ON HYPERGEOMETRIC FUNCTIONS

CONNECTED WITH QUANTUM COHOMOLOGY OF FLAG SPACES

Vadim Schechtman Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, 53225 Bonn

Introduction

In the Givental’s work on the Gromov-Witten invariants for projective complete intersections, [G1], the principal role is played by certain formal power series connected with the quantum cohomology of a manifold. One has a manifold X with a natural torus action, with the finite number of fixed points, and one has a power series Z_{w} associated with each fixed point x_{w}. The coefficients of these series are certain integrals over the spaces of stable maps of genus 0 curves with two marked points to X. These series form a fundamental system of solutions of a certain lisse D-module on a power of the punctured disk. The small quantum cohomology of X coincides with the algebra of functions on its characteristic variety. The series Z_{w} are uniquely determined by certain recursion relations relating Z_{w} with all Z_{w′} if x_{w} is connected with x_{w′} by a fixed line.

The present work consists of three parts. The First part contains nothing new. Here we present the Givental’s computations from [G1], in the simplest case of a projective space, in more detail than in [G1]. In the Second part, we write down the above recursion relations for the flag spaces X = G/B (G being a simple algebraic group), see Theorem II.3.8, which is the main result of this paper. Here X is equipped with the natural action of the maximal torus of G.

In the work [G2], Givental gave another beautiful set of relations which also determines the above mentioned series completely. Namely, these are Toda lattice differential equations (more precisely, we need the equivariant version of the results of [G2]). This set of relations has completely different nature, and it is highly nontrivial fact that both sets of relations determine the same series. In the Third part, we check this by a direct computation for G = SL(3). It turns out that in this case the series Z_{w} admit a nice explicit expression, see III.1.2, III.2.2. (A posteriori it is not surprising, since in this case X admits the Plücker embedding into ℙ^{1} × ℙ^{1}, and one can use another computation by Givental, dealing with the toric complete intersections).

This paper arose from the author’s attempt to make an exercise on the beautiful fixed point technique of Givental and Kontsevich. I am grateful to Yu. I. Manin for many stimulating discussions, to A. Goncharov for the stimulating interest on the first stage of this work, and to V. Batyrev who communicated to me formula III.1.4. This work was done in the highly stimulating atmosphere of Max-Planck-Institut für Mathematik, and I am very grateful to MPI for the hospitality.

Part I. PROJECTIVE SPACES

§1. Equivariant cohomology of ℙ^{n}

1.1. Let X denote the n-dimensional projective space ℙ^{n} = {(z_{0} : …: z_{n})}, the space of lines L ⊂ V = ℂ^{n+1}. The torus T = (ℂ^{∗})^{n+1} acts on X by the rule (α_{0}, …, α_{n}) · (z_{0} : …: z_{n}) = (α_{0}z_{0} : …: α_{n}z_{n}); this action has n + 1 fixed points x_{i} = (0 : … 0 : 1 : 0 … 0) (one on i-th place), i = 0, …, n.

Let L denote the line bundle over X whose fiber over L ⊂ V is L; L has an obvious T -equivariant structure.

Let 𝔱 denote the Lie algebra of T, 𝔱_{ℤ}^{∗} := Hom(T, ℂ^{∗}) ⊂ 𝔱^{∗} the lattice of characters. For λ ∈ 𝔱_{ℤ}^{∗}, let L_{λ} be the T -equivariant line bundle over the point, equal to ℂ, with T acting by means of the character λ. Assigning to λ the first Chern class c_{1}(L_{λ}) ∈ H_{T}^{2} (pt), we identify the graded ring A := H_{T}^{∗} (pt) with ℂ[t_{ℤ}^{∗}] = ℂ[λ_{0}, …, λ_{n}], λ_{i} being the projection on i-th factor.

The graded A-algebra R := H_{T}^{∗} (X) is identified with ℂ[p, λ_{0}, …, λ_{n}]/(∏_{i}(p − λ_{i})) where p := c_{1}(L) ∈ H_{T}^{2} (X). It is computed using the Bott’s fixed point theorem, [AB].

Since our cohomologies will be even anyway, it is convenient to use the grading of the rings A, R, etc. by assigning to p and λ_{i} the degree 1.

1.2. The Euler classes of the tangent spaces at the fixed points x_{i} are equal to

e_{i} := e(T_{X;x_{i}}) = ∏_{b≠i} (λ_{i} − λ_{b}) ∈ H_{T}^{2n}(pt) = A^{n}  (1.1)

Let us consider the restriction map i_{b}^{∗} : H_{T}^{∗} (X) ⟶ H_{T}^{∗} (x_{b}). We have

i_{b}^{∗}(p) = i_{b}^{∗}(c_{1}(L)) = c_{1}(L_{x_{b}})

The fiber L_{x_{b}} is the line ℂ·(0, …, 1, … 0) ⊂ ℂ^{n+1} (1 on b-th place); the Lie algebra 𝔱 acts on this line by means of the character λ_{b}. Therefore,

i_{b}^{∗}(p) = λ_{b},

whence

i_{b}^{∗}(f(p)) = f(λ_{b})  (1.2)

Let A′ be the ring obtained from A by inverting all elements e_{i}, i.e. by inverting all the differences λ_{a} − λ_{b} (a ≠ b). Bott’s theorem says that the restriction map

i^{∗} : H_{T}^{∗} (X) ⟶ H_{T}^{∗} (X^{T}) = ⊕_{b=1}^{n} A · 1_{x_{b}}

becomes an isomorphism after the base change A ⟶ A′. We denote R′ = R⊗_{A} A′.

Let us introduce the elements

ϕ_{i}(p) := ∏_{b≠i} (p − λ_{b}) ∈ R^{n}  (1.3)

Obviously,

ϕ_{i}(λ_{j}) = e_{i}δ_{ij}  (1.4)

It follows from the Bott’s theorem that the set

{ϕ_{i}(p)/e_{i}^{1/2}; i = 0, …, b}  (1.5)

is the basis of orthonormal idempotents of the algebra R′ (to be precise, we should adjoin the square roots of e_{i} to R′).

One can express this in a slightly different way. We have the integration map

∫_{X} : R ⟶ A

of degree −2n, given by

∫_{X} f(p) = ∑_{i} res_{p=λ_{i}} (\frac{f(p)}{∏_{b}(p − λ_{b})}dp) = ∑_{i} \frac{f(λ_{i})}{e_{i}}  (1.6)

We have the Poincaré pairing ⟨·, ·⟩ : R × R ⟶ A,

⟨f, g⟩ = ∫_{X} fg

We have

⟨f, ϕ_{i}⟩ = f(λ_{i}),  (1.7)

hence

⟨ϕ_{i}, ϕ_{j}⟩ = e_{i}δ_{ij}  (1.8)

For each f ∈ R,

f = ∑_{i} \frac{f(λ_{i})}{e_{i}}ϕ_{i}  (1.9)

Note that the rhs a proiori lies in R′ but in fact it belongs to R ⊂ R′, since the lhs does.

§2. Partition function

2.1. Let X_{d} (d ≥ 0) denote the stack of stable maps {f : (C; y_{1}, y_{2}) ⟶ X} of genus 0 curves with two marked points, such that f_{∗}([C]) = d · β where β ∈ H_{2}(X) is the generator dual to p. It is a Deligne-Mumford stack for d ≥ 1.

Let L_{1} be the line bundle over X_{d} whose fiber at a point (f, …) is the tangent space T_{C;y_{1}}; denote c(d) = c_{1}(L_{1}) ∈ H_{T}^{2} (X_{d}).

We want to calculate the following formal power series

Z(q, p) = Z(q, p, λ, h) = 1 + ∑_{d≥1} e_{1∗} (\frac{1}{h + c(d)}) q^{d}  (2.1)

Here

e_{1} : X_{d} ⟶ X

is the evaluation map sending (f, …) to f(y_{1}). This is the same as to compute the series

Z_{i}(q) = ⟨Z(q), ϕ_{i}⟩ = ∑_{d} ∫_{X} ϕ_{i}e_{1∗} (\frac{1}{h + c(d)}) q^{d} = ∑_{d} (∫_{X_{d}} \frac{e_{1}^{∗}ϕ_{i}}{h + c(d)}) q^{d} (2.2)

(i = 0, …, n). First, let us formulate the answer.

2.2. Define the series

S(q, p) = ∑_{d≥0} \frac{1}{∏_{b=0}^{n} ∏_{m=1}^{d}(p − λ_{b} + mh)} · q^{d}  (2.3)

and

S_{i}(q) := S(q, λ_{i}) = ∑_{d≥0} \frac{1}{d! ∏_{b≠i} ∏_{m=1}^{d}(λ_{i} − λ_{b} + mh)} · \frac{q^{d}}{h^{d}}  (2.4)

2.3. Theorem. (a) Z(q, p) = S(q, p).

(b) For all i = 0, …, n, Z_{i}(q) = S_{i}(q).

Of course, (a) and (b) are equivalent, due to the remarks of the previous section. The theorem will be proven in Section 5, after preliminaries in Sections 3, 4.

2.4. Dimension count. The dimension of X_{d} is equal to

dim(X_{d}) = nd + n + d − 1  (2.5)

Indeed, one sees easily that the dimension of the space of maps ℙ^{1} ⟶ ℙ^{n} of degree d is equal to (d + 1)(n + 1) − 1. To get the dimension of X_{d}, we have to subtract from this number 3 (reparametrizations of ℙ^{1}) and add 2 (marked points).

The theorem says that for each d ≥ 0,

e_{1∗} (\frac{1}{h + c(d)}) = \frac{1}{∏_{b=0}^{n} ∏_{m=1}^{d}(p − λ_{b} + mh)}  (2.6)

Let us assign to the variable h the degree 1. The map e_{1∗} decreases the degree by dim(X_{d}) − dim(X) = nd + d − 1. Therefore, the degree of the lhs of (2.6) is equal to

−1 − (nd + d − 1) = −(n + 1)d,

which equals the degree of the rhs. Note that this is true for d = 0 as well (the map e_{1∗} has a positive degree 1 in this case!).

§3. Recursion relation

In this section we will study the series S(q, p) and S_{i}(q).

3.1. As a first remark, note that the normalized series

S_{i}^{norm} := \frac{⟨S, ϕ_{i}⟩}{⟨ϕ_{i}, ϕ_{i}⟩}  (3.1)

may be written as

S_{i}^{norm}(q) = ∑_{d≥0} \frac{1}{∏_{b≠i} ∏_{m=0}^{d}(λ_{i} − λ_{b} + mh)} · \frac{q^{d}}{d!h^{d}} =

= ∑_{d≥0} \frac{1}{∏_{b=0}^{n} ∏_{m=0;(b,m)≠(i,0)}^{d}(λ_{i} − λ_{b} + mh)} · q^{d}  (3.2)

3.2. Define the series

s_{i}(q) := S_{i}(qh) = ∑_{d≥0} b_{i}(d)q^{d}  (3.3)

where

b_{i}(d) = b_{i}(d, λ, h) = \frac{1}{d! ∏_{j≠i} ∏_{m=1}^{d}(λ_{i} − λ_{j} + mh)}  (3.4)

The next theorem is the main result about our series.

3.3. Theorem. For each i = 0, …, n have

s_{i}(q, h) = 1 + ∑_{k>0} ∑_{j≠i} \frac{c_{i}^{j}(k) · q^{k}}{λ_{i} − λ_{j} + kh} · s_{j}(q; \frac{λ_{j} − λ_{i}}{k})  (3.5)

where

c_{i}^{j}(k) = \frac{1}{k! ∏_{b≠i} ∏_{m=1; (b,m)≠(j,k)}^{k} (λ_{i} − λ_{b} + \frac{m(λ_{j}−λ_{i})}{k})}  (3.6)

The relations (3.5), (3.6) uniquely determine the series s_{i}.

The theorem is a variant of a simple fractions decomposition. We use the following elementary fact.

3.4. Lemma. Let f(h) be a non-constant polynomial with distinct roots α_{1}, …, α_{N}. Then

\frac{1}{f(h)} = ∑_{k=1}^{N} \frac{1}{(h − α_{k})f′(α_{k})}  (3.7)

We have

f′(α_{k}) = (\frac{f(h)}{h − α_{k}} ) |_{h=α_{k}}  (3.8)

Indeed, the difference of the rhs and lhs of (3.7) does not have singularities on h ∈ ℙ^{1}, hence it is a constant; but the value of both sides at ∞ is equal to 0, hence they are equal. The formula (3.8) is evident. △

Now let us apply this to the coefficients b_{i}(d), (3.4); we get

b_{i}(d) = \frac{1}{d!} ∑_{j≠i}

∑_{k=1}^{d} \frac{1}{λ_{i} − λ_{j} + kh} · \frac{1}{∏_{b≠i} ∏_{m=1; (b,m)≠(j,k)}^{d}(λ_{i} − λ_{b} + m · \frac{λ_{j}−λ_{i}}{k})}  (3.9)

Let us split the product in the denominator in the two parts:

∏_{m=1}^{d} = ∏_{m=1}^{k} · ∏_{m=k+1}^{d}

The second product is equal to

∏_{m=k+1}^{d} := ∏_{b≠i} ∏_{m=k+1}^{d}(λ_{i} − λ_{b} + m\frac{λ_{j} − λ_{i}}{k}) =

= ∏_{b≠i} ∏_{m′=1}^{d−k}(λ_{i} − λ_{b} + λ_{j} − λ_{i} + m′ · \frac{λ_{j} − λ_{i}}{k}) = ∏_{b≠i} ∏_{m′=1}^{d−k}(λ_{j} − λ_{b} + m′ · \frac{λ_{j} − λ_{i}}{k})

Hence

\frac{1}{∏{d \atop m=k+1}} = \frac{d!}{k!}b_{j}(d − k, \frac{λ_{j} − λ_{i}}{k})  (3.10)

Therefore,

b_{i}(d, h) = ∑_{j≠i} ∑_{k=1}^{d} \frac{1}{λ_{i} − λ_{j} + kh} · \frac{b_{j}(d − k, \frac{λ_{j}−λ_{i}}{k})}{k! ∏_{b≠i} ∏_{m=1; (b,m)≠(j,k)}^{k}(λ_{i} − λ_{b} + m · \frac{λ_{j}−λ_{i}}{k})} =

= ∑_{j≠i} ∑_{k=1}^{d} \frac{1}{λ_{i} − λ_{j} + kh} · c_{i}^{j}(k) · b_{j}(d − k, \frac{λ_{j} − λ_{i}}{k})  (3.11)

Obviously, (3.11) is equivalent to (3.5). This proves (3.5).

The uniqueness is obvious since s_{i}(0) = 1, and the recursion relations determine s_{i}(q) modulo q^{k+1} once we know s_{j}(q) modulo q^{k} for all j ≠ i.

The theorem is proved. △

3.5. In order to get a better feeling what is going on, let us consider some examples. First, the case n = 0 is almost trivial, but gives a nice answer.

We have in this case A = ℂ[λ] (λ := λ_{0}); R = ℂ[p, λ]/(p − λ) = A;

S(q) = S_{0}(q) = e^{q/h}  (3.12)

There are no recursion relations.

3.6. The case n = 1. We have

R = ℂ[p, λ_{0}, λ_{1}]/((p − λ_{0})(p − λ_{1}));

ϕ_{0} = p − λ_{1}, ϕ_{1} = p − λ_{0}; e_{0} = λ_{0} − λ_{1}, e_{1} = λ_{1} − λ_{0}.

It is convenient to introduce the ”root” α := λ_{0} − λ_{1}. We have

s_{0}(q) = s_{0}(q; α) = ∑_{d≥0} \frac{q^{d}}{d! ∏_{m=1}^{d}(α + mh)};  (3.13)

s_{1}(q) = s_{1}(q; α) = ∑_{d≥0} \frac{q^{d}}{d! ∏_{m=1}^{d}(−α + mh)} = s_{0}(q; −α)  (3.13)′

The recursion relations look as follows

s_{0}(q; α, h) = 1 + ∑_{k>0} \frac{c(k; α)q^{k}}{α + kh} · s_{1}(q; α; −\frac{α}{k})  (3.14)

and

s_{1}(q; α, h) = 1 + ∑_{k>0} \frac{c(k; −α)q^{k}}{−α + kh} · s_{0}(q; α; \frac{α}{k})  (3.14)′

where

c(k; α) = \frac{k^{k}}{(k!)^{2}} · \frac{1}{α^{k−1}}  (3.15)

The first few values of c(k; α):

c(1; α) = 1; c(2; α) = \frac{1}{α}; c(3; α) = \frac{3}{4α^{2}}  (3.16)

The relation (3.14)′ is obtained from (3.14) by switching α to −α.

Now let us make a little computation: start building up the series s_{0}, s_{1} using (3.14), (3.14)′. We have

s_{0}(h) = 1 + \frac{q}{α + h}s_{1}(−α) + \frac{q^{2}}{α + 2h} · \frac{1}{α}s_{1}(−\frac{α}{2}) + …

s_{1}(h) = 1 + \frac{q}{−α + h}s_{0}(α) + …

Thus,

s_{0} = 1 + \frac{q}{α + h} + …; s_{1} = 1 + \frac{q}{−α + h} + …

hence

s_{0} = 1 + \frac{q}{α + h} + (\frac{1}{α + 2h} · \frac{1}{α} + \frac{1}{α + h} · \frac{1}{−2α}) q^{2} =

= 1 + \frac{q}{α + h} + \frac{q^{2}}{2(α + h)(α + 2h)} + …  (3.17)

which is the correct answer, up to this order.

3.7. As the last example, assume that n is arbitrary and let us check (3.3), (3.4) up to the first order(sic!) using (3.5), (3.6).

We have

c_{i}^{j}(1) = \frac{1}{∏_{b≠i,j}(λ_{j} − λ_{b})}  (3.18)

Therefore,

s_{i}(h) = 1 + ∑_{j≠i} \frac{q}{λ_{i} − λ_{j} + h} · \frac{1}{∏_{b≠i,j}(λ_{j} − λ_{b})} + … =

= 1 + \frac{q}{∏_{j≠i}(λ_{i} − λ_{j} + h)} + …  (3.19)

where we have used the formula

\frac{1}{∏_{j≠i}(λ_{i} − λ_{j} + h)} = ∑_{j≠i} \frac{1}{∏_{b≠i,j}(λ_{j} − λ_{b})} · \frac{1}{λ_{i} − λ_{j} + h}  (3.20)

§4. First reduction.

4.1. We start proving Theorem 2.3.

Let us define the series z_{i} by

z_{i}(q) = Z_{i}(qh)  (4.1)

(cf. (3.3)). According to Theorem 3.3, in order to prove Theorem 2.3, it suffices to show that z_{i}(q) satisfy the relations (3.5).

4.2. Let us define the coefficients B_{i} by

S_{i}(q) = ∑_{d} B_{i}(d)q^{d}  (4.2)

(cf. (2.4)). Thus, we have

B_{i}(d) = \frac{b_{i}(d)}{h^{d}} = \frac{1}{∏_{b=0}^{n} ∏_{m=1}^{d}(λ_{i} − λ_{b} + mh)}  (4.3)

As we have already noted, (3.5) is equivalent to the identities

b_{i}(d, h) = ∑_{j≠i} ∑_{k=1}^{d} \frac{c_{i}^{j}(k)}{λ_{i} − λ_{j} + kh} · b_{j}(d − k, \frac{λ_{j} − λ_{i}}{k})  (4.4)

or

b_{i}(d, h) = ∑_{j≠i} ∑_{k=1}^{d} \frac{c_{i}^{j}(k)}{λ_{i} − λ_{j} + kh} · (\frac{λ_{j} − λ_{i}}{k} )^{d−k} · B_{j}(d − k, \frac{λ_{j} − λ_{i}}{k})  (4.5)

If we assign to λ_{i} and h the degree 1 then we have

deg b_{i}(d) = −dn; deg B_{i}(d) = −dn − n; deg c_{i}^{j}(k) = −kn + 1  (4.6)

and the identities (4.4), (4.5) are homogeneous, cf. 2.4.

4.3. Let us denote

U_{i}(d) := ∫_{X_{d}} \frac{e_{1}^{∗}ϕ_{i}}{h + c(d)}  (4.7)

Thus,

Z_{i}(q) = ∑_{d≥0} U_{i}(d)q^{d}  (4.8)

We have

U_{i}(d) = \frac{1}{h} ∑_{a≥0} \frac{(−1)^{a}}{h^{a}} ∫_{X_{d}} e_{1}^{∗}ϕ_{i} · c(d)^{a}

We have U_{i}(0) = 1. Assume now that d ≥ 1. The degree of the integral ∫_{X_{d}} e_{1}^{∗}ϕ_{i} · c(d)^{a} is equal to n + a − (nd + n + d − 1) = −nd + a − d + 1 which is less than zero for a < d, hence this integral is zero for these a. Therefore,

U_{i}(d) = \frac{1}{h} ∑_{a≥d} \frac{(−1)^{a}}{h^{a}} ∫_{X_{d}} e_{1}^{∗}ϕ_{i} · c(d)^{a} = \frac{1}{h^{d}} ∫_{X_{d}} \frac{e_{1}^{∗}ϕ_{i} · (−c(d))^{d}}{h + c(d)}

Let us denote

u_{i}(d) := ∫_{X_{d}} \frac{e_{1}^{∗}ϕ_{i} · (−c(d))^{d}}{h + c(d)}  (4.9)

(for d = 0 we set u_{i}(0) := 1). Thus,

z_{i}(q) = ∑_{d≥0} u_{i}(d)q^{d}; u_{i}(d) = U_{i}(d)h^{d}  (4.10)

We have to prove that z_{i}(d) = b_{i}(d). Therefore, Theorem 2.3 is equivalent to

4.4. Theorem. The integrals u_{i}(d) satisfy the relations

u_{i}(d, h) = ∑_{j≠i} ∑_{k=1}^{d} \frac{c_{i}^{j}(k)}{λ_{i} − λ_{j} + kh} · u_{j}(d − k, \frac{λ_{j} − λ_{i}}{k})  (4.10)

This is what we are going to prove in the next section, using the Bott’s localization theorem.

§5. Fixed point formula.

5.1. We will compute the integrals u_{i}(d) (see (4.9)) by means of the Bott’s fixed point formula. It says that

u_{i}(d) = ∑_{P} u_{i}(P),  (5.1)

the summation running over all connected components P ⊂ X_{d}^{T}. Here u_{i}(P) denotes the integral

u_{i}(P) := ∫_{P} (\frac{e_{1}^{∗}ϕ_{i} · (−c(d))^{d}}{h + c(d)} ) |_{P} ·\frac{1}{e(N_{P/X_{d}})}  (5.2)

Here N denotes the normal bundle, e the Euler (top Chern) class.

What do the connected components look like (cf. [K])? Let l_{ij} ⊂ X denote the straight line connecting the points x_{i} and x_{j}. These are the curves in X stable under the action of T.

A point in X_{d}^{T} is a stable map

f : (C; y_{1}, y_{2}) ⟶ X  (5.3)

such that f(y_{i}) ∈ X^{T} = {x_{0}, …, x_{n}} and each irreducible component C_{1} ⊂ C is mapped either to one of the points x_{i} — in this case we call C_{1} vertical, or to one of the lines l_{ij} — in this case we call C_{1} horizontal. The map

f|_{C_{1}} : C_{1} = ℙ^{1} ⟶ l_{ij}  (5.4)

is a finite covering ramified at points x_{i} and x_{j}. The sum of the degrees of these coverings over all horizontal C_{1} should be equal to d.

The connected component P to which the point (5.3) belongs, is specified by the combinatorial data: which irreducible components of C are vertical or horizontal, and the degrees of the coverings (5.4) for horizontal components.

5.2. Let us consider the integral u_{i}(P) (5.2). Let f as in (5.3) be a point in P. We have

e_{1}^{∗}ϕ_{i}|_{P} = ϕ_{i}(λ_{j})  (5.5)

if f(y_{1}) = x_{j}. Therefore, u_{i}(P) may be nonzero only if f(y_{1}) = x_{i}. We will suppose this is the case from now on.

Let C_{1} ⊂ C be the irreducible component containing y_{1}.

5.3. Claim. If u_{i}(P) is nonzero then C_{1} is horizontal.

Indeed, suppose that C_{1} is vertical. Let us call a special point a marked point or a point of intersection of two irreducible components.

The connected component P has the form M_{s+1}×? where M_{s+1} is the Deligne-Mumford moduli stack of genus 0 curves with s + 1 marked points, this whole component mapping to x_{i}. A generic curve in M_{s+1} contains the marked point y_{1}, maybe the marked point y_{2}, and s or s − 1 points of intersection with horizontal curves, depending on whether it contains y_{2} or not; s special points altogether.

We have dim(M_{s+1}) = s − 2. Since the total degree of f is d, the number of horizontal components does not exceed d; therefore, s − 1 ≤ d, hence

dim(M_{s+1}) < d  (5.6)

Consequently, c(d)^{d}|_{P} = 0, i.e. u_{i}(P) = 0. The claim is proven. △

5.4. Now let us consider the component P containing

a stable curve

(f : C = C_{1} ∪ C_{2} ⟶ X) ∈ P ⊂ X_{d}^{T}  (5.7)

where C_{1} is the irreducible component containing the marked point y_{1}, which is mapped with multiplicity k onto the line l_{ij}, and C_{2} is all the rest. The map

f_{1} := f|_{C_{1}}: C_{1} = ℙ^{1} ⟶ l_{ij}  (5.8)

is the k-fold covering ramified only over two points x_{i} and x_{j}, where 1 ≤ k ≤ d. The map

f_{2} := f|_{C_{2}}: C_{2} ⟶ X  (5.9)

belongs to a connected component P_{2} of the fixed point space X_{d−k}^{T}.

We want to compute the integral (5.2). Let us compute the terms under the integral. As we have already noted,

e_{1}^{∗}ϕ_{i}|_{P} = ϕ_{i}(λ_{i}) = e_{i} = ∏_{b≠i}(λ_{i} − λ_{b})  (5.10)

We have

c(d)|_{P} = \frac{λ_{i} − λ_{j}}{k}  (5.11)

5.5. Normal bundle. The most labourous job is to compute the Euler class of the normal bundle over P. We use the Kontsevich formula, (cf. [K], 3.3.1): the class of N_{P/X_{d}} in the Grothendieck group of T -equivariant bundles over P is equal to

[N_{P/X_{d}}] = [H^{0}(C_{1}; f_{1}^{∗}T_{X})_{0 at y}] − [H^{0}(C_{1}; T_{C_{1}})_{0 at y}]+
+[T_{y;C_{1}} ⊗ T_{y;C_{2}}] + [T_{y_{1};C_{1}}] + [N_{P_{2}/X_{d−k}}]  (5.12)

(we use the notations describing a bundle by the fiber at a point f). We have

[H^{0}(C_{1}; f_{1}^{∗}T_{X})_{0 at y}] = [H^{0}(C_{1}; f_{1}^{∗}T_{X})] − [(f_{1}^{∗}T_{X})_{y}]  (5.13)

and

[H^{0}(C_{1}; T_{C_{1}})_{0 at y}] = [H^{0}(C_{1}; T_{C_{1}})] − [T_{y;C_{1}}]  (5.14)

Therefore,

[N_{P/X_{d}}] = ([H^{0}(C_{1}; f_{1}^{∗}T_{X})] − [(f_{1}^{∗}T_{X})_{y})])+
+(−[H^{0}(C_{1}; T_{C_{1}})] + [T_{y_{1};C_{1}}] + [T_{y;C_{1}}]) + [T_{y;C_{1}} ⊗ T_{y;C_{2}}] + [N_{P_{2}/X_{d−k}}] (5.15)

We have

e([T_{y_{1};C_{1}}]) = \frac{λ_{i} − λ_{j}}{k}; e([T_{y;C_{1}}]) = \frac{λ_{j} − λ_{i}}{k}  (5.16)

and

e([H^{0}(C_{1}; T_{C_{1}})]) = \frac{λ_{i} − λ_{j}}{k} · [0] · \frac{λ_{j} − λ_{i}}{k}  (5.17)

so that the second bracket in (5.15) gives simply −[0]. All zeros in this game will cancel out in the final expression for e(N_{P/X_{d}}) (see (5.22) below)!

5.6. Lemma. We have

e([H^{0}(C_{1}; f_{1}^{∗}T_{X})]) = ∏_{b=0}^{n} ∏_{m=0}^{k}(\frac{k − m}{k}λ_{i} + \frac{m}{k}λ_{j} − λ_{b})/[0]  (5.18)

Note that in the product there two factors equal to [0]: they correspond to the values (m, b) = (0, i) or (k, j). One of these zeros is cancelled.

Proof. We have the exact sequence of vector bundles over X = ℙ^{n}:

0 ⟶ O_{X} ⟶ V^{∗} ⊗ O_{X}(1) ⟶ T_{X} ⟶ 0  (5.19)

We have

H^{0}(C_{1}; f_{1}^{∗}(V^{∗} ⊗ O_{X}(1))) = H^{0}(C_{1}; f_{1}^{∗}O_{X}(1)) ⊗ V^{∗}

The Lie algebra 𝔱 acts on H^{0}(C_{1}; f_{1}^{∗}O_{X}(1)) by the characters

\frac{k − m}{k}λ_{i} + \frac{m}{k}λ_{j}; m = 0, …, k,

and on V^{∗} by the characters −λ_{b}, b = 0, …, n. This implies the lemma. △

The remaining zero in (5.18) will cancel out with the zero from (5.17).

5.7. We have

e([(f_{1}^{∗}T_{X})_{y}]) = e([T_{x_{j};X}]) = e_{j} = ∏_{b≠j}(λ_{j} − λ_{b})  (5.20)

Finally, we have

e([T_{y;C_{1}}] ⊗ [T_{y;C_{2}}]) = \frac{λ_{j} − λ_{i}}{k} + c(d − k)  (5.21)

Combining everything together, we have proven

5.8. Lemma. We have

e(N_{P/X_{d}}) = \frac{∏_{b=0}^{n} ∏_{m=0; (b,m)≠(i,0),(j,k)}^{k}(\frac{k−m}{k}λ_{i} + \frac{m}{k}λ_{j} − λ_{b})}{e_{j}}·

·(\frac{λ_{j} − λ_{i}}{k} + c(d − k)) · e(N_{P_{2};X_{d−k}}) △  (5.22)

5.9. Now we can plug our computations in (5.2). We get

u_{i}(P) = \frac{1}{k} · \frac{e_{i} · (\frac{λ_{j}−λ_{i}}{k} )^{d}}{h + \frac{λ_{i}−λ_{j}}{k}} ·

·\frac{1}{∏_{b=0}^{n} ∏_{m=0; (b,m)≠(i,0),(j,k)}^{k}(\frac{k−m}{k}λ_{i} + \frac{m}{k}λ_{j} − λ_{b})}·

· ∫_{P_{2}} \frac{e_{j}}{c(d − k) + \frac{λ_{i}−λ_{j}}{k}} · \frac{1}{e(N_{P_{2}/X_{d−k}})}  (5.23)

The overall factor 1/k is due to the group of automorphisms of the covering f_{1}, having order k. In the big product, the terms with m = 0 give the contribution e_{i}, and the terms with b = i give the contribution

∏_{m=1}^{k}(−\frac{m}{k}λ_{i} + \frac{m}{k}λ_{j}) = k! (\frac{λ_{j} − λ_{i}}{k} )^{k}  (5.24)

The integral is by definition U_{j}(P_{2}) (remember that P_{2} is the connected component of the smaller space X_{d−k}!). Thus, we get

u_{i}(P) = \frac{1}{kh + λ_{i} − λ_{j}} ·

·\frac{1}{k! ∏_{b≠i} ∏_{m=1; (b,m)≠(j,k)}^{k} (λ_{i} − λ_{b} + \frac{m(λ_{j}−λ_{i})}{k})} · (\frac{λ_{j} − λ_{i}}{k} )^{d−k} · U_{j}(P_{2}) =

= \frac{c_{i}^{j}(k)}{kh + λ_{i} − λ_{j}} · (\frac{λ_{j} − λ_{i}}{k} )^{d−k} · U_{j}(P_{2})  (5.25)

This implies formula (4.10) (cf. 4.5). Theorem 4.4, and hence Theorem 2.3 are proven. △

Part II. FLAG SPACES

§1. Equivariant cohomology of flag spaces

1.1. Let V = ℂ^{n+1}. Let X be the variety of complete flags of linear subspaces 0 ⊂ V_{1} ⊂ … ⊂ V_{n} ⊂ V, dim(V_{i}) = i. We set

D := dim(X) = n + (n − 1) + … + 1 = \frac{n(n + 1)}{2}  (1.1)

The torus T = (ℂ^{∗})^{n+1} acts on V. Namely, if v_{0}, …, v_{n} is the standard basis of V, we put

α ∘ (∑ z_{i}v_{i}) = ∑ α_{i}z_{i}v_{i}, α = (α_{i}) ∈ T

By functoriality, T acts on X.

We denote by 𝔱 the Lie algebra of T, and denote by λ_{i} ∈ 𝔱^{∗} the character which is the projection on the i-th component.

1.2. Fixed points. Let F = (V_{1} ⊂ … ⊂ V_{n}) be a flag fixed under the action of T. Since T V_{1} = V_{1}, there exists i_{0} such that V = ℂ · e_{i_{0}}. Taking the quotient, we get the flag F′ = (V_{2}/V_{1} ⊂ … ⊂ V_{n}/V_{1}) in the space V′ = V/V_{1}, fixed under the action of the torus T′ = T/(ℂ^{∗})_{i_{0}}. By induction on n, we conclude that there exists the unique permutation (i_{0}, …, i_{n}) of the set {0, …, n} such that V_{k} is spanned by e_{i_{0}}, …, e_{i_{k−1}} (k = 1, …, n).

We shall denote the group of all bijections w : {0, …, n} ⟶^{∼} {0, …, n} by W, and we shall identify such a bijection with the permutation (w(0), …, w(n)).

The previous discussion identifies the set of fixed points X^{T} with W: to an element w ∈ W correponds the flag

x_{w} = (ℂ · e_{w(0)} ⊂ … ⊂ ⊕_{i=0}^{k}ℂ · e_{w(i)} ⊂ … ⊂ V) ∈ X^{T}  (1.2)

Tangent spaces. As a 𝔱-module, the tangent space T_{w} := T_{X;x_{w}} at the point x_{w}, has the set of characters {λ_{i_{p}} − λ_{i_{q}}, 0 ≤ p < q ≤ n}. Hence, its Euler class is given by

e_{w} := e(T_{w}) = ∏_{p<q} (λ_{i_{p}} − λ_{i_{q}})  (1.3)

Fixed lines. Given a permutation w = (i_{0}, …, i_{n}) and an integer p, (1 ≤ p ≤ n), let s_{p}w be the permutation with i_{p} and i_{p−1} transposed, the other entries remaining in place.

If x_{w} = (V_{1} ⊂ … ⊂ V_{n}), let ℓ_{w,s_{p}w} be the projective line inside X consisting of all flags of the form

V_{1} ⊂ … ⊂ V_{p−1} ⊂ V′ ⊂ V_{p+1} ⊂ … ⊂ V

where all the subspaces V_{i} are fixed, and V′ is varying. This line is fixed under the action of T, and contains exactly two fixed points: x_{w} and x_{s_{p}w}.

However, these are not all fixed lines. In fact, the fixed lines passing through one fixed point, correspond to all positive roots (and we have just described the lines corresponding to the simple roots), cf. [BGG].

Consider the case w = id (identity permutation); so x_{id} is the standard flag, with V_{i} spanned by {e_{0}, …, e_{i−1}}. Given p < q, let s_{pq} ∈ W be the permutation of p and q. For a matrix A = (a_{ij}) ∈ GL(2), set

e_{p;A} = a_{11}e_{p} + a_{21}e_{q}; e_{q;A} = a_{12}e_{p} + a_{22}e_{q}  (1.4)

Let x_{A} be the flag with the spaces V_{i;A} spanned by e_{0;A}, …, e_{i−1;A} where e_{i;A} = e_{i} for i ≠ p, q. When A runs through GL(2), the flags x_{A} form the projective line ℓ_{e;s_{pq}} = GL(2)/B, stable under the action of T, and passing through the fixed points x_{e} and x_{s_{pq}}.

In a similar manner, one defines the T -stable line ℓ_{w;s_{pq}w} passing through x_{w} and x_{s_{pq}w}.

1.3. Cohomology. Let L_{i} be the line bundle over X whose fiber over a flag V_{1} ⊂ … ⊂ V_{n} is equal to V_{i+1}/V_{i}; let u_{i} ∈ H^{2}(X) be its first Chern class. The cohomology algebra of X is equal to

H^{∗}(X) = ℂ[u_{0}, …, u_{n}]/(σ_{0}(u), …, σ(u))  (1.5)

Here σ_{i}(u) are the elementary symmetric functions defined by the rule

∏_{i=0}^{n}(Z + u_{i}) = Z^{n+1} + σ_{0}(u)Z^{n} + … + σ_{n}(u)  (1.6)

The T -equivariant cohomology R = H_{T}^{∗} (X) is an A = H_{T}^{∗} (pt) = ℂ[λ_{0}, …, λ_{n}]- algebra isomorphic to

H_{T}^{∗} (X) = ℂ[u_{0}, …, u_{n}; λ_{0}, …, λ_{n}]/(σ_{0}(u) − σ_{0}(λ), …, σ_{n}(u) − σ_{n}(λ)) (1.7)

For each w = (i_{0}, …, i_{n}) ∈ W, define an element of H_{T}^{2D}(X) by

ϕ_{w} = ∏_{p<q} (u_{i_{p}} − λ_{i_{q}})  (1.8)

Let i_{w} denote the embedding {x_{w}} ↪ X.

1.4. Lemma. We have i_{w}^{∗}ϕ_{w′} = e_{w} · δ_{ww′}.

Proof. It follows from definitions, that

i_{w}^{∗}u_{p} = λ_{w(p)}  (1.9)

On the other hand,

∏_{p<q} (λ_{w(p)} − λ_{q}) = 0  (1.10)

for w ≠ e. This implies the lemma. △ Set

A′ = A[e_{w}^{−1}]_{w∈W} = A[(λ_{p} − λ_{q})^{−1}]_{p<q}; R′ = R ⊗_{A} A′  (1.11)

1.5. Corollary. The elements {ϕ_{w}}_{w∈W} form an A′-basis of the algebra R′. The multiplication in R′ is recovered from the rule

ϕ_{w}ϕ_{w′} = e_{w}δ_{ww′}  (1.12)

Proof. This is a corollary of the Bott’s fixed point theorem, cf. I.1.2. △

§2. Partition function

2.1. It is convenient to switch to an arbitrary flag space X = G/B, G being the simple simply connected algebraic group G associated to a finite root system R. The manifold X is acted upon by T, the maximal torus of G. We identify H_{2}(X; ℤ) with the coroot lattice

H_{2}(X; ℤ) = Q = ⊕_{i∈I} ℤ · α_{i}^{∨},  (2.1)

where α_{i}^{∨} are the simple coroots; it contains the submonoid of positive coroots

Q^{+} = ⊕_{I} ℕ · α_{i}^{∨} ⊂ Q  (2.2)

The cohomology group H^{2}(X; ℤ) is the dual weight lattice

H^{2}(X; ℤ) = P = Char(T) = ⊕_{I} ℤ · ω_{i},  (2.3)

where ω_{i} are the fundamental weights; it contains the submonoid of dominant weights

P^{+} = ⊕_{I} ℕ · ω_{i} ⊂ P  (2.4)

The root system R lies inside P; we denote by R^{+} ⊂ R the subset of positive roots and by {α_{i}}_{i∈I} the set of simple roots. To each λ ∈ P = Char(T) corresponds the line bundle L_{λ} over X, with c_{1}(L_{λ}) = λ.

In the Grothendieck group of T -equivariant bundles, the class of the tangent bundle of X is equal to the sum

[T_{X}] = ⊕_{α∈R^{+}} [L_{α}]  (2.5)

In particular,

dim(X) = card(R^{+})  (2.6)

Let W be the Weyl group of R. The torus T acts on X with the finite set of fixed points {x_{w}}_{w∈W}, each x_{w} lying inside the corresponding Schubert cell (cf. [BGG]). The fixed lines pass through the pairs x_{w}, x_{s_{α}w} where s_{α} is the reflection corresonding to a positive root α.

We identify the cohomology ring A = H_{T}^{∗} (pt) with ℂ[P] = ℂ[α_{i}]_{I}. The Euler classes of the tangent spaces are given by

e_{w} = e(T_{x_{w};X}) = w(e_{id}); e_{id} = ∏_{α∈R^{+}} α  (2.7)

Set A′ = A[α^{−1}]_{α∈R}. Bott’s Theorem gives the A′-base {ϕ_{w}}_{w∈W} in H_{T}^{∗} (X)_{A′}, such that

i_{w}^{∗}ϕ′_{w} = δ_{ww′}e_{w}  (2.8)

2.2. To each β ∈ Q^{+} corresponds the space X_{β} of stable maps

f : (C; y_{1}, y_{2}) ⟶ X; f_{∗}([C]) = β; g(C) = 0  (2.9)

of curves of genus 0 with two marked points, to X. Set |β| = ∑ ν_{i} if β = ∑ ν_{i}α^{∨}.

2.3. Lemma. dim(X_{β}) = 2|β| + dim(X) − 1.

Proof. Let us choose a point (2.9) with C = ℙ^{1}. We have

dim(X_{β}) = dim(T_{f;X_{β}}) = dim(H^{0}(C; f^{∗}T_{X})) − dim(H^{0}(C; T_{C}))+
+ dim(T_{y_{1};C}) + dim(T_{y_{2};C})  (2.10)

Note that H^{1}(C; f^{∗}T_{X}) = 0. Therefore,

dim(H^{0}(C; f^{∗}T_{X})) = χ(C; f^{∗}T_{X}) = ∑_{α∈R^{+}} χ(C; f^{∗}L_{α}) =

= ∑_{α∈R^{+}} (⟨β, α⟩ + 1) = ⟨β, 2ρ⟩ + card(R^{+}) = 2|β| + dim(X),  (2.11)

since ⟨α_{i}^{∨}, ρ⟩ = 1 for each i. Here ρ denotes the half-sum of the positive roots. Obviously, dim(H^{0}(C; T_{C})) = 3. Plugging this into (2.10), we get the lemma. △

In (2.11), we have used the equalities

χ(C; f^{∗}L_{α}) = ⟨β, α⟩ + 1  (2.12)

This number may be negative if the root system is not simply laced.

2.4. Denote

c(β) = c_{1}(T_{1}) ∈ H_{T}^{2} (X_{β})  (2.13)

where T_{1} is the line bundle over X_{β} whose fiber over a point (2.9) is equal to T_{y_{1};C}.

Let e_{1} : X_{β} ⟶ X be the ”evaluation at y_{1}” map. Our aim is to calculate the formal power series

Z(q) = ∑_{β∈Q^{+}} e_{1∗}(\frac{1}{h + c(β)}) · q^{β}  (2.14)

in variables q = (q_{i})_{i∈I}, with coefficients in the ring H_{T}^{∗} (X)(h). Here q^{β} = ∏ q_{i}^{ν_{i}}, for β = ∑ ν_{i}α_{i}^{∨}.

This amounts to calculating |W | power series

Z_{w}(q) = ⟨ϕ_{w}, Z⟩ = ∑_{β∈Q^{+}} (∫_{X_{β}} \frac{e_{1}^{∗}ϕ_{w}}{h + c(β)}) · q^{β}  (2.15)

2.5. Example. For the root system A_{1}, the series Z(q) is given by the expression I, (2.3), with n = 1. The Weyl group has order 2; there are two series Z_{w}(q):

Z_{±}(q) = ∑_{d=0}^{∞} \frac{q^{d}}{d!h^{d} ∏_{m=1}^{d}(±α + mh)} = ∑_{d=0}^{∞} \frac{q^{d}}{d_{0;h}^{!}d_{±α;h}^{!}}  (2.16)

Here we have used the notation

d_{α;h}^{!} = ∏_{m=1}^{d} (α + mh)  (2.17)

The series Z_{+} (resp. Z_{−}) corresponds to the trivial (resp. non-trivial) element of the Weyl group, cf. I, (2.4) and I.3.6.

§3. Fixed point computation

3.1. Let us return to the series (2.15). Denote

I_{w}(β) = ∫_{X_{β}} \frac{e_{1}^{∗}ϕ_{w}}{h + c(β)}  (3.1)

Set

J_{w}(β) = ∫_{X_{β}} \frac{e_{1}^{∗}ϕ_{w} · (−c(β))^{|β|}}{h + c(β)}  (3.2)

The next lemma is proved in the same manner as I, (4.10).

3.2. Lemma. We have

I_{w}(β) = \frac{1}{h^{|β|}} J_{w}(β) △

3.3. Now, we want to compute the integral J_{w}(β) using the fixed point formula. We have

J_{w}(β) = ∑_{P} J_{w}(P)  (3.3)

where

J_{w}(P) = ∫_{P} (\frac{e_{1}^{∗}ϕ_{w} · (−c(β))^{|β|}}{h + c(β)} )|_{P} · \frac{1}{e(N_{P/X_{β}})}  (3.4)

Here P denotes a component of the fixed point space X_{β}^{T}.

Let us compute J_{w}(P). The picture of a connected component is the same as in I, §5. Thus, let f : (C; y_{1}, y_{2}) ⟶ X be a point in P ⊂ X_{β}. The integral J_{w}(P) may be non-zero only if f(y_{1}) = x_{w}, so we will assume this. Let C = C_{1} ∪C_{2} where C_{1} is the connected component containing y_{1} and C_{2} is all the rest. As in I.5.3, we prove that C_{1} is horizontal, i.e. it covers with some multiplicity k > 0 a fixed line ℓ_{w;s_{α}w}, for some α ∈ R^{+}.

To simplify the notations, assume that w = id. We have

e_{1}^{∗}ϕ_{id}|_{P} = e_{id} = e(T_{x_{id};X}) = ∏_{γ∈R^{+}} γ  (3.5)

and

c(β)|_{P} = \frac{α}{k}  (3.6)

3.4. Lemma. We have

e(N_{P/X_{β}}) = (−1)^{k}(k!)^{2} · (\frac{α}{k})^{2k−1} · ∏_{γ∈R^{+}; γ≠α}(\frac{∏_{m≥1} (γ − \frac{m}{k}α)}{∏_{m≥1} (s_{α}γ − \frac{m}{k}α)})·

·\frac{e_{id}}{e_{s_{α}}} · (−\frac{α}{k} + c(β − kα^{∨}))

Proof. We have (cf. I.5.5)

[N_{P/X_{β}}] = [H^{0}(C_{1}; f_{1}^{∗}T_{X})] − [(f_{1}^{∗}T_{X})_{y}]−
−[H^{0}(C_{1}; T_{C_{1}})] + [T_{y_{1};C_{1}}] + [T_{y;C_{1}}] + [T_{y;C_{1}} ⊗ T_{y;C_{2}}] + [N_{P_{2}/X_{β−kα^{∨}}}]  (3.7)

Note that P_{2} (containing the map f|_{C_{2}}) is the connected component of X_{β−kα^{∨}}. We have

e([(f_{1}^{∗}T_{X})_{y}]) = e([T_{x_{sα};X}]) = e_{s_{α}};  (3.8)

next,

e([T_{y_{1};C_{1}}]) = \frac{α}{k}; e([T_{y;C_{1}}] = −\frac{α}{k},  (3.9)

and

e([H^{0}(C_{1}; T_{C_{1}})]) = \frac{α}{k} · [0] · −\frac{α}{k}  (3.10)

We have

e([T_{y;C_{1}} ⊗ T_{y;C_{2}}]) = −\frac{α}{k} + c(β − kα^{∨})  (3.11)

3.5. Lemma. We have

e([H^{0}(C_{1}; f_{1}^{∗}T_{X})]) = ∏_{γ∈R^{+}}(\frac{∏_{m≥0} (γ − \frac{m}{k}α)}{∏_{m≥1} (s_{α}γ − \frac{m}{k}α)})  (3.12)

Proof. As we have already noted, [T_{X}] = ∑_{γ∈R^{+}} [L_{γ}] in the Grothendieck group. Also, H^{1}(C_{1}; f_{1}^{∗}T_{X}) = 0 (convexity of X), and our lemma is the corollary of the following one.

3.6. Lemma. We have

e([χ(C_{1}; f_{1}^{∗}L_{γ})]) = \frac{∏_{m≥0} (γ − \frac{m}{k}α)}{∏_{m≥1} (s_{α}γ − \frac{m}{k}α)}  (3.13)

All but finite number of terms in the fraction cancel out, and we are left with a finite product in the numerator (resp. denominator), if ⟨γ, α^{∨}⟩ is positive (resp. negative). △

Now, Lemma 3.4 follows from Lemma 3.5. Note that the product (3.12) contains one factor equal to zero: in the numerator, coresponding to γ = α, m = k. This zero cancels out with the zero in (3.10), and we are left with the well defined non-zero product.

This completes the proof of Lemma 3.4. △ Substituting this result into (3.4), we get 3.7. Lemma. We have

J_{w}(P; h) = \frac{C_{w}(α; k)}{kh + w(α)} · J_{s_{α}w}(P_{2}; −\frac{w(α)}{k})  (3.14)

where

C_{w}(α; k) = w(C_{id}(α; k))  (3.15)

and

C_{id}(α; k) = (−1)^{k(|α|+1)}α^{k|α|−2k+1} · \frac{k^{k(2−|α|)}}{(k!)^{2}} ·

· ∏_{γ∈R^{+}; γ≠α}(\frac{∏_{m≥1} (s_{α}γ − \frac{m}{k}α)}{∏_{m≥1} (γ − \frac{m}{k}α)}) △  (3.16)

Here we use the notation |α| = ∑ a_{i} for α = ∑ a_{i}α_{i}. Let us introduce the series

z_{w}(q) = ∑_{β} J_{w}(β)q^{β} = Z_{w}(hq)  (3.17)

3.8. Theorem. We have

z_{w}(q; h) = 1 + ∑_{α∈R^{+}; k>0} \frac{C_{w}(α; k)q^{kα^{∨}}}{kh + α} · z_{s_{α}w}(q; −\frac{w(α)}{k})  (3.18)

where C_{w}(a; k) are given by the formulas (3.15), (3.16).

This is an immediate corollary of Lemma 3.7. △ Let us look more attentively at the expression (3.16). Recall 3.9. Lemma ([B], Ch. VI, §1, 1.6, Prop. 17, Cor. 2). Let

w = s_{1} · . . . · s_{q}  (3.19)

be a reduced decomposition of an element of the Weyl group, where s_{i} is the reflection correposnding to a simple root α_{i}. Then the roots γ_{i} = s_{q}s_{q−1} · . . . · s_{i+1}(α_{i}) (i = 1, …, q) are all positive, distinct, and

R^{+} ∩ w^{−1}(−R^{+}) = {θ_{1}, …, θ_{q}} △  (3.20)

Obviously, if in the product in (3.16) both γ and s_{α}γ are positive, then the two factors corresponding to them cancel out. The previous lemma says that we must keep only l(s_{α}) − 1 terms in the product (we have already taken care of the term γ = α).

3.10. Corollary. If the root α is simple then

C_{id}(α; k) = \frac{k^{k}}{(k!)^{2}} α^{−k+1}  (3.21)

with

In fact, for a simple α, the big product disappears at all. The expression (3.21) coincides with I, (3.15).

Part III. COMPUTATIONS FOR SL(3)

§1. Formula

1.1. In this part, X will denote the flag manifold G/B, with G = SL(3). According to [G2], the quantum cohomology of X is given by the Fourier transform of the following D-module on the three-torus.

Let A = (a_{ij}) be the 3 × 3-matrix with a_{ii} = u_{i−1} (i = 1, 2, 3); a_{i,i−1} = v_{i} (i = 2, 3); a_{i,i+1} = −1 (i = 1, 2); a_{13} = a_{31} = 0. Consider the characteristic polynomial

det(λ + A) = λ^{3} + P_{1}λ^{2} + P_{2}λ + P_{3} = λ^{3} + (u_{0} + u_{1} + u_{2})λ^{2}+
+(u_{0}u_{1} + u_{0}u_{2} + u_{1}u_{2} + v_{1} + v_{2})λ + u_{0}u_{1}u_{2} + u_{0}v_{2} + u_{2}v_{1}  (1.1)

Thus, the polynomials P_{i} = P_{i}(u; v) are the deformed symmetric functions.

Consider the three-dimensional torus T, with multiplicative coordinates q_{0}, q_{1}, q_{2}. We define the differential operators D_{i} on T, where D_{i} is obtained from P_{i} by the substitution u_{j} = q_{j}∂_{q_{j}}, v_{j} = q_{j}/q_{j−1}. We are interested in the solutions of the system

D_{1}ϕ = D_{2}ϕ = D_{3}ϕ = 0,  (1.2)

ϕ = ϕ(q_{0}, q_{1}, q_{2}).

First of all, since D_{1}ϕ = 0, the function ϕ depends in fact only on the quotients v_{1} = q_{1}/q_{0}, v_{2} = q_{2}/q_{1}. It is useful to write up the expressions of the operators q_{i}∂_{q_{i}} acting on such functions, in coordinates v_{1}, v_{2}:

q_{0}∂_{q_{0}} = −v_{1}∂_{1}; q_{1}∂_{q_{1}} = v_{1}∂_{1} − v_{2}∂_{2}; q_{2}∂_{q_{2}} = v_{2}∂_{2}  (1.3)

(we set for brevity ∂_{i} = ∂_{v_{i}}).

In these coordinates, the remaining operators look as follows.

D_{2} = −(v_{1}∂_{1})^{2} + (v_{1}∂_{1})(v_{2}∂_{2}) − (v_{2}∂_{2})^{2} + v_{1} + v_{2}  (1.4)
D_{3} = −(v_{1}∂_{1})^{2}v_{2}∂_{2} + v_{1}∂_{1}(v_{2}∂_{2})^{2} − v_{2}(v_{1}∂_{1}) + v_{1}(v_{2}∂_{2})  (1.5)

1.2. Theorem. There exists a unique, up to a multiplicative constant, solution ϕ(v_{1}, v_{2}) of the system

D_{2}ϕ = D_{3}ϕ = 0  (1.6)

in the ring of formal power series ℚ[[v_{1}, v_{2}]].

If we normalize ϕ by the condition ϕ(0) = 1, it will be given by the formula

ϕ(v_{1}, v_{2}) = ∑_{i,j≥0} \frac{(i + j)!}{(i!)^{3}(j^{!})^{3}} v_{1}^{i}v_{2}^{j}  (1.7)

Here D_{2}, D_{3} are given by (1.4), (1.5).

Proof. Let us denote the power series (1.7) by ∑_{i,j≥0} a_{ij}v_{1}^{i}v_{2}^{j}, a_{00} = 1. Equation D_{2}ϕ = 0 is equivalent to the recursion

(i^{2} − ij + j^{2})a_{ij} = a_{i−1,j} + a_{i,j−1}  (1.8)

(we imply that a_{ij} = 0 if either i or j is negative). Equation D_{3}ϕ = 0 is equivalent to

ij(i − j)a_{ij} = −ia_{i,j−1} + ja_{i−1,j}  (1.9)

Both formulas are checked immediately for a_{ij} given by (1.6). Already (1.8) defines all a_{ij} uniquely from a_{00}. △

1.3. Remarks. (a) We have a_{ij} = a_{ji}.

(b) It follows from (1.8) that

a_{i0} = \frac{1}{(i!)^{2}}  (1.10)

The function

ϕ(v, 0) = ∑ \frac{v^{i}}{(i!)^{2}}  (1.11)

conicides with the hypergeometric function associated with G/B for G = SL(2). (c) The formulas (1.8) and (1.9) imply the identity

i^{3}a_{i,j−1} − j^{3}a_{i−1,j} = 0  (1.12)

This, together with (1.10), gives immediately (1.7).

1.4. Another formula. V. Batyrev communicated to me another formula for the solution of (1.6): ϕ̃ = ∑_{i,j} b_{ij}v_{1}^{i}v_{2}^{j} where

b_{ij} = \frac{1}{(i!)^{2}(j!)^{2}} ∑_{r} C_{i}^{r}C_{j}^{r}  (1.13)

where

C_{b}^{a} = \frac{b!}{a!(b − a)!}  (1.14)

for integers a, b such that 0 ≤ a ≤ b, and 0 if a < 0 or a > b.

Claim. The function ϕ̃ is equal to ϕ.

Indeed, our claim is equivalent to the identity

∑_{r} C_{i}^{r}C_{j}^{r} = C_{i+j}^{i}  (1.15)

To prove this, one remarks that to choose i elements from a set which is a disjoint union of two sets of cardinalities j and i, is the same as to choose some r elements from the first set, and i − r from the second one.

§2. Equivariant version

Below, we will deform equations (1.6) and the solution (1.7). In terms of quantum cohomology, this deformation corresponds to passing to the equivariant cohomology, with respect to a natural action of a three-torus on X.

2.1. Introduce differential operators

D_{i;h;λ} = P_{i}(hq_{0}∂_{q_{0}}, hq_{1}∂_{q_{1}}, hq_{2}∂_{q_{2}}; q_{1}/q_{0}, q_{2}/q_{1}) − σ_{i}(λ_{0}, λ_{1}, λ_{2})  (2.1)

Here P_{i}(u_{0}, u_{1}, u_{2}; v_{1}, v_{2}) are the polynomials (1.1); σ_{i}(λ) = P_{i}(λ; 0) are the elementary symmetric functions.

We are interested in the solutions of the system

D_{1;h;λ}ψ = D_{2;h;λ}ψ = D_{3;h;λ}ϕ = 0  (2.2)

of the form

ψ(q_{0}, q_{1}, q_{2}) = q_{0}^{λ_{0}/h}q_{1}^{λ_{1}/h}q_{2}^{λ_{2}/h}ϕ(q_{0}, q_{1}, q_{2}); ϕ(q) = ∑_{i,j,k} b_{ijk}q_{0}^{i}q_{1}^{j}q_{2}^{k}  (2.3)

The equation D_{1;h;λ}ϕ = 0 implies i + j + k = 0, thus, for a solution ψ, the factor ϕ(q) would depend only on v_{1} = q_{1}/q_{0} and v_{2} = q_{2}/q_{1}.

To formulate the answer, we introduce the notations α_{i} = λ_{i} − λ_{i−1};

p_{α}^{!} = (h + α)(2h + α) · . . . · (ph + α)  (2.4)

2.2. Theorem. There exists a unique, up to a multiplicative constant, solution of the system (2.2) having the form (2.3), with ϕ ∈ ℚ[[v_{1}, v_{2}]], v_{1} = q_{1}/q_{0}, v_{2} = q_{2}/q_{1}. If we normalize it by the condition ϕ(0) = 1, it will have the form

ϕ(v_{1}, v_{2}) = ∑_{i,j≥0} \frac{(i + j)_{α_{1}+α_{2}}^{!}}{i!j!i_{α_{1}}^{!}j_{α_{2}}^{!}i_{α_{1}+α_{1}}^{!}j_{α_{1}+α_{2}}^{!}} \frac{v_{1}^{i}v_{2}^{j}}{h^{i+j}}  (2.5)

Proof. Let us denote by a_{ij} the coefficient at v_{1}^{i}v_{2}^{j} of the unknown ϕ. The equation D_{2;h;λ}ϕ = 0 is equivalent to

(i^{2}h^{2} − ijh^{2} + j^{2}h^{2} + ihα_{1} + jhα_{2})a_{ij} = a_{i−1,j} + a_{i,j−1}  (2.6)

(cf. 1.8)). The equation D_{3;h;λ}ϕ = 0 is equivalent to

[(ih − λ_{0})(ih − jh + λ_{1})(jh + λ_{2}) + λ_{0}λ_{1}λ_{2}]a_{ij} =
= (−ih + λ_{0})a_{i,j−1} + (jh + λ_{2})a_{i−1,j}  (2.7)

(cf. 1.9)). These identities are checked directly. The uniqueness follows from (2.6). △ 2.3. Remarks. We have a_{ij}(α_{1}, α_{2}) = a_{ji}(α_{2}, α_{1}). Equation (2.6) implies

a_{i0} = \frac{1}{h^{i}i!i_{α_{1}}^{!}}  (2.7)

(cf. (1.10)). On the other hand, (2.6) and (2.7) together imply the identity

ih(ih + α_{1})(ih + α_{1} + α_{2})a_{i,j−1} − jh(jh + α_{2})(jh + α_{1} + α_{2})a_{i−1,j} = 0 (2.8)

(cf. (1.12)). This and (2.7) give the expression (2.5).

§3. Comparison with the fixed point method

3.1. Let us see what does Theorem II.3.8 give for G = SL(3), i.e. for the root system A_{2}. The simple roots are α_{1}, α_{2}, the positive ones are α_{1}, α_{2}, α := α_{1} + α_{2}.

The Weyl group W = Σ_{3} has two Coxeter generators s_{1}, s_{2} corresponding to the simple roots, and s_{α} = s_{1}s_{2}s_{1}. For example, s_{1} takes α_{1} to −α_{1} and α_{2} to α, etc.

We have six series z_{w} (w ∈ W):

z_{w} = z_{w}(q_{1}, q_{2}; α_{1}, α_{2}; h) = ∑_{i≥0,j≥0} a_{ij;w}q^{i}q^{j};  (3.1)

where

a_{ij;w} = w(a_{ij}); a_{ij} = a_{ij}(α_{1}, α_{2}; h) := a_{ij;id}  (3.2)

The element w acts only on the arguments α_{1}, α_{2}.

It is easy to see that the recursion relation of Theorem II.3.8 takes the following form.

3.2. Theorem. We have

z_{id}(q; h) = 1 + ∑_{k>0} \frac{q_{1}^{k}}{kh + α_{1}} · \frac{k^{k}}{(k!)^{2}} · \frac{1}{α_{1}^{k−1}} · z_{s_{1}}(q; −α_{1}/k)+

+ ∑_{k>0} \frac{q_{2}^{k}}{kh + α_{2}} · \frac{k^{k}}{(k!)^{2}} · \frac{1}{α_{2}^{k−1}} · z_{s_{2}}(q; −α_{2}/k) − ∑_{k>0} (\frac{q_{1}^{k}q_{2}^{k}}{kh + α_{1} + α_{2}} · \frac{k^{2(k−1)}}{(k!)^{2}} ·

·\frac{α_{1} + α_{2}}{α_{1}α_{2}} · \frac{1}{∏_{m=1}^{k−1} (mα_{1} − (k − m)α_{2})^{2}} · z_{s_{α}}(q; −(α_{1} + α_{2})/k)) △  (3.3)

Now, the main result of this section is 3.3. Theorem. The recursion relation (3.3) is satisfied, with

a_{ij} = \frac{(i + j)_{α_{1}+α_{2}}^{!}}{i!j!i_{α_{1}}^{!}j_{α_{2}}^{!}i_{α_{1}+α_{2}}^{!}j_{α_{1}+α_{2}}^{!}}  (3.4)

Proof. Assume that i ≤ j. Let us write a_{ij} in the form

a_{ij}(α_{1}, α_{2}; h) = \frac{[(i + j)h + α_{1} + α_{2}] · . . . · [(1 + j)h + α_{1} + α_{2}]}{(jh + α_{1} + α_{2}) · . . . · (h + α_{1} + α_{2})}·

·\frac{1}{i!(h + α_{1}) · . . . · (ih + α_{1}) · j!(h + α_{2}) · . . . · (jh + α_{2})}  (3.5)

The denominator (as a function of h) has the distinct roots:

h = −\frac{α_{1} + α_{2}}{k} (k = 1, …, i); h = −\frac{α_{1}}{k} (k = 1, …, i);

h = −\frac{α_{2}}{k} (k = 1, …, j).  (3.6)

Accordingly, we have the simple fraction decomposition

a_{ij}(α_{1}, α_{2}; h) = ∑_{k=1}^{i} \frac{^{1}b_{ij}^{k}(α_{1}, α_{2})}{kh + α_{1}} + ∑_{k=1}^{j} \frac{^{2}b_{ij}^{k}(α_{1}, α_{2})}{kh + α_{2}} + ∑_{k=1}^{i} \frac{^{3}b_{ij}^{k}(α_{1}, α_{2})}{kh + α_{1} + α_{2}} (3.7)

The theorem is equivalent to 3.4. Lemma. We have

(a) For 1 ≤ k ≤ i,

1b_{ij}^{k}(α_{1}, α_{2}) = \frac{k^{k}}{(k!)^{2}} · \frac{1}{α_{1}^{k−1}} · a_{i−k,j}(−α_{1}, α_{1} + α_{2}; −α_{1}/k);  (3.8)

(b) For 1 ≤ k ≤ j,

2b_{ij}^{k}(α_{1}, α_{2}) = \frac{k^{k}}{(k!)^{2}} · \frac{1}{α_{2}^{k−1}} · a_{i,j−k}(α_{1} + α_{2}, −α_{2}; −α_{2}/k);  (3.9)

(c) For 1 ≤ k ≤ i,

3b_{ij}^{k} = −\frac{k^{2(k−1)}}{(k!)^{2}} · \frac{α_{1} + α_{2}}{α_{1}α_{2}} · \frac{1}{∏_{m=1}^{k−1} [mα_{1} − (k − m)α_{2}]^{2}} ·

·a_{i−k,j−k}(−α_{2}, −α_{1}; −(α_{1} + α_{2})/k)  (3.10)

This lemma is established by a direct computation. The theorem is proved. △ 3.5. Corollary. The series Z_{id}(q_{1}, q_{2}) = z_{id}(q_{1}/h, q_{2}/h) coincides with the

series ϕ(q_{1}, q_{2}) from (2.5). △

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