Conformal blocks in the tensor product of vector representations and localization formulas

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

R. RIMÁNYI, A. VARCHENKO

Conformal blocks in the tensor product of vector representations and localization formulas

Tome XX, n^o1 (2011), p. 71-97.

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Conformal blocks in the tensor product of vector representations and localization formulas

R. Rim´anyi⁽¹⁾ and A. Varchenko⁽²⁾

ABSTRACT.— Using equivariant localization formulas we give a formula for conformal blocks at level one on the sphere as suitable polynomials.

R^´ESUM´E.— Using equivariant localization formulas we give a formula for conformal blocks at level one on the sphere as suitable polynomials.

1. Introduction

We consider conformal blocks on the Riemann sphere in thesl_m Wess- Zumino-Novikov-Witten conformal field theory. For a partition λ = (λ1, λ2, . . . , λm) ∈ N^m (λi λi+1)we denote |λ| =

λi, and d(λ) = λ1−λm. We fix the levelof the theory withd(λ)and distinct points z₁, . . . , z_|λ|,∞on the sphere. We assign to each finite pointz_a the standard m-dimensional vector representation ofsl_m, denoted byV, and to infinity – the irreduciblesl_mrepresentation with highest weight (−λ_m, . . . ,−λ₁). The associated space of conformal blocks CB_z(λ)can be realized as a vector sub- space of the tensor productV^⊗|λ|, and the tensor product can be realized as a suitable vector space of polynomials. The subspace of conformal blocks

(∗) Re¸cu le 04/04/2010, accept´e le 04/05/2010

The first author is supported by the Marie Curie Fellowship PIEF-GA-2009-235437 and NSA award CON:H98230-10-1-0171. He also thanks the hospitality of MPIM Bonn in November 2009. The second author is supported in part by NSF grant DMS-0555327.

(1) Department of Mathematics, University of North Carolina at Chapel Hill, USA [email protected]

(2) Department of Mathematics, University of North Carolina at Chapel Hill, USA [email protected]

CB_z(λ)⊂V^⊗|λ|is defined as the set of solutions to a system of differential equations due to the description of conformal blocks in [FSV1, FSV2]. We solve that system for= 1. In that case dim CB¹_z(λ)= 1 and we give a for- mula for one remarkable polynomial generating the one-dimensional space of conformal blocks (form= 2 a formula was given in [V]).

A striking property of the formula is its similarity to equivariant local- ization formulas. According to these formulas, if a torus acts on a compact manifold with a finite fixed point set, then the integral of an equivariant cohomology class on the manifold can be computed by collecting some data at the fixed points. From these data one writes down a rational function which will be equal to the integral of the equivariant cohomology class. Not only our conformal block has the structure of the rational function from a localization formula, but the proof showing that it is indeed a conformal block uses equivariant localization. We hope that this connection between conformal field theory and equivariant cohomology will be useful in both areas in the future. More conceptual discussion of this connection see in [RSV].

A Selberg-type integral is a multidimensional integral that can be calcu- lated explicitly, see [S, As, Op, Sp]. Conformal blocks in the Wess-Zumino- Novikov-Witten conformal field theory can be represented by multidimen- sional hypergeometric integrals due to [SV]. According to a general principle in [MV], if the space of conformal blocks is one-dimensional, then the hy- pergeometric integral representing the conformal block can be calculated explicitly giving a Selberg-type integral. See demonstrations of that princi- ple in [TV], [FSV], [W]. Our formula for the conformal block at level one can produce a new Selberg-type integral. For m = 2 such an integral was described in [V]. We plan to describe the corresponding integral for an ar- bitrarym in [RSV].

The main object of this paper are the functions Pz(λ)introduced in Section 4. The main results are Theorems 4.4-4.7 describing conformal block properties of functionsPz(λ). Theorem 5.1 says thatPz(λ)satisfies the KZ equation ifd(λ)1.

More comments on conformal blocks are collected in Sections 6-7. Taking suitable products of conformal blocks at level one, we construct in Section 6 elements in the space of conformal blocks CB_z(λ)at any level. We show in Section 7 that the constructed elements generate CB_z(λ)for genericz.

The proofs are based on our formula for conformal blocks at level one.

In this paper we assign the vector representations to all finite points z₁, . . . , z_|λ|. The case of more general representations assigned to finite points may be studied by fusion procedure.

The authors thank V. Schechtman for stimulating and useful discussions.

Conventions. The set of natural numbers is N = {0,1,2, . . .}. Certain expressions in this paper are parameterized by partitionsλ. In notation we put the partition in bracket, e.g. CB(λ),P(λ),s(λ). However, for concrete partitions, e.g. λ= (4,2,1), we do not repeat brackets, that is, we do not writeP( (4,2,1)), we will simply writeP(4,2,1).

2. Spaces of conformal blocks

2.1. The Lie algebras gl_m,sl_m and their representations

Let e_i,j,i, j = 1, . . . , m, be the standard generators of the complex Lie algebragl_msatisfying the relations [e_i,j, e_s,k] =δ_j,se_i,k−δ_i,ke_s,j. We identify the Lie algebrasl_m with the subalgebra in gl_m generated by the elements e_i,i−e_j,j ande_i,j fori=j,i, j= 1, . . . , m.

A vector uin a gl_m-module has weight λ = (λ1, λ2, . . . , λm) ∈ C^m, if ei,iu=λiufori= 1, . . . , m. The vectoruis calledsingularifei,ju= 0 for 1i < jm.

Letλ= (λ₁, λ₂, . . . , λ_m)∈N^m be a partition, i.e. letλ_i λ_j fori < j.

Throughout the paper we will use the following shorthand notations:|λ|= λ_i, andd(λ) =λ₁−λ_m. The irreducible finite dimensionalgl_m-module with highest weightλwill be denoted byL_λ. The moduleL(1,0,...,0) is the standard m-dimensional vector representation ofgl_m. We will denote it by V. The module V, and other gl_m-modules will also be considered as sl_m- modules.

The Lie algebragl_macts onC[y⁽¹⁾, . . . , y^(m)] by the differential operators ei,j → y⁽ⁱ⁾∂/∂y^(j). This action preserves the degree of polynomials. The standard gl_m-moduleV is identified with the subspace ofC[y⁽¹⁾, . . . , y^(m)] consisting of homogeneous polynomials of degree one.

For a given partitionλ∈N^m, the polynomial ring

R_m,λ=C[y₁⁽¹⁾, . . . , y_|⁽¹⁾_λ|, y⁽²⁾₁ , . . . , y⁽²⁾_|λ|, . . . , y^(m)₁ , . . . , y^(m)_|λ| ] (2.1)

admits a gl_m-module structure by the rule e_i,j → _|λ|

a=1ya⁽ⁱ⁾∂/∂y^(j)a . The gl_m-moduleV^⊗|λ|is identified with the subspace ofR_m,λconsisting of poly- nomials that have homogeneous degree 1 with respect to each m-tuple of variablesya⁽¹⁾, . . . , y^(m)a , fora= 1, . . . ,|λ|.

2.2. Conformal blocks

Consider the Lie algebragl_m, its vector representationV, a partitionλ, and the gl_m-moduleV^⊗|λ| as identified with a subspace of R_m,λ in (2.1).

Recall the action ofei,j on the latter, namely

e_i,j=

|λ_|

a=1

y_a⁽ⁱ⁾∂/∂y_a^(j). (2.2) The space of singular vectors of weightλis

SV(λ) ={p∈V^⊗|λ||ei,jp= 0, ei,ip=λipfor 1i < j m}.

Fixing distinct complex numbersz= (z1, . . . , z_|λ|)we define the differential operators

e^z_i,j=

|λ|

a=1

zay_a⁽ⁱ⁾ ∂

∂y^(j)a

, for 1i, jm. (2.3)

For a positive integer d(λ)we define the space of conformal blocks at level by

CB_z(λ) ={p∈SV(λ)|

e^z_1,m−d(λ)+1

p= 0}.

If−d(λ)+ 1 is greater than they^(m)-degree (namely, λ_m)of singular vectors fromSV(λ), then the defining equation of levelconformal blocks is vacuous, hence we have

CB^d_z^(λ)(λ)⊂CB^d(_z ^λ)+1(λ)⊂. . .⊂CB^λ_z¹(λ) = SV(λ).

Let us emphasize that in the definition above, as well as in the whole paper,zdenotes a collection ofdistinctcomplex numbers.

Remark 2.1. — This definition of conformal blocks is nonstandard. Usu- ally the space of conformal blocks in a WZW model is defined if one has a set of distinct points on a Riemann surface marked with irreducible rep- resentations of an affine Lie algebra, see [KL]. If the Riemann surface is

the Riemann sphere, then one can describe the space of conformal blocks in terms of finite dimensional representations of the corresponding finite di- mensional Lie algebra. That description is one of two main results of [FSV1]

and [FSV2]. We take that description as our definition for the case when the marked points of the Riemann sphere arez₁, . . . , z_|λ|,∞and the associated representations are the standard sl_m-modules assigned to all finite points za and the irreducibleslm highest weight module assigned to the point at infinity with the highest weight (−λm, . . . ,−λ1).

2.3. “Symmetry and vanishing” description of conformal blocks forλ= (N, . . . , N)

Assume thatλ= (N, . . . , N)∈N^m. In this case we can interpret the space of conformal blocks as follows.

LetC=C^m|λ|be the vector space with coordinatesya⁽ⁱ⁾fori= 1, . . . , m, a= 1, . . . ,|λ|. The polynomialsp∈V^⊗|λ|are functions onC. Consider the vector subspace

A(z) ={γ∈ C |y^(m)_a (γ) =z_ay⁽¹⁾_a (γ), a= 1, . . . ,|λ|}

of codimension|λ|inC.

For any integer 1k|λ|and a subsetBk ={1b1< . . . < bk |λ|}

introduce the differential operator

∂_Bk = k i=1

y⁽¹⁾_b

i

∂

∂y_b^(m)

i

.

The special linear group SLm acts diagonally on the polynomial ring R_m,λof (2.1)by substitution in each set ya⁽¹⁾, . . . , ya^(m)ofm-tuples of vari- ables. Hence it acts on the subspace identified with V^⊗|λ| too. If λ = (N, . . . , N), then the subspace of singular vectors SV(λ) ⊂ V^⊗|λ| is the subspace of SL_m-invariant polynomials.

The following theorem is theSLm analogue of [R, Thm. 4.3], also [LV, Lemma 1.3]. Its proof is straightforward calculation.

Theorem 2.2. — Letλ= (N, . . . , N) for someN ∈N. Then anSLm- invariant polynomialp∈V^⊗|λ| lies inCB_z(λ), if and only if

(∂Bkp)|_A(z)= 0 (2.4)

for all B_k = {1 b₁ < . . . < b_k |λ|} with k N −−1. In other words, anSL_m-invariant polynomialp∈V^⊗|λ|lies inCB_z(λ), the space of conformal blocks, if and only if it vanishes atA(z) to orderN−.

Corollary 2.3. — Ifλ= (N, . . . , N)for some N ∈N, then anSLm- invariant polynomial p∈V^⊗|λ| lies in CB_z(λ)if and only if p vanishes to orderN− at theSLm-orbit of A(z).

3. Rational functions: localizations, divided differences In this section we recall equivariant localization and divided difference formulas. Conformal blocks of Section 4 (eg. (4.1), (4.2)) have the structure of some expressions from these formulas. Moreover, some of these formulas will be used in the proofs of Theorems 4.4 and 4.5.

3.1. Rational function identities obtained from localization formulas

The resultant of the setsAandB of variables is defined to be

R(A|B) =

a∈A

b∈B

(a−b).

Later we will also drop the set signs, and write eg.R(a|b, c)forR({a}|{b, c}) = (a−b)(a−c). For sets A1, A2, . . . , Am of variables R(A1|A2|. . .|Am)we will denote the generalized resultant

i<jR(Ai|Aj). For a setI of indices, zI will denote the set of variables{zi}i∈I.

Lemma 3.1. — Let k < n be positive integers. Let p (resp. q) be sym- metric polynomials in k (resp. n−k) variables. Let _{1,...,n}

k

denote the set of k-element subsets of{1, . . . , n}. For such a subset I, let I¯denote the complement set{1, . . . , n} −I. Then for deg(p)+ deg(q)< k(n−k),

I∈(^{1,...,n}k )

p(zI)q(zI¯) R(zI|zI¯) = 0

is an identity of rational functions in the variablesz1, . . . , zn.

This lemma is well known in at least two areas of mathematics: the the- ory of symmetric functions (in terms of generalized Lagrange interpolation, see e.g. [L, Thm. 7.7.1]), and in the theory of equivariant localization. Let us recall this latter argument. If the torusT acts on the compact manifold

M with fixed pointsf₁, . . . , f_r, andα∈H_T^∗(M;Q)is an equivariant coho- mology class, then the Atiyah-Bott equivariant localization theorem [AB]

states that

M

α= r i=1

α|fi

e(TfiM).

Heree(TfM)is theT-equivariant Euler class of the representation ofT on the tangent space of M at f. Let us choose M to be the Grassmannian Grk(Cⁿ), with theT = (S¹)ⁿ action induced by the standard T-action on Cⁿ. Letαbe thep-value of the Chern roots of the universal subbundle over M, times, the q-value of the Chern roots of the universal quotient bundle overM. Since deg(p)+ deg(q)< k(n−k)= dim(Gr_kCⁿ)the integral _Mα is clearly 0. Then the localization theorem proves Lemma 3.1.

Consider the partition λ = (λ1, . . . , λm) ∈ N^m. Correspondingly, we can consider the partial flag manifoldFλ parameterizing the flags of linear subspaces

0 =V0⊂V1⊂V2⊂. . .⊂Vm−1⊂Vm=C^|λ|

inC^|λ|, whereVj/Vj−1 has dimensionλj forj= 1,2, . . . , m. Let the tauto- logical bundle over Fλ corresponding to thej’th linear space Vj be called Ej (j = 0,1, . . . , m). Forj= 1, . . . , mletγj be the collection of the Chern roots of the quotient bundle E_j/E_j−1. The standard action of (S¹)^|λ| on C^|λ|induces an action onFλ. Then for the symmetric polynomialspj inλj

variables (j= 1, . . . , m), the equivariant localization theorem gives

±

Fλ

m i=1

pj(γj) =

I

_m

i=jp_j(z_Ij)

R(zI1|zI2|. . .|zIm), (3.1) where I = (I1, I2, . . . , Im)is a partitioning of the integers {1, . . . ,|λ|}into mparts satisfying

∪jIj={1, . . . ,|λ|}, Ii∩Ij =∅(i=j), |Ij|=λj.

When

deg(pi)<dimFλ=

i<j(λi−λj), then the left hand side of (3.1)vanishes, establishing the identity that the right hand side is 0.

3.2. Localization formulas vs divided differences

For a polynomial p in variables z1, z2, . . ., the ith divided difference is defined by

∂ip=p−p(z_i↔z_i+1) zi−zi+1

. (3.2)

It is well known that the divided difference operators∂_i satisfy the relations

∂i∂j =∂j∂i if |i−j|2, ∂i∂i+1∂i=∂i+1∂i∂i+1. (3.3) Hence ∂_ω can be defined for a permutation of indexes, as∂_ir· · ·∂_i1, where s_i1· · ·s_ir is a reduced word forωin terms of the elementary transpositions s_i.

The algebra of divided differences is closely related with localization for- mulas. For example, letk < n, and letω_(k,n−k)be the following permutation of{1, . . . , n}:

i →(n−k) +iforik and i →i−kfori > k.

Letpandqbe symmetric polynomials inkandn−kvariables respectively.

Then

∂ω_(k,n−k)(p(z1, . . . , zk)q(zk+1, . . . , zn)) =

I∈(^{1,...,n}k )

p(zI)q(zI¯) R(zI|zI¯) .

More generally, consider the partitionλagain. We define the permuta- tionω_λ as follows. Forλ₁+. . .+λ_u−1< iλ₁+. . .+λ_u,

i →i+

j>u

λ_j−

j<u

λ_j.

[In plain language ω_λ is described by dividing the numbers from 1 to |λ|

into groups of cardinalityλi in order, then reversing the order of the groups without changing the relative positions of pairs of numbers in the same group.]

Then for the symmetric polynomialspj in λj variables (j = 1, . . . , m), we have

∂ωλ



^m

j=1

pj(zJj)



=

I

m

i=jpj(zIj) R(z_I1|zI2|. . .|zIm), where the summation runs for partitionsI as in (3.1).

3.3. A generalized divided difference.

In the next chapter we will mention an extended version of divided differ- ence operations. This applies to functions depending not only onz1, z2, . . .,

but on other sets of variables. We modify only the numerator in the def- inition (3.2)by applying the transposition i ↔ (i+ 1)to certain sets of variables (including thez’s). These sets of variables will be indicated in the upper index of the∂ sign, eg.

∂_i^x,y,zp(x₁, x₂,. . ., y₁, y₂,. . ., z₁, z₂,. . .) =p−p(z_i↔z_i+1, x_i↔x_i+1, y_i↔y_i+1) zi−zi+1

. Generalized divided differences with fixed upper indexes also satisfy the re- lations (3.3), hence∂_ω^x,y,z is defined for permutationsω. Observe, however, that our generalized divided difference operators do not preserve polynomi- als.

4. The P_z(λ) function 4.1. Definition, examples

Letm2, and letλ= (λ1, . . . , λm)∈N^mbe a partition. We will study various expressions in the variables

y_a^(j) andza, for j∈ {1, . . . , m}, a∈ {1, . . . ,|λ|}.

For a subsetU ⊂ {1, . . . ,|λ|}, we defineY_U^(j)=

a∈Uya^(j). Definition 4.1. — We define

P_z(λ) =

I

_m

j=1Y_I^(j)_j

R(z_I1|z_I2|. . .|z_Im), (4.1) where the summation runs for I = (I1, . . . , Im) with Ii∩Ij = ∅, ∪Ij = {1, . . . ,|λ|},|Ij|=λj.

Recall that associated with the partitionλwe defined the permutation ω_λ in Section 3.2, and recall the extended divided difference operation from Section 3.3. The functionPz(λ)can be written in the concise form

Pz(λ) =∂_ω^{y(1),...,y(m),z}

λ

y⁽¹⁾₁ · · ·y_λ⁽¹⁾₁y_λ⁽²⁾₁₊₁· · ·y⁽²⁾_λ1+λ2 · · · y_|^(m)_λ|−λ

m+1· · ·y_|^(m)_λ|

. (4.2) Example 4.2. —

Pz(1,1)= y⁽¹⁾₁ y⁽²⁾₂ z1−z2

+y₂⁽¹⁾y₁⁽²⁾ z2−z1

= det

y⁽¹⁾₁ y₂⁽¹⁾ y⁽²⁾₁ y₂⁽²⁾

z1−z2

.

P_z(2,1)= y₁⁽¹⁾y₂⁽¹⁾y⁽²⁾₃

(z₁−z₃)(z₂−z₃)+ y₁⁽¹⁾y₃⁽¹⁾y₂⁽²⁾

(z₁−z₂)(z₃−z₂)+ y⁽¹⁾₂ y₃⁽¹⁾y₁⁽²⁾ (z₂−z₁)(z₃−z₁). Pz(2,2)= y⁽¹⁾₁ y⁽¹⁾₂ y⁽²⁾₃ y₄⁽²⁾

R(z₁, z₂|z3, z₄)+ [5 similar terms]

= perm

y₁⁽¹⁾y₂⁽¹⁾ y₃⁽¹⁾y₄⁽¹⁾ y₁⁽²⁾y₂⁽²⁾ y₃⁽²⁾y₄⁽²⁾

R(z1, z2|z3, z4) +

perm

y⁽¹⁾₁ y⁽¹⁾₃ y⁽¹⁾₂ y⁽¹⁾₄ y⁽²⁾₁ y⁽²⁾₃ y⁽²⁾₂ y⁽²⁾₄

R(z1, z3|z2, z4)

+ perm

y₁⁽¹⁾y⁽¹⁾₄ y⁽¹⁾₂ y⁽¹⁾₃ y₁⁽²⁾y⁽²⁾₄ y⁽²⁾₂ y⁽²⁾₃

R(z1, z4|z2, z3) ,

[wherepermmeans the permanent of a matrix.]

Pz(1,1,1)=

(i,j,k)∈S3

y_i⁽¹⁾y_j⁽²⁾y⁽³⁾_k (z_i−z_j)(z_i−z_k)(z_j−z_k)

= det





y₁⁽¹⁾ y₂⁽¹⁾ y₃⁽¹⁾ y₁⁽²⁾ y₂⁽²⁾ y₃⁽²⁾ y₁⁽³⁾ y₂⁽³⁾ y₃⁽³⁾





(z₁−z₂)(z₁−z₃)(z₂−z₃).

Remark 4.3. — In [V, (3.2)] the rational function det

y_j+N⁽¹⁾ y⁽²⁾_i −y⁽¹⁾_i y⁽²⁾_j+N zi−zj+N

i,j=1,...,N

is considered (with the substitutiony⁽¹⁾_j = 1 for allj). One can show that this function is equal to

±





1i<jN

(z_i−z_j)(z_i+N−z_j+N)



·P_z(N, N).

4.2. Vanishing properties

The function Pz(λ)satisfies remarkable differential equations with re- spect to variablesy_j⁽ⁱ⁾. Recall the differential operators from (2.2).

Theorem 4.4. — We have

ek,lPz(λ) = 0 if λk λl.

Proof. — The effect of the terms of the differential operator e_k,l on a square-free y-monomial is that they replace a y^(l) variable with the corre- spondingy^(k)variable. Hence we have

e_k,l



^m

j=1

Y_I^(j)_j



=

v∈Il



^m

j=1

Y_I^(j)_j y^(k)v

yv^(l)



.

Therefore, they-monomials occurring ine_k,lPz(λ)are of the form_m

j=1Y_K^(j)_j forK_i∩K_j=∅,∪Kj={1, . . . ,|λ|}, and

|Kj|=







λ_j j=k, l λk+ 1 j=k

λ_l−1 j=l.

For such a monomial, and av∈Kk define

I_j^(v)=







Kj j=k, l

Kk− {v} j=k K_l∪ {v} j=l.

Then the coefficient ofm

j=1Y_K^(j)_j in ek,lPz(λ)is

v∈Kk

1 R(z_I(v)

1 |. . .|z_I^(v)

m )= ±1

R(zK1|zK2|. . .|zKm)

v∈Kk

R(zv|zKl)

R(zv|zKk−{v}). (4.3) The main observation in the last equality is that the sign ±1 does not depend on the choice ofv∈Kk.

In the factor

v∈Kk

R(z_v|z_Kl) R(zv|zKk−{v})

of the last expression we consider the variables zKl as parameters. In the remaining variables this expression is of the form of the identity in Lemma 3.1. Hence it is 0 if the numerator has smaller degree than the denominator, ie. if λl−1 =|Kl|<|Kk| −1 =λk. This inequality is satisfied ifλk λl.

Now recall the differential operators from (2.3).

Theorem 4.5. — We have

e^z_k,lPz(λ) = 0 if λk> λl, (4.4) e^z_k,l2

P_z(λ) = 0 if λ_k =λ_l. (4.5)

Proof. — The proof of (4.4)is analogous with the proof of Theorem 4.4.

The change is that formula (4.3)expressing the coefficients ofy-monomials ofe^z_k,lP_z(λ)has to be replaced with

v∈Kk

1 R(z_I^(v)

1 |. . .|z_I^(v)

m )= ±1

R(zK1|zK2|. . .|zKm)

v∈Kk

z_vR(z_v|zKl)

R(zv|zKk−{v}). (4.6)

The last factor

v∈Kk

zvR(zv|z_Kl)

R(zv|z_Kk−{v}) vanishes due to Lemma 3.1 if the numerator has smaller degree than the denominator, i.e. if 1 + (λ_l−1)=

1 +|K_l|<|K_k| −1 =λ_k. This holds ifλ_k > λ_l.

To prove (4.5)observe that the effect of the terms of the differential operator

e^z_k,l

2

on square-freey-monomials is replacing twoy^(l) variables with the corresponding y^(k) variables times the corresponding z variables.

That is, we have e^z_k,l2



^m

j=1

Y_I^(j)_j



=

v=w∈Il



^m

j=1

Y_I^(j)_j y^(k)v yw^(l)z_vz_w y^(k)v yw^(l)



.

Therefore, they-monomials occurring in (e^z_k,l)²P_z(λ)are of the formm j=1Y_K^(j)

j

forK_i∩K_j=∅,∪K_j={1, . . . ,|λ|}, and

|K_j|=







λj j=k, l λ_k+ 2 j=k

λl−2 j=l.

For such a monomial, andv=w∈Kk define

I_j^(v,w)=







Kj j=k, l

Kk− {v, w} j=k K_l∪ {v, w} j=l.

Then the coefficient ofm

j=1Y_K^(j)_j in (e^z_k,l)²Pz(λ)is

v=w∈Kk

zvzw

R(z_I(v,w)

1 |. . .|z_I^(v,w)

m ) = ±1

R(zK1|. . .|zKm)

v=w∈Kk

zvzwR(zv, zw|zKl) R(zv, zw|zKk−{v,w}). The main observation in the last equality is, again, that the sign ±1 does not depend on the choice ofv=w∈Kk.

The factor

v=w∈Kk

z_vz_wR(z_v, z_w|zKl) R(zv, zw|zKk−{v,w})

vanishes—because of Lemma 3.1—if the degree 2+2(λ_l−2)of the numerator is less than the degree 2λk of the denominator, i.e. ifλk λl. 4.3. P_z(λ) functions are conformal blocks

Let m ∈ N, let λ ∈ N^m be a partition, and let d(λ), as before.

Recall that a function in the variablesy^(j)a ,z_a(j= 1, . . . , m,a= 1, . . . ,|λ|) belongs to the space CB_z(λ)of conformal blocks of level, if

• it is a polynomial of homogeneous degree 1 with respect to eachm- tuple of variablesya⁽¹⁾, . . . , ya^(m),

• it vanishes under the action of the differential operator_|λ_|

a=1ya^(k)∂/∂y^(l)a

for allk < l,

• it vanishes under the action of the differential operator (_|λ_|

a=1zay⁽¹⁾a ∂/∂y^(m)a )^−d(λ)+1.

The properties of theP_z(λ)function presented in Sections 4.1, 4.2 trans- late to the following theorem.

Theorem 4.6. — We have

P_z(λ)∈CB^d(_z ^λ)(λ)if d(λ)>0, Pz(λ)∈CB¹_z(λ)if d(λ) = 0.

Dimensions of CB_z(λ)spaces are independent ofz. They are calculated, e.g. as structure constants in so-called fusion rings, see e.g. [Z], [GV]. A special case of these results is that dim CB¹_z(N + 1, . . . ,N + 1,N, . . . ,N)= 1 for anymand any number ofN’s. Hence the relevant part of Theorem 4.6 can be reformulated as follows.

Theorem 4.7. — Letm2and letλ= (N+ 1, . . . , N+ 1, N, . . . , N)∈ N^mfor any number ofN’s (possiblym). Then for any collection of pairwise distinct complex numbersz= (z1, . . . , z_|λ|), the polynomialPz(λ)is a basis in the one-dimensional space of conformal blocks CB¹_z(λ).

Form= 2 and λ= (N, N)a basis conformal block is given in [V], cf.

Remark 4.3.

4.4. Asymptotics

When concrete formulas are known for conformal blocks, one can often derive Selberg integral formula type applications, see e.g. [MV], [FSV], [V].

In these applications one needs certain asymptotic properties of conformal blocks. Now we show such an asymptotic property ofP_z(λ).

We define the discriminant of the ordered list of variablesx1, . . . , xn by D(x1, . . . , xn) =

1i<jn

(xi−xj).

Let

Pˇ(N) =Pz(N, N, . . . , N

m

)· N k=1

D(z(k−1)m+1, . . . , zkm).

Observe that

P(1)= detˇ

y⁽ⁱ⁾_j

i=1,...,m j=1,...,m

. Direct calculation gives the following theorem.

Theorem 4.8. — ForN2 the limit ofPˇ(N)as

z_(k−1)m+1, z_(k−1)m+2, . . . , z_km−1→z_km ∀k= 1, . . . , N is

− _N

k=1det

y_j⁽ⁱ⁾

i=1,...,m

j=(k−1)m+1,...,km

D(zm, z2m, . . . , zN m)^m(m−1) .

5. Pz(N, . . . , N) satisfies the KZ equations at level 1 Let

X_|λ| = {z= (z1, . . . , z_|λ|)∈C^|λ||z_a=z_b for alla=b}.

The trivial vector bundleη :V^⊗|λ|×X_|λ| →X_|λ|has the KZ connection at level,

∂

∂z_i − 1

m+

j=i

Ω^(i,j)

z_i−z_j , i= 1, . . . ,|λ|,

where Ω =⊕^m_a,b=1e_a,b⊗e_b,ais thegl_mCasimir operator and Ω^(i,j):V^⊗|λ|→ V^⊗|λ|is the linear operator acting as Ω on thei-th andj-th factors and as the identity on all the other factors.

Consider the subbundle of conformal blocks with fiber CB_z(N, . . . ,N)⊂ V^⊗|λ|. It is well known [KZ] that this subbundle is invariant with respect to the KZ connection at level.

For= 1 this subbundle is a line-bundle, withP :=P_z(N, . . . , N)being a nowhere zero section in it (see Theorem 4.7). Since the KZ connection has regular singularities, the function

P˜:=P·

1i<jmN

(zi−zj)^−aij,

for some choice of numbersa_ij, must be a solution of the KZ equations with

= 1, 

 ∂

∂zi − 1 m+ 1

j=i

Ω^(i,j) zi−zj



P˜= 0, i= 1, . . . ,|λ|. (5.1)

Equivalent to (5.1)is



 ∂

∂zi − 1 m+ 1

j=i

Ω^(i,j) zi−zj



P=P·

j=i

ai,j

zi−zj

. (5.2)

In our conventions Ω^(i,j) reduces to the following operator:

Ω^(i,j)(f) =f(y^(k)_i ↔y^(k)_j ∀k).

Theorem 5.1. — We haveai,j=−_m+1^m for alli=j, that is, the func- tion

P˜=P_z(N, . . . , N)·

1i<jmN

(z_i−z_j)^m+1m (5.3) satisfies the level 1 KZ equation (5.1).

Proof. — Without loss of generality we choosei= 1. The coefficient of m

k=1Y_{(k−1)N^(k) _+1,...,kN} on the left hand side of (5.2)is

−

mN

j=N+1

1 R

1

z₁−z_j − 1 m+ 1



^N

j=2

1 z₁−z_j · 1

R+

mN

j=N+1

1 z₁−z_j · 1

R^1,j



, (5.4)

where R=R(z₁, . . . , z_N|z_N+1, . . . , z_2N|. . .|z_(m−1)N+1, . . . , z_mN)is the de- nominator of the monomial_m

k=1Y_{^(k)_(k−1)N+1,...,kN}inP, andR^1,j =R(z1↔ z_j). According to (5.2) the expression (5.4) is equal to

1 R

mN

j=2

a1,j

z₁−z_j.

Multiplying withR, and checking the residue atzN+1=z1 we obtain

−1− 1

m+ 1(−1)=a_1,N+1,

which implies a1,N+1=−m/(m+ 1). For the otherai,j’s the result follows by symmetry.

Remark 5.2. — In proving that (5.3)satisfies the KZ equations we used a priori that (5.2)holds for some constants ai,j. Equivalently, we made a particularly lucky choice of a hyperplane to take the residue of the coefficient of a well-chosen monomial. The fact that the resulting ˜P satisfies the KZ equations, equivalently, that other residues would give the sameai,j’s encode remarkable identities. For example, one has

R·

mN

j=N+1

∂1,j

1 R

= (1−m) N j=2

1 z₁−z_j + 2

mN j=N+1

1 z₁−z_j (for the sameRas in the proof above). It would be interesting to see direct proofs of these properties; or even more interestingly, a combinatorial proof using the form (4.2)ofP.

Remark 5.3. — We may consider the slm KZ equations instead of the gl_m version (5.1)above. The only change in (5.1)is that the gl_m Casimir operator has to be replaced with the sl_m Casimir operator. In our conven- tions thesl_mCasimir operator is

Ω^(i,j)_slm (f) =f(y_i^(k)↔y^(k)_j ∀k)− 1 mf.

Arguments analogous to the above prove that Pz(N, . . . , N)·

1i<jmN

(zi−zj)^m−1m is a solution to theslmKZ equations.

6. Products of level one conformal blocks are higher level conformal blocks

Recall that conformal blocks of level 1 have dimension 1, namely dim CB¹_z(λ) = 1, ford(λ)= 0 or 1,

and that the functions Pz(λ)of Section 4.1 form a basis in these spaces.

Recall also that for theseλ’sei,jPz(λ) = 0 ifi < j, as well as e^z_1,m2

P_z(λ) = 0 if d(λ) = 0,

e^z_1,mP_z(λ) = 0 if d(λ) = 1. (6.1) Now we are going to use these Pz(λ)functions as building blocks to produce elements of CB_z(λ)for1 and anyλ. In Section 7 we will show that these elements span the space of conformal blocks for genericz.

Definition 6.1. — Let λ ∈ N^m be a partition, and let k₁, . . . , k_|λ| be pairwise different integers. We define P^{(k1,...,k|λ|)}(λ) to be obtained from P_z(λ)by substituting

y_k^(j)_a for ya^(j) and z_ka for z_a

for allj = 1, . . . , m anda= 1, . . . ,|λ|.

For example

P^(3,5)(1,1)= y₃⁽¹⁾y₅⁽²⁾ z3−z5

+y₅⁽¹⁾y⁽²⁾₃ z5−z3

.

Definition 6.2. — Let λ ∈ N^m be a partition, and let d(λ). Put formally λ₀ = +λ_m. Let U = {U₁, . . . , U} be a partitioning of the set {1,2, . . . ,|λ|}into subsets such that for every j = 0, . . . , m−1 we have exactly λ_j −λ_j+1 of the U_j’s satisfying |U_j| ≡ jmod(m). Consider each U_j as an ordered set, ordered by the natural ordering of integers. Define the partition µ^(j) ∈ N^m to be the unique partition with |µ^(j)| = |Uj| and d(µ^(j)) = 0 or1. Define

Q(U) = j=1

P^(Uj)(µ^(j)).

Proposition 6.3. — We have

Q(U)∈CB_z(λ) for any choice ofU.