Schubert calculus and singularity theory

Journal of Geometry and Physics 62 (2012) 352–360

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Schubert calculus and singularity theory Vassily Gorbounov a,∗ , Victor Petrov b a

University of Aberdeen, Mathematics, King’s College, Aberdeen, AB24 3UE, United Kingdom

b

Johannes Gutenberg-Universität, Institut für Mathematik, Staudingerweg 9 55099, Mainz, Germany

article

info

Article history: Received 26 April 2011 Received in revised form 27 August 2011 Accepted 22 October 2011 Available online 4 November 2011 Keywords: Schubert calculus Homogeneous spaces Jacobi rings Landau–Ginzburg model Mirror symmetry Frobenius manifolds

abstract Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces, it has to be redesigned when applied to other generalized cohomology theories such as the equivariant, the quantum cohomology, K -theory, and cobordism. All this cohomology theories are different deformations of the ordinary cohomology. In this note, we show that there is, in some sense, the universal deformation of Schubert calculus which produces the above mentioned by specialization of the appropriate parameters. We build on the work of Lerche Vafa and Warner. The main conjecture these authors made was that the classical cohomology of a Hermitian symmetric homogeneous manifold is a Jacobi ring of an appropriate potential. We extend this conjecture and provide a simple proof. Namely, we show that the cohomology of the Hermitian symmetric space is a Jacobi ring of a certain potential and the equivariant and the quantum cohomology and the K -theory is a Jacobi ring of a particular deformation of this potential. This suggests to study the most general deformations of the Frobenius algebra of cohomology of these manifolds by considering the versal deformation of the appropriate potential. The structure of the Jacobi ring of such potential is a subject of well developed singularity theory. This gives a potentially new way to look at the classical, the equivariant, the quantum and other flavors of Schubert calculus. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved.

1. Introduction The connection between the cohomology of the homogeneous space G/Hand the quantum field theory called the coset model G/H was stated in the influential paper by Lerche et al. [1]. The key conjectural claim was that the ring H∗ (G/H ) is the Jacobi ring of some function, the so called potential. In [1] and subsequent publications, this conjecture was supported by a number of non-trivial examples. Later Gepner in [2] had calculated the potential for the cohomology of any Grassmann manifold Gr(k, n) and discovered that the cohomology of the Grassmann manifolds and the fusion rings of the unitary group are Jacobi rings of the same potential. This work was followed by Witten in [3], where it was pointed out that this connection extends naturally to the quantum cohomology of the Grassmann manifold if one considers the deformed Gepner potential. Also in [4], it was argued that a certain deformation of the potential provides the connection with the affine Toda lattice type field theory. This gave a beautiful link between the representation theory of the loop groups, the quantum Schubert calculus, the integrable systems and the singularity theory. The connection between the representation theory and singularity theory was the subject of the conjecture by Zuber, solved in [5]. We take on the task of extending the conjecture of Lerche, Vafa and Warner about the cohomology of Hermitian symmetric spaces as defined in [6] and providing simple proof of it. We show that the classical cohomology of these

∗

Corresponding author. Tel.: +44 0 1224 273745; fax: +44 0 1224 272607. E-mail addresses: [email protected] (V. Gorbounov), [email protected] (V. Petrov).

0393-0440/$ – see front matter Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2011.10.016

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353

homogeneous spaces are Jacobi rings of an appropriate Taylor polynomial, the potential, which we calculate explicitly, and the equivariant cohomology with respect to the torus action, the small quantum cohomology and K -theory are the Jacobi rings of a particular deformation of the potential. Our approach offers a possibly new way to look on the classical as well as the equivariant and quantum formulas from the Schubert calculus like the Littlewood–Richardson rule from the point of view of singularity theory [7,8]. It is interesting to explore the connection between these two ways of describing the same algebra. In the case of the Grassmann manifold such an attempt is available in the literature [9]. Next we give a description of the small quantum cohomology of these spaces as a scheme by solving the equations defining the small quantum cohomology as an algebra with the help of Kostant’s work on cyclic elements [10]. Although such a result had appeared already in [11] we find it worthwhile to point the connection with Kostant’s work. The suggested description agrees nicely with the Hori–Vafa conjecture for the quantum cohomology of the Grassmannian [12,13]. We also make explicit the Vafa–Intriligator formula by calculating the denominators in it. They turn out to be the values of the Hessian of the appropriate potential at the critical points. The Gromov–Witten invariants can be calculated as a sum over the vertices of an appropriate polytope with the explicitly calculated weights. The vertices are identified using Kostant’s description of cyclic elements. Further using the description of the equivariant cohomology with respect to the torus action of these homogeneous spaces as a Jacobi ring we give the description of the scheme Spec H∗T (G/P ) in the spirit of the work [14] where the similar result was obtained for toric varieties and some other spaces. In the modern language of Mirror Symmetry such a presentation of cohomology as a Jacobi ring is called a Landau–Ginzburg model of the homogeneous manifold. The quantum cohomology of an algebraic variety is a certain deformation of the cohomology ring. It carries a structure of a Frobenius manifold and hence a structure of an F-manifold, introduced in [15]. If the appropriate Frobenius manifold is semisimple and satisfies some additional natural conditions, the underlying F -structure is, thanks to a theorem of Hertling on singularities [16], quite well understood. Namely it comes from the so called Saito setup of a miniversal deformation of a singularity [8]. For the above spaces we have found such a singularity, so that we have a conceptual understanding of this sort of mirror symmetry. We would like to point out that our potential is a Taylor polynomial and is different from the potential obtained out of toric degeneration of the homogeneous space, like in the approach developed in [17,12,18,19]. The modern treatment of these ideas is given for example in [20] where such a potential is called the weak Landau–Ginzburg model. The connection of our potential to the weak Landau–Ginzburg model and to the quantum cohomology is now being investigated. 2. Summary of Gepner’s work Let us remind the Gepner result on the cohomology of the Grassmannian. The cohomology of the Grassmannian Gr(k, n) = U(n)/ U(k) × U(n − k) have the following presentation: let C = 1 + c1 + · · · + ck + · · · is the full class of the tautological bundle and C¯ = 1 + c¯1 + · · · + c¯n + · · ·, the relation between them is C C¯ = 1. For our purposes we need the following description of H∗ (Gr(k, n), Z): H∗ (Gr(k, n), Z) = Z[c1 , . . . , ck ]/¯cn−k+1 = · · · = c¯n = 0. The existence of the potential can be seen using the following easy observation: consider the relations in the cohomology of the Grassmannian

• • • • •

c1 c2 c3 c4 c5

+ c¯1 + c1 c¯1 + c¯2 + c2 c¯1 + c1 c¯2 + c¯3 + c3 c¯1 + c2 c¯2 + c1 c¯3 + c¯4 + c4 c¯1 + c3 c¯2 + c2 c¯3 + c1 c¯4 + c¯5 .

Proposition 1. Consider c¯i as a function in c1 , c2 , . . . defined by the above equation C C¯ = 1. The following identity holds:

∂ c¯i ∂ c¯i+l = . ∂ cj ∂ cj+l Proof. C¯ is a function in c1 , c2 , . . . . Differentiating the identity C C¯ = 1 with respect to ci produces

∂ ¯ C¯ C =− . ∂ ci C In other words the function C¯ satisfies the system of linear partial differential equations:

∂ ¯ ∂ ¯ C = C. ∂ ci ∂ cj Taking the graded components of this identity proves the proposition.



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This implies that the ideal defining the cohomology ring of the Grassmann manifold is a Jacobi ideal. Figuring out the potential is easy now. The function C¯ has the form C1 . Therefore the generating function for the potentials is V = log(C ). Taking now the graded component of the weight n + k − 1 and setting ci = 0 for i > k we obtain the Landau–Ginzburg potential for Gr(k, n). Here is the potential for Gr(2, 5) V =−

c16

3

+ c14 c2 − c12 c22 +

c23

.

6 2 3 It is easy now to obtain the Gepner description of the generating function for the potential: V = log(1 + c1 + c2 + · · ·) = log Π (1 + ti ) =

−

log(1 + ti ) =

−

ti −

i

− t2 i

i

2

+

− t3 i

i

3

− ···.

3. The cohomology of the orthogonal Grassmann manifold and the Catalan numbers The cohomology of the orthogonal Grassmannian OGr(n, 2n) = SO(2n)/ U(n) can be presented as follows [21]. Consider a polynomial ring Z[x1 , x2 , . . . , x2n−2 ], deg xi = 2i, and factor out the set of elements Pi = x2i + 2

− (−1)k xi+k xi−k + (−1)j x2i . k=1

As usual we set xi to be zero if the number i is outside of the interval [1, 2n − 2]. A few of the elements Pi are listed below:

• • • • •

x21 x22 x23 x24 x25

− x2 − 2x1 x3 + x4 − 2x2 x4 + 2x1 x5 − x6 − 2x3 x5 + 2x2 x6 − 2x1 x7 + x8 − 2x4 x6 + 2x3 x7 − 2x2 x8 + 2x1 x9 − x10 .

The meaning of the variables xi is as follows: it is the Chern classes of the tautological bundle divided by 2 [21]. Introduce two generating functions X+ = 1 + x2 + x4 + · · · and X− = x1 + x3 + x5 + · · ·. Considering the even x2n as functions of the odd x2m+1 ’s the above relations can be written as (X+ + X− )(X+ − X− − 1) = X− . Proposition 2. One has

∂ x2(n+k) ∂ x2n = . ∂ x2(m+k)+1 ∂ x2m+1 This implies the existence of the potential for the ideal generated by the Pi ’s. Let us calculate it. Solving for X+ one obtains X+ = −

1 2

 +

1 4

+ X−2 .

Since

∫ 

1 + u2 dx =

u 



1 + u2 + ln( 1 + u2 + u)



2 we obtain the generating function for the potentials V =

1

−X− +





2 1 + X− + ln( 1 + X−2 + X− ) .



2 The component of a given degree 2n − 1 of V gives the potential for the cohomology of H∗ (OGr(n, 2n)). Here is the Landau–Ginsburg potentials for OGr(4, 8), OGr(5, 10), and OGr(6, 12). V7 = x1 x23 − x41 x3 +

2 7

x71

V9 = −2x31 x23 + 2x61 x3 +

1 3

x33 −

5 9

x91

V11 = 2x61 x5 + 6x51 x23 − 2x21 x33 + x1 x25 + x23 x5 − 5x81 x3 − 4x31 x3 x5 +

14 11

x11 1 .

Remark 1. It is interesting to note that the above relations are related to the Catalan numbers. Namely the expansion of the function g ( u) = −

1 2

 +

1 4

+ u2−

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355

is the generating function for the Catalan numbers g (u) =

− (−1)n Cn−1 u2n .

Indeed setting the odd xi for i > 1 in the defining relations equal to zero one obtains Segner’s recurrence formula for the Catalan number. Remark 2. Since Bn and Cn has the same Weyl group, and their torsion index is 2, one has

 H

 



1

Sp(n)/ U(n), Z

∗

≃H

2

∗

  OGr(n + 1, 2n + 2), Z

1 2

,

so the result applies to the symmetric Hermitian space Sp(n)/ U(n) as well. 4. The quadrics In this section we consider the case of even-dimensional quadrics Q2n−2 = SO(2n)/ SO(2) × SO(2n − 2). Note that for odd-dimensional quadrics we have

 H

∗

  Q2n−1 , Z

 



1

Gr(1, 2n), Z

∗

=H

2

1

2

,

the case already considered above. It is well-known that H∗ (Q2n−2 , Z) has a presentation

Z[h, l]/(hn − 2hl, l2 ) when n is even and

Z[h, l]/(hn − 2hl, l2 − hn−1 l) when n is odd, where h is the class of a hyperplane section and l is the class of a maximal totally isotropic subspace. Consider first the case of even n. The function n h2n−1 V (h, l) = −hn l + hl2 + 2(2n − 1) satisfies the equations

∂l V = −(hn − 2hl); n

∂h V = l2 + hn−2 (hn − 2hl), 2

so V is a potential. Similarly, in the case of odd n we set V (h, l) = −hn l + hl2 +

n−1 2(2n − 1)

h2n−1 .

Then

∂l V = −(hn − 2hl); ∂h V = l2 − hn−1 l +

n−1 2

hn−2 (hn − 2hl),

and V is a potential. 5. Cohomology of E6 /Spin(10) × SO(2), E7 /E6 × SO(2) and E8 /E7 × SO(2) In [22] (see also [23]) the following presentation was obtained: H∗ (E6 / Spin(10) × SO(2), Z) = Z[y1 , y4 ]/(r9 , r12 ), where r9 = 2y91 + 3y1 y24 − 6y51 y4 ; r12 = y34 − 6y41 y24 + y12 1 .

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Setting V (y1 , y4 ) = 2y91 y4 + y1 y34 − 3y51 y24 −

5 13

y13 1

we see that

∂y4 V = r9 ; ∂y1 V = r12 − 3y31 r9 , so V is a potential. Similarly, a presentation for H∗ (E7 /E6 × SO(2), Z) looks as follows:

Z[y1 , y5 , y9 ]/(r10 , r14 , r18 ), where r10 = y25 − 2y1 y9 ; r14 = 2y5 y9 − 9y41 y25 + 6y91 y5 − y14 1 ; r18 = y29 + 10y31 y35 − 9y81 y25 + 2y13 1 y5 . Setting V (y1 , y5 , y9 ) = −y25 y9 + y1 y29 + 3y41 y35 − 3y91 y25 + y14 1 y5 −

2 19

y19 1

gives

∂y9 V = −r10 ; ∂y5 V = −r14 ; ∂y1 V = r18 + 2y31 y5 r10 + 2y41 r14 , so V is a potential. Consider now H∗ (E8 /E7 × SO(2)). The existence of the potential for this ring has been stated in [1] as a problem. From [22] we have: H∗ (E8 /E7 × SO(2)) = Z[y1 , y6 , y10 , y15 ]/(r15 , r20 , r24 , r30 ), where r15 = 2y15 − 16y51 y10 − 10y31 y26 + 10y91 y6 − y15 1 14 r20 = 3y210 + 10y21 y36 + 18y41 y6 y10 − 2y51 y15 − 8y81 y26 + 4y10 1 y10 − y1 y6 2 14 r24 = 5y46 + 30y21 y26 y10 + 15y41 y210 − 2y91 y15 − 5y12 1 y6 + y1 y10 2 r30 = y215 − 8y310 + y56 − 2y31 y26 y15 + 3y41 y6 y210 − 8y51 y10 y15 + 6y91 y6 y15 − 9y10 1 y10 3 14 15 20 24 30 − y12 1 y6 − 2y1 y6 y10 − 3y1 y15 + 8y1 y10 + y1 y6 − y1 .

It is clear that the grading on our algebra and the degrees of the relations prevent the existence of a potential whose gradient would give the above relations. But one can check that the following is true Proposition 3. The ring

Q[y1 , y6 , y10 , y15 ]/(r15 , r20 , r24 , r30 )





is a Jacobi ring with respect to the ‘modified gradient’ y1 ∂∂y , ∂∂y , ∂ y∂ , ∂ y∂ . The potential is 1 6 10 15 9 3 2 5 2 14 8 2 V = y215 − y15 y15 1 + 10y15 y1 y6 − 10y15 y1 y6 − 16y15 y1 y10 + 8y1 0y10 − 80y1 y6 y10 + 80y1 y6 y10 2 24 18 2 12 3 6 4 + 64y10 1 y10 + −5y1 y6 + 30y1 y6 − 50y1 y6 + 25y1 y6 +

y30 1 4

.

Remark 3. One can check a similar statement for all extra-special homogeneous spaces, that is corresponding to a parabolic subgroup whose unipotent radical has a one-dimensional commutator subgroup.

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357

6. Equivariant cohomology and Bott denominators Let X = G/H be any projective homogeneous variety. The rational cohomology of X has the Borel presentation H∗ (X , Q) = Q[I1 , . . . , In ]/(R1 , . . . , Rn ), where Ii are the fundamental invariants of WH in the space of characters of a maximal torus T of WG , and Rj are the fundamental invariants of G on the same space considered as functions in Ii . The T -equivariant cohomology of X considered as an algebra over H∗ (BT ) = Z[y1 , . . . , yn ] are produced by the following deformation H∗T (X , Q) = Q[y1 , . . . , yn , I1 , . . . , In ]/(R1 − R1 (y1 , . . . , yn ), . . . , Rn − Rn (y1 , . . . , yn )). Specialize (y1 , . . . , yn ) to any point ‘‘in general position’’ (more precisely, lying in the interior of some Weyl chamber). Then the common zeros of all relations considered as functions in weights t1 , . . . , tn of T form WG -orbit of (y1 , . . . , yn ) that will be denoted by Q . Note that Ii take constant values on WH -orbits in Q , therefore the image P of Q under (I1 , . . . , In ) consists of WG /WH elements. ∂R By Bourbaki [24, Ch. IV, Section 5, n. 4] the Jacobian det( ∂ t i ) coincides with the product of positive roots of G (considered j

∂I

as linear functions in ti ) up to a constant. Similarly, det( ∂ ti ) is the product of positive roots of H up to a constant. It follows j

that



∂ Ri det ∂ Ij

 = c · chdim X (TX ),

the top Chern class of the tangent bundle of X , up to a constant c. The general theory of Artin rings as in [5] implies that the intersection form on H∗T (X ) looks as follows:

⟨f ⟩ = C ·

f (p)

−

chdim X (TX )(p) p∈P

,

(∗)

where f is a polynomial in Ii and C is some constant. Substituting f = chdim X (TX ) we get the Euler characteristic of X on the left-hand side and C · |WG /WH | on the right-hand side, so actually C = 1. Remark 4. A similar formula was obtained by Akyildiz and Carrell in [25], and for extraordinary cohomology by Bressler and Evens in [26, Theorem 1.8]. There is a more general formula for the Gysin map H∗ (G/H ) → H∗ (G/K ) involving summation over WK /WH ; details are to appear in a joint paper of the second author and Baptiste Calmés. For example, consider the case of Grassmannian X = Gr(k, n). If f is a polynomial in c1 , . . . , ck considered as a symmetric function in t1 , . . . , tk , we have f (yi1 , . . . , yik )

−

⟨f ⟩ =

(yi − yj )

∏ {i1 ,...,ik }∈Λk {1,...,n}

.

i∈{i1 ,...,ik },j∈{1,...,n}\{i1 ,...,ik }

In particular, this expression is always a polynomial in yi , which is constant if deg f = k(n − k) and 0 if deg f < k(n − k). Remark 5. ∏ The following observation is due to Fedor Petrov. Applying the Combinatorial Nullstellensatz [27, Theorem 4] to g = k1! f · 1≤i
Another example is the maximal orthogonal Grassmannian OGr(n, 2n). If f is a polynomial in x1 , . . . , xn considered as a symmetric function in t1 , . . . , tn , we have

⟨f ⟩ =

f (ε1 y1 , . . . , εn yn )

− εi =±1,

∏

εi =1

i

∏ 1≤i
(εi yi + εj yj )

.

Remark 6. The Bott localization theorem says that the pull-back map H∗T (X , Q) →



H∗T (p, Q)

p∈P

becomes an isomorphism after inverting all roots. The map from the left to the right can be easily computed. In terms of generators Ii it corresponds to evaluation at p, and in terms of Schubert additive generators it can be described inductively using the divided difference operators invented by Demazure (see [28] for the case of Grassmannians; the general case is

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similar). Now formula (∗) together with the fact that the intersection form is nondegenerate allows to find explicitly the inverse map (that of course requires denominators in general). In particular, this gives a new algorithm for computing (equivariant) Littlewood–Richardson coefficients. The method applies as well for other cohomology theories (like K -theory or cobordism), details are to appear. Remark 7. The result of [29] suggests the relation of the quantum cohomology of homogeneous spaces and the twisted K theory via the Verlinde algebra. It would be interesting to investigate it. Also following the approach of [30,31] one may be able to obtain the Chiral de Rham complex of the homogeneous spaces as the ‘‘Jacobi ring’’ of the ‘‘chiralisation’’ of the potential. We plan to return to this point in future work. 7. Spectrum of the ring of equivariant cohomology The construction above allows to describe the equivariant cohomology of the Hermitian homogeneous spaces as an affine scheme in the spirit of work [14]. We consider the case of the Grassmann manifolds as an illustration. The compact maximal torus T = SO(2)n inside U(n) acts naturally on Gr(k, n). The equivariant cohomology ring H∗T (Gr(k, n)) is an algebra over the cohomology of the classifying space of T . The latter is isomorphic to Z[y1 , . . . , yn ], deg yi = 2, where yi are the first∏ Chern classes of the appropriate line n bundles over T . The relations in the cohomology ring come from the relation C C¯ = i=1 (1 + yi ). First few relations look like this:

• • • • • •

+ c¯1 − e1 (y) + c1 c¯1 + c¯2 − e2 (y) + c2 c¯1 + c1 c¯2 + c¯3 − e3 (y) + c3 c¯1 + c2 c¯2 + c1 c¯3 + c¯4 − e4 (y) + c4 c¯1 + c3 c¯2 + c2 c¯3 + c1 c¯4 + c¯5 − e5 (y) .... c1 c2 c3 c4 c5

Here ej (y) is the j-th elementary symmetric functions in yi . This allows to view c¯i as functions in cj and yj . Then H∗T (Gr(k, n), Z) = Z[c1 , . . . , ck ]/¯cn−k+1 = · · · = c¯n = 0. Describing Spec H∗T (Gr(k, n)) amounts to solving the equations c¯n−k+1 = · · · = c¯n = 0 in c1 , . . . , ck with respect to y1 , . . . , yn . Applying the description of zeros of these equations given in Section 6 to our case we obtain the following. Take any k-element subset of the set {y1 , . . . , yn }, say {yi1 , . . . , yik }. Then the appropriate solution is given as follows: c1 = e1 (yi1 , . . . , yik ) c2 = e2 (yi1 , . . . , yik )

··· ck = ek (yi1 , . . . , yik ). We take now the union of all these subvarieties (over k-element subsets). Proposition 4. The scheme Spec H∗T (Gr(k, n), C) is the union above considered as a reduced subscheme of (Gr(k, n), C).

k

i=1

H2i T

Proof. Section 6 shows that these two schemes have the same C-points, but H∗T (Gr(k, n), C) is a subring of a reduced Indeed, ∗ ring  p∈P HT (p, C) and so is reduced. 8. Quantum cohomology, Hori–Vafa conjecture and Vafa–Intriligator formula Let X = G/H be a Hermitian symmetric space. The main result of [23] says that the quantum cohomology of X are given by the following deformation of the ordinary cohomology (the notation is explained in Section 6): QH∗ (X , Q) = Q[q, I1 , . . . , In ]/(R1 , . . . , Rn−1 , Rn + q). If Rj = Ii , one can drop Ii and Rj from both lists, and in this way all presentations considered above are obtained. It follows that QH∗ (X ) is also a Jacobi ring with the potential V [q] = V + qh, V being the potential of H∗ (X ). Let us look at the solution set of the system of equations R1 = 0

··· Rm−1 = 0 Rm = −q.

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359

This system describes cyclic elements in the Cartan subalgebra of the Lie algebra of G, and by [10] the Weyl group WG acts on the set of solutions simply transitively. Let us describe the solutions explicitly. In the case of G = U(n) one can choose Ri =

n 1−

tji ,

i j =1

and we can take

(y1 , . . . , yn ) =

√ n

√ √  −q, ζn n −q, . . . , ζnn−1 n −q ,

where ζn is n-th root of 1. The Weyl group W (An−1 ) = Sn acts by permutations. In the case of G = SO(2n + 1) or Sp(2n) we have Ri =

n 1 −

2i j=1

tj2i ,

and we can take the same (y1 , . . . , yn ) as in the case U(n). The Weyl group W (Bn ) = W (Cn ) acts by permutations and changes of signs at any positions. In the case of G = SO(2n) we have Ri =

Rn =

n 1 −

2i j=1 n ∏

tj2i ,

i < n;

tj ,

j =1

and we can take yn = 0 and (y1 , . . . , yn−1 ) as in the case of U(n − 1). The Weyl group |W (Dn )| acts by permutations and changes of signs at even number of positions. As in Section 6, we denote the image of Q under the map (I1 , . . . , In ) by P; it consists of |WG /WH | points. Remark 8. The method of solving the quantum equations can be related to the Hori–Vafa conjecture stated for the Grassmannian. It says that the quantum cohomology of the Grassmannian Gr(k, n) can be ‘‘recovered’’ from the quantum cohomology of the product (Pn−1 )k of projective spaces by some sort of symmetrization procedure. To obtain the set P for Gr(k, n) we take the k-element subsets of the set n-th roots of unity, that is the corresponding set for Pn−1 , and apply the first k elementary symmetric functions, which in our situation is the Hori–Vafa symmetrization. A similar method works for other homogeneous spaces. Say, in the case of OGr(n, 2n) the set P is obtained from the corresponding set for Pn−2 by changing signs at arbitrary elements and computing the sums of odd powers. Using [32, Theorem 4.5] we obtain the following formula for the Gromov–Witten invariants which is due to Vafa and Intriligator [33,34,11,35]:

⟨f ⟩g =

−

chdim X (TX )g −1 (p)f (p),

p∈P

where f is a polynomial in I1 , . . . , In . In particular, we recover a formula in [9] for the case of Grassmannians. Acknowledgments The authors visited the Max Planck Institute for Mathematics in Bonn while working on this paper. The inspiring discussions with V. Golyshev, Yu. I. Manin and especially with F. Hirzebruch are gratefully acknowledged. The second author is supported by RFBR 09-01-00878, 09-01-91333, 10-01-90016, 10-01-92651. References [1] W. Lerche, C. Vafa, N.P. Warner, Chiral rings in N = 2 superconformal theories, Nuclear Phys. B 324 (2) (1989) 427–474. [2] D. Gepner, Fusion rings and geometry, Comm. Math. Phys. 141 (1991) 381–411. [3] E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, in: Geometry, Topology, & Physics, in: Conf. Proc. Lecture Notes Geom. Topology, vol. IV, Int. Press, Cambridge, MA, 1995, pp. 357–422.

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