Vertex Algebras and the Landau-Ginzburg/Calabi-Yau Correspondence Vassily Gorbounov, Fyodor Malikov To Borya Feigin on his 50th birthday Introduction Introduced in [MSV] for any smooth (algebraic, analytic, etc.) manifold X th* *ere is a sheaf of vertex algebras chX. For example, the vertex algebra of global s* *ections over CN , chCN(CN ) or simply ch(CN ), is well known as "bc - fifl-system", an apparently unsophisticated object. Despite various important contributions [B, BD, BL, KV1], however, very little is known about the cohomology vertex algebra H*(X, chX) even if X is toric - except perhaps the case of P2n where at least * *the character of the space of global sections, H0(P2n, chP2n), has been computed: * *it was shown to be equal to the elliptic genus of P2n in [MS]. Let F = {f = 0} PN-1 . (1) be a Calabi-Yau hypersurface. The present paper is devoted to an interplay betw* *een ch(CN ) and closely related algebras on the one hand, and H*(F, chF) on the o* *ther. Let us now formulate the main result. Preparations. Set = ZN (ZN )*. Associated to this lattice in the standard manner there are the lattice vertex * *algebra V and the fermionic vertex algebra ( bc-system), F , which is none other than* * the vacuum representation of the infinite-dimensional Clifford algebra Cl(C ZN ).* * In the important paper [B] Borisov introduces the vertex algebra B = V F and the vertex algebra embedding ch(CN ) ,! B . Define the ZN -action ZN x CN ! CN , (m, ~x) 7! exp (2ßi m__N)~x. There arises the vertex subalgebra of ZN -invariants ( ch(CN ))ZN ch(CN ). ______________ partially supported by National Science Foundation 1 2 Let us now warp the lattice : Pdenote by ZNorbthe sublattice of ZN consisting (m0, m1, ..., mN-1 ) such that j mj is divisible by N and define orb = ZNorb (ZNorb)*. Just as above, there arises the vertex algebra B orb. Note that B and B orb ha* *ve a non-empty intersection, which contains ( ch(CN ))ZN ; thus ( ch(CN ))ZN ,! B orb. Now we extend ( ch(CN ))ZN inside B orb to a bi-differential vertex algebra. Let {Xi} ZN , {X*i} (ZN )* be the standard dual bases.*Associated to them inside V there are fields, such as, Xi(z), X*i(z), eXi (z). Let the correspo* *nding tilded letters denote their superpartners inside F , e.g. X~i(z), X~*i(z). Denote X*orb= _1_N(X*0+ X*1+ . .X.*N-1) 2 (ZNorb)*. Form 1 M (n) (n) * fLG= fLG , LfG = ( ch(CN ))ZN enXorb. n=0 This is clearly a Z+ -graded subalgebra of B orb (but not of B ). Now define two operators N-1X * Dorb = ( eXorb(z)X~*j(z))(0), dLG = df(z)(0)2 End (LfG), j=0 where f is the polynomial appearing in (1) and df(z) is computed by using the definition P X d(xm00xm11. .x.mN-1N-1)(z) = e jmjXj (z) mjX~j(z). j These are commuting, square zero derivations of LfG; thus we have obtained the bi-differential vertex algebra (LfG; Dorb, dLG ). Note that it is filtered* * by the bi-differential vertex ideals M1 (m) LfG n = LfG ; m=n hence there arises the projective system of bi-differential vertex algebras fLG)ZN is the ZN -invariant part of the Mi* *lnor ring. Remarks. (i) The sign ~ in item (i) means that rather than being isomorphic the compl* *exes are filtered and the corresponding graded complexes are naturally isomorphic. T* *his is not too serious a complication; in fact, ~ is a genuine isomorphism if i < N* * - 1 and there is a 1-step filtration if i = N - 1. (ii) The reader familiar with previous work might expect H*(F, *F) -~! H*(F, chF)0 instead of the first isomorphism in the item (ii) o* *f the theorem. We have indeed changed the conformal grading and find this important; so much so that in the main body of the text (see especially 2.3.3) we change t* *he terminology and notation: we write chTF for chFwith the changed grading and call it the the algebra of chiral polyvector fields. Now we would like to make two points. First, let us demonstrate how this res* *ult works. Application: an elliptic genus formula. Let EllF(ø, s) be the 2-variable e* *lliptic genus of F as defined, for example, in [BL] or [KYY]. Introduce Y1 1 - e2ßi((n+1)ø+(1-1=N)s)N 1 - e2ßi(nø+(-1+1=N) s)N E(ø, s) = ______________________________________________________NN. n=0 1 - e2ßi((n+1)ø+s=N) 1 - e2ßi(nø-s=N) It follows easily from Theorem 1 (see 4.11- 4.12) that N-1XN-1X 2 2 EllF(ø, s) = _1_N eßi(N-2){ -js+(j -j)ø+j }E(ø, s - jø - l). (3) l=0 j=0 The structure of this formula is rather clear: the infinite product E(ø, s) ref* *lects the polynomial nature of the space ch(CN ) of which it is indeed an Euler characte* *r, 4 the summation with respect to l extracts the ZN -invariants, and the summation w.r.t. j reminds of the summation over "twisted sectors" because the change of variable s 7! s - jø is reminiscent of the spectral flow. All of this smacks o* *f an orbifold and indeed formula (3) was proposed in [KYY] as the elliptic genus of * *the Landau-Ginzburg orbifold. This brings about the 2nd point we would like to make. Landau-Ginzburg orbifold interpretation. The Landau-Ginzburg model is asso- ciated to an affine manifold and a function over it known as superpotential. In* * the case where the manifold is CN and the function is a homogeneous polynomial f with a unique singularity at 0, Witten's discovery [W2] can perhaps be formulat* *ed in the language accessible to us as follows: 1) There is an action of the N = 2 superconformal algebra on ch(CN ) such that it commutes with the differential df(z)(0). 2) The cohomology vertex algebra Hdf(z)(0)( ch(CN )) with thus defined action of the N = 2 superconformal algebra is the chiral algebra attached to the Landa* *u- Ginzburg model with superpotential f. We find it convenient not to pass to the cohomology but to declare the Landa* *u- Ginzburg model to be the differential vertex algebra ( ch(CN ), df(z)(0)) with * *the above fixed N = 2 superconformal algebra action. Remark. We shall argue in 5.1.5 that alternatively one can think of ( ch(CN ), df(z)(0)) as the "right definitionö f the chiral de Rham complex c* *hSpecMf over the spectrum of the Milnor ring. Next consider the space ch(CN )enX*orb. It does not belong to B orb but car* *ries a canonical structure of a twisted ch(CN )-module and this is synonimous to be* *ing a twisted sector. Therefore, taking the direct sum of these, 0 n N - 1 and then extracting ZN -invariants corresponds accurately with what physicists call* * the Landau-Ginzburg orbifold, see e.g. [V]. In the notation we have adopted, the fo* *r- > 0. (1.1.* *1) 1.2. Definition. A vertex algebra is a supervector space V with a distinguis* *hed element 1 2PV called vacuum and a parity preserving map Y (., z) : V ! Field(V * *), Y (a, z) = n2Z a(n)z-n-1 , such that the following axioms hold: (i) vacuum: Y (1 , z) = IdV , a(-1)1 = a; (1.2.* *1) (ii) Borcherds identity: for any a, b 2 V and any rational function F (z, w) i* *n z, w with poles only at z = 0, w = 0, z - w = 0 Res z-w Y (Y (a, z - w)b, w)iw,z-w F (z, w) i * * j = Resz Y (a, z)Y (b, w)iz,wF (z, w) - (-1)par(a)par(b)Y (b, w)Y (a, z)iw,zF.* *(z, w) (1.2.* *2) In the latter formula the standard notation is used: Res tmeans the coeffici* *ent of t-1 in the indicated formal Laurent expansion; io,ospecifies exactly which Laur* *ent expansion is to be used, e.g. iz,w stands for the expansion in the domain |w| <* * |z|, iw,z-w for that in the domain |z - w| < |w|, etc. One thinks of a(n) as the " n-th multiplication by a", so there arises a fam* *ily of multiplication (n): V V ! V. (1.2.* *3) 1.3. W- and cohomology vertex algebras. If F = 1, then (1.2.2) gives (a(0)b)(n)= [a(0), b(n)], n 2 Z. (1.3.* *1) In other words, for any a 2 V , a(0)2 End V is a derivation of all the multipli* *cations. Hence, Ker a(0) V (1.3.* *2) is a vertex subalgebra known as a W -algebra. Furthermore, suppose a 2 V is odd and a(0)a = 0. Then (1.3.1) implies that a2(0)= 0 and Im a(0) Kera(0)is an ideal. Therefore, the cohomology Ha(0)(V ) def=Kera(0)=Im a(0) (1.3.* *3) 8 carries a canonical vertex algebra structure. The vertex algebras to be used in* * this text will mostly be either W- or cohomology vertex algebras. 1.4. Chiral rings. Suppose that V is graded so that V = 1n=0Vn and Vn (r)Vm Vn+m-r-1 . (1.4.* *1) The grading satisfying this condition will be called conformal. One can show that if (1.4.1) is valid, then (-1): V0 V0 ! V0 is associative and supercommutative. In the context of the unitary N = 2 supers* *ym- metry, supercommutative associative algebras attached to graded vertex algebras in this way are often called chiral rings [LVW] - not to be confused with chiral algebras although these rings are indeed algebras. We shall take the liberty to* * call these rings chiral in any case. 1.5. Remarks. P (i) Let ffi(z-w) = n2Z zn w-n-1 . It follows from (1.2.2) that [Y (a, z), * *Y (b, w)] is local, that is, equals a linear combination of the delta-function derivatives, * *@nwffi(z - w), over fields in w. (ii) A vertex algebra V is said to be generated by a collection of fields Y * *(aff, z), {aff} V if V is the linear span of (non-commutative) monomials in (aff)(j)app* *lied to vacuum 1 2 V . The important reconstruction theorem, e.g. [K, Theorem 4.5], says, and we are omitting some details, that this can be reversed: if there is* * a collection of mutually local fields vff(z) which generate V from a fixed vector* *, then V carries a unique vertex algebra structure such that vff(z) = Y (vff, z) for * *some vff2 V . Because of this we will allow ourselves in our list of well-known exam* *ples, which we are about to begin, to fix only a space V and a collection of mutually* * local fields that generate this space. Typically, we shall have a Lie algebra, a coll* *ection of fields with values in this algebra, and a representation of this algebra suc* *h that nilpotency condition (1.1.1) is satisfied by the fields. (iii) It should be clear what a vertex algebra homomorphism is. As in (ii), speaking of homomorphisms we shall often specify only images of generating fiel* *ds. Now to some basic excamples. 1.6. bc-system. Let Cl be the Lie superalgebra with basis b(i), c(i), i 2 Z* * (all odd) and C (even) and commutation relations [b(i), c(j)] = ffii,-j-1C, [C, ci] = [C, bi] = 0, i, j 2 Z.(1.6* *.1) Let F = IndCl+ClC, (1.6.* *2) where Cl+ is the Lie subalgebra spanned by x(i), C, i 0, and C is an Cl+ - modulePwhere x(i)'s act by 0 and C as multiplication by 1. In terms of fields x(z) i2Zx(i)z-i-1 , x = b or c, (1.6.1) becomes: [b(z), c(z)] = ffi(z - w). (1.6.* *3) 9 Hence the vertex algebra structure on F , see 1.5. A little more generally, to any purely odd C-vector W with a non-degenerate symmetric form (., .) one can attach the LiePsuperalgebra Cl(W ) = W C[t, t-1* * ] CC with the following bracket: if x(z) = i2Z(x ti)z-i-1 , then [x(z), y(z)] = ffi(z - w)(x, y)C, [x(z), C] = 0. (1.6.* *4) The corresponding vertex algebra is FW = IndCl(W)Cl(W)+C, (1.6.* *5) where Cl(W )+ = W C[t] CC, W C[t] operates on C by 0, and C by 1. 1.7. fifl- and bc - fifl-system. The fifl-system is obtained by the äp rity change" functor applied to the beginning of 1.6: the even Lie algebra a is span* *ned by fi(i), fl(i), C, the bracket is [fi(z), fl(w)] = -[fl(z), fi(w)] = ffi(z - w)C, [C, x(z)] = 0.(1.7* *.1) The vertex algebra, B, is defined in the same way as F , see (1.6.2). If V and W are vertex algebras, then V W carries the standard vertex algeb* *ra structure. Denote F B = F B. (1.7.* *2) This algebra and its modifications are local models for the chiral de Rham comp* *lex. 1.8. Heisenberg algebra. Let, as in the end of 1.6, h be a purely even vector space with a non-degenerate symmetric form (., .). There arises then the Heisen* *berg Lie algebra ^h= h C[t, t-1 ] C . C P with bracket defined by letting fields be x(z) = i2Z(x ti)z-i-1 and then se* *tting [a(z), b(w)] = (a, b)@w ffi(z - w), [C, a(z)] = 0. (1.8.* *1) The vertex algebra attached to this Lie algebra is V (h) = Ind^h^h+C, (1.8.* *2) where ^h+= h C[t] C . C, ^h+= h C[t] perates on C by zero, C by 1. 1.9. Lattice vertex algebras. We shall neeed a lattice L, that is, a free abelian group with integral bilinear form (., .) and a 2-cocycle ffl : L x L ! C*. (1.9.* *1) There arise the group algebra C[L] with multiplication eff.efi= eff+fi, ff, fi * *2 L, and the twisted group algebra, Cffl[L], equal to C[L] as a vector space but with tw* *isted multiplication: eff.fflefi= ffl(ff, fi)eff+fi. (1.9.* *2) 10 Let hL = C Z L. There arises the Heisenberg vertex algebra V (hL ), see (1.8.2* *). As a vector space, the lattice vertex algebra is defined by VL = V (hL ) Cffl[L], par(V (hL ) eff) (ff, ff) mod 2,(1.9* *.3) where par means the parity, see 1.1. P This vertex algebra is generated by the familiar fields x(z) = i2Z(x ti)* *z-i-1 attached to x(-1) 1 and the celebrated vertex operators X ff(j) X ff(j) eff(z) = effexp ( ____z-j )exp( ____z-j )zff(0), (1.9.* *4) j<0 -j j>0 -j attached to 1 eff. The action of x(i), i 6= 0 ignores Cffl(L), the action of x(0)is uniquely de* *termined by x(0)(1 eff) = (ff, x) eff. (1.9.* *5) The following commutator and OPE formulas are valid: [x(z), eff(w)] = ffi(z - w)(x, ff)eff(w), (1.9.* *6) eff(z)efi(w) = (z - w)(ff,fi): eff(z)efi(w) :, (1.9.* *7) and the reader is advised to consult [K, (5.4.5b)] for the meaning of the two-v* *ariable field : eff(z)efi(w) :. Note that (1.9.7) allows to compute all operations (ef* *f)(n)efi. We shall need the following particular cases: (eff)(-1)efi=limz!w(z - w)(ff,fi): eff(z)efi(w) := ( 0 if(ff, fi) > 0 (1.9.* *8) ffl(ff, fi)eff+fiif(ff, fi) = 0, [eff(z), efi(w)] = 0 if(ff, fi) 0. (1.9.* *9) For the future use let us mention that for any sub-semigroup M L there ari* *ses the vertex subalgebra VM,L def=V (hL ) Cffl[M] VL . (1.9.1* *0) naturally graded by M. 1.10. N = 2 super-Virasoro algebra. The celebrated N = 2 super-Virasoro algebra, to be denoted N2 following [K], is a supervector space with basis G(n), Q(n), n 2 Z (all odd), L(n), J(n), n 2 Z, C (all even), and bracket [L(z), L(w)]= 2@w ffi(z - w)L(w) + ffi(z - w)L(w)0 (1.10.1* *a) [J(z), J(w)]= @w ffi(z - w)C=3, 11 [L(z), G(w)]= 2@w ffi(z - w)G(w) + ffi(z - w)G(w)0, (1.10.1* *b) [J(z), G(w)]= ffi(z - w)G(w), [L(z), Q(w)] = @w ffi(z - w)Q(w) + ffi(z - w)Q(w)0, (1.10.1* *c) [J(z), Q(w)] = -ffi(z - w)Q(w), [L(z), J(w)] = @2wffi(z - w) C_6+ @w ffi(z - w)J(w) + ffi(z - w)J(w)0,(1.10* *.1d) [Q(z), G(w)] = @2wffi(z - w) C_6- @w ffi(z - w)J(w) + ffi(z - w)L(w).(1.10.* *1e) The vertex algebra structure is carried by the following N2-module: V (N2)c = IndN2N2Cc. (1.10.* *2) where N2 is the subalgebra linearly spanned by G(n), L(n), Q(n), J(n), C, n * * 0, and on Cc G(n), L(n), Q(n), J(n) operate by 0, and C as multiplication by c. 1.10.1. Definition. An N2-structure on a vertex algebra W is a vertex algebra homomorphism V (N2)c ! W . Note that the field L(z) generates the Virasoro algebra; thus an N2-structure on a vertex algebra induces a conformal structure, and the grading by eigenvalu* *es of L(1)is conformal, cf. (1.4.1). 1.11. Automorphisms. It is obvious that if W is a vector space with a symmetric non-degenerate bilinear form, then there is an embedding O(W ) ,! Aut FW , (1.11.* *1) where O (W ) is the orthogonal group and FW a vertex algebra defined in (1.7.2* *); this comes from the standard action of O (W ) on the Clifford Lie algebra Cl(W * *). The analogous construction with O (W ) replaced with Aut (L) and FW with VL does not quite work because of the cocycle (1.9.1). Here is one trivial observa* *tion: if we let Aut ffl(L) be the subgroup of Aut (L) stabilizing ffl, then there is * *an (obvious) embedding: Aut ffl(L) ,! Aut VL . (1.11.* *2) The Lie algebra N2 affords an exceptional, mirror symmetry automorphism Q(z) 7! G(z), G(z) 7! Q(z), J(z) 7! -J(z), L(z) 7! L(z) + J(z)0. (1.11.* *3) 1.12. Spectral flow. In all our examples except VL vertex algebras came from infinite dimensional Lie algebras. One feature these Lie algebras have in common is that they admit a spectral flow. Define for any n 2 Z a linear transformation: Cl ! Cl, s.t. b(z) ! b(z)z-n , c(z) ! c(z)zn , a ! a, s.t.fi(z) ! fi(z)z-n , fl(z) ! fl(z)zn , N2 ! N2, s.t. Q(z) ! Q(z)zn , G(z) ! G(z)z-n , Sn : 1 nC (1.12.* *1) J(z) 7! J(z) - __z___3, L(z) 7! L(z) - 1_znJ(z) + _1_z2n(n - 1) C_6, 12 where we abused the notation by letting the same letter stand for the maps of different spaces, Cl, a, N2, defined in 1.6,7,10 resp. We hope this will not l* *ead to confusion. An untiring reader will check that in each of the cases, Sn is an automorphism of the Lie algebra in question. Maps (1.12.1) generate a Z-action on each of the algebras known as the spect* *ral flow. In each of the cases, therefore, there arises a family of functors on the* * category of modules Sn : Mod ! Mod , M 7! Sn(M), (1.12.* *2) action on Sn(M) being defined by precomposing that on M with Sn of (1.12.1). The origin of spectral flows (1.12.1) belongs to lattice vertex algebras. Le* *t VL be a lattice vertex algebra and LieVL the linear span inside End VL of the coeffic* *ients of the fields v(z), v 2 V . It is well known [K, F-BZ] that LieVL is a Lie suba* *lgebra of End VL . If M L is a sub-semigroup, then we have, cf. (1.9.5), VM,L VL , LieVM,L LieVL , (1.12.* *3) Let eff: VL ! VL (1.12.* *4) be multiplication by eff. If the restriction of the cocycle ffl(., .) to M L * *is trivial, that is, ffl(M, .) = 1, (1.12.* *5) then the conjugation by map (1.12.4) defines an automorphism Sff: LieVM,L ! LieVM,L , X 7! (eff)-1 O X O eff; ff 2 L. (1.12.* *6) For example, under this map efi(z) 7! efi(z)z(ff,fi), fi 2 M (1.12.* *7) cf. (1.12.1); the desired power of z ows its appearance to the factor zff(0)in * *(1.9.4). Likewise, x(z) 7! x(z) + (ff,_x)_z. (1.12.* *8) Spectral flows (1.12.1) are all obtained as follows: embed the corresponding ve* *rtex algebra into an appropriate VL , thus obtain a morphism of the corresponding Lie algebra Lie(o) ! LieVL , and then restrict "spectral flow in the direction ff" * *(1.12.6) to the image. This operation will be of importance for us in 3.10, 4.6, 5.2.15. 1.13. Boson-fermion correspondence. Let Z be the standard 1-dimensional lattice: this means that if we let Ø be the generator, then (Ø, Ø) = 1. There a* *rise VZ, where ffl(., .) = 1, and the famous vertex algebra isomorphism: F -~! VZ, b(z) 7! eØ(z), (1.13.* *1) c(z) 7! e-Ø (z), : b(z)c(z) :7! Ø(z) 13 The interested reader will check that SN |Cl of (1.12.1) is indeed implemented * *by conjugation with e-nØ . Likewise ~ F n -! VZn, bi(z) 7! eØi(z), (1.13.* *2) ci(z) 7! e-Øi (z), : bi(z)ci(z) :7! Øi(z), where the cocycle on VZn is chosen so as to ensure that VZn -~! VZ n. 2. The algebra of chiral polyvector fields This section is a reminder on the chiral de Rham complex. Our exposition is close to [MSV] but has been influenced by [GMS]. We would like to single out 2.* *3.3, where the title is clarified, and 2.3.5, where a simple cohomology computation * *is carried out; the results of this computation will play an important role in the* * proof of Theorem 4.7. Sect. 2.4 is an exposition of a result of [B]. 2.1. Suppose we have a family of sheaves of vector spaces AX , one for each smooth algebraic manifold X. We shall call AX natural if for any 'etale morphi* *sm OE : Y ! X there is a sheaf embedding A(OE) : OE-1 AX ,! AY such that given a diagram: X !OEY !_ Z the following associativity condition holds: A(_ O OE) = A(OE) O OE-1 (A(_)), where OE-1 is understood as the inverse image functor on the category of sheave* *s of vector spaces. It should be clear what a natural sheaf morphism AX ! BX means. Here are some obvious examples: OX , 1X, TX , and sheaves obtained as a res* *ult of all sorts of tensor operation performed on these. Note that all these sheave* *s are sheaves of OX -modules and our definition ignores this extra structure. Howeve* *r, one talks about natural sheaves AX of different classes of algebras, such as, * *commu- tative, associative, Lie, vertex, etc., by requiring that AOEpreserve this stru* *cture. Constructed in [MSV] for any smooth algebraic manifold X there is a sheaf of vertex algebras, chX. It satisfies the following conditions. (i) chXis natural as a sheaf of vertex algebras and it carries a bi-grading* * chX= m,n ch,mX,nsuch that each homogeneous component ch,mX,nis a natural sheaf of vector spaces. (ii) There are natural morphisms: *X ,! chX- *TX ; (2.1.* *1) (iii) chXis not a sheaf of OX -modules, but it carries a filtration such th* *at there is the following family of natural sheaf isomorphisms O i j Gr chX-~! S*qn+1(TX ) S*qn(TX*) *qn+1y(TX ) *qny-1(TX*). (2.1.* *2) n 0 14 where we habitually use the following " generating functions of families of she* *aves": M ch,m M1 M1 Gr chX= qn ym Gr X,n , S*t(A) = tnSn (A), *t(A) = tn n(A). m,n n=0 n=0 (iv) it follows from (i) that for any X there is a canonical group embedding æX : Aut X ! Aut chX(X). (2.1.* *3) Explicit formulas for the latter appeared in [MSV, (3.1.6)] as a result of gues* *swork and were used to define chXsatisfying (i-iii). In 2.2 we shall look at some examples that serve as a local model and are ne* *eded later; in 2.3 we shall very briefly discuss how these local models are glued to* *gether and what effect the gluing has on N2-structures and chiral rings. 2.2. A local model. It is easiest to begin with a local situation in the presence of a coordinat* *e system. 2.2.1. Let U be a smooth affine manifold with a coordinate system ~xby which we mean a collection of functions x1, ..., xn 2 O(U), n = dim U, such that the differential forms dx1, ..., dxn form a basis of the space of 1-forms T *(U) ov* *er O(U). A coordinate system determines a collection of vector fields @x1, ..., @xn such* * that @xixj =< @xi, dxj >= ffiij. It follows that [@xi, @xj] = 0 for all i, j, and @x1, ..., @xn form a basis of * *the space of vector fields T *(U). Let ch(U, ~x) be the following superpolynomial ring over O(U). ch(U, ~x) = O(U)[xi,(-j-1), @xi,(-j); dxi,(-j)@dxi,(-j), 1 i n, j (1],2* *.2.1) the generators xi,(-j-1), @xi,(-j)being even, dxi,(-j)@dxi,(-j)odd. Identifying* * xi,(-j-1), @xi,(-j)with different even copies of dxi and @xi resp., and dxi,(-j), @dxi,(-j* *)with different odd copies thereof, we obtain an identification of superalgebras O ch(U, ~x) -~! (S*(T (U)) S*(T *(U)) *(T (U)) *(T *(U))). (2.2.* *2) n 0 This is a local version of (2.1.2) and the images of embeddings (2.1.1) are gen* *erated over O(U) by dxi,(-1), @dxi,(-1). The ring ch(U, ~x) carries a canonical vertex algebra structure [MSV]. To f* *or- mulate the result introduce an even derivation T 2 End ch(U, ~x) determined by the conditions X T (f) = xi,(-2)@xif, T (a(-n)) = na(-n-1) , (2.2.* *3) i where f 2 O(U), a = xi, @xi, or @dxi, and n 1. Note that under identification (2.2.2) the first of these conditions says that T (O(U)) T *(U) and the restr* *iction T |O(U) equals the de Rham differential. 15 2.2.2. Lemma. There is a unique vertex algebra structure on ch(U, ~x) Y : ch(U, ~x) ! Field( ch(U, ~x)) determined by the conditions: (i) ch(U, ~x) is generated by the fields Y (f, z), f 2 O(U), Y (@xi,(-1), z* *), Y (@dxi,(-1), z), Y (dxj,(-1), w), the list of non-zero brackets amongst them b* *eing as follows: [Y (@xi,(-1), z), Y (f, z)] = ffi(z - w)Y (@xif, w), (2.2.4* *a) [Y (@dxi,(-1), z), Y (dxj,(-1), w)] = ffiijffi(z - w); (2.2.4* *b) (ii) T -covariance: [T, a(z)] = a(z)0, (2.2.* *5) a(z) = Y (f, z), Y (@xi,(-1), z), Y (@dxi,(-1), z), Y (dxj,(-2), w); (iii) vacuum: Y (1, z) = Id, Y (f, z)g|z=0 = fg, Y (a, z)1|z=0 = a, (2.2.* *6) where 1, f, g 2 O(U), a = @xi,(-1), dxi,(-1)or @dxi,(-1). The uniqueness assertion of this lemma is an immediate consequence of Theo- rem 4.5 in [K]. While proving the existence assertion in general is something o* *f a problem, in many examples, sufficient for our present purposes, this is easy. B* *efore we begin discussing these examples, let us unburden the notation by setting: f(z) = Y (f, z),dxi(z) = Y (dxi,(-1), z), (2.2.* *7) @xi(z) = Y (@xi,(-1), z),@dxi(z) = Y (@dxi,(-1), z). 2.2.3. Example: an affine space. If U = C with the canonical coordinate x = * *~x, then ch(C, x) is nothing but F B of (1.7.2). Indeed, since in this case O(U) =* * C[x], ch(C, x) is generated by the fields x(z), @x(z), dx(z), @dx(z), which accordin* *g to (2.2.4a,b) satisfy [@x(z), x(w)] = ffi(z - w), [@dx(z), dx(w)] = ffi(z - w). A quick glance at (1.6.3, 1.7.1) shows that b(z) 7! @dx(z), c(z) 7! dx(z), fl(z) 7! x(z), fi(z) 7! @x(z) identifies ch(C, x) with F B. Likewise, ch(CN , ~x) = F B n . (2.2.* *8) Incidentally, the same formulas define a vertex algebra morphism F B N ,! ch(U, ~x). (2.2.* *9) 16 The nature of this morphism is this: a coordinate system ~xdetermines an 'etale map U ! CN ; hence (2.2.9) is a manifestation of the naturality of chX, see 2.* *1. 2.2.4. Example: localization of an affine space. Let f 2 C[x1, ..., xn],* * Uf = CN - {~x: f(~x) = 0}, and C[x1, ..., xn]f the corresponding localization. To ex* *tend the vertex algebra structure from ch(CN , ~x) to ch(Uf, ~x) = C[x1, ..., xn]f C[x1,...,xn] ch(CN , ~x) it suffices to define the field f-1 (z). In [MSV] an explicit formula for this * *field was written down using Feigin's insight. Lemma 2.2.2 is a convenient alternative to* *ol to compute the action of this (and similar) fields. Indeed, in view of the commuta* *tion relations (2.2.4a-b) it suffices to know f-1 (z)(n)g, g 2 O(U). Due to (2.2.6)* * we have 8 < 0 ifn 0 f-1 (z)(n)g = : _g_ifn = -1. f The values f-1 (z)(n)g, n -2, are determined by using (2.2.5). The case where g = 1 suffices and the repeated application of (2.2.5) gives f-1 (z)1 = ezT f-1 . For example, X @x f f-1 (z)(-2)1 = T ( 1_f) = - xi,(-2)_i__2. i f We shall mostly need localization to the complements of hyperplanes. The cor- responding vertex algebras can be realized, thanks to [B], inside lattice verte* *x al- gebras; this will be reviewed in some detail in sect. 3. 2.2.5. Two N2-structures and two chiral rings. We shall need two morphisms of vertex algebras æ1, æ2 : V (N2)3n ! ch(U, ~x), n = dim U. The first was used in [MSV] and in terms of fields is defined by X X Q(1)(z) = dxi(z)@xi(z), G(1)(z) = : xi(z)0@dxi(z), iX i X J(1)(z) = - : dxi(z)@dxi(z) :, L(1)(z) = : xi(z)0@xi(z) : + : dxi(z)0@d* *xi(z) :, i i (2.2.1* *0) where we let A(1)= æ1(A), A=Q, G, J, or L. The second is obtained by composing the first with automorphism (1.11.3); the result is this: X X Q(2)(z)= : xi(z)0@dxi(z), G(2)(z) = dxi(z)@xi(z), Xi iX J(2)(z)= : dxi(z)@dxi(z) :, L(2)(z) = : xi(z)0@xi(z) : - : dxi(z)@dxi* *(z)0: . i i (2.2.1* *1) 17 As was noted in 1.10.1, the operators L(i)(1)give two conformal gradings and a * *simple computation shows that the corresponding chiral rings, 1.4, are as follows: KerL(1)(1)= O(U)[dx1,(-1), ..., dxn,(-1)], KerL(2)(1)= O(U)[@dx1,(-1), ..., * *@dxn,(-1)]. (2.2.1* *2) 2.3. Gluing the local models. 2.3.1. Localisation procedure explained in 2.2.4 carries over to any ch(U,* * ~x), see 2.2.1, and defines, in the presence of a coordinate system, a sheaf of vert* *ex algebras U V 7! chU,~x(V ) def= ch(V, ~x) over U. By using the action of the group of coordinate changes [MSV, (3.1.6)] o* *ne obtains canonical identifications chU,~x~-! chU,~y for any two coordinate systems ~x, ~y. This defines a family of sheaves U 7! * *chU, where U is 'etale over CN , natural w.r.t. to 'etale morphisms. Finally covering a smooth manifold X by charts {Uff} 'etale over CN one def* *ines a sheaf chXby gluing over intersections according to the diagram chUff,! chUff\Ufi- chUfi. Let us recall, briefly but in some more detail, the effect of this procedure on* * the N2-structure. 2.3.2. Two N2-structures, the chiral de Rham complex and algebra of chiral polyvector fields. It was computed in [MSV] that a coordinate change ~x7! ~y, * *via (2.1.3), induces the following transformation of fields (2.2.10): Q(1)(z)7! Q(1)(z) + (dDR (Tr log{(@xiyj)}))(z)0 G(1)(z)7! G(1)(z) (2.3.* *1) J(1)(z)7! J(1)(z) + (Tr log{(@xiyj)})(z)0 L(1)(z)7! L(1)(z), and of course similar transformation formulas can be written for fields (2.2.11* *). It follows that L(i)(z)(1), J(i)(z)(0), i = 1, 2 is a well-defined quadruple of operators acting on chX. Since [L(i)(z)(1), J(i* *)(z)(0)] = 0, there arise two competing bi-gradings by öc nformal weight, fermionic charge* *": chX= n 0,m2Z (i) ch,mX,n, i = 1, 2, (i) ch,m (i) (i) (2.3.* *2) X,n = Ker(L (z)(1)- nId) \ Ker(J (z)(0)- mId). 18 Formula (2.2.12) shows that the chiral ring, 1.4, now technically a sheaf of* * chiral rings, associated to the first is the algebra of differential forms: C(1)= *X : U 7! O(U)[dx1,(-1), ..., dxn,(-1)]; (2.3.3* *a) the chiral ring associated to the second is the algebra of polyvector fields C(2)= *TX , U 7! O(U)[@dx1,(-1), ..., @dxn,(-1)]. (2.3.3* *b) This is how morphisms (2.1.1) come about. 2.3.3. Terminology. From now on we shall call chXequipped with grading (2.3* *.2) where i = 2 the algebra of chiral polyvector fields and re-denote it by chTX .* * The sheaf chXequipped with grading (2.3.2) where i = 1 will retain the name of the chiral de Rham complex. Equations (2.3.3a,b) is one justification of this terminology. Note that (2* *.1.2) uses bi-grading (2.3.2,i=1); the i = 2 analogue is as follows: O i j Gr chTX -~! S*qn(TX ) S*qn+1(TX*) *qny-1(TX ) *qn+1y(TX*).(2.3* *.5) n 0 Transformation formulas (2.3.1) imply that L(1)(z) is preserved; hence chXalwa* *ys carries a conformal structure, see 1.10.1 for the definition. The situation is * *different with chTX : it does not carry a conformal structure compatible with its confor* *mal grading unless X is Calabi-Yau. Indeed, as follows from the last of formulas (2* *.2.11), L(2)(z) = L(1)(z) + J(1)(z)0 and the latter picks the 1st Chern class as a resu* *lt of transformation (2.3.1). If, however, X is a projective Calabi-Yau manifold, then it can be derived f* *rom (2.3.1), [MSV], that the quadruple of fields Q(i)(z), G(i)(z), J(i)(z), Q(i)(z)* *, i = 1, 2, can be made sense of globally, and both chX, chTX acquire an N2-structure, s* *ee 1.10.1 for the definition. What is especially clear is 2.3.4. Lemma. If ! is a non-vanishing holomorphic form over X, and X admits an atlas consisting of charts {(U, ~x)} such that locally ! = dx1 ^ . .^* *.dxn, then formulas (2.2.10,11) define an N2-structure on chXand chTX resp. Indeed, in this case the jacobian det(@xiyj) equals 1, and the correction te* *rms in (2.3.1) vanish. 2.3.5. An example: X = CN - 0. As an illsutration, let us compute the cohomology vertex algebra H*(CN - 0, chTCN -0), an example that will prove important later on. The manifold CN - 0 is quasiaffine and, therefore, it has the standard glob* *al coordinate system xi, @xi, 0 i N - 1 inherited from CN . This places us in * *the situation of Lemma 2.2.2 and we obtain a morphism of bi-graded sheaves M chCN -0~-! (S*qn(TCN -0) S*qn+1(TC*N-0) *qny-1(TCN -0) *qny(TC*N-0* *)), n 0 (2.3.* *6) 19 cf. (2.3.5). For the same reason, the sheaf on the R.H.S. of (2.3.6) is free, h* *ence it suffices to compute H*(CN - 0, OCN -0). It is a nice excersise in ~Cech cohomol* *ogy to prove that 8 >>> C[x0, ..., xN-1i]fn = 0 > C[xi1 ]=C[xi] ifn = N - 1 (2.3.* *7) >>>i=0 : 0 otherwise. The first line of (2.3.7) says that all the global sections of chTCN -0 are re* *strictions from CN . Therefore, H0(CN - 0, chTCN -0) = chT (CN ) = F B N , (2.3.* *8) cf. (2.2.8). Similarly, it follows from the 2nd line of (2.3.7) that _ N-1 ! O HN-1 (CN - 0, chTCN -0) = C[xi1 ]=C[xi] C[~x] chT (CN ), (2.3.* *9) i=0 where we use the notation of (2.2.1) with (U, ~x) = (CN , ~x). A moment's thought shows that as an chT (CN )-module, HN-1 (CN -0, chTCN -* *0) is obtained from chT (CN ) by spectral flow (1.12.2): HN-1 (CN - 0, chTCN -0) = S1( chT (CN )) = S1(F B N ). (2.3.1* *0) Indeed, by definition 1.7, chT (CN ) = F B N is generated by a vector 1 an- nihilated by xi,(j), @xi,(j), j 0, (and we identify xi = xi,(-1)); according* * to (2.3.9), HN-1 (CN - 0, chTCN -0) is generated by a vector annihilated by xi,(j* *-1), @xi,(j+1), j 0; the latter annihilating subalgebra is mapped onto the former * *by S1 of (1.12.1). The odd variables are treated similarly; but notice also that * *the Clifford algebra has only one irreducible module and so the spectral flow on it* * is inessential. Of course, the 3rd line of (2.3.7) implies Hi(CN - 0, chTCN -0) = 0 ifi 6= 0, N - 1. (2.3.1* *1) 2.4. The algebra of chiral polyvector fields over hypersurfaces. This is an exposition of a result of [B]. 2.4.1. Let L ! X (2.4.* *1) be a line bundle, L* ! X (2.4.* *2) 20 its dual, t : X ! L (2.4.* *3) its section with smooth zero locus Z(t). Following [B] we shall relate chTL* a* *nd chTZ(t)as follows. Identify t with a fiberwise linear function on L*. We have the de Rham diffe* *r- ential of t, dt 2 H0(L*, 1L*); via (2.1.1), dt 2 H0(L*, chTL* ). It is clear * *that dt(0): chTL* ! chTL* (2.4.* *4) is a derivation with zero square, cf. 1.3. L* carries an action of C* defined by fiberwise multiplication. By the natur* *ality, 2.1 (i), this action lifts to an action on chTL* . Hence there arises the grad* *ing M chTL* = Rn( chTL* ). n2Z The operator dt(0)has degree 1 with respect to this grading and, therefore, (2.* *4.4) is actually a complex of sheaves R* chTL* = ( chTL* , dt(0)) such that dt(0)-1 ch dt(0)0 ch dt(0)1 ch dt(0) . .-.! R ( TL* ) -! R ( TL* ) -! R ( TL* ) -! . . . (2.4.* *5) 2.4.2. Lemma ([B]) The cohomology sheaf Hndt(0)( chTL* ) of complex (2.4.5) is zero unless n = 0. The sheaf H0dt(0)( chTL* ) is supported on Z(t) and natur* *ally isomorphic to chTZ(t). 2.4.3. Observe that dt 2 H0(L*, chTL* ) is of conformal weight 1, as follo* *ws e.g. from (2.3.5). Therefore, the differential of complex (2.4.5) preserves con* *formal weight, and the conformal weight 0 component of (2.4.5) is the following classi* *cal complex: . . .dt-! i+1TL* -dt! iTL* -dt! i-1TL* -dt!. . ., (2.4.* *6) with differential equal to the contraction with the 1-form dDR t. Lemma 2.4.2 s* *ays, in particular, that this complex computes the algebra of polyvector fields on Z* *(t), a well-known result perhaps. Note that in the chiral de Rham complex setting, cf. [B], this classical construction is somewhat harder to discern because the* *re conformal weight is not preserved by dt(0). 2.4.4. The N2-structure. Let L* be the canonical line bundle. Then both L* and Z(t) are Calabi-Yau - both have a nowhere zero global holomorphic volume form - and both chTL* and chTZ(t)carry an N2-structure, see the end of 2.3.3. Let us write down some explicit formulas. Suppose there is a nowhere zero global holomorphic volume form ! and X can be covered by charts s, y1, ..., yN-1 , s being the coordinate along the fiber,* * such 21 that ! = ds ^ y1 ^ . .^.yN-1 . Then, as follows from Lemma 2.3.4, N-1X N-1X Q(z) 7! s(z)0@ds(z) + yj(z)0@dyj(z),G(z) 7! ds(z)@s(z) + dyj(z)@yj(z* *), j=1 j=1 N-1X N-1X J(z) 7! -ds(z)@ds(z) - dyj(z)@dyj(z),L(z) 7! s(z)0@s(z) + yj(z)0@yj(z)- j=1 j=1 N-1X - ds(z)@ds(z)0- dyj(z)@dyj(z)0 j=1 (2.4.* *7) well defines an N2-structure on chTL* . [B, Proposition 5.8] says that, via Lemma 2.4.2, the N2-structure on chTZ(t* *)is determined by N-1X G(z) 7! ds(z)@s(z) + dyj(z)@yj(z), j=1 N-1X (2.4.* *8) Q(z) 7! s(z)0@ds(z) + yj(z)0@dyj(z) - (s(z)@ds(z))0. j=1 3. The lattice vertex algebra realization and applications to toric varieties. This section is an exposition of part of Borisov's free field realization [B* *]. It does not contain any new results except perhaps Lemma 3.8, and in order to construct the spectral sequence appearing in the latter the entire section had to be writ* *ten up. 3.1. Let M be a rank N free abelian group, M* = HomZ(M, Z) its dual. Give = M M* a lattice structure by defining the symmetric bilinear form x ! Z, (3.1.* *1) induced by the natural pairing M* x M ! Z, (X*, X) 7! X*(X). There arises the lattice vertex algebra V , 1.9, where we fix the following cocycle, cf.(1.9.1), *(Y ) * * * ffl(X + X*, Y + Y *) = (-1)X , X, Y 2 M, X , Y 2 M . (3.1.* *2) Next, consider the complexification C = C Z onto which form (3.1.2) carries over. There arises the fermionic vertex algebra F C which we re-denote by F , * *see (1.6.5). Finally, following [B] introduce Borisov's vertex algebra B = V F . (3.1.* *3) 22 Notation. The notational problem one faces here is that the lattice twice manifests itself inside B : first, as an ingredient of V ; second, as that of* * F . We attempt to resolve this issue by letting capital latin letters, X, Y, Z, .. ( X* **, Y *, Z*, .. resp.), denote elements of M (M* resp.) in the context of V ; and let the til* *ded letters, X~, ~Y, ~Z, .. or X~*, ~Y,*~Z*, .. denote their respective copies in t* *he context of F . Later on this will be related to geometry and then we shall let the lowerc* *ase letters denote the respective coordinates. Note that the assignment M* 7! B is functorial. Indeed, if g 2 Hom (M*1, M* **2) is an isomorphism of abelian groups, then (g-1 , g*) 2 Hom (M*2, M*1) x Hom (M2, M1) ,! Hom ( 2, 1) is an isomorphism of lattices preserving form (3.1.1) and cocycle (3.1.2). Acco* *rding to (1.11.1,2), this isomorphism induces the following isomorphism of vertex alg* *ebras ^g: B 2 ! B 1 (3.1.4* *a) * g-1X* X*(z) 7! g-1 X*(z), ~X*(z) 7! g-1 ~X*(z), eX (z) 7! e (z) *)X (3.1.4* *b) X(z) 7! (g*)X(z), ~X(z) 7! (g*)X~(z), eX (z) 7! e(g (z). Therefore, if we introduce the category of lattices morphisms being the de- scribed isomorphisms, then M* 7! B , g 7! ^g (3.1.* *5) is a contravariant functor. Later we shall have to work with g such that g(M*1) M*2but after the exten* *sion of scalars to Q the induced g 2 Hom Q((M*1)Q , (M*2)Q ) is an isomorphism. In * *this case the functorial nature of M* 7! B is a little more subtle because g-1 may * *have non-integer entries. There are two ways around. Consider a vertex subalgebra*BM, B , see definition (1.9.10). (This si* *mply means that all the fields eX (z) are not allowed.) It is clear that B.,.: M* 7! BM, , g 7! ^g|BM, (3.1.* *6) is a contravariant functor because the indicated restriction of (3.1.4a) makes * *sense for any lattice embedding g. Second, naturally associated to g there is a map ^g: B 2 ,! Bg-1 2, (3.1.* *7) still defined by (3.1.4a), where the lattice g-1 2 is defined to be g*M2 g-1* * M*2 ( 1)Q . Note that (3.1.6) is a üs bfunctorö f (3.1.7). 3.2. Let us introduce the following terminology and notation pertaining to t* *oric variety theory: by a basic cone oe M* we shall mean a sub-semigroup spanned over Z+ by part of a basis (over Z) of M* . Let < oe > denote the (uniquely 23 determined) spanning set of oe . Given a basic cone oe M* , let ~oe M be its* * dual cone defined by ~oe= {X 2 M s.t.oe(X) 0}. A smooth toric variety will always be defined by fixing a lattice as in 3.* *1 and a regular fan . (Regular means that is a collection of basic cones in M* .) * *If we define Uoe= SpecC[~oe], oe 2 , (3.2.* *1) where C[~oe] is the semigroup algebra of ~oe, then there arises a canonical emb* *edding Uoe0 Uoe, oe0 oe. (3.2.* *2) The toric variety X attached to is defined by declaring that U = {Uoe, oe 2 } (3.2.* *3) is its covering and by gluing the charts over intersections Uff- Uff\fi,! Ufi according to (3.2.2). Note that the assignment (oe, M* ) 7! Uoeis functorial. Indeed, if we intro* *duce the category whose objects are pairs (oe, M* ) and morphisms (oe1, M*1) ! (oe2,* * M*2) are abelian group morphisms g : M*1! M*2such that g(oe1) oe2, then g*(~oe2) ~oe1. Hence g* induces a ring homomorphism C[~oe2] ! C[~oe1] and thus a morphi* *sm ~g: Uoe1! Uoe2. Of course, (oe, M* ) 7! Uoe, g 7! ~g (3.2.* *4) is a covariant functor. This can be globalized: given ( 1, M*1)and ( 2, M*2) with a lattice morphism g : M*1! M*2such that for each oe1 2 1 there is oe2 2 2 containing g(oe1), th* *ere arises a morphism ~g: X 1 ! X 2. (3.2.* *5) 3.3. We would like to define Borisov's realization [B], that is, a vertex al* *gebra embedding chT (Uoe) ,! BM, for each basic cone oe 2 M* . To write down an explicit formula for this map, let us choose a basis of M* , X*0, ..., X*N-1, s* *uch that oe is spanned by X*0, ..., X*m-1. This fixes the dual basis X0, ..., XN-1 of M. In order to conform to the notation of sect.2, let xj def=eXj , @xi, 0 i * * m - 1, be a coordinate system on Uoe. Thus Uoe= SpecC[x0, ..., xm-1 , xm1 , ..., xN1-1]. Borisov proves that there is a vertex algebra homomorphism B(oe) : chT (Uoe) ! BM, (3.3.* *1) determined by the assignment xi1 (z) 7! e Xi (z), xj(z) 7! eXj (z), m i N - 1, j m - 1, (3.3.2* *a) dxi(z) 7!: eXi (z)X~i(z) : . 24 @xi(z) 7!: (X*i(z)- : X~i(z)X~*i(z) :)e-Xi (z) :, @dxi(z) 7!: e-Xi (z)X~*i(z)(* *:,3.3.2b) cf. 3.1, Notation. (Note that (3.3.2a-b) are exactly (2.1.3) specialized to t* *he ex- ponential change of variables xi ! eXi ; this remark is also borrowed from [B].) Formulas (3.3.2a) are manifestly independent of the choice of variables; it is * *easy then to infer that so are (3.3.2b). This embedding naturally depends on oe. To make a precise statement, give 3.3.1. Definition. Fix a number N. C is a category whose objects are pairs (oe, M* ), dim M* = N, all and morphisms (oe1, M*1) ! (oe2, M*2) are abelian gr* *oup embeddings g : M*1,! M*2such that g(oe1) = oe2. Note that the conditions imposed on morphisms in this definition strengthen those used in (3.2.4). In fact, a short computation shows that the map ~g: Uoe1! Uoe2associated with a morphism g 2 Mor C((oe1, M*1), (oe1, M*1)) in (3.2.4) is * *'etale. Hence, by virtue of the naturality of chTX , see 2.1(i), the composition (oe, M* ) 7! Uoe7! chT (Uoe) defines a contravariant functor chT (.) : (oe, M* ) 7! chT (Uoe). (3.3.* *3) By forgetting oe, we regard contravariant functor (3.1.6) as defined on C. The * *main property of (3.3.1) is formulated as follows. 3.4. Lemma. The assignment (oe, M* ) 7! B(oe), see (3.3.1), is a functor mo* *r- phism B(oe) : chT (.) ! B.,., where B.,.is the functor defined in (3.1.6). Notational convention. Using this fact we shall not distinguish between chT (Uoe) and B(oe)( chT (Uoe)) BM, . 3.5. Let S* 2< oe > be one of the generators of oe and let oe \ S* denote th* *e cone spanned by < oe > \S*. There arises then the restrtiction morphism res(oe, S*) : chT (Uoe) ,! chT (Uoe\S*), and one would like to extend it to a resolution. Let * J *(oe, S*) = 1n=0J n(oe, S*), J n(oe, S*) = chT (Uoe\S*)enS , (3.5.* *1) where chT (Uoe\S*) is thought of as a vertex subalgebra of BM, , see 3.4, Not* *ational convention, and this makes sense out of chT (Uoe\S*)enS* as a subspace of B . J *(oe, S*) is evidently a Z+ -graded vertex subalgebra of B . 25 Let * D(oe, S*) = (eS (z)S~*(z))(0). (3.5.* *2) It is evidently a square zero derivation of J *(oe, S*), see 1.3. Thus (J *(oe,* * S*), D(oe, S*)) is a differential graded vertex algebra. Now look upon chT (Uoe) as a differential graded vertex algebra with chT (* *Uoe) placed in degree 0, zero spaces placed everywhere else, and zero differential. * *Then, by taking the composition *) chT (Uoe) res(oe,S! chT (Uoe\S*) = J 0(oe, S*) ,! J *(oe, S*), res(oe, S*) can be interpreted as a morphism of differential graded vertex alge* *bras res(oe, S*) : ( chT (Uoe), 0) ,! (J *(oe, S*), D(oe, S*)). (3.5.* *3) This construction is natural. To explain this, let us give the following defin* *ition, cf. 3.3.1. 3.5.1. Definition. Cpnt is a category whose objects are triples (S*, oe, M* * *) with S* 2< oe >, (oe, M* ) 2 Ob (C), and morphisms (S*1, oe1, M*1) ! (S*2, oe2, M*2)* * are abelian group morphisms g : M*1! M*2such that g(oe1) = oe2 and g(S*1) = S*2. It is clear that both ( chT (.), 0) : (S*, oe, M* ) 7! ( chT (Uoe), 0) (3.5.* *4) and (J *(.), D(.)) : (S*, oe, M* ) 7! (J *(oe, S*), D(oe, S*))(3.5.* *5) are contravariant functors from Cpnt to the category of differential graded ver* *tex algebras. Indeed, the former is essentially (3.3.3), as to the latter, one has * *to apply map (3.1.7) restricted to J *(oe, S*). 3.6. Lemma. (i) The assignment (S*, oe) 7! res(oe, S*), see (3.5.3), is a fu* *nctor morphism res(.) : ( chT (.), 0) 7! (J *(.), D(.)). (ii) For each (S*, oe, M* ) map (3.5.3) is a quasiisomorphism. 3.7. Let us apply Lemma 3.6 to the situation where the fan satisfies the following condition: there is S* 2 M* such that S* 2< oe > for all highest dime* *nsion cones oe 2 . Let \ S* = {oe \ S* : oe 2 }, cf. the beginning of 3.5. This * *means that the morphism X ! X =ZS* (3.7.* *1) induced by the canonical projection M* ! M* =ZS* is a line bundle, and the map X \S* ,! X (3.7.* *2) induced by the tautological inclusion \S* is the embedding of the total sp* *ace of the line bundle without the zero section. We would like to relate the cohomo* *logy groups H*(X , chTX ) and H*(X \S* , chTX \S* ). 26 The nerve of the covering U is a simplicial object in the category Cpnt. * *Ap- plying to it functor (3.5.4) one gets the complex chT (U ) commonly known as the ~Cech complex. (A complex, not a bi-complex, because we ignore the trivial differential on ( chT (.), 0).) We denote it by ~C*(U , chTX ; dC~), where d* *C~is the ~Cech differential. Likewise applying functor (3.5.5) to the nerve of U , we obtain the bi-comp* *lex C~*(U , J *; dC~, D(S*)). By definition C~p(U , J q; dC~, D(S*)) = ~Cp(U \S* , chTX \S* eqS*; dC~, D(S*)).(3.7.* *3) According to Lemma 3.6, res(U ) : ~C*(U , chTX ; dC~) ! ~C*(U , J *; dC~, D(S*))(3.7.* *4) is a quasiisomorphism. More precisely, the bi-complex ~C*(U , J *; dC~, D) giv* *es rise to two spectral sequences both converging to its total cohomology. The one where the vertex differential D(.) is used first degenerates in the first term to the* * ~Cech complex C~*(U , chTX ; dC~) - this follows at once from Lemma 3.6 (ii). Hen* *ce both the sequences abut to H*(X , chTX ). By definition, the first and the s* *econd terms of the second spectral sequence are as follows: * * (Ep,q1, d1) = (Hp(X \S* , chTX \S* eqS ), D(S )), * * (3.7.* *5) D(S*) = (eS (z)S~ (z))(0). * Ep,q2= HqD(S*)(Hp(X \S* , 1n=0 chTX \S* enS )). (3.7.* *6) Let us summarize our discussion. 3.8. Lemma. There is a spectral sequence {Ep,qr, dr} ) H*(X , chTX ) that satisfies (3.7.5,6). 3.9. The formation of bi-complex (3.7.3), and hence of the corresponding spe* *c- tral sequences, is functorial in X . To make this precise, let a triple S*, 2* * and M*2 satisfy the conditions imposed on S*, and M* in 3.7, and let us give ourselves another pair 1 and M*1, dim M*1= dim M*2along with a lattice embedding g : M*2! M*1 s.t.g( 1) = 2 \ S*. (3.9.* *1) According to (3.2.5) this induces a map ~g: X 1 7! X 2, (3.9.* *2) which is 'etale, cf. 3.3. Map (3.9.1) gives rise to the lattice g-1 *2and the * *embedding of Borisov's algebras ^g: B 2 ,! Bg-1 2 (3.9.* *3) due to (3.1.7). It is rather obvious that maps (3.9.2,3) allow to pull the bi-c* *omplex C~p(U 2\S* , chTX eqS*; d , D(S*)) (3.9.* *4) 2\S* C~ 27 back onto X 1. Let us write down the relevant formula. The bi-complex (3.9.4) consists of the family of elements * foe2 {TX 2\S* (Uoe)}eqS , oe 2 2. Define C~p(U 1, ^g chTX 1 eqg-1S*; dC~, g-1 D(S*)) (3.9.* *5) to consist of the family of elements -1S* foe2 ^g{TX 1 (Ugoe)}eqg , oe 2 1. By construction, the map foe7! ^gf(g-1 oe), oe 2 2 delivers an isomorphism of (3.9.5) and (3.9.4): C~p(U 2\S* , chTX eqS*; d , D(S*)) -~! 2\S* C~ (3.9.* *6) C~p(U 1, ^g chTX 1 eqg-1S*; dC~, g-1 D(S*)). 3.10. Digression: Borisov's realization and the spectral flow. We are now ma* *king good on our promise to show how spectral flow (1.12.1) is realized via lattice * *vertex algebras in the case of the bc - fifl-system. We shall show in 5.2.15 that a si* *mple version of this construction does the same for N2. According to our conventions chT (CN ) BM, , and by definition (3.1.2) M L satisfies (1.12.1). Therefore any ff 2 M* generates on LieBM, the spectral * *flow in the direction of ff, see (1.12.6). A glance at (3.3.2a,b) shows that xi(z) 7! xi(z)z, dxi(z) 7! dxi(z)z, SP jX*j: -1 -1 (3.10.* *1) @xi(z) 7! @xi(z)z , @dxi(z) 7! @dxi(z)z , cf. 1.12.7-8, and this does coincide with S1 of (1.12.1). It follows that the map P * P * P * e- j Xj : e jXj chT (CN ) ! chT (CN ); v 7! e- j Xj v (3.10.* *2) P * identifies e jXj chT (CN ) with the spectral flow transform of chT (CN ): P * ~ e jXj chT (CN ) -! S1( chT (CN )), (3.10.* *3) see definition of the spectral flow transform (1.12.2). Therefore, results of 2.3 can be rewritten as follows: 8 ch N >< TP(C ) ifn = 0 Hn (CN - 0, chTCN -0) = > chT (CN )e jX*j ifn = N - 1 (3.10.* *4) : 0 otherwise. 28 4. Chiral polyvector fields over hypersurfaces in projective spaces Having put all the preliminaries out of the way we can tackle our main probl* *em - computation of H*(F, chTF ) for a smooth hypersurface F PN-1 . 4.1. Let L ! PN-1 be the degree N line bundle over PN-1 and ß : L* ! PN-1 (4.1.* *0) its dual. Let us give the spaces C-N = C and CN - 0 a C*-space structure as follows: C-N x C* ! C-N , u . t = st-N , (4.1.* *1) C* x (CN - 0) ! CN - 0, t . (x0, ..., xN-1 ) = (tx0, ..., txN-1 ). One has the quotient realization L* = C-N xC* (CN - 0), (4.1.* *2) where we impose the relation (u; x0, ..., xN-1 ) ~ (ut-N ; tx0, ..., txN-1 ), t 6= 0. (4.1.* *3) Let ZN act on CN as follows ZN x (CN - 0) ! (CN - 0), p ___ p ___ (4.1.* *4) ~m. (x0, ..., xN-1 ) = (e2ß -1m=N x0, ..., e2ß -1m=N xN-1 ). Crucial for our purposes is the following isomorphism of smooth algebraic varie* *ties p : (CN - 0)=ZN -~! L* - 0, (4.1.* *5) class of(x0, ..., xN-1 ) 7! class of(1; x0, ..., xN-1 ), where L* - 0 denotes L* without the zero section. (This is well known and obvio* *us: deleting the zero section means requiring that u 6= 0; then one uses the C*-act* *ion to make u = 1; this leaves only the classes of (1; x0, ..., xN-1 ) and simultan* *eously breaks the C*-action down to the ZN -action defined in (4.1.4).) We wish to study Calabi-Yau hypersurfaces in PN-1 . To define any such hy- persurface, take x0, ..., xN-1 to be, in accordance with (4.1.1), the homogene* *ous coordinates on PN-1 . Let F = {(x0 : . .:.xN-1 ) s.t.f(x0, ..., xN-1 ) = 0}, (4.1.* *6) where f is a degree N homogeneous polynomial with a unique singularity at 0. Such an f can be regarded as a section of L. The corresponding fiberwise linear function t = t(u; x0, ..., xN-1 ) on L*is, in terms of coordinates on C-p x (Cp* * - 0), t = uf(x0, ..., xN-1 ), (4.1.* *7) 29 cf. (4.1.2,3). The pull-back of this function onto (CN - 0)=ZN under (4.1.5* *) is literally f(x0, ..., xN-1 ): p*(t) = f(x0, ..., xN-1 ) (4.1.* *8) as follows from (4.1.5). 4.2. L* carries the standard affine covering U = {Uj, 0 j N - 1} defined by Uj = {class of(u; x0, ..., xN-1 ) s.t.xj 6= 0}. (4.2.* *0) Then ßU = {ß(Uj), 0 j N - 1} is (also standard) affine covering of PN-1 and ßU \ F = {ß(Uj) \ F} is an affine covering of F. By definition, the cohomology of the ~Cech complex C~*(ßU \ F, chTF) equals the cohomology H*(F, chTF). A more practical way to compute the latter is provided by Lemma 2.4.2. Indeed, being currently in the situation of this lemma we obtain the bi-complex C~*(U, R* chTL* ; dC~, dt(0)), that is, the C~ech comp* *lex with coefficients in the complex ( chTL* , dt(0)) defined in (2.4.5). Associat* *ed to this bi-complex there are two standard spectral sequences, 0Ep,qrand 00Ep,qr, s* *uch that (0Ep,q1,0d1)= (Hp(L*, Rq( chTL* )), dt(0)), 0Ep,q2= Hq p * ch (4.2.1* *a) dt(0)(H (L , TL* )), (00Ep,q1,00d1) = ~Cp(U, Hqdt(0)(Rq( chTL* )). (4.2.1* *b) In the latter Hqdt(0)(R* chTL* ) denotes the q-th cohomology sheaf of complex (* *2.3.5). 4.3. Lemma. Both 0Ep,qrand 00Ep,qrabut to H*(F, chTF). 4.4. Proof. Observe that even though sheaf complex (2.4.5) appears to be inf* *nite in both directions, its differential preserves conformal weight (cf. 2.4.3) an* *d it is easy to see that each fixed conformal weight component of (2.4.5) is finite. (I* *ndeed, the differential dt(0)changes fermionic charge by one and it follows from (2.3.* *5) that for a fixed conformal weight fermionic charge may acquire only a finite number * *of values.) This implies, in a standard manner, that both the spectral sequences a* *but to the cohomology of the total complex C*(U, R* chTL* ; dC~+ dt(0)). Lemma 2.4.2 implies that ( 00Ep,q1-~! 0 ifq 6= 0 Cp(ßU \ F, chTF) otherwise . Hence the second spectral sequence degenerates to the ~Cech complex over F and the lemma follows. 4.5. We now wish to compute the 1st term of the 1st spectral sequence record* *ed in (4.2.1a). Ignoring the double grading we rewrite (4.2.1a) as (0E**1,0d1) = (H*(L*, chTL* ), dt(0)), (4.5.* *1) We have L* - 0 ,! L*; this places us in the set-up of 3.7, see e.g. (3.7.1,2),* * and in order to compute (H*(L*, chTL* )) we employ the spectral sequence of Lemma 3.8. - in this particular case we shall be able to compute all its terms. Begin* * with 30 4.5.1. Toric description of L*. Choose s = xN0, yj = xj_x, 1 j N (4.5.* *2) 0 to be coordinates of L* over the "big cellÜ 0, see (4.2.0). Let M N-1M ML* = ZS { ZYi}, i=1 M N-1M (4.5.* *3) M*L*= ZS* { ZYi*}, i=1 L* = ML* M*L*, and the bases {S, Y1, ..., YN-1 } and {S*, Y1*, ..., YN*-1} be dual to each oth* *er. It follows from (4.5.2) and the identifications s = eS , yj = eYj that a fan* * L* that defines L* can be chosen as follows: the set of its 1-dimensional generat* *ors consists of N-1 X S*, NS* - Yi*, Y1*, Y2*, ..., YN*-1. (4.5.* *4) i=1 The set of the highest dimension cones consists of N cones, each generated by S* and the rest of the vectors in (4.5.4) except one of them. These data determine L* uniquely. 4.5.2. Computation of (E**1, d1), (E**2, d2),..., (E**1, d1 ). Proceeding along the lines of 3.7 we obtain (E(L*)***, d*) ) H*(L*, chTL* ). (4.5.41=* *2) (3.7.5) reads (we skip L*): * qS* * (Ep,q1, d1) = (Hp(L* - 0, chTL*-0 )eqS , (e (z)S~ (z))(0)). (4.5.* *5) Thanks to (4.1.5) and the naturality of chTX , 2.1 (i), there are canonical is* *mor- phisms Hp(L* - 0, chTL*-0 ) -~! Hp(CN - 0=ZN , chTCN \0=ZN) -~! Hp(CN - 0, chT ZN (4.5.* *6) CN -0) . The latter has been already computed, see 2.3.5, formulas (2.3.8,10,11). Theref* *ore, (Ep,q1, d1) is as follows: 8 ch N ZN qS* >< T (C ) e ifp = 0 Ep,q1= > S1( chT (CN )ZN )eqS* ifp = N - 1 (4.5.7* *a) : 0 otherwise * * d1 = (eS (z)S~ (z))(0). (4.5.7* *b) 31 Note that (E0,*1, d1) is canonically a differential graded vertex algebra and (* *EN-1,*1, d1) is canonically a differential graded (E0,*1, d1)-module. Therefore, as a vertex* * alge- bra, (E*,*1, d1) = (E0,*1, d1) (EN-1,*1, d1) (4.5.* *8) is the abelian extension of the former by the latter. Therefore, (3.7.6) reads* * as follows: E*,*2= Hd1(E0,*1, d1) Hd1(EN-1,*1, d1), (4.5.* *9) and again Hd1(E0,*1, d1) is a vertex algebra, Hd1(EN-1,*1, d1) its module, and * *E*,*2 is the abelian extension of the former by the latter. Thanks to (4.5.7a), the dimension consideration shows that d2 = d3 = . .d.N-1 = 0, (4.5.1* *0) E*,*2= E*,*3= . .=.E*,*N-3= E*,*N. The same argument shows that the non-zero components of the last non-zero dif- ferential are the following maps: d(i)N: EN-1,i2! E0,i+N2. (4.5.1* *1) As follows from 4.5.1, L* is covered by N open affine sets. Therefore, Hn (L*, * * chTL* ) = 0 if n > N - 1. This implies that the maps d(i)Nare isomorphisms if i > 0 and d* *(0)N is an epimorphism. Hence E*,*1= E0,02 E0,12 . . .E0,N-22 E0,N-12 Kerd(0)N. (4.5.1* *2) ___-z__" EN-1,01 By the spectral sequence definition, (4.5.12) implies Hi(L*, chTF) -~! Hd1(E0,i1, d1), 0 i N - 2, 0 ! Hd1(E0,N-11, d1) ! HN-1 (L*, chTF) ! Kerd(0)N! 0. (4.5.1* *3) Now let us perform a change of coordinates that will reveal so far invisible structure of result (4.5.7ab, 4.5.13). 4.6. We have used the coordinates attached to L* by definition. Now let us employ the map CN - 0 ! L* - 0, (4.6.* *1) that is, the composition of (4.1.5) and the natural projection CN -0 ! (CN -0)=* *ZN . We would like to recast the argument of 4.5 in terms inherent in CN - 0. By invoking (4.6.1) we have placed ourselves in the situation of 3.9. Let us* * make this explicit. CN - 0 has the standard coordinate system xi, @xi and has, therefore, the f* *ol- lowing toric description: M = N-1i=0ZXi, M* = N-1i=0ZX*i, = M M* , (4.6.2* *a) 32 so that X*i(Xj) = ffiij. (4.6.2* *b) = {oe0, ..., oeN-1 }, oej = {xj 6= 0}. (4.6.2* *c) Map (4.6.1) is induced by the lattice (cf. 3.2) embedding g : M* ! M*L*(= M*L*-0) (4.6.* *3) dual to the lattice embedding g* : ML* ! M, S 7! NX0, Yj 7! Xj - X0, 1 j N - 1. (4.6.4* *a) Indeed, comparing (4.1.5) and (4.5.2) one obtains: under (4.6.1), s 7! xN0, yj* * 7! xj=x0 and it remains to use xi = eXi , s = eS , yj = eYj in order to obtain (4.* *6.4a). It is immediate to see that N-1X g(X*0) = NS* - Yj*, g(X*j) = Yj*, j 1; (4.6.4* *b) j=1 thus g( ) = L* . Therefore, we are indeed in the situation of 3.9, and if we w* *rite down isomorphism (3.9.6) explicitly, we will have all the assertions of 4.5 rec* *ast in terms pertaining to CN - 0. According to (3.1.7), there arises an isomorphism ^g: B L* ! Bg-1 L* , (4.6.* *5) where g-1 L* = g*M L* g-1 M* L* Q . It follows from (4.6.4a) that N-1X N-1X g*M L* = { miXi s.t.N| } M. (4.6.* *6) i=0 i=0 Inverting (4.6.4b) we obtain that g-1 M* L*is spanned over Z by g-1 (S*) = _1_N(X*0+ X*1+ . .+.X*N-1), g-1 (Yj*) = X*j, 1 j N - 1. (4.6.* *7) Let us introduce the notation X*orb= _1_N(X*0+ X*1+ . .+.X*N-1). (4.6.* *9) The first line of (4.5.7a) and (4.5.7b) rewrites as follows: * (E0,*1, d1) = ( 1n=0 chT (CN )ZN enXorb, Dorb) N-1X * (4.6.1* *0) Dorb = _1_N (X~i(z)eXorb(z))(0). i=0 33 (For the latter (3.1.4b) and (4.6.7) were used.) This is a differential (w.r.t. Dorb) graded vertex algebra and because of i* *ts importance and its relation to Landau-Ginzburg models to be discovered later on, we make a digression. 4.6.1. Vertex algebra fLGorb. Introduce the following notation: LfGorb = 1n=0fLG(n)orb, LfG(n)orb= chT (CN )ZN enX*orb Bg-1 L* .(4.6.1* *1) This vertex algebra is filtered by the system of differential vertex ideals fLGomrb= n m LfG (n)orb. (4.6.1* *2) Hence there arises a projective system of differential vertex algebras fLG)N. (4.9.* *3) N-1 4.9.1. Remark. For the same reason that was indicated in Remark 4.7.1, isomor- phisms (4.9.1) have a good chance of being valid for p = N - 1 as well. Were th* *is the case, (4.9.2) would be unnecessary. Beginning of the proof. Item (i) is indeed simply (4.2.1a), Lemma 4.3, and (* *4.7.1- 2) of Theorem 4.7 put together. The isomorphism in (ii) is well known classical* *ly and the degeneration assertion follows very easily. We prefer, however, to emph* *asize 36 some additional structure hidden in the spectral sequence and then use it to gi* *ve, among other things, a self-contained proof of (ii), see 4.13. 4.10. Addendum: N2-structure. Condition (4.1.0) ensures that L* is the canonical line bundle and this places us in the situation of 2.4.4: coordinates s, y1, ..., yN-1 of (4.5.2) satisfy the conditions imposed in 2.4.4, formulas * *(2.4.7) define an N2-structure on chTL* and (2.4.8) define that on chTF. In order to compute this structure in terms of LfGorb one has to do the following: first, c* *om- pute the images of (2.4.8) in B L* under Borisov's map (3.3.2a,b); second, app* *ly map (4.6.5) to the result. This is straightforward and tedious but rewarding, * *the reward being the coincidence of the result with Witten's description of the Lan* *dau- Ginzburg model as we shall see in the next section, Lemma 5.2.14. In order to record the result it is convenient to use the boson-fermion corr* *espon- dence, 1.13. Let us introduce the standard lattice ZN so that the standard ba* *sis Ø0, ..., ØN-1 is orthonormal. Then, see 1.13, one can make identifications X~i(z) = eØi(z), X~*i(z) = e-Øi (z). (4.10.* *1) 4.10.1. Lemma. The N2-structure on H*(F, chTF) comes from the following N2-structure on fLGorb: N-1X N-1X ` 1 ' G(z) 7! : X*j(z) - Øj(z) eØj(z) :, Q(z) 7! : Xj(z) + ___Øj(z) e-Øj* * (z) :, j=0 j=0 N N-1X` 1 ' J(z) 7! - ___X*j(z) + Xj(z) + Øj(z) :, j=0 N N-1X` 1 1 ' L(z) 7! : Xj(z)X*j(z) : + __: Øj(z)2 : - __Øj(z)0- Xj(z)0. j=0 2 2 Proof. N2 is generated as a Lie algebra by 2 fields, G(z) and Q(z). Let us do Q(z), the field that acquires the geometrically mysterious factor 1=N. We have 37 starting with (2.4.8) N-1X Q(z) 7! s(z)0@ds(z) + yj(z)0@dyj(z) - (s(z)@ds(z))0= j=1 N-1X : yj(z)0@dyj(z) : - : s(z)(@ds(z))0:= j=1 N-1X : eYj(z)0(: e-Yj (z)Y~*j(z) :)- : eS (z)(: e-S (z)S~*j(z) :)0:= j=1 N-1X : Yj(z)Y~*j(z) : + : S(z)S~*(z) : -S~*(z)0= j=1 N-1X 1 N-1X 1 N-1X : (Xj(z) - X0(z))X~*j(z) : + : NX0(z) ___ X~*j(z) : - ___ X~*j(z)0= j=1 N j=0 N j=0 N-1X 1 N-1X : Xj(z)X~*j(z) : - ___ X~*j(z)0= j=0 N j=0 N-1X ` 1 ' : Xj(z) + ___Øj(z) e-Øj (z) : . j=0 N A brief guide to this computation is as follows: the 3rd line is (3.3.2a-b) app* *lied to the 1st line, in the 5th line transformation formulas (4.6.7) are used, and bos* *on- fermion correspondence (4.10.1) is used in the 7th. In addition, the well-known differentiation formula eff(z)0=: ff(z)eff(z) : has been repeatedly employed. 4.11. Character and Euler character formulas. Whenever one has a bi- graded vector space M V = Vnm, dim Vnm < 1, m,n one can define its character: X chV (s, ø ) = e2ßi(ms+nø)dim Vnm, (4.11.* *1) m,n and if in addition V is a supervector space, one can define its Euler character X Eu V (s, ø ) = e2ßi(ms+nø)sdim Vnm, (4.11.* *2) m,n where the super-dimension sdim Vnm is defined in the standard manner to be the dimension of the even component of Vnm minus the dimension of its odd component. Note that if V carries an odd differential preserving the bi-grading, then EuHd(V )(s, ø ) = Eu V (s, ø ). (4.11.* *3) 38 )N. (4.13.* *7) N-1 Finally, all the higher differentials vanish simply because the elements listed* * in (4.13.5,7) are genuine cocycles. This completes the proof of Theorem 4.9. It is amusing to note that we have obtained "vertex" representatives of all * *the classes of the cohomologyPH*(F, *TF). Since, as was explained in 4.8, the eige* *n- values of X*orb+ jXj,(0)give the cohomological grading, we have: * -P (X +Ø ) i-1 N-i-1 (class of)eiXorb j j j 2 H (F, TF), 1 i N - 1, (4.13.* *8) 40 Y m P 1 X (class of) xjj = e jmjXj 2 Hm (F, m TF), m = ___ mj, 0 mj N - 2. j N j (4.13.* *9) 4.13.2. Multiplicative structure. The multiplicative structure of the ring H* **(F, *TF) is well known, of course. But let us restore it by the "vertex" methods. According to (2.3.3b), the chiral ring of H*(F, chTF) is isomorphic to H*(F* *, *TF). , embeds into H*(F, *TF) as a ring, cf. (4.9.3). Let us now look at elements (4.13.8). The computation i * P j P P mj_ * P eiXorb- j(Xi+Øi) e jmjXj = lim(z - w)i j NeiXorb- j(Xj+Øj) = (-1) z!w 8 X >>> 0 if mj > 0 < j >>> iX*orb-P j(Xi+Øi) X : e if mj = 0, j (4.13.1* *0) (0) as follows from (1.9.8), is valid even in (LfGorb)0; hence in H*(F, *TF) as we* *ll. Likewise, if s + t < N, one obtains i * P j * P esXorb- j(Xj+Øj) etXorb- j(Xj+Øj) = (-1) * -2 P (X +Ø ) (4.13.1* *1) (-1)s limz!w(z - w)N-s-t e(s+t)Xorb j j j = 0 (0) inside (LfGorb)0, hence inside H*(F, *TF) as well. Finally, the same computation shows that i * P j * P * P esXorb- j(Xj+Øj) e(N-s)Xorb- j(Xj+Øj) = (-1)seNXorb-2 j(Xj+Øj) (-1) (4.13.1* *2) ) . Thus one is tempted to set chTSpecMf (Spec Mf) = Hdf(z)(0)(LG f ), thereby defining the sheaf chTSpecMf , where Mf = C[x0, ..., xN-1 ]= < df >. To put this somewhat differently, we have resolved the singularity of Mf by passing to the DGA K*f, see (5.1.3), and then chiralized the latter so as to ob* *tain (5.1.2). There seems to be a natural construction [KV2] allowing to chiralize i* *n a similar manner other free DGA's thereby extending algebras of chiral polyvector fields from smooth varieties to spectra of Milnor rings to a wider class of sch* *emes. 5.2. Landau-Ginzburg orbifold. 5.2.1. Familiar in vertex algebra theory is the following pattern: V is a v* *ertex algebra; g is its order N automorphism; V (i)is a än turally defined" gi-twiste* *d V - module, 1 i N - 1; assuming that the group {1, g, ..., gN-1 } also acts on * *each 43 V (i), naturally w.r.t to the V -action, one forms the vertex algebra of g-inva* *riants, V g, and its (untwisted) modules (V (i))g, 1 i N - 1. It sometimes so happe* *ns that the space V g (V (1))g . .(.V (N-1))g (5.2.* *1) carries an "interesting" vertex algebra structure compatible with the described* * V g- module structure. Vertex algebra (5.2.1) is often referred to as an orbifold or* * a V orbifold. The most famous example of an orbifold is undoubtedly the Monster vertex algebra V Mnstr[FLM]. Indeed, V Mnstr= VLg VLg,1, where L is the Leech lattice, g its involution, and VL,1 is an irreducible g-tw* *isted VL -module (unique by Dong's theorem [D2]). As suggested in a number of physics papers, an incomplete list including [V,VW,W2] and references therein, realization of this pattern in the case where V = LG f, degf = N, and g = exp (2ßiæJ0), æJ(z) defined in Lemma 5.1.1, should be closely related to "string vacua". We shall construct the candidates for LG* * (i)f and in order to do so we shall need a recollection on vertex algebra twisted mo* *dules. This notion was introduced in [FFR]. In our presentation we shall mostly follow [KR]. 5.2.2. Twisting data. Let G be an additive subgroup of C containing Z. A vertex algebra V is call* *ed G=Z-graded if V = m~2G=ZV [m~], (5.2.2* *a) so that V [m~](n)V [~l] V [m~ + ~l]. (5.2.2* *b) It is clear that V [0] V is a vertex subalgebra. To give an example, let g be an order N automorphism of a vertex algebra V . Let G = 1_NZ. We have G=Z -~! ZN . Then V = N-1m=0V [m] = {v 2 V : gv = e2ßim=N v} (5.2.* *3) is a ZN -grading. By definition, in this case V [0] is the vertex subalgebra o* *f g- invariants, V g. Let W be a vector space and m~ 2 G=Z, where G is as in 2.1.1. An m~-twisted End M-valued field is a series X a(z) = a(m)z-m-1 , m2m~ where a(m) 2 End M is such that for any v 2 W , a(m)v = 0 if Re m >> 0. Let FieldG (W ) be the linear space of m-twisted End W -valued fields for all m 2 G* *=Z. 44 5.2.3. Definition. (cf. Definition 1.2 and [KR, sect.5) A G-twisted V -module W is a parity preserving linear map X æ : V ! FieldG(W ), (æa)(z) = æa(m)z-m-1 m satisfying the following axioms: (i) if a 2 V [m~], then (æa)(z) is ~m-twisted; (ii) æ(1 ) = id; (iii) (twisted Borcherds identity) for any a 2 V [m~], b 2 V and F (z, w) = * *zm (z - w)l such that m 2 ~m, l 2 Z Res z-wæ(a(z - w)b, w)iw,z-w F (z, w) i * * j = Resz (æa)(z)(æb)(w)iz,wF (z, w) - (-1)par(a)par(b)(æb)(w)(æa)(z)iw,zF (z, * *w). (5.2.* *4) 5.2.4. Remarks. (i) Note that the l = 0 case of the twisted Borcherds identity is the follow* *ing commutator formula X1 ` m ' [æa(m), æb(k)] = æ(a(j)b)(m+k-j) . (5.2.* *5) j=0 j This shows that the coefficients of the fields (æa)(z) form a Lie algebra. (ii) A Z-twisted vertex algebra module is called simply a vertex algebra mod* *ule. In particular, the restriction of a twisted V -module W to the vertex subalgeb* *ra V [0] V is a V [0]-module. If G arises from an order N automorphism g as in (5.2.3), then a G-twisted module is called g-twisted or twisted by g. The restr* *iction of a g-twisted V -module to the vertex subalgebra V g is a V g-module. (iii) It should be clear what the phrases "W is an irreducible twisted V -mo* *dule" and " W is a twisted V -module generated by a collection of fields {(æaff)(z) f* *rom a given vector m 2 M". (iv) A vertex algebra is canonically a module over itself. Given an arbitrar* *y G=Z- graded vertex algebra, there is no obvious way to construct a G-twisted module, but let us consider some concrete examples. 5.2.5. The twisted module chT (CN )~~,~~. Given 2 n-tuples ~~= (~0, ..., ~N* *-1 ) 2 CN , ~~= (~0, ..., ~N-1 ) 2 CN , let G be the Z-span of ~i, ~i, and 1, 0 i * *N - 1. Define the G=Z grading on chT (CN ) by declaring that xi 2 chT (CN )[-~~i], @xi 2 chT (CN )[~~i], dxi 2 chT (CN )[-~~i], @dxi 2 chT (CN )[-~~i]. 5.2.6. Lemma. (i) There is a unique up to isomorphism structure of a G-twisted chT (CN )- module æ~~,~~: chT (CN ) ! FieldGW 45 generated by vac 2 W such that (æ~~,~~xi)(-~i+j)vac = (æ~~,~~@xi)(~i+j)vac = (5.2.* *6) (æ~~,~~(dxi))(-~i+j)vac = (æ~~,~~@dxi)(~i+j)vac = 0. (ii) This module can be constructed by letting W = chT (CN ) as a vector space and (æ~~,~~xi)(z) = xi(z)z~i, (æ~~,~~@xi)(z) = @xi(z)z-~i , (5.2.* *7) (æ~~,~~dxi)(z) = dxi(z)z~i, (æ~~,~~@dxi)(z) = @dxi(z)z-~i . Proof. The commutation relations (5.2.5) applied to the quadruple of fields (æ~~,~~xi)(z) (æ~~,~~@xi)(z), (æ~~,~~dxi)(z), (æ~~,~~@xi)(z) imply that their c* *oefficients span the Lie algebra isomorphic to Cl a, see 1.6, 1.7. For example, [(æ~~,~~@xi)(ff), (æ~~,~~xi)(fi)] = ffiff,-fi-1. Conditions (5.2.6) become then the vacuum vector conditions for Cl a which of course determine W uniquely even as Cl a-module, and uniqueness follows. This fixes recipe (5.2.7), and a standard argument lucidly explained e.g. in [* *KR, sect. 5] allows one to extend (5.2.7) naturally to a twisted module structure æ* *~~,~~: chT (CN ) ! FieldGW . Notation. We shall let chT (CN )~~,~~denote the twisted chT (CN )-module occurring in Lemma 5.2.6. 5.2.7. The twisted module V +j . While constructing chT (CN )~~,~~require* *s a little effort, a family of twisted modules over a lattice vertex algebra seems * *to be built into the definition of the latter. For simplicity we shall only consider * *the case where the lattice is that defined in 3.1. (The following should be regarded * *as known even though we failed to find the needed results in the literature, but s* *ee e.g. a similar and more involved discussion in [D1].) Fix j 2 C Z M* and consider the abelian group Lj = +Cj C . Let C[ j] be its group algebra. By analogy with the lattice vertex algebra V = V (hL ) C* *ffl[ ] introduce V j = V (hL ) C[ j]. Let æ : VL ! FieldV (5.2.* *8) be the vertex algebra structure map. Note that V j = V (hL ) C[ j] is natural* *ly a V (hL )-module: indeed x(z)1 efi, x 2 hL , makes perfect sense if fi 2 Lj, or* * indeed if fi 2 C Z L, see (1.9.5). Similarly, formula (1.9.2) may be extended without* * any changes to define a Cffl[ ]-action on V j = V (hL ) C[ j]; indeed, cocycle ff* *l(ff, fi) defined in (3.1.2) makes perfect sense for any fi 2 L + Cj - it, so to say, ign* *ores j. Since any field æ a(z), a 2 VL , is written in terms of the operators we ha* *ve just described, cf (1.9.4-5), the very formula for æ a(z) defines it as a "fieldö * *perating on V j = V (hL ) C[ j]. For example, ` ' x(z)ej = x(z) + 1_z(x, j)1ej, x 2 hL (5.2.9* *a) 46 i j eff(z)ej = eff(z)z(ff,j)1ej, (5.2.9* *b) where 1 = e0 is the vacuum vector of V . (Note, by the way, that (5.2.9a-b) a* *re analogous to spectral flow formulas (1.12.7-8).) In order to fix the twist, see* * 5.2.2, let us restrict such fields to the subspace V +j V j where only eff+j, ff 2* * L, are allowed. Formulas (5.2.9a-b) allow us to conclude that there is a tautolog* *ical embedding æ V ,! FieldGjV +j , (5.2.1* *0) where Gj is the grading on V that assigns degree -(ff, j) to eff, ff 2 . Hen* *ce there arises the composition æ +j : V æ! æ V ,! FieldGjV +j , (5.2.1* *1) 5.2.8. Lemma. Map (5.2.11) endows V +j with a Gj-twisted V -module struc- ture. Proof. Recall that the intuition behind twisted Borcherds identity (5.2.4) - as well as its untwisted version (1.2.2) - is that the 3 expressions appearing * *in it, æ(a(z -w), w), (æa)(z)(æb)(w), and (-1)par(a)par(b)(æb)(w)(æa)(z) are Laurent s* *eries expansions of the same function in 3 respective domains, see the short discussi* *on after Definition 1.2. This can be made precise: if W is a vector space graded by finite dimensional subspaces, then one can define matrix elements of fields * *and their products, such as, < v*, æ(a(z - w), w)v >, < v*, (æa)(z)(æb)(w)v >, and (-1)par(a)par(b)< v*, (æb)(w)(æa)(z)v >. If these are indeed Laurent series ex- pansions of the same rational function twisted by z-m , wk, m, k not necessarily integral, with poles on z = w, z = 0, w = 0, then (5.2.4) holds; cf. [K, Remark 4.9a]. In our case a suitable grading is easy to exhibit and a familiar argume* *nt along the lines of [K, sect.5.4] shows that matrix elements of the products of * *fields from æ +j (V ) are indeed such rational functions. Furthermore, it is easy to* * see that if we identify V +j -~! V , v 7! ve-j , then these matrix elements, with fixed v*, v 2 V , become analytic functions of j, see (5.2.9a-b). But if j 2 M* , then V +j = V by definition and the matrix elements are indeed equal to each other rational functions. Thanks to analytici* *ty, this must hold for all j and the lemma follows. 5.2.9. The constructions of 5.2.5 and 5.2.6 are related to each other. Inv* *oke Borisov's algebra B , see (3.1.3). Since B = V F , the space B +j def=V +j F (5.2.1* *2) is naturally a twisted B -module. Hence its pull-back w.r.t. Borisov's embeddi* *ng chT (CN ) ! B (5.2.1* *3) is a twisted chT (CN )-module. 5.2.10. Lemma. If N-1X j = ~jX*j, (5.2.1* *3) j=0 47 then the twisted chT (CN )-module generated by chT (CN ) from the vector ej 1* * 2 V +j is isomorphic to chT (CN )~~,~~, where ~~= (~0, ..., ~N-1 ). Proof. According to Lemma 5.2.5 in order to prove the lemma one only has to check that the vector ej satifies relations (5.2.6). But this is obvious. For e* *xample, (5.2.9b) gives i j æL+j xi(z)ej = eXi (z)z(Xi,j)1 ej = eXi (z)z~i1 ej, which is exactly the first of conditions (5.2.6). The remaining 3 fields are de* *alt with in exactly the same manner. 5.2.11. Notation. It is natural to denotePthe twisted module chT (CN )~~,* *~~ * realized as in Lemma 5.2.10 by chT (CN )e j~jXj . 5.2.12. Landau-Ginzburg orbifold. We now wish to orbifoldize the differen- tial graded vertex algebra LG f, (5.1.2), with respect to the Zp-action generat* *ed by the operator g = exp (æJ(z)(0)), where æJ(z)(0)is defined in Lemma 5.1.1. Accor* *d- ing to the pattern reviewed in 5.2.1, in order to do so one has to exhibit a gi* *-twisted differential LG f-module for each 1 i N - 1. (Remark 5.2.4 (ii) explains wh* *at "gi-twisted" means) As follows from (5.2.3), the ith twisting gradation is determined by the act* *ion of æJ(z)(0)on the generators. Formulas of Lemma 5.1.1 imply æJ(z)(0)xi = - 1_pxi, æJ(z)(0)@xi = 1_p@xi, (5.2.1* *4) æJ(z)(0)dxi = (1 - 1_p)dxi, æJ(z)(0)@dxi = ( 1_p- 1)@xi. Therefore, chT (CN )i~1_ ~1_, where ~1_= (1=p, ..., 1=p), is a natural candida* *te for the p,i p p exp i(æJ(z)(0))-twisted module. So we define, cf. 5.2.11, 1_P X* LG (i)fdef= chT (CN )e p j j . (5.2.1* *5) As mentioned in 5.2.4 (ii), LG (i)fis a (untwisted) chT (CN )g-module. (Recall* * that chT (CN ) = LG f = LG (0)fas vertex algebras.) Since the Landau-Ginzburg differential df(z)(0)comes from df 2 chT (CN )g, * *it operates naturally on LG (i)fthus making it a differential LG gf-module. Since (5.1.4) maps V (N2)N p-2_pinto LG gf, each LG (i)facquires an N2-struc* *ture. i jg i * * jg In particular, g operates on LG (i)fand LG (i)f is also a differential LG (* *0)f- module. 5.2.13. Definition. Define the Landau-Ginzburg orbifold to be the following differential LG gf-module: p-1Mi j g LG f,orb= LG (j)f . j=0 48 Note that if deg f = p = N, then LG f,orb simply coincides with (LfGorb, df) appearing in Theorems 4.7, 4.9. This space, however, carries two a priori diffe* *rent N2-structures: one computed in Lemma 4.10.1 and having purely geometric origin and another, Landau-Ginzburg structure, recorded in Lemma 5.1.1. 5.2.14. Lemma. The two N2-structures coincide with each other. Proof is, of course, a routine computation consisting in applying Borisov's * *for- mulas (3.3.2a-b) to the 4 fields of Lemma 5.1.1. Let us consider Q(z) and leave* * the rest as an exercise for the interested reader. We have (and recall that we are * *using boson-fermion correspondence (4.10.1)): N-1X 1 1 Q(z) 7! - ___xi(z)@dxi(z)0- ( ___- 1)xi(z)0@dxi(z) 7! i=0 N N N-1X 1 - ___: eXi (z) : (-Xi(z) - Øi(z)) e-Xi-Øi (z) :: - `i=0 N' _1_- 1 : : X (z)eXi (z) :e-Xi-Øi (z) := N i N-1X 1 ` ' ___: (Xi(z) + Øi(z)) e-Øi (z): + 1 - 1__ : Xi(z) e-Øi (z):= i=0 N N N-1X 1 ___: Øi(z)e-Øi (z) : + : Xi(z)e-Øi (z) :, i=0 N as it should, cf. Lemma 4.10.1. Note that this computation is parallel to that * *we performed in the proof of Lemma 4.10.1.