Fermionic screenings and chiral de Rham complex on CY manifolds with line bundles

0.12 MB·English

by  Sergei E. Parkhomenko

Fermionic screenings and chiral de Rham complex
on CY manifolds with line bundles.
S. E. Parkhomenko
Landau Institute for Theoretical Physics
2 142432 Chernogolovka, Russia
1
0
[email protected]
2

n
a
J
Abstract

1
2
We represent a generalization of Borisov’s construction of chiral de Rham complex for the

] case of line bundle twisted chiral de Rham complex on Calabi-Yau hypersurface in projective
h
t space. We generalize the differential associated to the polytope ∆ of the projective space Pd−1
-
p by allowing nonzero modes for the screening currents forming this differential. It is shown that
e
the numbers of screening current modes define the support function of toric divisor of a line
h
[ bundle on Pd−1 that twists the chiral de Rham complex on Calabi-Yau hypersurface.

3
”PACS: 11.25Hf; 11.25 Pm.”
v
9 Keywords: Strings, Conformal Field Theory.
3
4
1. Introduction
4
.
5
TheCalabi-Yaumanifoldswithbundlesappearinvarioustypesofcompactifications ofstring
0
1 theory. The first example is heterotic string compactification on Calabi-Yau (CY) manifolds.
1 This is currently most successfulapproach to the problem of stringmodels construction relevant
:
v to 4-dimensional particle physics. The main ingredients of the heterotic models is a Calabi-
Xi Yau three-fold and two holomorphic vector bundles on it. In the simplest case of ”standard
embedding” one of the bundles is taken to be trivial while the other one coincide with the
r
a
tangent bundle of the Calabi-Yau manifold. But explicit constructions of the bundles in general
case is hard to obtain. Nevertheless, in the series of papers [1] the monad bundles approach
has been developed for the systematic construction of a large class of vector bundles over the
Calabi-Yau manifolds defined as complete intersections in products of projective spaces.
Although the monad construction of [1] is quite general, it is purely classical, so the Gepner
models [2] (see also [3]) are the only known models of the quantum string compactification. It
rises the question what is the quantum version of monad bundle construction?
The second example is Type IIA compactification with D-branes wrapping the Calabi-Yau
manifold. In this case the Chan-Paton vector bundle appears [4], so that the similar question
make sense: what object describes the quantum strings on Calabi-Yau manifold with Chan-
Paton bundles?
In this note we analyze the most simple version of these questions when we have only line
bundle on the Calabi-Yau manifold and represent a construction of vertex operator algebra
starting from Calabi-Yau hypersurace in projective space and a line bundle defined on this
space.
1
Our approach is based essentially on the work of Borisov [5] where a certain sheaf of vertex
operator algebras endowed with N = 2 Virasoro superalgebra action has been constructed for
each pair of dual reflexive polytopes ∆ and ∆∗ defining CY hypersurface in toric manifold
P . Thus, Borisov constructed directly holomorphic sector of the CFT from toric dates of CY
∆
manifold. The main object of his construction is a set of fermionic screening currents associated
to the points of that pair of polytopes. Zero modes of these currents are used to build up a
differential D∆ + D∆∗ whose cohomology calculated in some lattice vertex algebra gives the
global sections of a sheaf known as chiral de Rham complex due to [7]. On the local sections
of the chiral de Rham complex the N = 2 Virasoro superalgebra is acting [7]. In CY case this
algebra survives the cohomology and hence, the global sections of the chiral de Rham complex
can be considered as a holomorphic sector of the space of states of the N = 2 superconformal
sigma-model on the CY manifold. The question how the costruction [5] is related to the Gepner
models has been clarified considerable in [8] and [9].
Borsov’s construction can also be generalized by allowing non-zero modes for the screening
currents forming the differential. We consider CY hypersurface in projective space Pd−1 and
generalize differential of Borisov by allowing non-zero modes only for screening currents associ-
ated tothepointsofPd−1 polytope∆. We thusgeneralize thedifferentialD leaving unchanged
∆
the differential D∆∗ defining CY hypersurface. We show that the numbers of screening current
modes from D define the support function of toric divisor [10], [11] of a line bundle on Pd−1.
∆
By this means, the chiral de Rham complex on Pd−1 appears to be twisted by the line bundle.
The paper is organized as follows. In Section 2 we briefly review the construction of paper
[5] for the case of CY hypersurface in Pd−1. In Section 3 we calculate first the cohomology
with respect to the generalized differential D and relate them to the sections of chiral de
∆
Rham complex twisted by the sheaf O(N). To do that we find the generalized bcβγ system
of fields generating the cohomology. The generalization appears only for the modes of vector
fields operators. They are replaced by the covariant derivative operators with U(1) connection.
Then we define the trivialization maps of the modules generated by these generalized bcβγ
fields to the modules of sections of the usual chiral de Rham complex over the affine space
and find the transition functions for different trivializations. Tese turn out to be the transition
functions of the O(N) bundle on Pd−1, where N is determined by the numbers of modes of
screening currents composing the differential D . Moreover, we establish the relation between
∆
the numbers of screening currents modes and toric divisor support function for the line bundle
O(N)onPd−1. Thesupportfunctionandtrivialization mapsareconsistentwiththelocalization
maps determined by the fan structure which allows to calculate the cohomology of the twisted
chiral de Rham complex by Cˇech complex of the covering. In complete analogy with [5], the
second differential D∆∗ is used to restrict the sheaf on the CY hypersurface.
In Section 4 we calculate the elliptic genus of the twisted chiral de Rham complex and
represent it in terms of theta functions. For a torus in P2 and K3 in P3 we find the limit as
q → 0 and relate the results to the Hodge numbers of the sheaf O(N) on the torus and K3.
Section 5 contains the concluding remarks.
2. Chiral de Rham complex on CY hypersurface in Pd−1.
In this section we review the construction [5] of chiral de Rham complex and its cohomology
for the case of CY hypersurface in projective space Pd−1.
Let {e ,...,e } be the standard basis in Rd and Λ ⊂ Rd be the lattice generated by the
1 d
2
vectors e = 1(e +...+e ), e , ...,e :
0 d 1 d 1 d
Λ= Ze ⊕Ze ⊕...⊕Ze (1)
0 1 d
We then consider the fan Σ⊂ Λ [10], [11] encoding the toric data of an O(d)-bundle
π :E → Pd−1 (2)
Themaximaldimensioncones of thefanared-dimensionalcones C ⊂ Λ,I = 1,2,...,d, spanned
I
by the vectors e ,...,eˆ,...,e , where the vector e omitted. The intersection of the maximal
0 I d I
dimension cones is also a cone in Σ
C ∩C ∩...∩C = C ∈ Σ. (3)
I J K IJ...K
All faces of the cone from Σ are the cones from Σ. (See [10], [11] for more detailed definition of
fan.)
Let{e∗,...,e∗}bethedualbasis tothestandardone{e ,...,e }andletΛ∗ betheduallattice
1 d 1 d
to Λ. For every cone C ∈ Σ, one considers the dual cone C∗ ∈ Λ∗ defined by
C∗ = {p∗ ∈ Λ∗|p∗(C)≥ 0} (4)
as well as the affine variety A = Spec(C[C∗]). If C∗ is a face of C˜∗ then C[C∗] is a localization
C
of C[C˜∗] by the monomials ap∗ ∈ C[C∗], where p∗ ∈ C˜∗ and p∗(C) = 0. It allows to glue A to
C
form the space E.
The polytope ∆ of Pd−1 is given by the points from Σ satisfying the equation
deg∗(Σ) = 1, (5)
where
deg∗ = e∗+...+e∗ (6)
1 d
Let X (z),X∗(z), i = 1,2,...,d be free bosonic fields and ψ (z),ψ∗(z), i = 1,2,...,d be free
i i i i
fermionic fields so that its OPE’s are given by
X∗(z )X (z ) = ln(z )δ +reg.,
i 1 j 2 12 i,j
ψ∗(z )ψ (z )= z−1δ +reg, (7)
i 1 j 2 12 i,j
where z = z −z .
12 1 2
The fields are expanded into the integer modes
∂X∗(z) = X∗[n]z−n−1, ∂X (z) = X [n]z−n−1,
i i i i
nX∈Z nX∈Z
ψi∗(z) = ψi∗[n]z−n−21, ψi(z) = ψi[n]z−n−12 (8)
nX∈Z nX∈Z
We therefore consider Ramond sector.
To the lattice Γ = Λ⊕Λ∗ we associate the direct sum of Fock spaces
ΦΓ = ⊕(p,p∗)∈ΓF(p,p∗), (9)
3
whereF(p,p∗) istheFockmodulegeneratedbyXi[n],Xi∗[n],ψi[n],ψi∗[n]fromthevacuum|p,p∗ >
determined by
X∗[n]|p,p∗ >= X [n]|p,p∗ >= ψ [n]|p,p∗ >= ψ∗[n−1]|p,p∗ >= 0,n > 0,
i i i i
X∗[0]|p,p∗ >=p∗|p,p∗ >, X [0]|p,p∗ >= p |p,p∗ > (10)
i i i i
For each vector e , i = 0,1,...,d generating 1-dimensional cone from Σ, we define the
i
fermionic screening current and screening charge
S∗(z) = e ·ψ∗exp(e ·X∗)(z), Q∗ = dzS∗(z) (11)
i i i i I i
We form the BRST operators for each maximal dimension cone C
I
D∗ = Q∗+...+Qˆ∗ +...+Q∗ (12)
I 0 I d
where Q∗ is omitted. Then, one considers the space
I
ΦCI⊗Λ∗ =⊕(p,p∗)∈CI⊗Λ∗F(p,p∗) (13)
ThespaceofsectionsM ofthechiraldeRhamcomplexovertheA isgivenbythecohomology
CI CI
of ΦCI⊗Λ∗ with respect to the operator DI∗. It is generated by the following fields [5]
a (z) =exp[w∗ ·X](z), α (z) = w∗ ·ψexp[w∗ ·X](z),
Iµ Iµ Iµ Iµ Iµ
a∗ (z) = (e ·∂X∗−w∗ ·ψ e ·ψ∗)exp[−w∗ ·X](z), α∗ (z) = e ·ψ∗exp[−w∗ ·X](z) (14)
Iµ µ Iµ i µ i Iµ Iµ µ Iµ
where w∗ are the dual vectors to the basis of vectors e ,µ = 0,...,Iˆ,...d generating the cone
Iµ µ
n o
C :
I
< w∗ ,e >=δ (15)
Iµ ν µν
The singular operator product expansions of these fields are
a∗ (z )a (z ) = z−1δ +...,
Iµ 1 Iν 2 12 µν
α∗ (z )α (z ) = z−1δ +... (16)
Iµ 1 Iν 2 12 µν
An important property is the behavior of the bcβγ system under the local change of coordi-
nates on A [7]. For each new set of coordinates
CI
b = g (a ,...,a ), a = f (b ,...,b ) (17)
Iµ µ I1 Id Iµ µ I1 Id
the isomorphic bcβγ system of fields is given by
b (z) = g (a (z),...,a (z)),
Iµ µ I1 Id
∂g
µ
β (z) = (a (z),...,a (z))α (z),
Iµ I1 Id Iν
∂a
Iν
∂f
β∗ (z) = ν (a (z),...,a (z))α∗ (z),
Iµ ∂b I1 Id Iν
Iµ
∂f
b∗ (z) = ν (a (z),...,a (z))a∗ (z)+
Iµ ∂b I1 Id Iν
Iµ
∂2f ∂g
λ ν (a (z),...,a (z))α∗ (z)α (z) (18)
∂b ∂b ∂a I1 Id Iλ Iρ
Iµ Iν Iρ
4
Here the normal ordering of operators is implied. It is also understud, whenever necessary in
what follows.
On the space M the N=2 Virasoro superalgebra acts by the currents
CI
G− = α a∗ , G+ = −a ∂α∗ − α∗ ∂a , J = a a∗ + α∗ α ,
Iµ Iµ I0 I0 Iµ Iµ I0 I0 Iµ Iµ
X X X
µ µ6=0,I µ6=0,I
1 1
T = (a∗ ∂a −∂a∗ a )−α ∂α∗ + (a∗ ∂a + (∂α∗ α −α∗ ∂α )) (19)
2 I0 I0 I0 I0 I0 I0 Iµ Iµ 2 Iµ Iµ Iµ Iµ
X
µ6=0,I
This algebra defines the mode expansion of the fields in Ramond sector
aI0(z) = aI0[n]z−n−21, a∗I0(z) = a∗I0[n]z−n−21,
X X
n n
α (z) = α [n]z−n−1, α∗ (z) = α [n]z−n,
I0 I0 I0 I0
X X
n n
a (z) = a [n]z−n, a∗ (z) = a∗ [n]z−n−1,
Iµ Iµ Iµ Iµ
X X
n n
αIµ(z) = αIµ[n]z−n−12, α∗Iµ(z) = αIµ[n]z−n−12, µ 6= I (20)
X X
n n
Then M is generated by the creation operators acting on the Ramond vacuum state |0 >
CI
defined by the conditions
a [n]|0 >= a∗ [n−1]|0 >=α [n]|0 >= α∗ [n−1]|0 >=0, n> 0. (21)
Iµ Iµ Iµ Iµ
IftheconeC isafaceoftheconeC andC itisspannedbythevectors(e ,e ,...,eˆ ,...,eˆ ,...,e ).
KJ K J 0 1 K J d
We then consider the BRST operator
D∗ = Q∗+Q∗+...+Qˆ∗ +...+Qˆ∗ +...+Q∗ (22)
KJ 0 1 K J d
acting on ΦCKJ⊕Λ∗. The space of sections MCKJ of the chiral de Rham complex over the
ACKJ is given by the cohomology of ΦCKJ⊕Λ∗ with respect to the operator DK∗ J. It is a local-
ization of M (M ) with respect to the multiplicative system generated by (a [0])mµ
CK CJ µ Kµ
( (a [0])mµ), with m w∗ (C ) = 0 ( m w∗ (C ) = 0). AnalogouQsly, the local-
µ Jµ µ µ Kµ KJ µ µ Jµ KJ
izQation maps can be dePfined for the cones whichPare the intersections of an arbitrary number of
maximal dimension cones [5].
Thelocalization mapsdefinedaboveallow tocalculate thecohomology ofthechiraldeRham
complex on E as Cˇech cohomology of the covering by A , I=1,...,d [5]:
CI
0 → ⊕ M → ⊕ M → ···M → 0 (23)
CI CI CKJ CKJ C12...d
It finishes the calculation of D -cohomology.
∆
ThenextstepistorestrictthechiraldeRhamcomplexonE totheCYmanifoldCY ⊂ Pd−1.
Let us define the function W on E which is linear on the fibers of E, so that in the coordinates
a on A this function is given by
Iµ CI
W =a (1+ (a )d) (24)
I0 Iµ
X
µ6=0,I
5
Then one has to introduce the corresponding screening currents and screening charges
S (z) = α (z)(1+ (a )d(z)),
I0 I0 Iµ
X
µ6=0,I
S (z) = α a (z)(a )d−1(z), µ 6= 0,I,
Iµ Iµ I0 Iµ
Q = dzS (z), µ 6= I (25)
µ I Iµ
and the BRST operator
D = Q (26)
W µ
X
µ6=I
(It is D∆∗ differential in the notations of [5]). Thecohomology of MCI with respect to DW gives
the space of sections M | of chiral de Rham complex on A ∩CY determined by the system
CI W CI
of equations
a = 0,
I0
1+ (a )d = 0. (27)
Iµ
X
µ6=0,I
The cohomology of the chiral de Rham complex on CY is calculated by Cˇech complex of
the covering [5]:
0→ ⊕ M | → ⊕ M | → ···M | → 0 (28)
CI CI W CKJ CKJ W C12...d W
Thus, we get D∆+D∆∗-cohomology.
One can consider a more general function W and BRST operator (26) adding the monomial
which corresponds to the internal point from the dual polytope ∆∗, [5].
It finishes the review of the chiral de Rham complex and its cohomology construction on CY
hypersurface in Pd−1.
3. Line bundle twisted chiral de Rham complex.
In this section the generalization of Borisov’s construction producingO(N)-twisted chiral de
Rham complex on CY hypersurface is proposed.
We twist the fermionic screening charges Q∗:
i
Q∗ → S∗[N ]= dzzNiS∗(z), N ∈ Z (29)
i i i I i i
Then the old charges Q∗ can be considered zero modes of the screening currents S∗(z) =
i i
S∗[n]z−n−1 and the BRST operator (12) is a particular case of more general one
n i
P
D∗ = S∗[N ]+...+Sˆ∗ [N ]+...+S∗[N ] (30)
I 0 0 I I d d
Now the question is what are the cohomology of the space (13) with respect to this new BRST
operator?
It follows by the direct calculation that the fields a (z), α (z), α∗ (z) from (14) still
Iµ Iµ Iµ
commute with the new BRST operator (30) but instead of a∗ (z) we have to take
Iµ
∇ (z) = a∗ (z)+N z−1a−1(z) (31)
Iµ Iµ µ Iµ
6
The last term in this expression can be regarded as coming from U(1) gauge potential on A .
CI
We see in waht follows that this is indeed so and the modes of the fields ∇ (z) can be regarded
Iµ
as a string version of the covariant derivatives.
In terms of this new bcβγ fields the N = 2 Virasoro superalgebra currents are given by
G−I = αIµ∇Iµ = G−I [n]z−n−32,
X X
µ n
G+I = −aI0∂α∗I0− α∗Iµ∂aIµ = G+I [n]z−n−23,
X X
µ6=0,I n
J = a ∇ + α∗ α = J [n]z−n−1,
I I0 I0 Iµ Iµ I
X X
µ6=0,I n
1 1
T = (∇ ∂a −∂∇ a )−α ∂α∗ + (∇ ∂a + (∂α∗ α −α∗ ∂α )) =
I 2 I0 I0 I0 I0 I0 I0 Iµ Iµ 2 Iµ Iµ Iµ Iµ
X
µ6=0,I
L [n]z−n−2 (32)
I
X
n
To calculate the cohomology let us consider the vertex operator V(0,p∗)(z) = exp[p∗X](z),
where p∗ ∈ Λ∗. We find
Sµ∗[Nµ](z1)V(0,p∗)(z2)= z1N2µSµ∗(z1)V(0,p∗)(z2) =
zNµ+p∗(eµ)e ·ψ∗exp[e ·X∗+p∗·X](z )+..., µ 6= I (33)
12 µ µ 2
Hence, the state |(0,p∗) > corresponding to the vertex V(0,p∗)(0) is in Ker(Sµ∗[Nµ]) if p∗(eµ) ≥
−N . The (Ramond sector) state saturating the inequality is |(0,− N w∗ ) > and has
µ µ6=I µ Iµ
the properties P
∇ [k]|(0,− N w∗ ) >=0,k ≥ 0,
Iµ µ Iµ
X
µ6=I
a [k]|(0,− N w∗ ) >=
Iµ µ Iµ
X
µ6=I
α [k]|(0,− N w∗ ) >=α∗ [k−1]|(0,− N w∗ )>= 0, k > 0. (34)
Iµ µ Iµ Iµ µ Iµ
X X
µ6=I µ6=I
Proposition.
The cohomology MCI of ΦCI⊕Λ∗ with respect to the differential (30) is generated from the
vacuum state
|Ω >= |(0,− N w∗ ) > (35)
I µ Iµ
X
µ6=I
by the creation operators of the fields (14) and (31).
The proof is similar to the proof of Proposition 6.5. from [5].
The vacuum |Ω > defines the trivializing isomorphism of modules (over the chiral de Rham
I
complex on A )
CI
g : M → M (36)
I CI CI
7
by the rule
g |Ω >=|0 >,
I I
g (∇ [k])g−1 = a∗ [k], g (a [k])g−1 = a [k],
I Iµ I Iµ I Iµ I Iµ
g (α [k])g−1 = α [k], g (α∗ [k])g−1 = α∗ [k] (37)
I Iµ I Iµ I Iµ I Iµ
We therefore call the vacuum |Ω > the trivializing vacuum.
I
Let us consider the subspace M0 ⊂ M generated from |Ω > by the operators a [0] and
I I I Iµ
α [0]. The operator G−[0] acts on this subspace by δ = α [0]∇ [0]. It is natural to
Iµ I I µ6=I Iµ Iµ
think that M0 is holomorphic de Rham complex over A wPith coefficients in holomorphic line
I CI
bundle.
On the intersections A ∩A the relations between the coordinates
CI CJ
a = a (a )d, a = a a−1, µ 6= I,J, a = a−1 (38)
I0 J0 JI Iµ Jµ JI IJ JI
can be used to find the relations between the trivializing vaccua
g |Ω >= (a [0])Nµ|Ω >= |0 >,
I I Iµ I
Y
µ6=I
g−1g |Ω >≡ g |Ω >=|Ω >,
I J J IJ J I
g =(a [0])N1+N2+...+Nd−dN0 (39)
IJ (J)I
as well as between the sections
g : M0 → M0 (40)
IJ J I
Thefunctionsg from(39) arethetransition functionsof thelinebundleon E which isinduced
IJ
from the O(N)-bundle on Pd−1, where
N = N +...+N −dN . (41)
1 d 0
By this means, the set of modules M0 with the differentials δ and the transition functions (39)
CI I
define the holomorphic de Rham complex on E with coefficients in the line bundle π∗O(N).
Onecanextendthisfinitedimensionaldiscussiontotheinfinitedimensionalone. Weconsider
the relation between the currents G−(z) and G−(z) on the intersection A ∩A . Because of
I J CI CJ
(38) and (18) we find
G−(z) = G−(z)+Nz−1α (z)a−1(z) ⇔
I J JI JI
G−[k] = G−[k]+N α [m]a−1[k−m] (42)
I J JI JI
X
m
In the finite-dimensional case the differentials δ are consistent on the intersections A ∩A :
I CI CJ
thedifferenceδ −δ comingfromthedifferenttrivializationsiscanceledbygaugetransformation
I J
of the gauge potential: A = A − g−1∂gIJ. A similar event should occur in the infinite-
Iµ Jµ IJ ∂aJµ
dimensional situation. Because the first Chern class on E is zero, the second term in the
expression (42) is due to different trivializations defined on A ∩A and has to be canceled
CI CJ
by the gauge transformation of the gauge potential:
∂g
G−[k] = G−[k]+N α [m]a−1[k−m]− (g−1 IJα )[k] = G−[k] (43)
I J JI JI IJ ∂a Jν J
X X Jν
m ν6=J
8
Hence, thecurrent G− ≡ G− as well as the N = 2 Virasoro superalgebra will beglobally defined
I
if one takes into account the transformation of gauge potential and extend the map (40) to the
map
g (z) = (a (z))N :M → M . (44)
IJ (J)I CJ CI
IftheconeC isafaceoftheconeC (C )andspannedbythevectors(e ,...,eˆ ,...,eˆ ,...e ),
IJ I J 0 I J d
one can consider the BRST operator
D∗ = S∗[N ]+...+Sˆ∗ [N ]+...+Sˆ∗ [N ]+...+S∗[N ] (45)
IJ 0 0 I I J J d d
acting on ΦCIJ⊗Λ∗.
The cohomology MCIJ of ΦCIJ⊗Λ∗ with respect to the differential (45) is the localization of
M (M )withrespecttothemultiplicativesystemgeneratedby (a [0])mµ ( (a [0])mµ),
CI CJ µ Iµ µ Jµ
with m w∗ (C )= 0 ( m w∗ (C ) = 0). Q Q
µ µ Iµ IJ µ µ Jµ IJ
ThPus, the module M Pis generated from the vacuum vector
CIJ
|Ω >= |(0,− N w∗ ) > (46)
IJ µ Iµ
X
µ6=I,J
by the creation operators of the fields a (z), ∇ (z), µ 6= I,J, a (z), a−1(z), a∗ (z), α (z),
Iµ Iµ IJ IJ IJ Iµ
α∗ (z), µ 6= I. M can also be generated from the vacuum
Iµ CIJ
|Ω˜ >=|(0,− N w∗ ) >= (a [0])N−NI−NJ|Ω > (47)
IJ µ Jµ IJ IJ
X
µ6=I,J
by the creation operators of the fields a (z), ∇ (z), µ 6= I,J, a (z), a−1(z), a∗ (z), α (z),
Jµ Jµ JI JI JI Jµ
α∗ (z), µ 6= J.
Jµ
Analogously, the modules M and localization maps can be defined for the cones
CIJ...K
C = C ∩C ∩...∩C .
IJ...K I J K
Relation (47) is a particular case of compatibility conditions localization maps to besatisfied
for localization maps. They are as follows. For each maximal-dimension cone C the trivializing
I
vacuum |Ω > defines a linear function ω∗ ∈ Λ∗ on this cone:
CI I
a−Nµ(0) = exp[−ω∗X](0), ω∗ = dN w∗ + N w∗ . (48)
(I)µ I I 0 I0 µ Iµ
Y X
µ6=I µ6=0,I
It is easy to see that the collection of ω∗ satisfies the obvious compatibility condition. Namely,
I
on the cone C = C ∩ C the functions ω∗ and ω∗ coincide and are given by the function
IJ I J I J
ω∗ ∈Λ∗ of the trivializing vacuum |Ω >:
IJ IJ
a−Nµ(0) = exp[−ω∗ X](0), ω∗ = dN w∗ + N w∗ . (49)
(I)µ IJ IJ 0 I0 µ Iµ
Y X
µ6=I,J µ6=0,I,J
It can be verified that similar compatibility conditions are also satisfied for the functions ω∗
IJ...K
on the cones C = C ∩C ∩...∩C . Then, the numbers N ,...,N of screening currents
IJ...K I J K 0 d
modes define the support function ω∗ on Σ [10], [11] of the toric divisor of the bundle π∗O(N)
on E.
Hence, similar to (23) we have the Cˇech complex of the covering by A , I = 1,...,d
CI
0→ ⊕ M → ⊕ M → ···M → 0 (50)
CI CI CKJ CKJ C12...d
9
which gives the cohomology of chiral de Rham complex on E twisted by π∗O(N).
Therestriction of thetwisted chiraldeRhamcomplex onCYhypersurfaceis straightforward
because BRST operator (26) commutes with the operators (30) and acts within each of the
modules M . Therefore, the complex
CIJ...K
0 → ⊕ M | → ⊕ M | → ···M | → 0 (51)
CI CI W CKJ CKJ W C12...d W
gives the cohomology of O(N)-twisted chiral de Rham complex on CY hypersurface. This
completes the construction.
4. The elliptic genus calculation.
In this section we calculate the elliptic genus of the twisted chiral de Rham complex closely
following [6]. Theq0 coefficient of the elliptic genus is related to the Hodge numbersof the sheaf
O(N) on CY manifold. To justify the construction in Section 3, we calculate it for the case of
torus T2 ⊂P2 and K3⊂ P3.
The discussion of Section 3 and the arguments of paper [6] allow extending the Definition
6.1. from [6] to the case under discussion: the elliptic genus is given by the supertrace over the
Cˇech cohomology space of the twisted chiral de Rham complex.
The calculation is greatly simplified using the torus (C∗)d that acts on E [6]. We compute
the function
d d
ρN(CY,t1,...,td,y,q) = X(−1)k−1 X superTrMCI1...Ik(YtKi iyJ[0]qL[0]−2c4), (52)
k=1 CI1,...,CIk i=1
where t are the formal variables grading the torus action with the help of generators K (whose
i i
explicit form is obvious) and then take the limit t → 1, i = 1,...,d to get the elliptic genus
i
Ell (CY,y,q). It is quite helpful for subsequent computations to write the N = 2 Virasoro
N
superalgebra acting on M in coordinates (7)
CIJ...K
G− = −z−1ω∗ ·ψ−deg∗ ·∂ψ+ψ·∂X∗, G+ = −deg∗·∂ψ∗+ψ∗·∂X,
IJ...K IJ...K IJ...K
J = −z−1deg·ω∗ +deg·∂X∗−deg∗ ·∂X +ψ∗ ·ψ,
IJ...K IJ...K
1 deg
T = (∂ψ∗ ·ψ−ψ∗·∂ψ)+∂X ·(∂X∗ −z−1ω∗ )− ·∂(∂X∗ −z−1ω∗ )−
IJ...K 2 IJ...K 2 IJ...K
deg∗
·∂2X (53)
2
where
1
deg = e = (e +...+e ). (54)
0 1 d
d
The supertraces over the modules M can be calculated as the supertraces over the spaces
CIJ...K
ΦCIJ...K⊗Λ∗ which are the complexes with respect to the differentials DI∗J...K = S0∗[N0]+...+
Sˆ∗ [N ]+...+Sˆ∗ [N ]+...+Sˆ∗ [N ]+...+S∗[N ]. Because of (53) we obtain
I I J J K K d d
ρ (CY,t ,...,t ,y,q) =
N 1 d
d
y−d−22+d−1 t<w∗,ei> (−1)codimC y−<deg∗,k>+<w∗−ω∗,deg>q<w∗−ω∗,k>G(y−1,q)d(55)
i
wX∗∈Λ∗Yi=1 CX⊂Σ kX∈C
10