Quantization and holomorphic anomaly

0.23 MB·English

by  Albert Schwarz

Quantization and holomorphic anomaly
Albert Schwarz∗ and Xiang Tang†
February 2, 2008
7
0
0
2 Abstract

n WestudywavefunctionsofB-modelonaCalabi-Yauthreefoldinvariouspolariza-
a tions.
J

4
1 Introduction

2
v Inpresentpaper,weconsiderwavefunctionsofB-modelonaCalabi-Yauthreefoldinvarious
1
polarizations and relations between these wave functions.
8
Onecaninterpretgenus0B-modelonaCalabi-YauthreefoldX asatheoryofvariations
2
1 ofcomplexstructures. The extendedmodulispaceofcomplex structures,the spaceofpairs
1 (complex structure, holomorphic 3-form), can be embedded into the middle-dimensional
6 cohomology H3(X,C) as a lagrangian submanifold. The B-model for arbitrary genus cou-
0
pled to gravity (the B-model topological string) can be obtained from genus 0 B-model by
/
h means of quantization; the role of Planck constant is played by λ2 where λ is the string
t coupling constant. (This is a general statement valid for any topological string. It was
-
p derived by Witten [12] from worldsheet calculation of [2]. ) The partition function of B-
e model is represented by wave function depending on choices of polarization in H3(X,C).
h
If the polarization does not depend holomorphically on the points of the moduli space of
:
v
complex structures,thenthe dependence ofwavefunctionofthe points ofthe moduli space
i
X is notnecessarilyholomorphic. (The t¯-dependence isgovernedbythe holomorphicanomaly
equation.) This happens, in particular, for a polarization that we call complex hermitian
r
a polarization. Otherpapersusetheterm“holomorphicpolarization”foracomplexhermitian
polarization in the sense of present paper; we reserve the term “holomorphic polarization”
forapolarizationthatdependsholomorphicallyonthepointsofthemodulispaceofcomplex
structures. The holomorphicpolarizationinoursensewaswidely usedinmirrorsymmetry;
thispolarizationanditsp-adicanalogwereusedtoanalyzeintegralityofinstantonnumbers
(genus 0 Gopakumar-Vafa invariants) [8].
The main goal of present paper is to study wave functions in various polarizations,
especially in holomorphic polarization. We believe that Gopakumar-Vafa invariants for
any genus can be defined by means of p-adic methods and this definition will have as a
consequence integrality of these invariants. The present paper is a necessary first step in
the realization of this program. It served as a basis for a conjecture about the structure of
Frobeniusmaponp-adicwavefunctionsformulatedin[10];thisconjectureimpliesintegrality
of Gopakumar-Vafa invariants.
∗partlysupportedbyNSFgrantNo. DMS0505735.
†partlysupportedbyNSFgrantNo. DMS0505735and0604552.
1
We begin with a short review of quantization of symplectic vector space (Section 2). In
Section 3, 4 and 5 we use the general results of Section 2 to obtain relations between the
wave functions of B-model in real, complex hermitian and holomorphic polarizations. In
Section 6 we compare these wave functions with worldsheet calculations of [2].
Theholomorphicanomalyequationswererecentlystudiedandappliedin[1],[5],[6],[9],
[11]. Someofequationsinourpaperdifferslightlyfromcorrespondingequationsin[1],[11].
However, this difference does not affect any conclusions of these papers.
Acknowledgments: WewouldliketothankM.AganagicandM.Kontsevichforhelpful
discussions.
2 Quantization
We consider a real symplectic vector space V and a symplectic basis of V. (A symplectic
structure can be considered as a skew symmetric non-degenerate bilinear form < , > on
V; we say that eα, e , α,β = 1,··· ,n = dim(V) is a symplectic basis if < eα,eβ >=<
β
e ,e >= 0, < eα,e >= δα.) It is well known that for every symplectic basis e =
α β β β
{eα,e }, one can construct a Hilbert space H ; these spaces form a bundle over the space
β e
M of all bases and one can construct a projectively flat connection on this bundle1. The
situationdoes notchange if we consider,insteadof a realbasisin V, a basis {eα,e } in the
α
complexification of V requiring that e be complex conjugate to eα.
α
The picture we described above is the standard picture of quantization of a symplectic
vectorspace. ThechoiceofabasisinV specifiesarealpolarization;thechoiceofabasisinits
complexificationdeterminesacomplexpolarization. ThequantummechanicslivesinHilbert
space of functions depending on n = 1dimV variables. To construct this Hilbert space, we
2
should fix a polarization,but Hilbert spaces correspondingto different polarizationscan be
identified up to a constant factor. In semiclassical approximation, vectors in Hilbert space
correspond to lagrangian submanifolds of V.
Let us describe the Hilbert space H for the case when e = {eα,e } is a symplectic
e β
basis of V. An element of V can be represented as a linear combination of vectors eα,e
β
with coefficients x ,xβ. After quantization, x and xβ become self-adjoint operators xˆ ,
α α α
xˆβ obeying canonical commutation relations(CCR):
~
[xˆ ,xˆ ]=[xˆα,xˆβ]=0, [xˆ ,xˆβ]= δβ (1)
α β α i α
We define H as the space of irreducible unitary representation of canonical commutation
e
relations.
A (linear) symplectic transformation transforms a symplectic basis {eα,e } into sym-
α
plectic basis {e˜α,e˜ }:
α
e˜α =Mαeβ +Nαβe
β β (2)
e˜ =R eβ +Sβe .
α αβ α β
1One says that a connection ∇ on a vector bundle over a space B is projectively flat if [∇X,∇Y] =
∇[X,Y]+C, where C is a constant depending on X,Y. For an infinite dimensional vector bundle over a
compactmanifoldB withaunitaryconnection,thismeansthatforeverytwopointse,e˜ofB connectedby
acontinuouspathinB,thereisanisomorphismbetweenthefibersHe andHe˜defineduptomultiplication
byaconstant;thisisomorphismdependsonthehomotopyclassofthepath. WesaythatasectionΦofthe
vector bundleisprojectivelyflatif∇XΦ=CXΦwhereCX isascalarfunctiononthebase.
2
This transformation acts on xˆ ,xˆα as a canonical transformation, i.e. the new operators
α
xˆ˜ ,xˆ˜α also obey CCR; they are related to xˆ ,xˆα by the formula:
α α
xˆα =Nβαxˆ˜ +Sαxˆ˜β
β β (3)
xˆ =Mβxˆ˜ +R xˆ˜β.
α α β βα
It follows from the uniqueness of unitary irreducible representation of CCR that there
exists a unitary operator T obeying
xˆ˜α =TxˆαT−1,
(4)
xˆ˜ =Txˆ T−1.
α α
This operator T is defined up to a constant factor relating H and H . In the case when
e e˜
{e˜α,e˜ } is an infinitesimal variation of {eα,e }, i.e. e˜=e+δe where
α α
δeα =mαeβ +nαβe ,
β β (5)
δe =r eβ +sβe ,
α αβ α β
we can represent the operator T as 1+δT, where
1 1 1
δT =− nαβxˆ xˆ + mβxˆαxˆ − r xˆαxˆβ +C. (6)
2~ α β ~ α β 2~ αβ
ThisformuladeterminesaprojectivelyflatconnectiononthebundlewithfibersH andthe
e
base consisting of all symplectic bases in V. A quantum state specifies a projectively flat
section of this bundle.
Theirreducible unitaryrepresentationofCCR canbe realizedby operatorsofmultipli-
cationand differentiationon the space of squareintegrablefunctions of x1,··· ,xn; one can
take xˆαΨ=xαΨ and xˆ Ψ= ~ ∂Ψ . Then a projectively flat connection takes the form:
α i ∂xα
~ ∂2Ψ ∂Ψ 1
δΨ= nαβ +mβxα − r xαxβΨ+CΨ. (7)
2 ∂xα∂xβ α ∂xβ 2~ αβ
We will call elements of H and corresponding functions of x1,··· ,xn wave functions.
e
It is important to notice that in Equation (7) instead of square integrable functions, we
can consider functions Ψ(x1,··· ,xn) from an almost arbitrary space E; the only essential
requirement is that the multiplication by xα and differentiation with respect to xα should
be defined on a dense subset of E and transform this set into itself.
Sometimes it is convenient to restrict ourselves to the space of functions of the form
Ψ = exp(Φ) where Φ = ϕ ~n is a formal series with respect to ~ (semiclassical wave
~ n
functions). Rewriting (7) on this space we obtain
P
1 ∂2Φ ∂Φ ∂Φ ∂Φ 1
δΦ= nαβ(~ + )+mβxα − r xαxβ +~C.
2 ∂xα∂xβ ∂xα∂xβ α ∂xβ 2 αβ
LetBbethesetofallsymplecticbasesinthecomplexificationofV. Weconsiderthetotal
space ofa bundle overB asthe directproduct B×E. One canuse Equation(7) to define a
projectivelyflatconnectiononthisvectorbundle. (Thecoefficientsofinfinitesimalvariation
(2) of the basis in V must be real; if we consider {eα,e } as a basis of complexification of
α
V, the coefficients of infinitesimal variation obey the same conditions nαβ = nβα,r =
αβ
r ,mα+sβ =0, but they can be complex.)
βα β α
3
Notice,howeverthatintherealcasewearedealingwithunitaryconnection;theoperator
T that identifies two fibers (up to a constant factor) always exists. In complex case,
e,e˜
the equation for projectively flat section can have solutions only over a part of the set of
symplectic bases. (Recall that the fibers of our vector bundles are infinite-dimensional.)
It is easy to write down simple formulas for the operatorT in the case when Nαβ =0
e,e˜
or R =0. In the first case we have
αβ
1
T (Ψ)(xα)=exp − (RM−1) xαxγ Ψ(Sαxβ), (8)
e,e˜ 2~ αγ β
(cid:0) (cid:1)
in the second case
~ ∂2
T (Ψ)(xα)=exp − (MNT)αγ Ψ(Sαxβ). (9)
e,e˜ 2 ∂xα∂xγ β
(cid:0) (cid:1)
Combining Equations (8) and (9), we obtain an expression for T that is valid when M
e,e˜
and S are non-degenerated matrices,
1 ~ ∂2
T Ψ(xα)=exp − (RM−1) xαxγ exp − (MNT)αγ Ψ((M−1)αxβ) .
e,e˜ 2~ αγ 2 ∂xα∂xγ β
(10)
(cid:0) (cid:1)(cid:8) (cid:0) (cid:1)(cid:2) (cid:3)(cid:9)
Usingtheexpression(10)andWick’stheorem,itiseasytoconstructdiagramtechniques
to calculate T eF.
e,e˜
RecallthatWick’stheorempermitsustorepresentanexpressionoftheform eAeV(x)dx
whereAisaquadraticformandV(x)doesnotcontainlinearandquadratictermsasasum
R
of Feynman diagrams :
1
exp( a xixj)eV(x)dx=eW (11)
ij
2
Z
where W is a sum of connected Feynman diagrams with propagator aij (inverse to a )
ij
and with vertices determined by V(x). Using this fact and Fourier transform we obtain a
diagram technique for T eF.
e,e˜
It follows from this statement that the action of T on the space of semiclassical wave
e,e˜
functions is given by rational expressions. This means that the action can be defined over
an arbitrary field (in particular, over p-adic numbers).
The above statements can be reformulated in the language of representation theory.
Assigning to every symplectic transformation (4) a unitary operator T defined by (7) we
obtainamultivaluedrepresentationofthesymplecticgroupSp(n,R)andcorrespondingLie
algebrasp(n,R)(metaplecticrepresentation). TherepresentationoftheLiealgebrasp(n,R)
can be extended to a representation of its complexification sp(n,C) in an obvious way.
However,themetaplecticrepresentationofSp(n,R)cannotbeextendedtoarepresentation
of Sp(n,C) because Sp(n,C) is simply connected and therefore it does not have any non-
trivial multivalued representations. (See Deligne [4] for more detailed analysis.)
3 B-model
Fromthe mathematicalviewpoint, the genus0 B-modelona compactCalabi-Yauthreefold
X is a theory of variations of complex structures on X. Let us denote by M the moduli
space of complex structures on X. For every complex structure, we have a non-vanishing
holomorphic(3,0)-formΩ onX,definedup toa constantfactor. Assigningthe setofforms
4
λΩ to every complex structure we obtain a line bundle L over M. The total space of this
bundle,i.e. thespaceofallpairs(complexstructureonX,formλΩ),willbedenotedbyM.
EveryformΩ specifies anelement ofH3(X,C)(middle-dimensionalcohomologyofX)that
willbedenotedbythesamesymbol. NoticethatΩdependsonthecomplexstructureonfX,
but H3(X,C) does not depend on complex structure. More precisely, the groups H3(X,C)
formavectorbundle overMandthis bundle isequippedwithaflatconnection∂ (Gauss-
a
Manin connection). In other words, the groups H3(X,C) where X runs over small open
subset of M are canonically isomorphic. However, the bundle at hand is not necessarily
trivial: the Gauss-Manin connection can have non-trivial monodromies. Going around a
closedhomotopicallynon-trivialloopγinM,weobtaina(possibly)non-trivialisomorphism
M :H3(X,C)→H3(X,C). The set of all elements of H3(X,C) correspondingto forms Ω
γ
constitutesalagrangiansubmanifoldLofH3(X,C). (The cupproductonH3(X,C)taking
values in H6(X,C) = C specifies a symplectic structure on H3(X,C). The fact that L is
lagrangian follows immediately from the Griffiths transversality.) We can also say that we
have a family of lagrangian submanifolds L ⊂ H3(X ,C) where H3(X ,C) denotes the
τ τ τ
third cohomology of the manifold X equipped with the complex structure τ ∈ M. Notice
that the Lagrangiansubmanifold L is invariant with respect to the monodromy group (the
group of monodromy transformations M ).
γ
The B-model on X for an arbitrary genus can be obtained by means of quantization
of genus 0 theory, the role of the Planck constant is played by λ2, where λ is the string
coupling constant. (More precisely, we should talk about B-model coupled to gravity or
aboutB-modeltopologicalstring.) Letusfixasymplecticbasis{e ,eA}inthevectorspace
A
H3(X ,C). Everyelementω ∈H3(X ,C)canberepresentedintheformω =xAe +x eA,
τ τ A A
where the coordinates xA,x can be represented as xA =< eA,ω >,x = − < e ,ω >.
A A A
Quantizing the symplectic vector space H3(X ,C) by means of polarization {e ,eA}, we
τ A
obtain a vector bundle H with fibers H . (It would be more precise to denote the fiber by
e
H stressingthata pointofthe base ofthe bundleH is a pair(τ,e)where τ ∈M ande is
τ,e
a symplectic basisinH3(X ,C), however,wewilluse the notationH ,havinginmind that
τ e
the notatione for the basis alreadyincludes the informationabout the correspondingpoint
τ = τ(e) of the moduli space M.) As usual, we have a projectively flat connection on the
bundleH. LetusdenotebyBthespaceofallsymplecticbasesinthecohomologyH3(X ,C)
τ
where τ runs over the moduli space M. Then the total space of the bundle H can be
identifiedwiththedirectproductB×E,whereE standsforthespaceoffunctionsdepending
on xA. Let us suppose that the basis {e ,eA} depends on the parameters σ1,··· ,σK and
A
∂ eA =mAeB+nABe
i B B (12)
∂ e =r eB+sBe ,
i A AB A B
where in the calculation of the derivatives ∂ = ∂ , we identify the fibers H by means of
i ∂σi e
Gauss-Maninconnection. ThenaprojectivelyflatsectionΨ(xA,σi,λ)satisfiesthefollowing
equation
∂Ψ 1 ∂ 1 ∂2
= − λ−2r xAxB +mAxB + λ2nBC +C (σ) Ψ(xA,σi,λ). (13)
∂σi 2 AB B ∂xA 2 ∂xB∂xC i
(cid:2) (cid:3)
This follows immediately from Equation (7). (Recall that the wave function Ψ depends on
half of coordinates on the symplectic basis {e ,eA}.)
A
ThewavefunctionoftheB-modeltopologicalstringisaprojectivelyflatsectionΨofthe
bundle H that in semiclassicalapproximationcorrespondsto the lagrangiansubmanifold L
5
comingfromgenus0theory. Ofcourse,suchasectionisnotuniqueandoneneedsadditional
assumptions to determine the wave function.
NoticethattherightobjecttoconsiderinB-modelisthewavefunctionΨ(x,e,λ)defined
asafunctionofxA andpolarizatione={e ,eA}. However,itisconvenienttoworkwithΨ
A
restricted to certain subset of the set of polarizations. In particular, we can fix an integral
basis {g (τ),gA(τ)} in H3(X ,C) that varies continuously with τ ∈ M. (The integral
A τ
vectors of H3(X ,C) are defined as vectors in the image of integralcohomology H3(X ,Z)
τ τ
inH3(X ,C).) Itis obviousthat the vectors{g ,gA}arecovariantlyconstantwithrespect
τ A
the Gauss-Manin connection, therefore we can assume that in this polarization the wave
function does not depend on the point of moduli space. It can be represented in the form
∞
Ψ (xA,λ)=exp λ2g−2F (xA) , (14)
real g
g=0
(cid:2)X (cid:3)
where F is the contribution of genus g surfaces. The leading term in the exponential as
g
alwaysspecifiesthesemiclassicalapproximation;itcorrespondstothegenuszerofreeenergy
F (xA). In the next section, we will calculate the transformation of the wave function Ψ
0
from the real polarization to some other polarizations. It is important to emphasize that
the Gauss-Manin connection can have non-trivial monodromies, hence the integral basis
{g (τ),gA(τ)} is a multivalued function of τ ∈M. The quantum state represented by the
A
wavefunctionΨ shouldbeinvariantwithrespecttothemonodromytransformationM ;
real γ
in other words, one can find such numbers c that
γ
M Ψ =c Ψ , (15)
γ real γ real
where Mγ stands for the transformaftion of wave function corresponding to the symplectic
transformation M (in other words M corresponds to M under metaplectic representa-
γ γ γ
tion). fThe condition (15) imposes severe restrictions on the state Ψ , but it does not
real
determine Ψ uniquely. f
real
In the next section, we will calculate the transformation of the wave function Ψ from
the real polarization to some other polarizations.
4 Complex hermitian polarization
Let us introduce special coordinates on M and M. We fix an integral symplectic basis
g0,ga,g ,g in H3(X,C). (This means that the vectors of symplectic basis gA,g belong
a 0 A
to the image of cohomology with integral coefficienfts H3(X,Z) in H3(X,C). We use small
Romanlettersforindicesrunningovertheset{1,2,··· ,r =h2,1}andcapitalRomanletters
for indices running over the set {0,1,··· ,r = h2,1}.) Then special coordinates of M are
defined by the formula
XA =<gA,Ω>. f
Recall that dimCM = h2,1, dimCM = dimCM+1 = 1dimCH3(X,C). Hence, we have
2
the right number of coordinates.
The functions xA =< gA, > anfd xA = − < gA, > define symplectic coordinates on
H3(X,C); on the lagrangiansubmanifold L we have
∂F (xA)
x = 0 ,
A ∂xA
6
where the function F (the generating function of the lagrangian submanifold L) has the
0
physical meaning of genus 0 free energy. Notice that the lagrangian submanifold L is in-
variant respect to dilations (this is a consequence of the fact that Ω is defined only up to a
factor), and it follows that F is a homogeneous function of degree 2.
0
Identifying Mwith the lagrangiansubmanifoldLwe seethat the functions xA onLare
special coordinates on M .
IftwopointsfofMcorrespondto the same pointofM(to the samecomplexstructure),
then the forms Ω are pfroportional; the same is true for the special coordinates XA. This
means that XA canfbe regarded as homogeneous coordinates on M. We can construct
inhomogeneous coordinates t1,··· ,tr by taking ti = Xi, i=1,··· ,h2,1. One can consider
X0
the free energy as a function f (t1,··· ,tr); then
0
X1 Xr
F (X0,··· ,Xr)=(X0)2f ( ,··· , ).
0 0 X0 X0
Let us work with the special coordinates XA =< gA,Ω > on M and coordinates ta =
Xa on M. One can say that we are working with homogeneous coordinates X0,··· ,Xr
X0
assumingthatX0 =1. WedefinecohomologyclassesΩa onaCalabfi-YaumanifoldX using
the formula
Ω =∂ Ω+ω Ω,
a a a
where ω are determinedfromthe conditionΩ ∈H2,1 and∂ (a=1,··· ,r =h2,1)stands
a a a
for the Gauss-Manin covariant derivatives with respect to the special coordinates ta = Xa
X0
onM. RepresentingΩ as XAg +∂ F gA and taking into the accountthat gA and g are
A A 0 A
covariantly constant, we obtain
∂ Ω=g +∂ ∂ F gB =g +τ gB.
a a a B 0 A aB
(Recall that it follows from the Griffiths transversality that ∂ Ω ∈ H3,0 +H2,1. Every
a
element of H3,0 is represented in the form ωΩ; this follows from the relation H3,0 =C.)
NowwecandefineabasisofH3(X,C)consistingofvectors(Ω,Ω ,Ω ,Ω). (Itisobvious
a a
that (Ω,Ω ) span H3,0 +H2,1. Similarly, Ω ,Ω span H2,1+H0,3.) Let us introduce the
a a
notation
e−K =−i<Ω,Ω>=i(XA ∂F0 −XA ∂F0 ). (16)
∂XA ∂XA
(The function K can be considered as a potential of a K¨ahler metric on M.) Then we can
calculate Ω using the relation that <Ω ,Ω>=0; we obtain
a a
Ω =∂ Ω−∂ KΩ.
a a a
We can relate the basis {Ω,Ω ,Ω ,Ω} to the integral symplectic basis {gA,g } by the
a a A
following formulas
Ω=XAg + ∂F0 gA, Ω =g + ∂2F0 gB−∂ K(XAg + ∂F0 gA), (17)
A ∂XA a a ∂Xa∂XB a A ∂XA
Ω=XAg + ∂F0 gA, Ω =g + ∂2F0 gB−∂ K(XAg + ∂F0 gA). (18)
A ∂XA a a ∂Xa∂XB a A ∂XA
The symplectic pairings between {Ω,Ω ,Ω ,Ω} are
i i
<Ω,Ω>=−ie−K, <Ωi,Ωj >=−iG¯ije−K,
7
where Gi¯j is a K¨ahler metric on M defined by Gi¯j =∂j∂iK. As the commutationrelations
are not the standard one, we introduce the following cohomology classes
Ωi =iGi¯jeKΩ , Ω=ieKΩ.
j
Due to the relations e e
<Ω,Ω>=1, <Ωi,Ω >=δj,
j i
we can say that {Ω,Ω ,Ωa,Ω} constitutes a symplectic basis, which specifies a complex
a e e
hermitian polarization.
Directly differentiatingethee above expressions with respect to the parameters ta and t¯a,
we have
∂ Ω=Ω −∂ KΩ, ∂ Ω =−∂ KΩ +ΓkΩ +iC Ωk,
i i i i j i j ij k ijk
∂ Ω=∂ KΩ, ∂ Ωj =∂ KΩj − ΓkΩk−Ωδ ; (19)
i i i i ij ije
k
X
e e e e e e
where Γkij is the Christoeffel symbol for the K¨ahler metric Gi¯j. And
∂iΩ=0, ∂iΩj =G¯ijΩ,
(20)
∂iΩ=−G¯ijΩj, ∂iΩj =−ie2KC¯jikΩk.
Applying (13), we obtain freom (19), (e20) equateions governing the dependence of the state
Ψ(xI,ti,t¯i,λ) on ti,ti (holomorphic anomaly equation).
∂∂Ψt¯i = 12λ2e2KC¯i¯jk¯G¯jjGk¯k∂x∂i∂2xj +G¯ijxj∂∂x0 +Ci Ψ, (21)
∂Ψ (cid:2) (cid:3)
=[x0 ∂ −∂ K(x0 ∂ +xj ∂ )−Γkxj ∂ − 1λ−2C xjxk+D ]Ψ. (22)
∂ti ∂xi i ∂x0 ∂xj ij ∂xk 2 ijk i
NoticethatusuallytheseequationsarewrittenwithC =0,D =0.Thisispossibleifwe
i i
consider only one of these equations; fixing C or D corresponds to (physically irrelevant)
i i
choiceofnormalizationofthe wavefunction. However,ingeneralitis impossibleto assume
that C = 0,D = 0. (We can eliminate C or D changing the normalization of the wave
i i i i
function, but we cannot eliminate both of them.) Let us emphasize that C and D are
i i
constraint by the requirement that (21) has a solution.
5 Holomorphic polarization
Let us start again with the integral symplectic basis {g ,g ,ga,g0}. We will normalize the
0 a
form Ω requiring that < g0,Ω >= X0 = 1. We would like to define a symplectic basis in
the middle dimensional cohomology that depends holomorphically on the points of moduli
space. Namely, we will consider the following basis in H3(X,C),
e0 =g0,
ea =ga−tag0,
e =∂ Ω,
a a
e =Ω=g +Xag +∂ F ga+∂ F g0,
0 0 a a 0 0 0
8
where ∂ stands for the Gauss-Manin connection. It is easy to check that this basis is
a
symplectic.
Using the above relations, we obtain an expression of the new basis in terms of the
integral symplectic basis gA,g ,
A
e0 =g0
ea =ga−tag0,
e =g +tag + ∂f0ga+(2f −ta∂f0)g0, (23)
0 0 a ∂ta 0 ∂ta
e =g + ∂2f0 gb+(∂f0 −tb ∂2f0 )g0.
a a ∂ta∂tb ∂ta ∂ta∂tb
Let q ,qA denote the coordinates on the basis gA,g , and ǫ ,ǫA the coordinates on the
A A A
basis eA,e . Equation(8) permits us to relate the wavefunction in realpolarizationto the
A
wave function in our new basis (in holomorphic polarization)
Ψhol(ǫI,ti,λ)=e−12λ−2RABǫAǫBΨreal(ǫ0,ǫi+tiǫ0,λ), (24)
where R is a the matrix
2f ∂f0
0 ∂ta .
∂f0 ∂2f0
∂ta ∂ta∂tb !
Notice that Ψ is defined up to a t-dependent factor; we use Equation (24) to fix this
hol
factor.
Using that g0,ga,g ,g are covariantly constant with respect to the Gauss-Manin con-
a 0
nection ∂ , we see that
a
∂ e =e , ∂ e =C ec
b 0 b b a abc (25)
∂ ea =δ e0, ∂ e0 =0,
b ab b
where C =∂ ∂ ∂ f . Applying Equation (13), we obtain from Equation (25) the depen-
abc a b c 0
dence of the state Ψ (ǫA,ti,λ) on the coordinates t1,··· ,th
hol
∂Ψ (ǫA,ti,λ) ∂ 1
hol =(ǫ0 − λ−2C ǫbǫc+σ (t))Ψ (ǫA,ti,λ). (26)
∂ta ∂ǫa 2 abc a hol
ThefunctionΨ definedbytheEquation(24)obeysEquation(26)withσ =0. Weremark
hol a
that because our basis is holomorphic, the state Ψ does not depend on antiholomorophic
variables t¯i. Therefore Equation (26) is the only equation the state Ψ (ǫ ,ti,λ) has to
hol i
satisfy. This equation can be easily solved. The solution can be written as follows,
Ψ =exp(W +W ),
1 2
W =W(ǫ0,ǫ0ta+ǫa), (27)
1
W =−λ−2(1 ∂af0 ǫiǫj + ∂f0ǫiǫ0+f (ǫ0)2),
2 2∂ti∂tj ∂ti 0
where W is an arbitrary function of h2,1+1 variables.
Comparing the above expression with Equation (24), we obtain
exp(W)=Ψ . (28)
real
LetusconsidernowtheB-modelintheneighborhoodofthemaximallyunipotentbound-
ary point. We choose g0 as covariantly constant cohomology class that can be extended to
the boundary point and we define ga as covariantly constant cohomology classes having
9
logarithmic singularities at the boundary point. The special coordinates coincide with the
canonicalcoordinates and the basis {eA,e } coincides with the basis that is widely used in
A
the theory of mirror symmetry. (See [3], Section 6.3). This can be derived, for example,
fromthefactthattheGauss-Maninconnectiondescribedbytheformula(25)hasthe same
form in both bases.
6 Partition function of B-model
The partition function Ψ of the topologicalsigma model on a Calabi-Yau threefold X (and
more generally of twisted N = 2 superconformal theory) can be represented as Ψ = eF,
where
F = λ2g−2F (t,t¯), (29)
g
g
X
and F has a meaning of contribution of surfaces of genus g to the free energy. The cor-
g
relation functions C(g) can be obtained from F by means of covariant differentiation.
NoticethatinEquatii1o,·n··,(i2n9)wecanconsidertandt¯gasindependentcomplexvariables. The
covariant derivatives with respect to t coincide with ∂ in the limit when t¯→ ∞ and t
∂ti
remains finite.
It is convenient to introduce the generating functional of correlationfunctions
∞ ∞
1 χ
W(λ,x,t,t¯)= λ2g−2C(g) xi1···xin +( −1)log(λ), (30)
n! i1,···,in 24
g=0n=1
XX
where C(g) = 0 for 2g−2+n ≤ 0. The number χ is defined as the difference between
i1,···,in
the numbers of the bosonic and fermionic modes; in the case of topological sigma-model it
coincides with the Euler characteristic of X (up to a sign).
The function W obeys the following holomorphic anomaly equations (Equation 3.17,
3.18, [2])
∂ λ2 ∂2 ∂ ∂
∂t¯i exp(W)= 2 Cijke2KGj¯jGkk¯∂xj∂xk −G¯ijxj(λ∂λ +xk∂xk) exp(W), (31)
(cid:2) (cid:3)
and
∂ ∂ χ ∂
+Γkxj +∂ K( −1−λ ) exp(W)
∂ti ij ∂xk i 24 ∂λ
(32)
(cid:2) ∂ 1 (cid:3)
= −∂ F − C xjxk exp(W).
∂xi i 1 2λ2 ijk
(cid:2) (cid:3)
One can modify the definition of W by introducing a new function W,
∞ ∞
1
W(λ,xi,ρ,t,¯t)= λ2g−2C(g) xi1···xinρ−nf−(2g−2)+
n! i1,···,in
g=1n=1
XX
f χ λ x χ
+( −1)logρ=W( , ,t,t¯)−( −1)log(λ).
24 ρ ρ 24
The function W (we will call it BCOV wave function) satisfies the equations
f ∂∂t¯i exp(W)= λ22C¯jik∂x∂j∂2xk +G¯ijxj∂∂ρ exp(W), (33)
(cid:2) (cid:3)
f f
10