Special geometry on Calabi--Yau moduli spaces and Q --invariant Milnor rings
aa r X i v : . [ h e p - t h ] A ug Special geometry on Calabi–Yau modulispaces and Q –invariant Milnor rings Alexander Belavin
L.D. Landau Institute for Theoretical PhysicsSemenov av. 1-AChernogolovka, 142432 Moscow region, Russia Department of Quantum PhysicsInstitute for Information Transmission ProblemsBolshoy Karetny per. 19, 127994 Moscow, Russia
E-mail: [email protected]
Abstract:
The moduli spaces of Calabi–Yau (CY) manifolds are the special Kählermanifolds. The special Kähler geometry determines the low-energy effective theory whicharises in Superstring theory after the compactification on a CY manifold. For the cases,where the CY manifold is given as a hypersurface in the weighted projective space, anew procedure for computing the Kähler potential of the moduli space has been proposedin [1–3]. The method is based on the fact that the moduli space of CY manifolds is amarginal subspace of the Frobenius manifold which arises on the deformation space of thecorresponding Landau–Ginzburg superpotential. I review this approach and demonstrateits efficiency by computing the Special geometry of the 101-dimensional moduli space ofthe quintic threefold around the orbifold point [3]. † Contribution to Proceedings of International Congress of Mathematicians 2018, Rio deJaneiro,(2018) ontents σ µ ( φ )
66 Computation of the Kähler potential 97 Real structure on the cycles Γ ± µ
118 Conclusion 12
To compute the low-energy Lagrangian of the string theory compactified on a CYmanifold [4], one needs to know the Special geometry of the corresponding CY modulispace [5–8].More precisely, the effective Lagrangian of the vector multiplets in the superspacecontains h , supermultiplets. Scalars from these multiplets take value in the target space M , which is a moduli space of complex structures on a CY manifold and is a specialKähler manifold. Metric G a ¯ b and Yukawa couplings κ abc on this space are given by thefollowing formulae in the special coordinates z a : G a ¯ b = ∂ a ∂ ¯ b K, e − K = − i Z X Ω ∧ ¯Ω ,κ abc = Z X Ω ∧ ∂ a ∂ b ∂ c Ω = ∂ F∂z a ∂z b ∂z c , (1.1)where z a = Z A a Ω , ∂F∂z a = Z B a Ω are the period integrals of the holomorphic volume form Ω on X . Here A a and B a formthe symplectic basis in H ( X, Z ) .We can rewrite the expression (1.1) for the Kähler potential using the periods as e − K = − i ΠΣΠ † , Π = ( ∂F, z ) , – 1 –here matrix (Σ) − is an intersection matrix of cycles A a , B a equal to the symplecticunit.The computation of periods in the symplectic basis appears to be very non-trivial. Itwas firstly performed for the case of the quintic CY manifold in the distinguished paper[9].Here I present an alternative approach to the computation of Kähler potential for thecase where CY manifold is given by a hypersurface W ( x, φ ) = 0 in a weighted projectivespace. The approach is based on the connection of CY manifold with a Frobenius ringwhich arises on the deformations of the singularity defined by the superpotential W ( x ) [10–12].Let a CY manifold X be given as a solution of an equation W ( x, φ ) = W ( x ) + h , X s =1 φ s e s ( x ) = 0 in some weighted projective space, where W ( x ) is a quasihomogeneous function in C ofdegree d that defines an isolated singularity at x = 0 . The monomials e s ( x ) also havedegree d and are in a correspondence to deformations of the complex structure of X .Polynomial W ( x ) defines a Milnor ring R . Inside R there exists a subring R Q whichis invariant under the action of the so-called quantum symmetry group Q that acts on C diagonally, and preserves W ( x, φ ) . In many cases dim R Q = dim H ( X ) and the ringitself has a Hodge structure R Q = ( R Q ) ⊕ ( R Q ) ⊕ ( R Q ) ⊕ ( R Q ) in correspondence withthe elements of degrees , d, d, d .Another important group is the subgroup of phase symmetries G , which acts diag-onally on C , commutes with the quantum symmetry Q and preserves W ( x ) . It actsnaturally on the invariant ring R Q , and this action respects the Hodge decomposition of R Q . This allows to choose a basis e µ ( x ) in each of the Hodge decomposition componentsof R Q to be eigenvectors for the G group action.On the ring R Q we introduce the invariant pairing η . The pairing turns the ring toa Frobenius algebra [13]. The pairing η plays an important for our construction of theexplicit expression for the volume of the Calabi-Yau manifold.Using the invariant ring R Q and differentials D ± = d ± d W ∧ we construct two Q − invariant cohomology groups H D ± ( C ) inv . These groups inherit the Hodge structurefrom R Q . We can choose in H D ± ( C ) inv the eigenbasises e µ ( x ) d x which are also invariantunder the phase symmetry action.As shown in [14], elements of these cohomology groups are in correspondence with theharmonic forms of H ( X ) . This isomorphism allows to define the antilinear involution ∗ on the invariant cohomology H D ± ( C ) inv that corresponds to the complex conjugation onthe space of the harmonic forms of H ( X ) .It turns out, that in the basis e µ ( x ) it reads ∗ e µ ( x ) d x = M νµ e ν ( x ) d x, M νµ = δ e µ · e ν ,e ρ A µ (1.2)where e ρ ( x ) is the unique element of degree d in R Q , and δ e µ · e ν ,e ρ is 1 if e µ · e ν = e ρ and– 2 – otherwise.Having H D ± ( C ) inv we define the relative invariant homology subgroups H ± ,inv := H ( C , W = L, Re L → ±∞ ) inv inside the relative homology groups H ( C , W = L, Re L → ±∞ ) . To do this we will use the oscillatory integrals and their pairing withelements of H D ± ( C ) inv . Using this pairing we define a cycle Γ ± µ in the basis of relativeinvariant homology to be dual to e µ ( x ) d x .At last we define periods σ ± µ ( φ ) to be oscillatory integrals over the basis of cycles Γ ± µ .They are equal to periods of the holomorphic volume form Ω on X in a special basis ofcycles of H ( X, C ) with complex coefficients.It follows from the phase symmetry invariance that in the chosen basis of cycles Γ ± µ the formula for Kähler potential has the diagonal form: e − K ( φ ) = X µ ( − | ν | σ + µ ( φ ) A µ σ − µ ( φ ) . On the other hand, as shown in [1], matrix A = diag { A µ } is equal to the product ofthe matrix of the invariant pairing η in the Frobenius algebra R Q and the real structurematrix M such that e − K ( φ ) = X µ,ν σ + µ ( φ ) η µλ M νλ σ − µ ( φ ) . The real structure matrix is nothing but matrix M from (1.2). Using this we are able toexplicitly compute the diagonal matrix elements A µ and to obtain the explicit expressionfor the whole e − K . It was shown in in [5–8] that the moduli space M of complex (or Kähler) structuresof a given CY manifold is a special Kähler manifold.Namely on M there exist so-called special (projective) coordinates z · · · z n +1 and a holo-morphic homogeneous function F ( z ) of degree 2 in z , called a prepotential, such that theKähler potential K ( z ) of the moduli space metric is given by e − K ( z ) = Z X Ω ∧ ¯Ω = z a · ∂ ¯ F∂ ¯ z ¯ a − ¯ z ¯ a · ∂F∂z a (2.1)To obtain this formula, we choose Poincare dual symplectic basises α a , β b ∈ H ( X, Z ) and A a , B b ∈ H ( X, Z ) and define the periods as z a = Z A a Ω , F b = Z B b Ω . Then using the Kodaira Lemma ∂ a Ω = k a Ω + χ a , we can show that F a ( z ) = 12 ∂ a ( F ( z )) , – 3 –here F ( z ) = 1 / z b F b ( z ) .Therefore, according to the definition (2.1) metric G a ¯ b = ∂ a ¯ ∂ ¯ b K ( z ) is a special Kählermetric with prepotential F ( z ) and with the special coordinates given by the period vector Π = (cid:0) F α , z b (cid:1) we write the expression for the Kähler potential as e − K ( z ) = Π µ Σ µν ¯Π ν , (2.2)where Σ is a symplectic unit, which is an inverse intersection matrix for cycles A a and B b .Using formula (2.2), we can rewrite this expression in a basis of periods defined asintegrals over arbitarary basisis of cycles q µ ∈ H ( X, Z ) ω µ = Z q µ Ω . Such that e − K = ω µ C µν ¯ ω ν , where C µν is the inverse marix of the intersection of the cycles q µ .So to find the Kähler potential, we must compute the periods over a basis of cycleson CY manifold and find their intersection matrix. Now let us specialize to the case where X is a quintic threefold: X = { ( x : · · · : x ) ∈ P | W ( x, φ ) = 0 } , and W ( x, φ ) = W ( x ) + X t =0 φ t e t ( x ) , W ( x ) = x + x + x + x + x (3.1)and e t ( x ) are the degree 5 monomials such that each variable has the power that is anon-negative integer less then four. Let us denote monomials e t ( x ) = x t x t x t x t x t by itsdegree vector t = ( t , · · · , t ) . Then there are precisely 101 of such monomials, which canbe divided into sets in respect to the permutation group S : (1 , , , , , (2 , , , , , (2 , , , , , (3 , , , , , (3 , , , , . In these groups there are correspondingly 1, 20,30, 30, 20 different monomials. We denote e ( x ) := e (1 , , , , ( x ) = x x x x x to bethe so-called fundamental monomial, which will be somewhat distinguished in our picture.– 4 –or this CY dim H ( X ) = 204 and period integrals have the form ω µ ( x ) = Z q µ x d x d x d x ∂W ( x, φ ) /∂x = Z Q µ d x · · · d x W ( x, φ ) , where q µ ∈ H ( X, Z ) and the corresponding cycles Q µ ∈ H ( C \ ( W ( x, φ ) = 0) , Z ) .Cohomology groups of the Kähler manifold X possess a Hodge structure H ( X ) = H , ( X ) ⊕ H , ( X ) ⊕ H , ( X ) ⊕ H , ( X ) . Period integrals measure variation of the Hodgestructure on H ( X ) as the complex structure on X varies with φ .This Hodge structure variation is in correspondence with a Frobenius ring which wewill now describe. Now we will consider W ( x ) as an isolated singularity in C and the associated withit Milnor ring R = C [ x , · · · , x ] h ∂ i W i . We can choose its elements to be unique smallest degree polynomial representatives. Forthe quintic threefold X its Milnor ring R is generated as a vector space by monomialswhere each variable has degree less than four, and dim R = 1024 .Since the polynomial W ( x ) is homogeneous one of the fifth degree it follows that W ( αx , . . . , αx ) = W ( x , . . . , x ) for α = 1 . This action preserves W ( x ) and is trivialin the corresponding projective space and on X . Such a group with this action is calleda quantum symmetry Q , in our case Q ≃ Z . Q obviously acts on the Milnor ring R .We define a subring R Q to be a Q -invariant part of the Milnor ring R Q := { e µ ( x ) ∈ R | e µ ( αx ) = e µ ( x ) } , α = 1 .R Q is multiplicatively generated by 101 fifth-degree monomials e t ( x ) from (3.1) and con-sists of elements of degree , , and . The dimensions of the corresponding subspacesare , , and .This degree filtration defines a Hodge structure on R Q . Actually, R Q is isomorphic to H ( X ) and this isomorphism sends the degree filtration on R Q to the Hodge filtration on H ( X ) [14].Let us denote χ i ¯ j = g i ¯ k χ ¯ k ¯ j as an extrinsic curvature tensor and g i ¯ k is a metric for thehypersurface W ( x, φ ) = 0 in P . Then the isomorphism above can be written as a mapfrom R Q to closed differential forms in H ( X ) : → Ω ijk ∈ H , ( X ) ,e µ ( x ) → e µ ( x ( y )) χ l ¯ i Ω ljk ∈ H , ( X ) if | µ | = 5 ,e µ ( x ) → e µ ( x ( y )) χ l ¯ i χ m ¯ j Ω lmk ∈ H , ( X ) if | µ | = 10 ,e ρ ( x ) = x x x x x → χ l ¯ i χ m ¯ j χ p ¯ k Ω lmp = κ ¯Ω ∈ H , ( X ) (4.1)– 5 –he details of this map can be found in [14]. We also introduce the notation e µ ( x ) for ele-ments of the monomial basis of R Q , where µ = ( µ , · · · , µ ) , µ i ∈ Z , e µ ( x ) = Q i x µ i i andthe degree of e µ ( x ) µ = P µ i is equal to zero module . In particular, ρ = (3 , , , , , that is e ρ ( x ) is the unique degree 15 element of R Q .The phase symmetry group Z acts diagonally on C : α · ( x , · · · , x ) =( α x , · · · , α x ) , α i = 1 . This action preserves W = P i x i . The mentioned abovequantum symmetry Q is a diagonal subgroup of the phase symmetries. Basis { e µ ( x ) } consits of the eigenvectors of the phase symmetry and each e µ ( x ) has a unique weight.Note that the action of the phase symmetry preserves the Hodge decomposition.Another important fact is that on the invariant ring R Q there exists a natural invariantpairing turning it into a Frobenius algebra [13]: η µν = Res e µ ( x ) e ν ( x ) Q i ∂ i W ( x ) . Up to an irrelevant constant for the monomial basis it is η µν = δ µ + ν,ρ . This pairing playsa crucial role in our construction.Let us introduce a couple of Saito differentials as in [1] on differential forms on C : D ± = d ± d W ( x ) ∧ . They define two cohomology groups H ∗ D ± ( C ) . The cohomologies areonly nontrivial in the top dimension H D ± ( C ) J ≃ R . The isomorphism J has an explicitdescription J ( e µ ( x )) = e µ ( x ) d x, e µ ( x ) ∈ R . We see, that Q = Z naturally acts on H D ± ( C ) and J sends the elements of Q -invariantring R Q to Q -invariant subspace H D ± ( C ) inv . Therefore, the latter space obtains theHodge structure as well. Actually, this Hodge structure naturally corresponds to theHodge structure on H ( X ) .The complex conjugation acts on H ( X ) so that H p,q ( X ) = H q,p ( X ) , in particular H , ( X ) = H , ( X ) . Through the isomorphism between R Q and H ( X ) the complexconjugation acts also on the elements of the ring R Q as ∗ e µ ( x ) = p µ e ρ − µ ( x ) , where p µ p ρ − µ = 1 and p µ is a constant to be determined. In particular, differential form builtfrom the linear combinations e µ ( x ) + p µ e ρ − µ ( x ) ∈ H ( X, R ) is real. σ µ ( φ ) Relative homology groups H ( C , W = L, Re L → ±∞ ) have a natural pairing with Q -invariant cohomology groups H D ± ( C ) inv defined as h e µ ( x )d x, Γ ± i = Z Γ ± e µ ( x ) e ∓ W ( x ) d x, H ( C , W = L, Re L → ±∞ ) . Using this we introduce two Q -invariant homology groups H ± ,inv as quotient of H ( C , W = L, Re L → ±∞ ) with respect to the subgroups orthogonal to H D ± ( C ) inv . We are grateful to V. Vasiliev for explaining to us the details about these homology groups and theirconnection with the middle homology of X . – 6 –ow we introduce basises Γ ± µ in the homology groups H ± ,inv using the duality with thebasises in H D ± ( C ) inv : Z Γ ± µ e ν ( x ) e ∓ W ( x ) d x = δ µν and the corresponding periods σ ± αµ ( φ ) := Z Γ ± µ e α ( x ) e ∓ W ( x,φ ) d x,σ ± µ ( φ ) := σ ± µ ( φ ) (5.1)which are understood as series expansions in φ around zero.The periods σ ± µ ( φ ) satisfy the same differential equation as periods ω µ ( φ ) of the holomor-phic volume form on X . Moreover, these sets of periods span same subspaces as functionsof φ . Therefore we can define cycles Q ± µ ∈ H ± ,inv such that Z Q ± µ e ∓ W ( x,φ ) d x = Z q µ Ω = Z Q µ d xW ( x, φ ) . (5.2)So the periods ω ± αµ ( φ ) are given by the integrals over these cycles analogous to (5.1).With these notations the idea of computation of periods [15] σ ± µ ( φ ) = Z Γ ± µ e ∓ W ( x,φ ) d x (5.3)can be stated as follows.To explicitly compute σ ± µ ( φ ) , first we expand the exponent in the integral (5.3) in φ representing W ( x, φ ) = W ( x ) + P s φ s e s ( x ) σ ± µ ( φ ) = X m Y s ( ± φ s ) m s m s ! ! Z Γ ± µ Y s e s ( x ) m s e ∓ W ( x ) d x. (5.4)We note, that σ − µ ( φ ) = ( − | µ | σ + µ ( φ ) , so we focus on σ µ ( φ ) := σ + µ ( φ ) . For each of the summands in (5.4) the form Q s e s ( x ) m s d x belongs to H D ± ( C ) inv , becauseit is Q − invariant. Therefore, we can expand it in the basis e µ ( x ) d x ∈ H D ± ( C ) inv . Namely we can find such a polynomial − form U, that Y s e s ( x ) m s d x = X ν C ν ( m ) e ν ( x ) d x + D + U. In result we obtain for the integral in (5.4) Z Γ ± µ Y s e s ( x ) m s e ∓ W ( x ) d x = C µ ( m ) . – 7 –o from (5.4) we have σ µ ( φ ) = X m Y s φ m s s m s ! ! Z Γ + µ Y s,i x P s m s s i i e − W ( x ) d x. (5.5)We can rewrite the sum in the exponent of x i as P s m s s i = 5 n i + ν i , ν i < .Therefore we need to compute the coefficients c mν in the equations Y x n i + ν i i d x = X ν c mν e ν ( x ) d x + D + U. Note that D + (cid:18) x n + k − f ( x , · · · , x ) d x ∧ · · · ∧ d x (cid:19) == (cid:20) x n + k + (cid:18) n + k − (cid:19) x n − k (cid:21) f ( x , · · · , x ) d x (5.6)Therefore in D + cohomology we have Y i x n i + ν i i d x = − (cid:18) n + ν − (cid:19) x n − ν Y i =2 x n i + ν i i d x, ν i < . (5.7)By induction we obtain Y i x n i + ν i i d x = ( − P i n i Y i (cid:18) ν i + 15 (cid:19) n i Y i x ν i i d x, ν i < . (5.8)where ( a ) n = Γ( a + n ) / Γ( a ) .Using (5.6) once again, we see that if any ν i = 4 then the differential form is trivial andthe integral is zero. Hence, rhs of (5.8) is proportional to e ν ( x ) and gives the desiredexpression. Plugging (5.8) into (5.5) and integrating over Γ + µ we obtain the answer σ µ ( φ ) = σ + µ ( φ ) = X n i ≥ Y i (cid:18) µ i + 15 (cid:19) n i X m ∈ Σ n Y s φ m s s m s ! , where Σ n = { m | X s m s s i = 5 n i + µ i } Further we will also use the periods with slightly different normalization, which turn outto be convenient ˆ σ µ ( φ ) = Y i Γ (cid:18) µ i + 15 (cid:19) σ µ ( φ ) = X n i ≥ Y i Γ (cid:18) n i + µ i + 15 (cid:19) X m ∈ Σ n Y s φ m s s m s ! . (5.9)– 8 – Computation of the Kähler potential
Pick any basis Q ± µ of cycles with integer or real coefficients as in (5.2). Then for theKähler potential we have the formula e − K = ω + µ ( φ ) C µν ω − ν ( φ ) (6.1)in which the matrix C µν is related with the Frobenius pairing η as η αβ = ω + αµ (0) C µν ω − βν (0) . (6.2)The derivation of the last relation is given in [16, 17].Let also T ± be the matrix that connects the cycles Q ± µ and Γ ± ν .That is Q ± µ = ( T ± ) νµ Γ ± ν . Then M = ( T − ) − T − is a real structure matrix, that is M ¯ M = 1 and by construction M doesn’t depend on the choice of basis Q ± µ . M is only defined by our choice of Γ ± µ .In [1] we deduced from (6.1) and (6.2) the formula e − K ( φ ) = σ + µ ( φ ) η µλ M νλ σ − ν ( φ ) = σ µ A µν σ ν , (6.3)where η µν = η µν = δ µ,ρ − ν .Now we show that the matrix A µν in (6.3) is diagonal. To do this we extend theaction of the phase symmetry group to the action A on the parameter space { φ s } suchthat W = W + P s φ s e s ( x ) is invariant under this new action. It easy to see that each e s ( x ) has an unique weight under this group action. Action A can be compensated usingthe coordinate tranformation and therefore is trivial on the moduli space of the quintic(implying that point W = W is an orbifold point of the moduli space).In particular, e − K = R X Ω ∧ ¯Ω is A invariant. Consider e − K = σ µ A µν σ ν as a series in φ s and φ t . Each monomial has a certain weight under A . For the seriesto be invariant, each monomial must have weight 0. But weight of σ µ σ ν equals to µ − ν and due to non-degeneracy of weights of σ µ only the ones with µ = ν have weight zero.Thus, (6.3) becomes e − K = X µ A µ | σ µ ( φ ) | . Moreover, the matrix A should be real and, because A = η · M, M ¯ M = 1 and η µν = δ µ + ν,ρ , we have A µ A ρ − µ = 1 . (6.4) Monodromy considerations
To fix finally the real numbers A µ we use monodromyinvariance of e − K around φ = ∞ . Pick some t = ( t , t , t , t , t ) with | t | = 5 and let φ s | s = t, = 0 , . We will consider only the first order in φ t .– 9 –hen the condition that period σ µ ( φ ) contains only non-zero summands of the form φ m φ t implies that µ = t + const · (1 , , , , mod 5. For each t from the table below the onlysuch possibilities are µ = t and µ = ρ − t ′ = (3 , , , , − t ′ , where t ′ denotes a vectorobtained from t by permutation (written explicitly in the table below) of its coordinates.Therefore, in this setting (6.3) becomes e − K = X k =0 a k | ˆ σ ( k,k,k,k,k ) | + a t | ˆ σ t | + a ρ − t ′ | ˆ σ ρ − t ′ | + O ( φ t ) , here we use periods ˆ σ from (5.9) and denote a t = A t / Q i Γ(( t i +1) / . And the coefficients a k , k = 0 , , , are already known from [9]. This expression has to be monodromyinvariant under the transport of φ around ∞ . From the formula (5.9) we have F = ˆ σ k ( φ t , φ ) = g t φ k F ( a, b ; a + b | ( φ / ) + O ( φ t ) ,F = ˆ σ ρ − t ′ ( φ t , φ ) = g ρ − t ′ φ t φ − a − b F (1 − a, − b ; 2 − a − b | ( φ / ) + O ( φ t ) , where g t , g ρ − t ′ are some constants. Explicitly for all different labels tt ρ − t ′ (a, b)(2,1,1,1,0) (3,2,2,2,1) (2/5,2/5)(2,2,1,0,0) (3,3,2,1,1) (1/5,3/5)(3,1,1,0,0) (0,3,3,2,2) (1/5,2/5)(3,2,0,0,0) (1,0,3,3,3) (1/5,1/5)When φ goes around infinity (cid:18) F F (cid:19) = B · (cid:18) F F (cid:19) , where B = 1 is ( a + b ) (cid:18) c ( a − b ) − e iπ ( a + b ) s ( a ) s ( b )2 e πi ( a + b ) s ( a ) s ( b ) e πi ( a + b ) [ e πia + e πib − / (cid:19) . Here c ( x ) = cos( πx ) , s ( x ) = sin( πx ) . It is straightforward to show the following Proposition 1. a t | ˆ σ t | + a ρ − t ′ | ˆ σ ρ − t ′ | = a t Y i Γ (cid:18) t i + 15 (cid:19) | σ t | + a ρ − t ′ Y i Γ (cid:18) − t i (cid:19) | σ ρ − t ′ | is B -invariant iff a t = − a ρ − t ′ . Due to symmetry we have a ρ − t ′ = a ρ − t in each case. From (6.4) it follows that theproduct of the coefficients at | σ µ | and | σ ρ − µ | in the expression for e − K should be 1: A ρ − t ′ · A t = a ρ − t ′ · a t Y i Γ (cid:18) t i + 15 (cid:19) Γ (cid:18) − t i (cid:19) = 1 . – 10 –ue to reflection formula a t = ± Q i sin( π ( t i + 1) / up to a common factor of π . The signturns out to be minus for Kähler metric to be positive definite in the origin. Therefore A µ = ( − deg( µ ) / Y γ (cid:18) µ i + 15 (cid:19) . Finally the Kähler potential becomes e − K ( φ ) = X µ =0 ( − deg( µ ) / Y γ (cid:18) µ i + 15 (cid:19) | σ µ ( φ ) | , (6.5)where γ ( x ) = Γ( x )Γ(1 − x ) . Γ ± µ Let cycles γ µ ∈ H ( X ) be the images of cycles Γ + µ under the isomorphism H + ,inv ≃ H ( X ) .Complex conjugation sends (2 , -forms to (1 , -forms. Similarly it extends to a mappingon the dual homology cycles γ µ . Lemma 1.
Conjugation of homology classes has the following form: ∗ γ µ = p µ γ ρ − µ , where ρ = (3 , , , , is a unique maximal degree element in the Milnor ring.Proof. We perform a proof for the cohomology classes represented by differential forms.For one-dimensional H , ( X ) and H , ( X ) it is obvious. Let Ω , := e t ( x ) χ l ¯ i Ω ljk ∈ H , ( X ) . Any element from H , ( X ) is representable by a degree 10 polynomial P ( x ) as followsfrom (4.1) as Ω , = Ω , := P ( x ) χ l ¯ i χ m ¯ j Ω lmk ∈ H , ( X ) . The group of phase symmetries modulo common factor acts by isomorphisms on X .Therefore, it also acts on the differential forms. Lhs and rhs of the previous equationshould have the same weigth under this action, and weight of the lhs is equal − t modulo (1 , , , , . It follows that P ( x ) = p t e ρ − t ( x ) with some constant p t .Using this lemma and applying the complex conjugation of cycles to the formula (6.3)to obtain e − K = X µ A µ | σ µ | = X µ p µ A µ | σ ρ − µ | , it follows that A µ = ± /p µ . Now formula (6.5) implies p µ = Y γ (cid:18) − µ i (cid:19) . – 11 – Conclusion
I am grateful to K. Aleshkin for the interesting collaboration; the talk is based on thejoint work with K. Aleshkin. Also I am thankful to M. Bershtein, V. Belavin, S.Galkin,D. Gepner, A. Givental, M. Kontsevich, A. Okounkov, A. Rosly, V. Vasiliev for the usefuldiscussions. The work has been performed for FASO budget project No. 0033-2018-0006.
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