)})} \frac{χ_{ρ}(η)}{z_{η}}p_{η} = \frac{1}{∏_{i,j}(1 − tx_{i}q^{j−1})}. (10)

a proof of a conjecture of mariño-vafa 297

Proof. Recall the following facts about Schur polynomials:

(11)  s_{ρ}(x) = ∑_{|η|=|ρ|} \frac{χ_{ρ}(η)}{z_{η}}p_{η}(x),
(12)  s_{ρ}(1, q, q^{2}, …) = \frac{q^{n(ρ)}}{∏_{e∈ρ}(1 − q^{h(e)})},
(13)  ∑_{n≥0} t^{n} ∑_{|ρ|=n} s_{ρ}(x)s_{ρ}(y) = \frac{1}{∏_{i,j}(1 − tx_{i}y_{j})}.

Combining the last two identities, one gets:

∑_{n≥0} t^{n} ∑_{|ρ|=n} \frac{q^{n(ρ)}}{∏_{e∈ρ}(1 − q^{h(e)})}s_{ρ}(x) = \frac{1}{∏_{i,j}(1 − tx_{i}q^{j−1})}.

The proof is completed by (11). q.e.d.

Lemma 2.5. For any partition ρ we have

\frac{1}{2} ∑_{e∈ρ} h(e) − n(ρ) = \frac{1}{4}κ_{ρ} + \frac{1}{2}|ρ|. (14)

Proof.

\frac{1}{2} ∑_{e∈ρ} h(e) − n(ρ)

= \frac{1}{2}(n(ρ′) − n(ρ) + |ρ|)

= \frac{1}{2} (∑_{i} (\begin{matrix}ρ_{i} \\ 2 \end{matrix}) − ∑_{i}(i − 1)ρ_{i} + |ρ|)

= \frac{1}{4} (∑_{i} ρ_{i}(ρ_{i} − 1) − 2 ∑_{i} iρ_{i} + 4|ρ|)

= \frac{1}{4}κ_{ρ} + \frac{1}{2}|ρ|.

q.e.d.

298 c.-c.m. liu, k. liu & j. zhou

Proof of Proposition 2.3. Let q = e^{−\sqrt{−1}λ}, and t = \sqrt{−1}q^{1/2}. Then

∑_{n≥0} t^{n} ∑_{|ρ|=n} ∑_{|η|=|ρ|} \frac{q^{n(ρ)}}{∏_{e∈ρ}(1 − q^{h(e)})} \frac{χ_{ρ}(η)}{z_{η}}p_{η}

= ∑_{n≥0} \sqrt{−1}^{n}q^{n/2} ∑_{|ρ|=n} ∑_{|η|=|ρ|} \frac{q^{n(ρ)− \frac{1}{2} ∑_{e∈ρ} h(e)}}{∏_{e∈ρ}(q^{−h(e)/2} − q^{h(e)/2})} \frac{χ_{ρ}(η)}{z_{η}}p_{η}

= ∑_{n≥0} \sqrt{−1}^{n}q^{n/2} ∑_{|ρ|=n} ∑_{|η|=|ρ|} \frac{q^{− \frac{1}{4} κ_{ρ}− \frac{1}{2} n}}{∏_{e∈ρ}(q^{−h(e)/2} − q^{h(e)/2})} \frac{χ_{ρ}(η)}{z_{η}}p_{η}

= ∑_{n≥0} ∑_{|ρ|=n} ∑_{|η|=|ρ|} \frac{e^{\frac{1}{4} κ_{ρ}\sqrt{−1}λ}}{∏_{e∈ρ} 2 sin(h(e)λ/2)} \frac{χ_{ρ}(η)}{z_{η}}p_{η}.

Hence by (10),

log (∑_{n≥0} ∑_{|ρ|=n} ∑_{|η|=|ρ|} \frac{e^{\frac{1}{4} κ_{ρ}\sqrt{−1}λ}}{∏_{e∈ρ} 2 sin(h(e)λ/2)} \frac{χ_{ρ}(η)}{z_{η}}p_{η})

= log (∑_{n≥0} t^{n} ∑_{|ρ|=n} ∑_{|η|=|ρ|} \frac{q^{n(ρ)}}{∏_{e∈ρ}(1 − q^{h(e)})} \frac{χ_{ρ}(η)}{z_{η}}p_{η})

= log \frac{1}{∏_{i,j}(1 − tx_{i}q^{j−1})} = ∑_{i,j≥1} ∑_{d≥1} \frac{1}{d}t^{d}q^{d(j−1)}x_{i}^{d}

= ∑_{j≥1} ∑_{d≥1} \frac{1}{d}t^{d}q^{d(j−1)}p_{d} = ∑_{d≥1} \frac{p_{d}}{d} \frac{t^{d}}{1 − q^{d}}

= − ∑_{d≥0} \frac{\sqrt{−1}^{d+1}p_{d}}{2d sin(dλ/2)}.

q.e.d.

By Proposition 2.1 and Proposition 2.3, we have

R(λ; 0; p) = log (∑_{n≥0} ∑_{|ρ|=n} ∑_{|η|=|ρ|} \frac{χ_{ρ}(η)}{z_{η}}e^{\frac{1}{4} κ_{ρ}\sqrt{−1}λ}V_{ρ}(λ)p_{η})

= log (∑_{n≥0} ∑_{|ρ|=n} ∑_{|η|=|ρ|} \frac{e^{\frac{1}{4} κ_{ρ}\sqrt{−1}λ}}{∏_{e∈ρ} 2 sin(h(e)λ/2)} \frac{χ_{ρ}(η)}{z_{η}}p_{η})

= − ∑_{d≥1} \frac{\sqrt{−1}^{d+1}p_{d}}{2d sin(dλ/2)}.

By Proposition 2.2, we have

C(λ; 0; p) = − ∑_{d≥1} \frac{\sqrt{−1}^{d+1}p_{d}}{2d sin(dλ/2)}.

So the initial condition (b) holds.

3. Proof of Theorem 1

Let μ, η be two partitions, both represented by Young diagrams. We write η ∈ J_{i,j}(μ) and μ ∈ C_{i,j}(η) if η is obtained from μ by removing a row of length i and a row of length j, then adding a row of length i + j. It is easy to see that

\frac{m_{i+j}(η)}{∏_{k} m_{k}(η)!} = \begin{cases}\frac{m_{i}(μ)m_{j}(μ)}{∏_{k} m_{k}(μ)!}, & i ≠ j, \\ \frac{m_{i}(μ)(m_{i}(μ)−1)}{∏_{k} m_{k}(μ)!}, & i = j. \end{cases} (15)

Recall

c_{μ} = ∑_{g∈C_{μ}} g

lies in the center of the group algebra ℂS_{d}, hence it acts as a scalar f_{ν}(μ) on any irreducible representation R_{ν}. In other words, let ρ : S_{d} → End R_{ν} be the representation indexed by ν, then

∑_{g∈C(μ)} ρ_{ν}(g) = f_{ν}(μ) id .

We need the following interpretation of κ_{ν} in terms of character.

Lemma 3.1. We have

κ_{ν} = 2f_{ν}(C(2)),

where we use C(2) to denote the class of transpositions.

Proof. By [22, p. 118, Example 7],

f_{ν}(C(2)) = |C(2)|\frac{χ_{ν}(C(2))}{dim R_{ν}} = n(ν′) − n(ν)

= ∑_{i=1}^{l(ν)} (\begin{matrix}ν_{i} \\ 2 \end{matrix}) − ∑_{i=1}^{l(ν)}(i − 1)ν_{i}

= \frac{1}{2} ∑_{i=1}^{l(ν)}(ν_{i}^{2} − 2iν_{i} + ν_{i}) = \frac{1}{2}κ_{ν}.

In the above we have used (7). q.e.d.

We need the following result:

Lemma 3.2. Suppose h ∈ S_{d} has cycle type μ. The product c_{(2)} · h is a sum of elements of S_{d} whose type is either a cut or a join of μ. More precisely, there are ijm_{i}(μ)m_{j}(μ) (when i < j) or i^{2}m_{i}(μ)(m_{i}(μ) − 1)/2 (when i = j) elements obtained from h by joining an i-cycle in h to a j-cycle in h, and there are (i + j)m_{i+j}(μ) (when i < j) or im_{2i}(μ) (when i = j) elements obtained from h by cutting an (i + j)-cycle into an i-cycle and a j-cycle.

Proof. Denote by [s_{1}, …, s_{k}] a k-cycle. Then

[s, t] · [s, s_{2}, …, s_{i}, t, t_{2}, …, t_{j}] = [s, s_{2}, …, s_{i}][t, t_{2}, …, t_{j}],

i.e., an i + j-cycle is cut into an i-cycle and a j-cycle. Conversely,

[s, t] · [s, s_{2}, …, s_{i}][t, t_{2}, …, t_{j}] = [s, s_{2}, …, s_{i}, t, t_{2}, …, t_{j}],

i.e., an i-cycle and a j-cycle is joined to an i + j-cycle. Hence, for a permutation h of type μ, c_{(2)} · h is a sum of all elements obtained from h by either a cut or a join. Fix a pair of i-cycle and j-cycle of h, there are i · j different ways to join them to an (i + j)-cycle. Taking into the account of m_{i}(μ) choices of i-cycles, and m_{j}(μ) choices of j-cycles, we get the number of different ways to obtain an element from h by joining an i-cycle in h to a j-cycle in h is

\begin{cases}ijm_{i}(μ)m_{j}(μ), & i < j \\ i^{2}m_{i}(μ)(m_{i}(μ) − 1)/2, & i = j. \end{cases}

Similarly, fix an (i + j)-cycle of h, there are i + j different ways to cut it into an i-cycle and a disjoint j-cycle when i < j. When i = j, there are only i different ways to cut it into two i-cycles. And taking into account the number of (i + j)-cycles in h, we get the number of different ways to obtain an element from h by cutting an (i + j)-cycle into an i-cycle and a j-cycle is

\begin{cases}(i + j)m_{i+j}(μ), & i < j, \\ im_{2i}(μ), & i = j. \end{cases}

q.e.d.

For any h ∈ S_{d} of cycle type μ we have

∑_{μ} f_{ν}(2)\frac{χ_{ν}(μ)}{z_{μ}}p_{μ}

= ∑_{μ} tr[f_{ν}(2) id ·ρ_{ν}(h)] · ∏_{i} \frac{p_{i}^{m_{i}(μ)}}{i^{m_{i}(μ)}m_{i}(μ)!}

= ∑_{μ} tr [ ∑_{g∈C(2)} ρ_{ν}(g) · ρ_{ν}(h)] · ∏_{i} \frac{p_{i}^{m_{i}(μ)}}{i^{m_{i}(μ)}m_{i}(μ)!}

= ∑_{μ} tr ρ_{ν} ( ∑_{g∈C(2)} g · h) · ∏_{i} \frac{p_{i}^{m_{i}(μ)}}{i^{m_{i}(μ)}m_{i}(μ)!}

= ∑_{μ} (∑_{i<j} ( ∑_{η∈J_{i,j}(μ)} ijm_{i}(μ)m_{j}(μ)χ_{ν}(η)

+ ∑_{η∈C_{i,j}(μ)}(i + j)m_{i+j}(μ)χ_{ν}(η))

+ ∑_{i} ( ∑_{η∈J_{i,i}(μ)} \frac{1}{2}i^{2}m_{i}(μ)(m_{i}(μ) − 1)χ_{ν}(η)

+ ∑_{η∈C_{i,i}(μ)} im_{2i}(μ)χ_{ν}(η))) ∏_{i} \frac{p_{i}^{m_{i}(μ)}}{i^{m_{i}(μ)}m_{i}(μ)!}

= \frac{1}{2} ∑_{i,j} ((i + j)p_{i}p_{j} \frac{∂}{∂p_{i+j}} + ijp_{i+j} \frac{∂}{∂p_{i}} \frac{∂}{∂p_{j}} ) ∑_{η} \frac{χ_{ν}(η)}{z_{η}}p_{η}.

In the last equality we have used (15). It follows that

\frac{∂R(λ; τ; p)^{•}}{∂τ}

= \frac{\sqrt{−1}λ}{2} ∑_{μ,ν} (f_{ν}(2)\frac{χ_{ν}(C(μ))}{z_{μ}}p_{μ}) e^{\sqrt{−1}(τ+ \frac{1}{2} )κ_{ν}λ/2}V_{ν}(λ)

= \frac{\sqrt{−1}λ}{2} (ijp_{i+j} \frac{∂}{∂p_{i}} \frac{∂}{∂p_{j}} + (i + j)p_{i}p_{j} \frac{∂}{∂p_{i+j}} )

· ∑_{η} \frac{χ_{ν}(η)}{z_{η}}p_{η}e^{\sqrt{−1}(τ+ \frac{1}{2} )κ_{ν}λ/2}V_{ν}(λ).

This finishes the proof of Theorem 1.

4. Moduli spaces of relative stable morphisms

In this section, we introduce the geometric objects involved in the proof of Theorem 2.

4.1 Moduli space of relative morphisms

We first describe the moduli space of relative stable morphisms to ℙ^{1} used in [19]. The moduli spaces of algebraic relative stable morphisms are constructed by J. Li [16].

We introduce some notations. For any nonnegative integer m, let

ℙ^{1}[m] = ℙ_{(0)}^{1} ∪ ℙ_{(1)}^{1} ∪ ⋯ ∪ ℙ_{(m)}^{1}

be a chain of m + 1 copies ℙ^{1}, where ℙ_{(l)}^{1} is glued to ℙ_{(l+1)}^{1} at p_{1}^{(l)} for 0 ≤ l ≤ m − 1. The irreducible component ℙ_{(0)}^{1} will be referred to as the root component, and the other irreducible components will be called the bubble components. A point p_{1}^{(m)} ≠ p_{1}^{(m−1)} is fixed on ℙ_{(m)}^{1}. Denote by π[m] : ℙ^{1}[m] → ℙ^{1} the map which is identity on the root component and contracts all the bubble components to p_{1}^{(0)}. For m > 0, let

ℙ^{1}(m) = ℙ_{(1)}^{1} ∪ ⋯ ∪ ℙ_{(m)}^{1}

denote the union of bubble components of ℙ^{1}[m].

Let μ be a partition of d > 0. Let \overline{M}_{g,0}(ℙ^{1}, μ) be the moduli space of morphisms

f : (C, x_{1}, …, x_{l(μ)}) → (ℙ^{1}[m], p_{1}^{(m)}),

such that:

1. (C, x_{1}, …, x_{l(μ)}) is a prestable curve of genus g with l(μ) marked points. For convenience, we assume the marked points are unordered.

2. f^{−1}(p_{1}^{(m)}) = ∑_{i=1}^{l(μ)} μ_{i}x_{i} as Cartier divisors, and deg(π[m] ∘ f) = d.

3. The preimage of each node in ℙ^{1}[m] consists of nodes of C. If f(y) = p_{1}^{(l)} and C_{1} and C_{2} are two irreducible components of C which intersect at y, then f|_{C_{1}} and f|_{C_{2}} have the same contact order to p_{1}^{(l)} at y.

4. The automorphism group of f is finite.

Two such morphisms are isomorphic if they differ by an isomorphism of the domain and an automorphism of the pointed curve (ℙ^{1}(m), p_{1}^{(0)}, p_{1}^{(m)}). In particular, this defines the automorphism group in the stability condition (4) above.

In [16, 17], J. Li showed that \overline{M}_{g,0}(ℙ^{1}, μ) is a separated, proper Deligne-Mumford stack with a perfect obstruction theory of virtual dimension

r = 2g − 2 + d + l(μ),

so it has a virtual fundamental class of degree r.

4.2 Torus action

Consider the ℂ^{∗}-action

t · [z^{0} : z^{1}] = [tz^{0} : z^{1}]

on ℙ^{1}. It has two fixed points p_{0} = [0 : 1] and p_{1} = [1 : 0]. This induces an action on ℙ^{1}[m] by the action on the root component induced by the isomorphism to ℙ^{1}, and the trivial actions on the bubble components. This in turn induces an action on \overline{M}_{g,0}(ℙ^{1}, μ).

4.3 The branch morphism

There is a branch morphism

Br : \overline{M}_{g,0}(ℙ^{1}, μ) → Sym^{r}ℙ^{1} ≅ ℙ^{r}.

Note that ℙ^{r} can be identified with ℙ(H^{0}(ℙ^{1}, O(r)), and the isomorphism

ℙ(H^{0}(ℙ^{1}, O(r)) ≅ Sym^{r}ℙ^{1}

is given by [s] ↦ div(s). The ℂ^{∗}-action on ℙ^{1} induces a ℂ^{∗}-action on H^{0}(ℙ^{1}, O(r)) by

t · (z^{0})^{k}(z^{1})^{r−k} = t^{−k}(z^{0})^{k}(z^{1})^{r−k}.

So ℂ^{∗} acts on ℙ^{r} by

t · [a_{0} : a_{1} : ⋯: a_{r}] = [a_{0} : t^{−1}a_{1} : ⋯: t^{−r}a_{r}],

where [a_{0} : a_{1} : ⋯: a_{r}] corresponds to ∑_{k=0}^{r} a_{k}(z^{0})^{k} (z^{1})^{r−k} ∈ H^{0}(ℙ^{1}, O(r)). With this action, the branch morphism is ℂ^{∗}-equivariant. See [6, 9] for more details.

The ℂ^{∗}-action on ℙ^{r} has r + 1 fixed points p_{0}, …, p_{r}, where p_{k} ∈ ℙ^{r} corresponds to the complex line ℂ(z^{0})^{k}(z^{1})^{r−k} ⊂ H^{0}(ℙ^{1}, O(r)).

4.4 The obstruction bundle

In [19], J. Li and Y. Song constructed an obstruction bundle over the stratum where the target is ℙ^{1}[0] = ℙ^{1}, and proposed an extension over the entire \overline{M}_{g,0}(ℙ^{1}, μ). Here we use a different extension which is defined in [1, Section 3].

Let

π : U_{g,μ} → \overline{M}_{g,0}(ℙ^{1}, μ)

be the universal domain curve, and let

P : T_{g,μ} → \overline{M}_{g,0}(ℙ^{1}, μ)

be the universal target. There is an evaluation map

F : U_{g,μ} → T_{g,μ}

and a contraction map

π̃ : T_{g,μ} → ℙ^{1}.

Let D_{g,μ} ⊂ U_{g,μ} be the divisor corresponding to the l(μ) marked points. Define

V_{D} = R^{1}π_{∗}(O_{U_{g,μ}}(−D_{g,μ}))
V_{D_{d}} = R^{1}π_{∗}F̃^{∗}O_{ℙ^{1}}(−1),

where F̃ = π̃ ∘ F : U_{g,μ} → ℙ^{1}. The fibers of V_{D} and V_{D_{d}} at

[f : (C, x_{1}, …, x_{l(μ)}) → ℙ^{1}[m] ] ∈ \overline{M}_{g,0}(ℙ^{1}, μ)

are H^{1}(C, O_{C}(−D)) and H^{1}(C, f̃^{∗}O_{ℙ^{1}}(−1)), respectively, where D = x_{1} + ⋯ + x_{l(μ)}, and f̃ = π[m] ∘ f. Note that

H^{0}(C, O_{C}(−D)) = H^{0}(C, f̃^{∗}O_{ℙ^{1}}(−1)) = 0,

so V_{D} and V_{D_{d}} are vector bundles of ranks l(μ) + g − 1 and d + g − 1, respectively. The obstruction bundle

V = V_{D} ⊕ V_{D_{d}}

is a vector bundle of rank r = 2g − 2 + d + l(μ).

We lift the ℂ^{∗}-action on \overline{M}_{g,0}(ℙ^{1}, μ) to V_{D} and V_{D_{d}} as follows. The action on V_{D_{d}} comes from an action on O_{ℙ^{1}}(−1) → ℙ^{1} with weights −τ − 1 and −τ at the two fixed points p_{0} and p_{1}, respectively, where τ ∈ ℤ. The fiber of V_{D} does not depend on the map f, so the fibers over two points in the same orbit of the ℂ^{∗}-action can be canonically identified. The action of λ ∈ ℂ^{∗} on V_{D} is multiplication by λ^{τ}.

5. Fixed points of torus action

5.1 Graph notation

Similar to the case of \overline{M}_{g,0}(ℙ^{1}, d), the connected components of the ℂ^{∗} fixed points set \overline{M}_{g,0}(ℙ^{1}, μ)^{ℂ^{∗}} are parameterized by labeled graphs.

Given a morphism

f : (C, x_{1}, …, x_{l(μ)}) → ℙ^{1}[m]

which represents a fixed point of the ℂ^{∗}-action on \overline{M}_{g,0}(ℙ^{1}, μ), let

f̃ = π[m] ∘ f : C → ℙ^{1}.

The restriction of f̃to an irreducible component of C is either a constant map to one of the ℂ^{∗} fixed points p_{0}, p_{1} or a cover of ℙ^{1} which is fully ramified over p_{0} and p_{1}. We associate a labeled graph Γ to the ℂ^{∗} fixed point

[f : (C, x_{1}, …, x_{l(μ)}) → ℙ^{1}[m] ]

as follows:

1. We assign a vertex v to each connected component C_{v} of f̃^{−1}({p_{0}, p_{1}}), a label i(v) = i if f̃(C_{v}) = p_{i}, where i = 0, 1, and a label g(v) which is the arithmetic genus of C_{v} (We define g(v) = 0 if C_{v} is a point). Denote by V (Γ)^{(i)} the set of vertices with i(v) = i, where i = 0, 1. Then the set V (Γ) of vertices of the graph Γ is a disjoint union of V (Γ)^{(0)} and V (Γ)^{(1)}.

2. We assign an edge e to each rational irreducible component C_{e} of C such that f̃|_{C_{e}} is not a constant map. Let d(e) be the degree of f̃|_{C_{e}}. Then f̃|_{C_{e}} is fully ramified over p_{0} and p_{1}. Let E(Γ) denote the set of edges of Γ.

3. The set of flags of Γ is given by

F (Γ) = {(v, e) : v ∈ V (Γ), e ∈ E(Γ), C_{v} ∩ C_{e} ≠ ∅}.

4. For each v ∈ V (Γ), define

d(v) = ∑_{(v,e)∈F (Γ)} d(e),

and let ν(v) be the partition of d(v) determined by {d(e) : (v, e) ∈ F (Γ)}. When the target is ℙ^{1}[m], where m > 0, we assign an additional label for each v ∈ V (Γ)^{(1)}: let μ(v) be the partition of d(v) determined by the ramification of f|_{C_{v}} : C_{v} → ℙ^{1}(m) over p_{1}^{(m)}.

Note that for v ∈ V (Γ)^{(1)}, ν(v) coincides with the partition of d(v) determined by the ramification of f|_{C_{v}} : C_{v} → ℙ^{1}(m) over p_{1}^{(0)}.

5.2 Fixed points

Let G_{g,0}(ℙ^{1}, μ) be the set of all the graphs associated to the ℂ^{∗} fixed points in \overline{M}_{g,0}(ℙ^{1}, μ), as described in Section 5.1. In this section, we describe the set of fixed points associated to a given graph Γ ∈ G_{g,0}(ℙ^{1}, μ).

5.2.1 The target is ℙ^{1}

Any ℂ^{∗} fixed point in \overline{M}_{g,0}(ℙ^{1}, μ) which is represented by a morphism to ℙ^{1} is associated to the graph Γ^{0}, where

V (Γ^{0})^{(0)} = {v_{0}}, V (Γ^{0})^{(1)} = {v_{1}, …, v_{l(μ)}}, E(Γ^{0}) = {e_{1}, …, e_{l(μ)}}, and

g(v) = g, g(v_{i}) = 0, d(e_{i}) = μ_{i}

for i = 1, …, l(μ). The two end points of the edge e_{i} are v_{0} and v_{i}. Let Aut(μ) denote the automorphism group of the partition μ of d. Any morphism associated to the graph Γ^{0} has automorphism group A_{Γ^{0}}, where

1 → ∏_{i=1}^{l(μ)} ℤ_{μ_{i}} → A_{Γ^{0}} → Aut(μ) → 1.

Let

\overline{M}_{Γ^{0}} = \begin{cases}\{point\}, & (g, l(μ)) = (0, 1), (0, 2), \\ \overline{M}_{g,l(μ)}, & (g, l(μ)) ≠ (0, 1), (0, 2). \end{cases}

There is a morphism

i_{Γ^{0}} : \overline{M}_{Γ^{0}} → \overline{M}_{g,0}(ℙ^{1}, μ)

whose image is the fixed locus F_{Γ^{0}} associated to Γ_{0}. The morphism i_{Γ_{0}} induces an isomorphism

\overline{M}_{Γ^{0}}/A_{Γ^{0}} ≅ F_{Γ^{0}}.

The dimension of F_{Γ^{0}} is

d_{Γ^{0}} = \{ \begin{matrix}0, & (g, l(μ)) = (0, 1), (0, 2), \\ 3g − 3 + l(μ), & (g, l(μ)) ≠ (0, 1), (0, 2). \end{matrix}

5.2.2 The target is ℙ^{1}[m], m > 0

Let Γ ∈ G_{g,0}(ℙ^{1}, μ) be a graph associated to a ℂ^{∗} fixed point represented by some morphism to ℙ^{1}[m], m > 0. Let \overline{M}_{Γ}^{(1)} denote the moduli space of morphisms

f̂: Ĉ → ℙ^{1}(m)

such that:

(a) Ĉ is the disjoint union of {C_{v} : v ∈ V (Γ)^{(1)}}.

(b) (C_{v}, x_{v,1}, …, x_{v,l(μ(v))}, y_{v,1}, …, y_{v,l(ν(v))}) is a prestable curve of genus g(v) with l(μ(v)) + l(ν(v)) marked points. Here the marked points are ordered.

(c) As Cartier divisors,

(f̂|_{C_{v}})^{−1}(p_{1}^{(0)}) = ∑_{i=1}^{l(μ(v))} μ(v)_{i} x_{v,i},

(f̂|_{C_{v}})^{−1}(p_{1}^{(m)}) = ∑_{j=1}^{l(ν(v))} ν(v)_{i} y_{v,j}.

The morphism (f̂|_{C_{v}})^{−1}(B) → B is of degree d(v) for each irreducible component B of ℙ^{1}(m).

(d) The automorphism group of f̂ is finite.

Two such morphisms are isomorphic if they differ by an isomorphism of the domain and an automorphism of the pointed curve (ℙ^{1}(m), p_{1}^{(0)}, p_{1}^{(m)}), which is an element of (ℂ^{∗})^{m}. In particular, this defines the automorphism group in the stability condition (d) above.

The moduli space \overline{M}_{Γ}^{(1)} is a variant of J. Li’s moduli spaces of stable relative morphisms [16, 17]. It is a separated, proper Deligne-Mumford stack with a perfect obstruction theory.

Given

f : (C, x_{1}, …, x_{l(μ)}) → ℙ^{1}[m]

associated to the graph Γ, let Ĉ be the disjoint union of {C_{v} : v ∈ V (Γ)^{(1)}}. Let f̂ be the restriction of f to Ĉ. Then

f̂: Ĉ → ℙ^{1}(m)

represents a point in \overline{M}_{Γ}^{(1)}.

Define

r_{0}(v) = 2g(v) − 2 + val(v),  v ∈ V (Γ)^{(0)},
r_{1}(v) = 2g(v) − 2 + l(μ(v)) + l(ν(v)), v ∈ V (Γ)^{(1)},

V^{I}(Γ)^{(0)} = {v ∈ V (Γ)^{(0)} : r_{0}(v) = −1},
V^{II}(Γ)^{(0)} = {v ∈ V (Γ)^{(0)} : r_{0}(v) = 0},
V^{S}(Γ)^{(0)} = {v ∈ V (Γ)^{(0)} : r_{0}(v) > 0},
V^{II}(Γ)^{(1)} = {v ∈ V (Γ)^{(1)} : r_{1}(v) = 0},
V^{S}(Γ)^{(1)} = {v ∈ V (Γ)^{(1)} : r_{1}(v) > 0}.

Note that V^{S}(Γ)^{(1)} ≠ ∅ by the stability condition (d).

Let Aut(Γ) denote the automorphism of the labeled graph Γ. The automorphism group of any morphism associated to the graph Γ is A_{Γ}, where

1 → ∏_{e∈E(Γ)} ℤ_{d(e)} → A_{Γ} → Aut(Γ) → 1.

Let \overline{M}_{Γ} = \overline{M}_{Γ}^{(0)} × \overline{M}_{Γ}^{(1)}, where

\overline{M}_{Γ}^{(0)} = ∏_{v∈V^{S}(Γ)^{(0)}} \overline{M}_{g(v),val(v)}.

There is a morphism

i_{Γ} : \overline{M}_{Γ} → \overline{M}_{g,0}(ℙ^{1}, μ)

whose image is the fixed locus F_{Γ} associated to the graph Γ. The morphism i_{Γ} induces an isomorphism \overline{M}_{Γ}/A_{Γ} ≅ F_{Γ}.

The dimension of \overline{M}_{Γ}^{(0)} is given by

d_{Γ}^{(0)} = ∑_{v∈V^{S}(Γ)^{(0)}}(3g(v) − 3 + val(v)),

and the virtual dimension of \overline{M}_{Γ}^{(1)} is given by

d_{Γ}^{(1)} = ( ∑_{v∈V (Γ)^{(1)}} r_{1}(v)) − 1.

So the virtual dimension of F_{Γ} is given by

d_{Γ} = d_{Γ}^{(0)} + d_{Γ}^{(1)}

= ∑_{v∈V^{S}(Γ)^{(0)}}(3g(v) − 3 + val(v)) + ∑_{v∈V (Γ)^{(1)}} r_{1}(v) − 1

= ∑_{v∈V (Γ)^{(0)}}(3g(v) − 3 + val(v)) + |V^{II}(Γ)^{(0)}| + 2|V^{I}(Γ)^{(0)}|

+ ∑_{v∈V (Γ)^{(1)}}(2g(v) − 2 + l(μ(v)) + l(ν(v))) − 1

= 3 ∑_{v∈V (Γ)^{(0)}} g(v) − 3|V (Γ)^{(0)}| + |E(Γ)| + |V^{II}(Γ)^{(0)}| + 2|V^{I}(Γ)^{(0)}|

+ 2 ∑_{v∈V (Γ)^{(1)}} g(v) − 2|V (Γ)^{(1)}| + l(μ) + |E(Γ)| − 1

= 2 ( ∑_{v∈V (Γ)} g(v) − |V (Γ)| + |E(Γ)| + 1) − 3 + l(μ)

+ ∑_{v∈V (Γ)^{(0)}}(g(v) − 1) + |V^{II}(Γ)^{0}| + 2|V^{I}(Γ)^{0}|

= 2g − 3 + l(μ) + ∑_{v∈V^{S}(Γ)^{(0)}}(g(v) − 1) + |V^{I}(Γ)^{0}|.

The last equality comes from the following identity:

g = ∑_{v∈V (Γ)} g(v) + b_{1}(Γ) = ∑_{v∈V (Γ)} g(v) − |V (Γ)| + |E(Γ)| + 1,

where b_{1}(Γ) is the first betti number of the graph Γ.

6. Proof of Theorem 2

6.1 Functorial localization

Let T = ℂ^{∗}. We have seen in Section 4.3 that the branch morphism

Br : \overline{M}_{g,0}(ℙ^{1}, μ) → ℙ^{r}

is T -equivariant. We will compute

Br_{∗}e_{T} (V) = ∑_{l=0}^{r} a_{l}(τ)H^{l}u^{r−l}.

by virtual functorial localization [20], where H ∈ H^{2}(ℙ^{r}; ℤ) is the hyperplane class, and a_{l}(τ) is a polynomial in τ. Recall that τ ∈ ℤ parametrizes torus actions on the obstruction bundle, as described in Section 4.4.

Let p_{0}, …, p_{r} ∈ ℙ^{r} be the torus fixed points defined as in Section 4.3, and let f_{k} : p_{k} → ℙ^{r} be the inclusion. From the torus action on ℙ^{r}

a proof of a conjecture of mariño-vafa 311

described in Section 4.3, one gets

\frac{f_{k}^{∗}Br_{∗}e_{T} (V)}{e_{T} (T_{p_{k}}ℙ^{r})} = \frac{F (τ, k)}{(−1)^{r−k}k!(r − k)!},

where

F (τ, x) = ∑_{l=0}^{r} a_{l}(τ)x^{l}.

By functorial localization, we have

∫_{p_{k}} \frac{f_{k}^{∗}Br_{∗}e_{T} (V)}{e_{T} (T_{p_{k}}ℙ^{r})} = ∑_{F_{Γ}⊂Br^{−1}(p_{k})} \frac{1}{|A_{Γ}|} ∫_{[\overline{M}_{Γ}]^{vir}} \frac{i_{Γ}^{∗}e_{T} (V)}{e_{T} (N_{Γ}^{vir})}

for k = 0, …, r, where N_{Γ}^{vir} → \overline{M}_{Γ} is the pull-back of the virtual normal bundle of F_{Γ} in \overline{M}_{g,0}(ℙ^{1}, μ). Note that Br(F_{Γ^{0}}) = p_{r}, and

Br(F_{Γ}) = p_{r−d_{Γ}^{(1)}−1}

for Γ ∈ G_{g,0}(ℙ^{1}, μ), Γ ≠ Γ^{0}. Recall that d_{Γ}^{(1)} is the virtual dimension of \overline{M}_{Γ}^{(1)}, and 0 ≤ d_{Γ}^{(1)} ≤ r − 1. So

F (τ, x) = ∑_{k=0}^{r} \frac{F (τ, k)}{(−1)^{r−k}k!(r − k)!}x(x − 1) ⋯
(x − k + 1)(x − k − 1) ⋯ (x − r)

= ∑_{k=0}^{r} I_{g,μ}^{r−k}(τ)x(x − 1) ⋯ (x − k + 1)(x − k − 1) ⋯ (x − r),

where

I_{g,μ}^{0}(τ) = \frac{1}{|A_{Γ^{0}}|} ∫_{\overline{M}_{Γ^{0}}} \frac{i_{Γ^{0}}^{∗}e_{T} (V)}{e_{T} (N_{Γ^{0}}^{vir})},

and

I_{g,μ}^{k}(τ) = ∑_{Γ∈G_{g,0}(ℙ^{1},μ),Γ≠Γ^{0},d_{Γ}+1=k} \frac{1}{|A_{Γ}|} ∫_{[\overline{M}_{Γ}]^{vir}} \frac{i_{Γ}^{∗}e_{T} (V)}{e_{T} (N_{Γ}^{vir})}

for k = 1, …, r.

6.2 Contribution from each graph

6.2.1 The target is ℙ^{1}

Consider the graph Γ^{0} ∈ G_{g,0}(ℙ^{1}, μ). We first consider the stable case, i.e., (g, l(μ)) ≠ (0, 1), (0, 2). Let d = |μ| as before. Using the Feynman rules derived in Appendix A, we obtain

I_{g,μ}^{0}(τ) = \frac{1}{|A_{Γ^{0}}|} ∫_{\overline{M}_{Γ^{0}}} \frac{i_{Γ^{0}}^{∗}e_{T} (V)}{e_{T} (N_{Γ^{0}}^{vir})}

= \frac{(−1)^{d−1}}{|Aut(μ)|}(τ(τ + 1))^{l(μ)−1} (∏_{i=1}^{l(μ)} \frac{∏_{a=1}^{μ_{i}−1}(μ_{i}τ + a)}{(μ_{i} − 1)!})

· ∫_{\overline{M}_{g,l(μ)}} \frac{Λ_{g}^{∨}(u)Λ_{g}^{∨}(τu)Λ_{g}^{∨}((−τ − 1)u)u^{2l(μ)−3}}{∏_{i=1}^{l(μ)}(u − μ_{i}ψ_{i})}

= \frac{(−1)^{d−1}}{|Aut(μ)|}(τ(τ + 1))^{l(μ)−1} (∏_{i=1}^{l(μ)} \frac{∏_{a=1}^{μ_{i}−1}(μ_{i}τ + a)}{(μ_{i} − 1)!})

· ∫_{\overline{M}_{g,l(μ)}} \frac{Λ_{g}^{∨}(1)Λ_{g}^{∨}(τ)Λ_{g}^{∨}(−τ − 1)}{∏_{i=1}^{l(μ)}(1 − μ_{i}ψ_{i})}

= \sqrt{−1}^{d−l(μ)}C_{g,μ}(τ).

In particular,

I_{0,μ}^{0}(τ) = \frac{(−1)^{d−1}}{|Aut(μ)|}(τ(τ + 1))^{l(μ)−1} (∏_{i=1}^{l(μ)} \frac{∏_{a=1}^{μ_{i}−1}(μ_{i}τ + a)}{(μ_{i} − 1)!})

· ∫_{\overline{M}_{0,l(μ)}} \frac{1}{∏_{i=1}^{l(μ)}(1 − μ_{i}ψ_{i})}

= \frac{(−1)^{d−1}}{|Aut(μ)|}(τ(τ + 1))^{l(μ)−1} (∏_{i=1}^{l(μ)} \frac{∏_{a=1}^{μ_{i}−1}(μ_{i}τ + a)}{(μ_{i} − 1)!}) d^{l(μ)−3}.

The contribution from Γ^{0} for the two unstable cases is also given by the above formula:

I_{0,(d)}^{0}(τ) = (−1)^{d−1} (\frac{∏_{a=1}^{d−1}(dτ + a)}{(d − 1)!}) \frac{1}{d^{2}} ,

I_{0,(μ_{1},μ_{2})}^{0}(τ) = \frac{(−1)^{d−1}}{|Aut((μ_{1}, μ_{2}))|}τ(τ + 1) (∏_{i=1}^{2} \frac{∏_{a=1}^{μ_{i}−1}(μ_{i}τ + a)}{(μ_{i} − 1)!}) \frac{1}{d}

Note that I_{g,μ}^{0}(τ) is a degree r = 2g − 2 + d + l(μ) polynomial in τ with rational coefficients, and

I_{g,μ}^{0}(−τ − 1) = (−1)^{d−l(μ)}I_{g,μ}^{0}(τ).

6.2.2 The target is ℙ^{1}[m], m > 0

Consider Γ ∈ G_{g,0}(ℙ^{1}, μ), Γ ≠ Γ_{0}. Using the Feynman rules derived in Appendix A, we obtain

I^{Γ} = \frac{1}{|A_{Γ}|} ∫_{[\overline{M}_{Γ}]^{vir}} \frac{i_{Γ}^{∗}e_{T} (V)}{e_{T} (N_{Γ}^{vir})} = \frac{1}{|A_{Γ}|} ∫_{[\overline{M}_{Γ}]^{vir}} \frac{∏_{v∈V (Γ)} B_{v}}{−u − ψ^{t}}

where

B_{v} = \begin{cases}A_{v}A_{v}^{V} ∏_{(v,e)∈F (Γ)}(A_{e}A_{e}^{V}), & v ∈ V (Γ)^{(0)}, \\ A_{v}A_{v}^{V} , & v ∈ V (Γ)^{(1)}. \end{cases}

More explicitly, in the notation of Appendix A,

B_{v} = ( ∏_{(v,e)∈F (Γ)} d(e)) (−1)^{d(v)−1}(τ(τ + 1))^{val(v)−1}

· ( ∏_{(v,e)∈F (Γ)} \frac{∏_{a=1}^{d(e)−1}(d(e)τ + a)}{(d(e) − 1)!})

· \frac{Λ_{g(v)}^{∨}(u)Λ_{g(v)}^{∨}(τu)Λ_{g(v)}^{∨}(−(τ + 1)u)u^{2 val(v)−3}}{∏_{(v,e)∈F (Γ)} (u − d(e)ψ_{(v,e)})} , v ∈ V^{S}(Γ)^{(0)},

d(v)(−1)^{d(v)−1} (\frac{∏_{a=1}^{d(v)−1}(d(v)τ + a)}{(d(e) − 1)!}) \frac{1}{d(v)^{2}} , v ∈ V^{I}(Γ)^{(0)},

d(e_{1})d(e_{2})(−1)^{d(v)−1}(τ(τ + 1)) (∏_{i=1}^{2} \frac{∏_{a=1}^{d(e_{i})−1}(d(e_{i})τ + a)}{(d(e_{i}) − 1)!}) \frac{1}{d(v)},

v ∈ V^{II}(Γ)^{(0)}, (v, e_{1}), (v, e_{2}) ∈ F (Γ),

(−1)^{g(v)+val(v)−1}(τu)^{r_{1}(v)} ∏_{(v,e)∈F (Γ)} d(e), v ∈ V (Γ)^{(1)}.

Recall that r_{1}(v) = 2g(v) − 2 + l(μ(v)) + val(v) for v ∈ V (Γ)^{(1)}, and

d_{Γ}^{(1)} = ( ∑_{v∈V (Γ)^{(1)}} r_{1}(v)) − 1

is the virtual dimension of \overline{M}_{Γ}^{(1)}.

We have

I^{Γ}(τ) = \frac{1}{|A_{Γ}|} ∏_{v∈V (Γ)^{(0)}} (|Aut(ν(v))|I_{g(v),ν(v)}^{0}(p)) ( ∏_{e∈E(Γ)} d(e))^{2}

· ∏_{v∈V (Γ)^{(1)}}(−1)^{g(v)+val(v)−1} ∫_{[\overline{M}_{Γ}^{(1)}]^{vir}} \frac{(τu)^{d_{Γ}^{(1)}+1}}{−u − ψ^{t}} .

Here ν(v) is the partition of d(v) determined by {d(e) : (v, e) ∈ F (Γ)}, as in Section 5.1.

Recall that

|A_{Γ}| = |Aut(Γ)| ∏_{e∈E} d(e),

so

I^{Γ}(τ) = \frac{1}{|Aut(Γ)|} ∏_{v∈V (Γ)^{(0)}}(|Aut(ν(v))|I_{g(v),ν(v)}^{0}(τ))

· ∏_{v∈V (Γ)^{(1)}} ((−1)^{g(v)+val(v)−1} ∏_{(v,e)∈F (Γ)} d(e))

· ∫_{[\overline{M}_{Γ}^{(1)}]^{vir}} \frac{(τu)^{d_{Γ}^{(1)}+1}}{−u − ψ^{t}}

= (−τ)^{d_{Γ}^{(1)}+1} \frac{1}{|Aut(Γ)|} ∏_{v∈V (Γ)^{(0)}}(|Aut(ν(v))|I_{g(v),ν(v)}^{0}(p))

· ∏_{v∈V (Γ)^{(1)}} ((−1)^{g(v)+val(v)−1} ∏_{(v,e)∈F} d(e)) ∫_{[\overline{M}_{Γ}^{(1)}]^{vir}}(ψ^{t})^{d_{Γ}^{(1)}}

= τ^{d_{Γ}^{(1)}+1}J^{Γ}(τ).

Note that J^{Γ}(τ) is a degree r − d_{Γ}^{(1)} − 1 polynomial in τ, and

J^{Γ}(−τ − 1) = (−1)^{d−l(μ)+d_{Γ}^{(1)}+1}J^{Γ}(τ).

6.3 Sum over graphs

We have

I_{g,μ}^{k}(τ) = τ^{k}J_{g,μ}^{k}(τ),

where J_{g,μ}^{0}(τ) = I_{g,μ}^{0}(τ), and

J_{g,μ}^{k}(τ) = ∑_{Γ∈G_{g,0}(ℙ^{1},μ),Γ≠Γ_{0}d_{Γ}^{(1)}+1=k}J^{Γ}(τ)

for k = 1, …, r. Note that J_{g,μ}^{k}(τ) is a degree r − k polynomial in τ, and

J_{g,μ}^{k}(−τ − 1) = (−1)^{|μ|−l(μ)+k}J_{g,μ}^{k}(τ).

For k = 1, we have

d_{Γ}^{(1)} = ∑_{v∈V (Γ)^{(1)}} r_{1}(v) − 1 = 0,

so V^{S}(Γ)^{(1)} consists of one single vertex v with r_{1}(v) = 1. In particular, g(v) = 0, and we have the following two cases:

Case 1: μ(v) = (μ_{i}), ν(v) = (j, k), where j + k = μ_{i}. In this case, we say ν ∈ C(μ) (cut) and define a_{μ,ν} = jk. Note that

∏_{v∈V (Γ)^{(1)}} ((−1)^{g(v)+val(v)−1} ∏_{(v,e)∈F} d(e)) ∫_{[\overline{M}_{Γ}^{(1)}]^{vir}} 1

= (−μ_{1} ⋯ μ_{l(μ)} \frac{jk}{μ_{i}} ) \frac{μ_{i}}{μ_{1} ⋯ μ_{l(μ)}}

= −a_{μ,ν}.

Case 2: μ(v) = (μ_{i}, μ_{j}), ν(v) = (μ_{i} + μ_{j}). In this case, we say ν ∈ J(μ) (join) and define b_{μ,ν} = μ_{i} + μ_{j}. Note that

∏_{v∈V (Γ)^{(1)}} ((−1)^{g(v)+val(v)−1} ∏_{(v,e)∈F} d(e)) ∫_{[\overline{M}_{Γ}^{(1)}]^{vir}} 1

= (μ_{1} ⋯ μ_{l(μ)} \frac{μ_{i} + μ_{j}}{μ_{i}μ_{j}} ) \frac{μ_{i}μ_{j}}{μ_{1} ⋯ μ_{l(μ)}}

= b_{μ,ν}.

316 c.-c.m. liu, k. liu & j. zhou

So

J_{g,μ}^{1}(τ) = − ∑_{ν∈J(μ)} \frac{| Aut(ν)|}{|Aut(Γ)|}b_{μ,ν}I_{g,ν}^{0}(τ) + ∑_{ν∈C(μ)} \frac{| Aut(ν)|}{|Aut(Γ)|}a_{μ,ν}I_{g−1,ν}^{0}(τ)

+ ∑_{g_{1}+g_{2}=g,ν^{1}∪ν^{2}∈C(μ)} \frac{| Aut(ν^{1})|| Aut(ν^{2})|}{| Aut(Γ)|}a_{μ,ν^{1}∪ν^{2}}I_{g_{1},ν^{1}}^{0}I_{g_{2},ν_{2}}^{0}(τ)

= − ∑_{ν∈J(μ)} I_{1}(ν)I_{g,ν}^{0}(τ) + ∑_{ν∈C(μ)} I_{2}(ν)I_{g−1,ν}^{0}(τ)

+ ∑_{g_{1}+g_{2}=g,ν^{1}∪ν^{2}∈C(μ)} I_{3}(ν^{1}, ν^{2})I_{g_{1},ν^{1}}^{0}I_{g_{2},ν_{2}}^{0}(τ),

where I_{1}, I_{2}, and I_{3} are defined as in [15].

We have

J_{g,μ}^{0}(τ) = \sqrt{−1}^{|μ|−l(μ)}C_{g,μ}(τ),
J_{g,μ}^{1}(τ) = \sqrt{−1}^{|μ|−l(μ)−1} ( ∑_{ν∈J(μ)} I_{1}(ν)C_{g,ν}(τ) + ∑_{ν∈C(μ)} I_{2}(ν)C_{g−1,ν}(τ)

+ ∑_{g_{1}+g_{2}=g,ν^{1}∪ν^{2}∈C(μ)} I_{3}(ν^{1}, ν^{2})C_{g_{1},ν^{1}}(τ)C_{g_{2},ν^{2}}(τ)) .

It follows from the definition that (6) in Theorem 2 is equivalent to

\frac{d}{dτ} J_{g,μ}^{0}(τ) = −J_{g,μ}^{1}(τ).

6.4 Final Calculations

We have

F (τ, x) = ∑_{k=0}^{r} I_{g,μ}^{r−k}(τ)x(x − 1) ⋯ (x − k + 1)(x − k − 1) ⋯ (x − r)

= ∑_{k=0}^{r} τ^{r−k}J_{g,μ}^{r−k}(τ)x(x − 1) ⋯ (x − k + 1)(x − k − 1) ⋯ (x − r)

= ∑_{k=0}^{r} τ^{k}J_{g,μ}^{k}(τ)x(x − 1) ⋯

(x − (r − k − 1))(x − (r − k + 1)) ⋯ (x − r).

Therefore,

Br_{∗}e_{T} (V) = ∑_{k=0}^{r} τ^{k}J_{g,μ}^{k}(τ)H(H − u) ⋯ (H − (r − k − 1)u)
· (H − (r − k + 1)u) ⋯ (H − ru).

For i = 0, …, r − 1, we have

H^{i}H(H − u) ⋯ (H − (r − k − 1)u)(H − (r − k + 1)u) ⋯ (H − ru)

= ((H − (r − k)u) + (r − k)u)^{i}H(H − u) ⋯ (H − (r − k − 1)u) · (H − (r − k + 1)u) ⋯ (H − ru)

= ((r − k)u)^{i}H(H − u) ⋯ (H − (r − k − 1)u)

· (H − (r − k + 1)u) ⋯ (H − ru)

since

H(H − u) ⋯ (H − ru) = 0.

Therefore,

∫_{ℙ^{r}} Br_{∗}e_{T} (V)H^{i} = u^{i} ∑_{k=0}^{r}(r − k)^{i}τ^{k}J_{g,μ}^{k}(τ).

Let J_{g,μ}^{k}(τ) = ∑_{j=0}^{r−k} a_{j}^{k}τ^{j}. We have

u^{−i} ∫_{ℙ^{r}} Br_{∗}e_{T} (V)H^{i} = ∑_{l=0}^{r} ( ∑_{j+k=l}(r − k)^{i}a_{j}^{k}) τ^{l}.

Here is a crucial observation: as a polynomial in τ, u^{−i} ∫_{ℙ^{r}} Br_{∗}e_{T} (V)H^{i} is of degree no more than i. Therefore,

∑_{j+k=l}(r − k)^{i}a_{j}^{k} = 0

for 0 ≤ i < l ≤ r. Now fix l such that 1 ≤ l ≤ r. We have

∑_{k=0}^{l}(r − k)^{i}a_{l−k}^{k} = 0, 0 ≤ i < l, (16)

which is a system of l linear equations of the l + 1 variables {a_{l−k}^{k} : k = 0, …, l}.

318 c.-c.m. liu, k. liu & j. zhou

Both

{(r − t)^{i} : i = 0, …, l − 1}

and

{1, t, t(t − 1), …, t(t − 1) … (t − l + 2)}

are bases of the vector space

{f(t) ∈ ℚ[t] : deg(f) ≤ l − 1},

so there exists an invertible l × l matrix (A_{ij})_{0≤i,j≤l−1} such that

t(t − 1) ⋯ (t − i + 1) = ∑_{j=0}^{l−1} A_{ij}(r − t)^{j}.

In particular,

k(k − 1) ⋯ (k − i + 1) = ∑_{j=0}^{l−1} A_{ij}(r − k)^{j}.

for k = 0, 1, …, l, so (16) is equivalent to

∑_{k=0}^{l} k(k − 1) ⋯ (k − i + 1)a_{l−k}^{k} = 0, 0 ≤ i < l,

i.e.,

∑_{k=i}^{l} \frac{k!}{(k − i)!}a_{l−k}^{k} = 0, 0 ≤ i < l.

The above equations can be rewritten as

( \begin{matrix}1 & 1 & ⋯ & ⋯ & ⋯ & ⋯ & 1 \\ 0 & 1! & 2 & ⋯ & ⋯ & ⋯ & l \\ 0 & 0 & 2! & 3 · 2 & ⋯ & ⋯ & l(l − 1) \\ 0 & 0 & 0 & 3! & ⋯ & ⋯ & l(l − 1)(l − 2) \\ 0 & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & ⋯ & ⋯ & ⋯ & 0 & (l − 1)! & l(l − 1) ⋯ 2 \end{matrix} ) ( a_{l}^{0} ; a_{l−1}^{1} ; ⋮ ; ⋮ ; a_{0}^{l} )

= ( 0 ⋮ ⋮ ⋮ 0 ) .

The kernel is clearly one dimensional. One can check that the kernel is given by

a_{l−k}^{k} = (−1)^{k} \frac{l!}{k!(l − k)!}a_{l}^{0}. (17)

Note that (17) for l = 1, …, r is equivalent to

J_{g,μ}^{k}(τ) = \frac{(−1)^{k}}{k!} \frac{d^{k}}{dτ^{k}} J_{g,μ}^{0}(τ) (18)

for k = 0, …, r. In particular,

J_{g,μ}^{1}(τ) = −\frac{d}{dτ} J_{g,μ}^{0}(τ)

which is equivalent to the cut-and-join equation (6) in Theorem 2. Equation (18) and the cut-and-join equation imply that J_{g,μ}^{k}(τ) can be obtained from J_{g,μ}^{0}(τ) by repeating the cut-and-join operation k-times.

7. Hurwitz numbers

In this section, inspired by T. Graber and R. Vakil [9], we use virtual localization on moduli spaces of relative stable morphisms to recover some results on Hurwitz numbers. We give a unified proof of the ELSV formula and the cut-and-join equation for Hurwitz numbers.

The Hirwitz numbers can be defined as

H_{g,μ} = ∫_{[\overline{M}_{g,0}(ℙ^{1},μ)]^{vir}} Br^{∗}H^{r}. (19)

We lift H^{r} ∈ H^{2r}(ℙ^{r}; ℤ) to ∏_{k=1}^{r}(H −w_{k}u) ∈ H_{T}^{2r}(ℙ^{r}; ℤ), where w_{k} ∈ ℤ, and compute

H_{g,μ} = ∫_{[\overline{M}_{g,0}(ℙ^{1},μ)]^{vir}} Br^{∗} (∏_{k=1}^{r}(H − w_{k}u))

by virtual localization.

Let p_{k} ∈ ℙ^{r} be the ℂ^{∗} fixed point defined as in Section 4.3, and f_{k} : p_{k} → ℙ^{r} be the inclusion. Then

f_{k}^{∗}(∏_{l=1}^{r}(H − w_{k}u)) = (∏_{l=1}^{r}(k − w_{l})) u^{r} ∈ H_{T}^{2r}(p_{k}).

Remark 7.1. In the definition of \overline{M}_{g,0}(ℙ^{1}, μ) given in Section 4.1, if we order the l(μ) marked points on the domain as in [16, 17], there will be an extra factor 1/| Aut(μ)| on the right-hand side of (19).

320 c.-c.m. liu, k. liu & j. zhou

7.1 Contribution from each graph

7.1.1 The target is ℙ^{1} Consider the graph Γ^{0} ∈ \overline{M}_{g,0}(ℙ^{1}, μ). We have Br(F_{Γ^{0}}) = \{p_{r}\}. We first consider the stable case. By the Feynman rules derived in Appendix A, the contribution from Γ^{0} is given by

∏_{l=1}^{r}(r − w_{l})Ĩ_{g,μ}^{0},

where

Ĩ_{g,μ}^{0} = \frac{1}{|A_{Γ}|} ∫_{\overline{M}_{g,l(μ)}} \frac{u^{r}}{e(N_{Γ}^{vir})}

= \frac{1}{|Aut(μ)|} ∏_{i=1}^{l(μ)} \frac{μ_{i}^{μ_{i}}}{μ_{i}!} ∫_{\overline{M}_{g,l(μ)}} \frac{Λ_{g}^{∨}(u)u^{2g+2l(μ)−3}}{∏_{i=1}^{l(μ)}(u − μ_{i}ψ_{i})}

= \frac{1}{|Aut(μ)|} ∏_{i=1}^{l(μ)} \frac{μ_{i}^{μ_{i}}}{μ_{i}!} ∫_{\overline{M}_{g,l(μ)}} \frac{Λ_{g}^{∨}(1)}{∏_{i=1}^{l(μ)}(1 − μ_{i}ψ_{i})}

In particular,

Ĩ_{0,μ}^{0} = \frac{1}{|Aut(μ)|} ∏_{i=1}^{l(μ)} \frac{μ_{i}^{μ_{i}}}{μ_{i}!} ∫_{\overline{M}_{0,l(μ)}} \frac{1}{∏_{i=1}^{l(μ)}(1 − μ_{i}ψ_{i})}

= \frac{1}{|Aut(μ)|} (∏_{i=1}^{l(μ)} \frac{μ_{i}^{μ_{i}}}{μ_{i}!} ) d^{l(μ)−3}.

The contribution from Γ^{(0)} for the two unstable cases is also given by the above formula:

Ĩ_{0,(d)}^{0} = \frac{d^{d}}{d!} \frac{1}{d^{2}} = \frac{d^{d−2}}{d!} .
Ĩ_{0,(μ_{1},μ_{2})}^{0} = \frac{1}{|Aut((μ_{1}, μ_{2}))|} \frac{μ_{1}^{μ_{1}}μ_{2}^{μ_{2}}}{μ_{1}!μ_{2}!} \frac{1}{d}.

7.1.2 The target is ℙ^{1}[m], m > 0 Consider Γ ∈ G_{g,0}(ℙ^{1}, μ), Γ ≠ Γ^{0}. We have Br(F_{Γ}) = {p_{r−d_{Γ}^{(1)}−1}}. The contribution from Γ is given by

∏_{l=1}^{r}(r − d_{Γ}^{(1)} − 1 − w_{l})Ĩ^{Γ},

a proof of a conjecture of mariño-vafa 321

where

Ĩ^{Γ} = \frac{1}{|A_{Γ}|} ∫_{[\overline{M}_{Γ}]^{vir}} \frac{u^{r}}{e_{T} (N_{Γ}^{vir})} = \frac{1}{|A_{Γ}|} ∫_{[\overline{M}_{Γ}]^{vir}} \frac{∏_{v∈V (Γ)} B̃_{v}}{−u − ψ^{t}} ,
B̃_{v} = \begin{cases}u^{2g(v)−2+val(v)}, A_{v} ∏_{(v,e)∈F (Γ)}(A_{e}u^{d(e)}) & v ∈ V (Γ)^{(0)}, \\ u^{r_{1}(v)}A_{v}, & v ∈ V (Γ)^{(1)}. \end{cases}

More explicitly, in the notation of Appendix A,

B̃_{v} = ( ∏_{(v,e)∈F (Γ)} d(e)) ( ∏_{(v,e)∈F (Γ)} \frac{d(e)^{d(e)}}{d(e)!} )

· \frac{Λ_{g(v)}^{∨}(u)u^{2g(v)+val(v)−3}}{∏_{(v,e)∈F (Γ)} (u − d(e)ψ_{(v,e)})}, v ∈ V^{S}(Γ)^{(0)},

d(e)\frac{d(e)^{d(e)−2}}{d(e)!} , v ∈ V^{I}(Γ)^{(0)}, (v, e) ∈ F (Γ),

d(e_{1})d(e_{2})\frac{d(e_{1})^{d(e_{1})}d(e_{2})^{d(e)_{2}}}{d(e_{1})!d(e_{2})!} \frac{1}{d(v)},

v ∈ V^{II}(Γ)^{(0)}, (v, e_{1}), (v, e_{2}) ∈ F (Γ),

u^{r_{1}(v)} ∏_{(v,e)∈F (Γ)} d(e), v ∈ V (Γ)^{(1)}.

So

Ĩ^{Γ} = \frac{1}{|A_{Γ}|} ∏_{v∈V (Γ)^{(0)}} (|Aut(ν(v))|Ĩ_{g(v),ν(v)}^{0})

· ( ∏_{e∈E(Γ)} d(e))^{2} ∫_{[\overline{M}_{Γ}^{(1)}]^{vir}} \frac{u^{d_{Γ}^{(1)}+1}}{−u − ψ^{t}} .

Recall that

|A_{Γ}| = |Aut(Γ)| ∏_{e∈E} d(e) ∏_{v∈V^{II}(Γ)^{(1)}} d(v),

so

Ĩ^{Γ} = \frac{(−1)^{d_{Γ}^{(1)}+1}}{|Aut(Γ)|} ∏_{v∈V (Γ)^{(0)}}(|Aut(ν(v))|I_{g(v),ν(v)}^{0})

· ( ∏_{v∈V^{S}(Γ)^{(1)}} ∏_{(v,e)∈F} d(e)) ∫_{[\overline{M}_{Γ}^{(1)}]^{vir}}(ψ^{t})^{d_{Γ}^{(1)}}.

7.2 Sum over Graphs

For k = 1, …, r define

Ĩ_{g,μ}^{k} = ∑_{Γ∈G_{g,0}(ℙ^{1},μ),Γ≠Γ^{0},d_{Γ}^{(1)}+1=k} Ĩ^{Γ}.

Then

H_{g,μ} = ∑_{k=0}^{r} ∏_{l=1}^{r}(r − k − w_{l})Ĩ_{g,μ}^{k}.

for any w_{1}, …, w_{r} ∈ ℤ.

Setting (w_{1}, …, w_{r}) = (0, 1, …, r − k − 1, r − k + 1, …, r) yields

H_{g,μ} = (−1)^{k}k!(r − k)!Ĩ_{g,μ}^{k}.

The k = 0 case implies the following formula due to Ekedahl, Lando, Shapiro, and Vainshtein [2], also proved by T. Graber and R. Vakil [9]:

H_{g,μ} = r!Ĩ_{g,μ}^{0} = \frac{r!}{|Aut(μ)|} ∏_{i=1}^{l(μ)} \frac{μ_{i}^{μ^{i}}}{μ_{i}!} ∫_{\overline{M}_{g,l(μ)}} \frac{Λ_{g}^{∨}(1)}{∏_{i=1}^{l(μ)}(1 − μ_{i}ψ_{i})}. (20)

For k = 1, we have

H_{g,μ} = −(r − 1)!Ĩ_{g,μ}^{1},

which is, by a derivation similar to that in Section 6.3, equivalent to

(21)  \frac{H_{g,μ}}{(r − 1)!} = ∑_{ν∈J(μ)} I_{1}(ν)Ĩ_{g,ν}^{0} + ∑_{ν∈C(μ)} I_{2}(ν)Ĩ_{g−1,ν}^{0}

+ ∑_{g_{1}+g_{2}=g} ∑_{ν^{1}∪ν^{2}∈C(μ)} I_{3}(ν^{1}, ν^{2})Ĩ_{g_{1},ν^{1}}^{0}Ĩ_{g_{2},ν_{2}}^{0}.

Combining (20) and (21), one obtains

\frac{H_{g,μ}}{(r − 1)!} = ∑_{ν∈J(μ)} I_{1}(ν)\frac{H_{g,ν}}{(r − 1)!} + ∑_{ν∈C(μ)} I_{2}(ν)\frac{H_{g−1,ν}}{(r − 1)!}

+ ∑_{g_{1}+g_{2}=g} ∑_{ν^{1}∪ν^{2}∈C(μ)} I_{3}(ν^{1}, ν^{2})

· \frac{H_{g_{1},ν^{1}}}{(2g_{1} − 2 + |ν^{1}| + l(ν^{1}))!} \frac{H_{g_{2},ν^{2}}}{(2g_{2} − 2 + |ν^{2}| + l(ν^{2}))!}

which is equivalent to the cut-and-join equation

(22) H_{g,μ} = ∑_{ν∈J(μ)} I_{1}(ν)H_{g,ν} + ∑_{ν∈C(μ)} I_{2}(ν)H_{g−1,ν}

+ ∑_{g_{1}+g_{2}=g} ∑_{ν^{1}∪ν^{2}∈C(μ)} ( \begin{matrix}r − 1 \\ 2g_{1} − 2 + |ν^{1}| + l(ν^{1}) \end{matrix} )

· I_{3}(ν^{1}, ν^{2})H_{g_{1},ν^{1}}H_{g_{2},ν^{2}}.

The above cut-and-join equation was first proved using combinatorics by Goulden, Jackson and Vainstein in [7] and later proved using symplectic sum formula by Li-Zhao-Zheng [15] and Ionel-Parker [11, Section 15.2].

Acknowledgements.

We wish to thank Jun Li for explaining his work, Jim Bryan, Bohui Chen, Tom Graber, Gang Liu, Ravi Vakil for helpful conversations, and Cumrun Vafa and Shing-Tung Yau for their interests in this work. The first author wishes to thank the hospitality of IPAM where she was a core participant of the Symplectic Geometry and Physics Program and did most of her part of this work. The second author is supported by an NSF grant. The third author is partially supported by research grants from NSFC and Tsinghua University.

Appendix A. Localization

In this appendix, we provide details of localization in our particular case. Related results are discussed in [10]. We first introduce some notation.

Let (w) be the 1-dimensional representation of ℂ^{∗} given by λ · z = λ^{w}z for λ ∈ ℂ^{∗}, z ∈ ℂ. We do calculations on \overline{M}_{Γ} which is a finite cover of F_{Γ} ⊂ \overline{M}_{g,0}(ℙ^{1}, μ), so the weight w of ℂ^{∗}-action can be fractional.

Given Γ ∈ G_{g,0}(ℙ^{1}, μ), let

[f : (C, x_{1}, …, x_{l(μ)}) → ℙ^{1}[m] ]

be a fixed point of the ℂ^{∗}-action on \overline{M}_{g,0}(ℙ^{1}, μ) associated to Γ. Given (v, e) ∈ F (Γ), where v ∈ V^{S}(Γ)^{(0)} ∪ V (Γ)^{(1)}, let q_{(v,e)} ∈ C denote the node at which C_{v} and C_{e} intersect. Let ψ_{(v,e)} denote the first Chern class of the line bundle over \overline{M}_{Γ} whose fiber at

[f : (C, x_{1}, …, x_{l(μ)}) → ℙ^{1}[m] ]

is T_{q_{(v,e)}}^{∗}C_{v}. Given v ∈ V^{II}(Γ)^{(0)}, let q_{v} denote the node at which C_{e_{1}} and C_{e_{2}} intersect, where {e_{1}, e_{2}} = {e ∈ E(Γ) : (v, e) ∈ F (Γ)}.

A.1 Virtual Normal Bundle

The tangent space T^{1} and the obstruction space T^{2} of \overline{M}_{g,0}(ℙ^{1}, μ) at

[f : (C, x_{1}, …, x_{l(μ)}) → ℙ^{1}[m] ] ∈ \overline{M}_{g,0}(ℙ^{1}, μ)

are given by the following two exact sequences [17, Section 5.1]:

0 → Ext^{0}(Ω_{C}(D), O_{C}) → H^{0}(D^{•}) → T^{1}
→ Ext^{1}(Ω_{C}(D), O_{C}) → H^{1}(D^{•}) → T^{2} → 0
0 → H^{0}(C, f^{∗}(ω_{ℙ^{1}[m]}(log p_{1}^{(m)}))^{∨}) → H^{0}(D^{•}) → ⊕_{l=0}^{m−1}H_{et}^{0}(R_{l}^{•})

→ H^{1}(C, f^{∗}(ω_{ℙ^{1}[m]}(log p_{1}^{(m)}))^{∨}) → H^{1}(D^{•}) → ⊕_{l=0}^{m−1}H_{et}^{1}(R_{l}^{•}) → 0

where D = x_{1} + ⋯ + x_{l(μ)}, ω_{ℙ^{1}[m]} is the dualizing sheaf of ℙ^{1}[m],

H_{et}^{0}(R_{l}^{•}) ≅ ⨁_{q∈f^{−1}(p_{1}^{(l)})} T_{q} (f^{−1}(ℙ_{(l)}^{1})) ⊗ T_{q}^{∗} (f^{−1}(ℙ_{(l)}^{1})) ≅ ℂ^{⊕n_{l}},

H_{et}^{1}(R_{l}^{•}) ≅ (T_{p_{1}^{(l)}}ℙ_{(l)}^{1} ⊗ T_{p_{1}^{(l)}}ℙ_{(l+1)}^{1})^{⊕(n_{l}−1)},

and n_{l} is the number of nodes over p_{1}^{(l)}.

Let f̃ = π[m] ∘ f : C → ℙ^{1}. Then

f^{∗}(ω_{ℙ^{1}[m]}(log p_{1}^{(m)}))^{∨} ≅ f̃^{∗}O_{ℙ^{1}}(1).

Let

B_{1} = Ext^{0}(Ω_{C}(D), O_{C}),  B_{2} = H^{0}(C, f̃^{∗}O_{ℙ^{1}}(1)),
B_{3} = ⊕_{l=0}^{m−1}H_{et}^{0}(R_{l}^{•}),  B_{4} = Ext^{1}(Ω_{C}(D), O_{C}),
B_{5} = H^{1}(C, f̃^{∗}O_{ℙ^{1}}(1)),  B_{6} = ⊕_{l=0}^{m−1}H_{et}^{1}(R_{l}^{•}).

We now assume that

[f : (C, x_{1}, …, x_{l(μ)}) → ℙ^{1}[m]] ∈ F_{Γ} ⊂ \overline{M}_{g,0}(ℙ^{1}, μ),

where Γ ∈ G_{g,0}(ℙ^{1}, μ), and F_{Γ} is the set of fixed points associated to Γ. The ℂ^{∗}-action on \overline{M}_{g,0}(ℙ^{1}, μ) induces ℂ^{∗}-actions on B_{1}, …, B_{6}. In particular, the ℂ^{∗}-actions on B_{2} and B_{5} come from the ℂ^{∗}-action on O_{ℙ^{1}}(1) which acts on O_{ℙ^{1}}(1)_{p_{0}} and O_{ℙ^{1}}(1)_{p_{1}} by (1) and (0), respectively. Let B̃_{i} denote the moving part of B_{i} under the ℂ^{∗}-action. Then each B̃_{i} form a vector bundle over \overline{M}_{Γ}. We will use the same notation B̃_{i} to denote the vector bundle.

Note that B̃_{3} = 0, and

B̃_{6} = \{ \begin{matrix}0, & m = 0, \\ H_{et}^{1}(R_{0}^{•}) = (T_{p_{1}^{(0)}}ℙ_{(0)}^{1} ⊗ T_{p_{1}^{(0)}}ℙ_{(1)}^{1})^{⊕(n_{0}−1)}, & m > 0. \end{matrix}

We have

\frac{1}{e_{T} (N_{Γ}^{vir})} = \frac{e_{T} (T̃^{2})}{e_{T} (T̃^{1})} = \frac{e_{T} (B̃_{1})e_{T} (B̃_{5})e_{T} (B̃_{6})}{e_{T} (B̃_{2})e_{T} (B̃_{4})}

where T̃^{1}, T̃^{2} are the moving parts of T^{1}, T^{2}, respectively.

A.1.1 The target is ℙ^{1}

We have seen that there is only one graph Γ^{0}. Recall that

\overline{M}_{Γ^{0}} = \{ \begin{matrix}\{point\} & (g, l(μ)) = (0, 1), (0, 2), \\ \overline{M}_{g,l(μ)} & (g, l(μ)) ≠ (0, 1), (0, 2). \end{matrix}

We first compute e_{T} (B̃_{1})/e_{T} (B̃_{4}). When (g, l(μ)) ≠ (0, 1), (0, 2). The domain is

C = C_{v_{0}} ∪ C_{e_{1}} ∪ ⋯ C_{e_{l(μ)}}.

We have

B̃_{1} = 0, B̃_{4} = ⨁_{i=1}^{l(μ)} T_{q_{i}}C_{v_{0}} ⊗ T_{q_{i}}C_{e_{i}},

where q_{i} = q_{(v_{0},e_{i})}. So

\frac{e_{T} (B̃_{1})}{e_{T} (B̃_{4})} = \frac{1}{∏_{i=1}^{l(μ)} (\frac{u}{μ_{i}} − ψ_{i})},

where ψ_{i} = ψ_{(v_{0},e_{i})}.

326 c.-c.m. liu, k. liu & j. zhou

When (g, l(μ)) = (0, 1), we have

B̃_{1} = (\frac{1}{d}) , B̃_{4} = 0,

so

\frac{e_{T} (B̃_{1})}{e_{T} (B̃_{4})} = \frac{u}{d}.

When (g, l(μ)) = (0, 2), we have

B̃_{1} = 0, B̃_{4} = (\frac{1}{μ_{1}} + \frac{1}{μ_{2}} ) .

so

\frac{e_{T} (B̃_{1})}{e_{T} (B̃_{4})} = \frac{1}{\frac{u}{μ_{1}} + \frac{u}{μ_{2}}} .

We next compute e_{T} (B̃_{5})/e_{T} (B̃_{2}). When (g, l(μ)) ≠ (0, 1), (0, 2), consider the normalization sequence

0 → f̃^{∗}O_{ℙ^{1}}(1) → (f̃|_{C_{v_{0}}} )^{∗}O_{ℙ^{1}}(1) ⊕ ⨁_{i=1}^{l(μ)}(f̃|_{C_{e_{i}}})^{∗}O_{ℙ^{1}}(1)

→ ⨁_{i=1}^{l(μ)} O_{ℙ^{1}}(1)_{p_{0}} → 0.

The corresponding long exact sequence reads

0 → H^{0}(C, f̃^{∗}O_{ℙ^{1}}(1))

→ H^{0}(C_{v_{0}}, (f̃|_{C_{v_{0}}} )^{∗}O_{ℙ^{1}}(1)) ⊕ ⨁_{i=1}^{l(μ)} H^{0}(C_{e_{i}}, (f̃|_{C_{e_{i}}})^{∗}O_{ℙ^{1}}(1))

→ ⨁_{i=1}^{l(μ)} O_{ℙ^{1}}(1)_{p_{0}} → H^{1}(C, f̃^{∗}O_{ℙ^{1}}(1))

→ H^{1}(C_{v_{0}}, (f̃|_{C_{v_{0}}} )^{∗}O_{ℙ^{1}}(1)) ⊕ ⨁_{i=1}^{l(μ)} H^{1}(C_{e_{i}}, (f̃|_{C_{e_{i}}})^{∗}O_{ℙ^{1}}(1)) → 0.

The representations of ℂ^{∗} are

0 → H^{0}(C, f̃^{∗}O_{ℙ^{1}}(1)) → H^{0}(C_{v_{0}}, O_{C_{v_{0}}} ) ⊗ (1) ⊕ ⨁_{i=1}^{l(μ)} (⨁_{a=1}^{μ_{i}} (\frac{a}{μ_{i}} ))

→ ⨁_{i=1}^{l(μ)}(1) → H^{1}(C, f̃^{∗}O_{ℙ^{1}}(1)) → H^{1}(C_{v_{0}}, O_{C_{v_{0}}} ) ⊗ (1) → 0

So we have

\frac{e_{T} (B̃_{5})}{e_{T} (B̃_{2})} = Λ_{g}^{∨}(u)u^{l(μ)−1} ∏_{i=1}^{l(μ)} (\frac{μ_{i}^{μ_{i}}}{μ_{i}!} u^{−μ_{i}}) . (23)

One can check that (23) is also valid for (g, l(μ)) = (0, 1), (0, 2).

Finally, B̃_{6} = 0, so e_{T} (B̃_{6}) = 1. We obtain the following Feynman rules:

\frac{1}{e_{T} (N_{Γ^{0}}^{vir})} = A_{v_{0}} ∏_{e∈E(Γ^{0})} A_{e},

where

A_{v_{0}} = \begin{cases}\frac{u}{d}, & (g, l(μ)) = (0, 1) \\ _{\frac{u}{\frac{u}{μ_{1}} + \frac{u}{μ_{2}}}} , & (g, l(μ)) = (0, 2) \\ \frac{Λ_{g}^{∨}(u)}{u} ∏_{i=1}^{l(μ)} _{\frac{u}{\frac{u}{μ_{i}} −ψ_{i}}} , & (g, l(μ)) ≠ (0, 1), (0, 2) \end{cases}

A_{e} = \frac{d(e)^{d(e)}}{d(e)!} u^{−d(e)}.

A.1.2 The target is ℙ^{1}[m], m > 0

Let Γ ∈ G_{g,0}(ℙ^{1}, μ), Γ ≠ Γ^{0}. We first compute e_{T} (B̃_{1})/e_{T} (B̃_{4}). We have

B̃_{1} = ⨁_{v∈V^{I}(Γ)^{(0)}} (\frac{1}{d(v)})

B̃_{4} = ⨁_{v∈V^{II}(Γ)^{(0)}} ( ∑_{(v,e)∈F (Γ)} \frac{1}{d(e)})

⊕ ⨁_{v∈V^{S}(Γ)^{(0)}} ( ⨁_{(v,e)∈F (Γ)} T_{q_{(v,e)}}C_{v} ⊗ T_{q_{(v,e)}}C_{e})

⊕ ⨁_{v∈V (Γ)^{(1)}} ( ⨁_{(v,e)∈F (Γ)} T_{q_{(v,e)}}C_{v} ⊗ T_{q_{(v,e)}}C_{e})

328 c.-c.m. liu, k. liu & j. zhou

So

\frac{e_{T} (B̃_{1})}{e_{T} (B̃_{4})} = ∏_{v∈V^{I}(Γ)^{(0)}} \frac{u}{d(v)} ∏_{v∈V^{II}(Γ)^{(0)}} ( ∑_{(v,e)∈F (Γ)} \frac{u}{d(e)})^{−1}

· ∏_{v∈V^{S}(Γ)^{(0)}} ( ∑_{(v,e)∈F (Γ)} \frac{1}{\frac{u}{d(e)} − ψ_{(v,e)}} )

· ∏_{v∈V (Γ)^{(1)}} ( ∏_{(v,e)∈F (Γ)} \frac{1}{\frac{−u}{d(e)} − ψ_{(v,e)}} )

We next compute e_{T} (B̃_{5})/e_{T} (B̃_{2}). Consider the normalization sequence

0 → f̃^{∗}O_{ℙ^{1}}(1) → ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}}(f̃|_{C_{v}})^{∗}O_{ℙ^{1}}(1) ⊕ ⨁_{e∈E(Γ)}(f̃|_{C_{e}})^{∗}O_{ℙ^{1}}(1)

→ ⨁_{v∈V^{II}(Γ)^{(0)}} O_{ℙ^{1}}(1)_{p_{0}} ⊕ ⨁_{v∈V^{S}(Γ)^{(0)}} ( ⨁_{(v,e)∈F} O_{ℙ^{1}}(1)_{p_{0}} )

⊕ ⨁_{v∈V (Γ)^{(1)}} ( ⨁_{(v,e)∈F} O_{ℙ^{1}}(1)_{p_{1}} ) → 0.

The corresponding long exact sequence reads

0 → H^{0}(C, f̃^{∗}O_{ℙ^{1}}(1))

→ ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}} H^{0}(C_{v}, (f̃|_{C_{v}})^{∗}O_{ℙ^{1}}(1))

⊕ ⨁_{e∈E(Γ)} H^{0}(C_{e}, (f̃|_{C_{e}})^{∗}O_{ℙ^{1}}(1))

→ ⨁_{v∈V^{II}(Γ)^{(0)}} O_{ℙ^{1}}(1)_{p_{0}} ⊕ ⨁_{v∈V^{S}(Γ)^{(0)}} ( ⨁_{(v,e)∈F} O_{ℙ^{1}}(1)_{p_{0}} )

⊕ ⨁_{v∈V (Γ)^{(1)}} ( ⨁_{(v,e)∈F} O_{ℙ^{1}}(1)_{p_{1}} )

→ H^{1}(C, f̃^{∗}O_{ℙ^{1}}(1))

a proof of a conjecture of mariño-vafa 329

→ ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}} H^{0}(C_{v}, (f̃|_{C_{v}})^{∗}O_{ℙ^{1}}(1))

⊕ ⨁_{e∈E(Γ)} H^{0}(C_{e}, (f̃|_{C_{e}})^{∗}O_{ℙ^{1}}(1)) → 0.

The representations of ℂ^{∗} are

0 → H^{0}(C, f̃^{∗}O_{ℙ^{1}}(1))

→ ⨁_{v∈V^{S}(Γ)^{(0)}} H^{0}(C_{v}, O_{C_{v}}) ⊗ (1) ⊕ ⨁_{v∈V (Γ)^{(1)}} H^{0}(C_{v}, O_{C_{v}}) ⊗ (0)

⊕ ⨁_{e∈E(Γ)} (⨁_{a=1}^{d(e)} (\frac{a}{d(e)}))

→ ⨁_{v∈V^{II}(Γ)^{(0)}}(1) ⊕ ⨁_{v∈V^{S}(Γ)^{(0)}} ( ⨁_{(v,e)∈F}(1)) ⊕ ⨁_{v∈V (Γ)^{(1)}} ( ⨁_{(v,e)∈F}(0))

→ H^{1}(C, f̃^{∗}O_{ℙ^{1}}(1))

→ ⨁_{v∈V^{S}(Γ)^{(0)}} H^{1}(C_{v}, O_{C_{v}}) ⊗ (1) ⊕ ⨁_{v∈V (Γ)^{(1)}} H^{1}(C_{v}, O_{C_{v}}) ⊗ (0) → 0

So

\frac{e_{T} (B̃_{5})}{e_{T} (B̃_{2})} = ∏_{v∈V (Γ)^{(0)}} (Λ_{g(v)}^{∨}(u)u^{val(v)−1}) ∏_{e∈E(Γ)} (\frac{d(e)^{d(e)}}{d(e)!} u^{−d(e)}) .

Finally,

B̃_{6} = (T_{p_{1}^{(0)}}ℙ_{(0)}^{1} ⊗ T_{p_{1}^{(0)}}ℙ_{(1)}^{1})^{|E(Γ)|−1} ,

so

e_{T} (B̃_{6}) = (−u − ψ^{t})^{|E(Γ)|−1},

where ψ^{t} is the first Chern class of the line bundle over \overline{M}_{Γ}^{(1)} whose fiber at

[f̂: Ĉ → ℙ^{1}(m)]

is T_{p_{1}^{(0)}}^{∗}ℙ^{1}(m). Note that ψ^{t} = d(e)ψ_{(v,e)} for v ∈ V (Γ)^{(1)}, (v, e) ∈ F.

We have the following Feynman rules:

\frac{1}{e_{T} (N_{Γ}^{vir})} = \frac{1}{−u − ψ^{t}} ∏_{v∈V (Γ)} A_{v} ∏_{e∈E(Γ)} A_{e},

where

A_{v} = \{ \begin{matrix}∏ & (v,e)∈F (Γ) & \frac{−u−ψ^{t}}{\frac{−u}{d(e)} −ψ_{(v,e)}} \\ = ∏_{(v,e)∈F (Γ)} d(e), & v ∈ V (Γ)^{(1)}, \\ \frac{Λ_{g(v)}^{∨}(u)}{u} ∏_{(v,e)∈F (Γ)} \frac{u}{\frac{u}{d(e)} −ψ_{(v,e)}} , & v ∈ V^{S}(Γ)^{(0)}, \\ \frac{u}{d(v)}, & v ∈ V^{I}(Γ)^{(0)}, \\ _{\frac{1}{\frac{1}{d(e_{1})} + \frac{1}{d(e_{2})}}} , & v ∈ V^{II}(Γ)^{(0)}, \\ & (v, e_{1}), (v, e_{2}) ∈ F (Γ), \end{matrix}

A_{e} = \frac{d(e)^{d(e)}}{d(e)!} u^{−d(e)}.

The degree of e_{T} (N_{Γ}^{vir}) is

1 + ∑_{v∈V^{S}(Γ)^{(0)}}(1 − g(v)) − |V^{I}(Γ)^{(0)}| + ∑_{e∈E(Γ)} d(e)

= 1 + ∑_{v∈V^{S}(Γ)^{(0)}}(1 − g(v)) − |V^{I}(Γ)^{(0)}| + d.

We have seen that the virtual dimension of F_{Γ} is

2g − 3 + l(μ) + ∑_{v∈V^{S}(Γ)^{(0)}}(g(v) − 1) + |V^{I}(Γ)^{0}|.

We have

2g − 3 + l(μ) + ∑_{v∈V^{S}(Γ)^{(0)}}(g(v) − 1) + |V^{I}(Γ)^{0}|

+ 1 + ∑_{v∈V^{S}(Γ)^{(0)}}(1 − g(v)) − |V^{I}(Γ)^{(0)}| + d

= 2g − 2 + d + l(μ)

as expected.

A.2 The bundle V_{D}

The short exact sequence 0 → O_{C}(−D) → O_{C} → O_{D} → 0

a proof of a conjecture of mariño-vafa 331

gives rise to a long exact sequence

0 → H^{0}(C, O_{C}(−D)) → H^{0}(C, O_{C}) → ⨁_{i=1}^{l(μ)} O_{x_{i}}
→ H^{1}(C, O_{C}(−D)) → H^{1}(C, O_{C}) → 0

The representations of ℂ^{∗} are

0 → (τ) → ⨁_{i=1}^{l(μ)}(τ) → H^{1}(C, O_{C}(−D)) ⊗ (τ) → H^{1}(C, O_{C}) ⊗ (τ) → 0.

So

e_{T} (V_{D}) = e_{T} (V_{0})(τu)^{l(μ)−1},

where

V_{0} = R^{1}π_{∗}O_{U_{g,μ}}.

Recall that π : U_{g,μ} → \overline{M}_{g,0}(ℙ^{1}, μ) is the universal curve. A.2.1 The target is ℙ^{1} There is only one graph Γ^{0} in this case. When (g, l(μ)) ≠ (0, 1), (0, 2), consider the normalization sequence

0 → O_{C} → O_{C_{v_{0}}} ⊕ ⨁_{i=1}^{l(μ)} O_{C_{e_{i}}} → ⨁_{i=1}^{l(μ)} O_{q_{i}} → 0.

The corresponding long exact sequence reads

0 → H^{0}(C, O_{C}) → H^{0}(C_{v_{0}}, O_{C_{v_{0}}} ) ⊕ ⨁_{i=1}^{l(μ)} H^{0}(C_{e_{i}}, O_{C_{e_{i}}}) → ⨁_{i=1}^{l(μ)} O_{q_{i}}

→ H^{1}(C, O_{C}) → H^{1}(C_{v_{0}}, O_{C_{v_{0}}} ) ⊕ ⨁_{i=1}^{l(μ)} H^{1}(C_{e_{i}}, O_{C_{e_{i}}}) → 0.

The representations of ℂ^{∗} are

0 → (τ) → (τ) ⊕ ⨁_{i=1}^{l(μ)}(τ) → ⨁_{i=1}^{l(μ)}(τ)
→ H^{1}(C, O_{C}) ⊗ (τ) → H^{1}(C_{v_{0}}, O_{C_{v_{0}}} ) ⊗ (τ) → 0.

So

i_{Γ}^{∗}e_{T} (V_{0}) = Λ_{g}^{∨}(τu)

which is clearly also valid for (g, l(μ)) = (0, 1), (0, 2).

We have

i_{Γ^{0}}^{∗}e_{T} (V_{D}) = A_{v_{0}}^{D},

where

A_{v_{0}}^{D} = Λ_{g}^{∨}(τu) · (τu)^{l(μ)−1}.

Note that the degree of i_{Γ^{0}}^{∗}e_{T} (V_{D}) is l(μ) + g − 1, as expected.

A.2.2 The target is ℙ^{1}[m], m > 0

Given a graph Γ ∈ G_{g,0}(ℙ^{1}, μ), Γ ≠ Γ^{0}, consider the normalization sequence

0 → O_{C} → ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}} O_{C_{v}} ⊕ ⨁_{e∈E(Γ)} O_{C_{e}}

→ ⨁_{v∈V^{II}(Γ)^{(0)}} O_{q_{v}} ⊕ ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}} ( ⨁_{(v,e)∈F} O_{q_{(v,e)}} ) → 0.

The corresponding long exact sequence reads

0 → H^{0}(C, O_{C}) → ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}} H^{0}(C_{v}, O_{C_{v}}) ⊕ ⨁_{e∈E(Γ)} H^{0}(C_{e}, O_{C_{e}})

→ ⨁_{v∈V^{II}(Γ)^{(0)}} O_{q_{v}} ⊕ ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}} ( ⨁_{(v,e)∈F} O_{q_{(v,e)}} )

→ H^{1}(C, O_{C})

→ ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}} H^{1}(C_{v}, O_{C_{v}}) ⊕ ⨁_{e∈E(Γ)} H^{1}(C_{e}, O_{C_{e}}) → 0.

The representations of ℂ^{∗} are

0 → (τ) → ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}}(τ) ⊕ ⨁_{e∈E(Γ)}(τ)

→ ⨁_{v∈V^{II}(Γ)^{(0)}}(τ) ⊕ ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}} ( ⨁_{(v,e)∈F}(τ))

→ H^{1}(C, O_{C}) ⊗ (τ) → ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}} H^{1}(C_{v}, O_{C_{v}}) ⊗ (τ) → 0.

a proof of a conjecture of mariño-vafa 333

So

i_{Γ}^{∗}e_{T} (V_{0}) = (τu)^{|E(Γ)|−|V (Γ)|+1} ∏_{v∈V (Γ)} Λ_{g(v)}^{∨}(τu)
i_{Γ}^{∗}e_{T} (V_{D}) = (τu)^{|E(Γ)|−|V (Γ)|+l(μ)} ∏_{v∈V (Γ)} Λ_{g(v)}^{∨}(τu).

We have the following Feynman rules:

i_{Γ}^{∗}e_{T} (V_{D}) = ∏_{v∈V (Γ)} A_{v}^{D},

where

A_{v}^{D} = \begin{cases}Λ_{g(v)}^{∨}(τu) · (τu)^{val(v)−1}, & v ∈ V (Γ)^{(0)}, \\ Λ_{g(v)}^{∨}(τu) · (τu)^{l(μ(v))−1} & v ∈ V (Γ)^{(1)}, \end{cases}

Note that the degree of i_{Γ}^{∗}e_{T} (V_{D}) is

∑_{v∈V (Γ)^{(0)}}(g(v) + val(v) − 1) + ∑_{v∈V (Γ)^{(1)}}(g(v) + l(μ(v)) − 1)

= ∑_{v∈V (Γ)} g(v) + |E(Γ)| − |V (Γ)| + l(μ(v))

= l(μ) + ( ∑_{v∈V (Γ)} g(v) + |E(Γ)| − |V (Γ)| + 1) − 1

= l(μ) + g − 1 as expected.

A.3 The bundle V_{D_{d}}

A.3.1 The target is ℙ^{1} There is only one graph Γ^{0} in this case. When (g, l(μ)) ≠ (0, 1), (0, 2), consider the normalization sequence

0 → f̃^{∗}O_{ℙ^{1}}(−1) → (f̃|_{C_{v_{0}}} )^{∗}O_{ℙ^{1}}(−1) ⊕ ⨁_{i=1}^{l(μ)}(f̃|_{C_{e_{i}}})^{∗}O_{ℙ^{1}}(−1)

→ ⨁_{i=1}^{l(μ)} O_{ℙ^{1}}(−1)_{p_{0}} → 0.

334 c.-c.m. liu, k. liu & j. zhou

The corresponding long exact sequence reads

0 → H^{0}(C, f̃^{∗}O_{ℙ^{1}}(−1))

→ H^{0}(C_{v_{0}}, (f̃|_{C_{v_{0}}} )^{∗}O_{ℙ^{1}}(−1)) ⊕ ⨁_{i=1}^{l(μ)} H^{0}(C_{e_{i}}, (f̃|_{C_{e_{i}}})^{∗}O_{ℙ^{1}}(−1))

→ ⨁_{i=1}^{l(μ)} O_{ℙ^{1}}(−1)_{p_{0}} → H^{1}(C, f̃^{∗}O_{ℙ^{1}}(−1))

→ H^{1}(C_{v_{0}}, (f̃|_{C_{v_{0}}} )^{∗}O_{ℙ^{1}}(−1)) ⊕ ⨁_{i=1}^{l(μ)} H^{1}(C_{e_{i}}, (f̃|_{C_{e_{i}}})^{∗}O_{ℙ^{1}}(−1)) → 0.

The representations of ℂ^{∗} are

0 → 0 → H^{0}(C_{v_{0}}, O_{C_{v_{0}}} ) ⊗ (−τ − 1) → ⨁_{i=1}^{l(μ)}(−τ − 1)

→ H^{1}(C, f̃^{∗}O_{ℙ^{1}}(−1))

→ H^{1}(C_{v_{0}}, O_{C_{v_{0}}} ) ⊗ (−τ − 1) ⊕ ⨁_{i=1}^{l(μ)} ( ⨁_{a=1}^{d(e)−1} (−τ − \frac{a}{d(e)})) → 0.

We have the following Feynman rules:

i_{Γ^{0}}^{∗}e_{T} (V_{D_{d}}) = A_{v_{0}}^{D_{d}} ∏_{e∈E(Γ)} A_{e}^{D_{d}},

where

A_{v_{0}}^{D_{d}} = Λ_{g}^{∨}((−τ − 1)u) · ((−τ − 1)u)^{l(μ)−1}
A_{e}^{D_{d}} = \frac{∏_{a=1}^{d(e)−1}(d(e)τ + a)}{d(e)^{d(e)−1}}(−u)^{d(e)−1}.

It is easily checked that the above Feynmann rules are also valid for (g, l(μ)) = (0, 1), (0, 2).

Note that the degree of i_{Γ}^{∗}e_{T} (V_{D_{d}}) is d + g − 1, as expected.

a proof of a conjecture of mariño-vafa 335

A.3.2 The target is ℙ^{1}[m], m > 0 Given a graph Γ ∈ G_{g,0}(ℙ^{1}, μ), Γ ≠ Γ^{0}, consider the normalization sequence

0 → f̃^{∗}O_{ℙ^{1}}(−1)

→ ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}}(f̃|_{C_{v}})^{∗}O_{ℙ^{1}}(−1) ⊕ ⨁_{e∈E(Γ)}(f̃|_{C_{e}})^{∗}O_{ℙ^{1}}(−1)

→ ⨁_{v∈V^{II}(Γ)^{(0)}} O_{ℙ^{1}}(−1)_{p_{0}} ⊕ ⨁_{v∈V^{S}(Γ)^{(0)}} ( ⨁_{(v,e)∈F} O_{ℙ^{1}}(−1)_{p_{0}} )

⊕ ⨁_{v∈V (Γ)^{(1)}} ( ⨁_{(v,e)∈F} O_{ℙ^{1}}(−1)_{p_{1}} ) → 0.

The corresponding long exact sequence reads

0 → H^{0}(C, f̃^{∗}O_{ℙ^{1}}(−1))

→ ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}} H^{0}(C_{v}, (f̃|_{C_{v}})^{∗}O_{ℙ^{1}}(1))

⊕ ⨁_{e∈E(Γ)} H^{0}(C_{e}, (f̃|_{C_{e}})^{∗}O_{ℙ^{1}}(−1))

→ ⨁_{v∈V^{II}(Γ)^{(0)}} O_{ℙ^{1}}(−1)_{p_{0}} ⊕ ⨁_{v∈V^{S}(Γ)^{(0)}} ( ⨁_{(v,e)∈F} O_{ℙ^{1}}(−1)_{p_{0}} )

⊕ ⨁_{v∈V (Γ)^{(1)}} ( ⨁_{(v,e)∈F} O_{ℙ^{1}}(−1)_{p_{1}} )

→ H^{1}(C, f̃^{∗}O_{ℙ^{1}}(−1))

→ ⨁_{v∈V^{S}(Γ)^{(0)}∪V (Γ)^{(1)}} H^{1}(C_{v}, (f̃|_{C_{v}})^{∗}O_{ℙ^{1}}(1))

⊕ ⨁_{e∈E(Γ)} H^{1}(C_{e}, (f̃|_{C_{e}})^{∗}O_{ℙ^{1}}(−1)) → 0.

336 c.-c.m. liu, k. liu & j. zhou

The representations of ℂ^{∗} are

0 → H^{0}(C, f̃^{∗}O_{ℙ^{1}}(−1))

→ ⨁_{v∈V^{S}(Γ)^{(0)}} H^{0}(C_{v}, O_{C_{v}}) ⊗ (−τ − 1) ⊕ ⨁_{v∈V (Γ)^{(1)}} H^{0}(C_{v}, O_{C_{v}}) ⊗ (−τ)

→ ⨁_{v∈V^{II}(Γ)^{(0)}}(−τ − 1) ⊕ ⨁_{v∈V^{S}(Γ)^{(0)}} ( ⨁_{(v,e)∈F}(−τ − 1))

⊕ ⨁_{v∈V (Γ)^{(1)}} ( ⨁_{(v,e)∈F}(−τ))

→ H^{1}(C, f̃^{∗}O_{ℙ^{1}}(−1))

→ ⨁_{v∈V^{S}(Γ)^{(0)}} H^{1}(C_{v}, O_{C_{v}}) ⊗ (−τ − 1) ⊕ ⨁_{v∈V (Γ)^{(1)}} H^{1}(C_{v}, O_{C_{v}}) ⊗ (−τ)

⊕ ⨁_{e∈E(Γ)} ( ∏_{a=1}^{d(e)−1} (−τ − \frac{a}{d(e)})) → 0.

We have the following Feynman rules:

i_{Γ}^{∗}e_{T} (V_{D_{d}}) = ∏_{v} A_{v}^{D_{d}} · ∏_{e∈E(Γ)} A_{e}^{D_{d}},

where

A_{v}^{D_{d}} = \begin{cases}Λ_{g(v)}^{∨}((−τ − 1)u) · ((−τ − 1)u)^{val(v)−1}, & v ∈ V (Γ)^{(0)}, \\ Λ_{g(v)}^{∨}(−τu) · (−τu)^{val(v)−1}, & v ∈ V (Γ)^{(1)}, \end{cases}

A_{e}^{D_{d}} = \frac{∏_{a=1}^{d(e)−1}(d(e)τ + a)}{d(e)^{d(e)−1}}(−u)^{d(e)−1}.

Note that the degree of i_{Γ}^{∗}e_{T} (V_{D_{d}}) is

∑_{v∈V (Γ)}(g(v) + val(v) − 1) + ∑_{e∈E(Γ)}(d(e) − 1)

= ∑_{v∈V (Γ)} g(v) + 2|E(Γ)| − |V (Γ)| + d − |E|

= d + ( ∑_{v∈V (Γ)} g(v) + |E(Γ)| − |V (Γ)| + 1) − 1

= d + g − 1

a proof of a conjecture of mariño-vafa 337

as expected.

A.4 The obstruction bundle

Combining results in Section A.3 and Section A.2, we obtain Feynman rules for the obstruction bundle

V = V_{D} ⊕ V_{D_{d}}.

A.4.1 The target is ℙ^{1} We have

i_{Γ^{0}}^{∗}(e_{T} (V)) = A_{v_{0}}^{V} ∏_{e∈E(Γ^{0})} A_{e}^{V} ,

where

A_{v_{0}}^{V} = A_{v_{0}}^{D}A_{v_{0}}^{D_{d}} = Λ_{g}^{∨}(τu)Λ_{g}^{∨}(−(τ + 1)u) · (τu(−τ − 1)u)^{l(μ)−1}
A_{e}^{V} = A_{e}^{D}A_{e}^{D_{d}} = \frac{∏_{a=1}^{d(e)−1}(d(e)p + a)}{d(e)^{d(e)−1}}(−u)^{d(e)−1}.

A.4.2 The target is ℙ^{1}[m], m > 0 For any Γ ∈ G_{g,0}(ℙ^{1}, μ), Γ ≠ Γ^{0}, we have

i_{Γ}^{∗}(e_{T} (V)) = ∏_{v∈V (Γ)} A_{v}^{V} · ∏_{e∈E(Γ)} A_{e}^{V} ,

where

A_{v}^{V} = A_{v}^{D}A_{v}^{D_{d}} = \begin{cases}Λ_{g(v)}^{∨}(τu)Λ_{g(v)}^{∨}(−(τ + 1)u) \\ ·(τu(−τ − 1)u)^{val(v)−1}, & v ∈ V (Γ)^{(0)}, \\ (−1)^{g(v)+val(v)−1}(τu)^{r_{1}(v)}, & v ∈ V (Γ)^{(1)} \end{cases}

A_{e}^{V} = A_{e}^{D}A_{e}^{D_{d}} = \frac{∏_{a=1}^{d(e)−1}(d(e)τ + a)}{d(e)^{d(e)−1}}(−u)^{d(e)−1}.

Recall that r_{1}(v) = 2g(v) − 2 + l(μ(v)) + val(v) for v ∈ V (Γ)^{(1)}.

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Department of Mathematics

Harvard University Cambridge, MA 02138

Department of Mathematics

University of California at Los Angeles Los Angeles, CA 90095-1555

Department of Mathematical Sciences Tsinghua University Beijing, 100084, China