Note on mirror symmetry and coisotropic D-branes on tori

0.17 MB·English

by  Christian van Enckevort

NOTE ON MIRROR SYMMETRY AND COISOTROPIC
D-BRANES ON TORI
CHRISTIANVANENCKEVORT
Abstract. Wedescribemirrorsymmetryonhigherdimensionaltori,paying
4 specialattentiontothebehaviourofD-branesundermirrorsymmetry. Tofind
0 the mirror D-branes the description of mirror symmetry on D-branes due to
0 Ooguri,OzenYinisused. Thismethodallowsustodealwiththecoisotropic
2 D-branesrecentlyintroducedbyKapustinandOrlov. Wecomparethistothe
descriptionofmirrorsymmetryonD-branesusingtheFourier-Mukaitransform
n
ofcharges.
a
J

9
1
1. Introduction
2
v Sinceitsfirstbreakthroughin1991(see[CdlOGP91])mirrorsymmetryhasbeen
3 developed in many directions. This topic has stimulated a lively exchange of ideas
4
betweenmathematicsandphysics. Sincetheformulationofthehomologicalmirror
0
symmetry conjecture by Kontsevich in [Kon95] and the introduction of D-branes
1
1 by Polchinsky (see [Pol95, PCJ96]), D-branes have started to play an increasingly
1 importantrole. WewillrestrictourattentiontoBPS-branesonCalabi-Yauspaces.
0 These come in two classes the (special) LagrangianD-branes and the holomorphic
/
h D-branes. These are also often called the A-branes and B-branes, because the
t Lagrangian branes can also be discussed in the topological A-model, whereas the
-
p B-branes can be included into the topological B-model.
e KontsevichproposestocombineeachoftheseclassesofD-branesintoacategory.
h
The holomorphic D-branes should lead to the category which is known among
:
v mathematicians as the derived category of coherent sheaves. The Lagrangian D-
i
X branesshouldgiverisetotheso-calledFukayacategory(orbetteraderivedversion
thereof). The homologicalmirror symmetry conjecture then says that for a mirror
r
a pairofCalabi-YaumanifoldsX andY the derivedFukayacategoryofX shouldbe
equivalent to the derived category of coherent sheaves of Y.
Recently these D-branecategorieshavebeen investigatedmoreclosely byphysi-
cists (see e.g. [Dou00] and references therein). These investigations led to inter-
esting conjectures and results about stability, the relation to Witten’s open string
field theory etc. Another interesting line of research was initiated by Kapustin
and Orlov in [KO01], where it was shown that the description of A-branes as La-
grangian branes is in general incomplete and extra objects corresponding to so-
called coisotropic manifolds have to be introduced. To find these objects Kapustin
and Orlov used the description of mirror symmetry introduced in [OOY96].
In this article I want to give a more concrete description of these extra objects.
In [KO01] they were studied from a general perspective. Here we will restrict
ourselves to even-dimensional tori as a special example of Calabi-Yau manifolds.
This restriction allows us to give a much more concrete description of these still
rather mysterious objects. We also investigate some general properties of these
objects and check that they match the expectations from mirror symmetry.
1
2 CHRISTIANVANENCKEVORT
2. Mirror symmetry on tori
2.1. Notation. Letusfirstbrieflydiscussthenotationthatwewillusethroughout
this text. We write X = Tn Tn for a 2n-dimensional torus and Y = Tn Tˇn
for its mirror. We will use coo×rdinates (x,y) Rn/Zn Rn/Zn on X and (x×,yˇ)
Rn/Zn Rn/Zn on Y. The data describing∈objects×on X will have a tilde, t∈o
×
distinguish them from the data describing objects on Y. The SYZ-fibration is
given by projection to the x-coordinates. We will discuss the A-model on X and
the B-model on Y.
The tangent bundle is trivial and can be written as TX = X Rn Rn. A
× ×
Ricci flat metric can then be identified with a positive definite 2n 2n-matrix g,
×
the complex structure with a 2n 2n-matrix squaring to I . The compatibility
2n
× −
ofthecomplexstructureandthemetriccanbewrittenasJtgJ =g. Itfollowsthat
ω =gJ is an antisymmetric matrix defining the K¨ahler form on X.
The complex structure can also be defined by requiring z := τx+y to define
complex coordinates on X. Here τ = τ +iτ is a complex n n-matrix. For
1 2
×
simplicity we will assume that τ = 0 and that τ be symmetric. This mirror
1 2
symmetric to requiring the B-field to vanish. The relation between these two ways
ofdefine the complex structureis thatintermofτ the matrix J canbe written as
0 τ−1
(2.1) J = 2 .
(cid:18) τ2 0 (cid:19)
−
On a Calabi-Yau manifold we also have a holomorphic (n,0)-form Ω. On a torus
this form can be written as Ω=dnz.
2.2. Gluing matrices. We will use the description of D-branes in Calabi-Yau
spacesthatwasgivenin[OOY96]. This descriptionwasalsousedby Kapustinand
Orlov to give a general description of the non-Lagrangian A-branes. In [OOY96]
everything is discussed in terms of local coordinates and it is not completely clear
how to formulate all of it in a coordinate independent way, especially because in
generalthemirrormanifoldandtheoriginalmanifoldarereallydifferentmanifolds.
Fortunately the mirror manifold of a torus is a torus again and in addition we can
useglobalflatcoordinatesonthetori(asdiscussedabove). Sowegiveadescription
tailored for tori and ignore subtleties that do not matter for tori.
Let X be a complex torus as discussed above. The worldsheet Σ is a two-
dimensional surface with metric h. The fields1 of the sigma model are the bosonic
field x which is a map Σ X and the fermionic fields ψ and ψ , which are
+ −
sections of x∗(TX) K1/2→and x∗(TX) K¯1/2 respectively. Here K is the bundle
⊗ ⊗
of (1,0)-forms on the surface Σ.
Let S be a submanifold of X. We will assume that S corresponds to an affine
subspace of Rn Rn (affine D-brane). This assumption will allow us to describe
×
the D-brane in terms of a matrix. We want to discuss boundary conditions for
open strings ending on S. For string theory on a Calabi-Yau manifolds there are
twospecialclassesofboundaryconditionsthatpreservehalfofthesupersymmetry,
which are called A- and B-type boundary conditions. For affine D-branes on a
torus they canbe describedusing an2n 2n-matrixR. Onthe boundary of Σ the
×
following boundary conditions should hold
∂xµ =Rµ ∂¯xν, ψµ = Rµ ψν.
ν + ± ν −
ThematrixRshouldbe orthogonalwithrespecttothe metricg onthe Calabi-Yau
manifold X. As explained in [OOY96] the codimension of the D-brane with gluing
matrix R is given by the dimension of the eigenspace E for eigenvalue 1 of
−1
−
1Iusethenotationfrom[Enc00]. Thefieldsx,ψ+ andψ− correspondtothefieldsX,ψL and
ψR in[OOY96].
NOTE ON MIRROR SYMMETRY AND COISOTROPIC D-BRANES ON TORI 3
R. The orthogonalcomplement E⊥ of that eigenspace is the tangent space to the
−1
D-brane. The restriction of R to that eigenspace can be written as
R|E−⊥1 =(g˜+F)−1(g˜−F).
Here F is an antisymmetric matrix defining the curvature of the connectiononthe
D-brane and g˜ is the restriction of the metric g to E⊥ .
−1
WhethertheseboundaryconditionsareofA-orB-typedependsontheadditional
conditions that we impose on R. The conditions for A-type boundary conditions
can be formulated as
RtωR= ω,
−
R∗Ω=Ω¯.
For B-type boundary conditions R should satisfy
RtωR=ω,
R∗Ω=eiθΩ,
for some θ R.
∈
2.3. T-duality and mirror symmetry. When the target manifold X has a fi-
bration with a torus as fibre, one can apply T-duality and replace the torus by
its dual. Because T-duality affects the boundary conditions, we should describe
what T-duality does with the gluing matrices discussed above. Following [OOY96]
we will describe T-duality using a matrix T. This matrix should square to I ,
2n
be symmetric (or equivalently orthogonal)with respect to the metric g and satisfy
TJT = J. In terms of this matrix mirror symmetry on D-branes should be de-
−
scribed as follows. If we start with a D-brane with gluing matrix R, then mirror
D-brane should have gluing matrix R′ = RT. However, for this description to be
valid there must be a relation between the choice of coordinates on X and on Y
(because R, T depend on the coordinates on X and R′ on the coordinates on Y).
WewillsupposethisdescriptionisvalidforcoordinatesonX suchthatg =I and
2n
J = 0 In . We will use appropriate coordinate transformationsto transformto
−In 0
thiss(cid:0)ituation(cid:1). Toseehowthatworks,wemusthaveacloserlookatthebackground
fields.
In the coordinates that we are using here T = diag(I , I ). For this matrix
n n
−
to be symmetric with respect to the metric g, we have to require that g has the
following block diagonal form g = diag(g ,g ), where g and g are positive
xx yy xx yy
definite symmetric n n-matrices. Recall that we consider a complex structure of
×
theblockform(2.1). NotethatJ automaticallysatisfiesTJT = J. Thecondition
−
that J be orthogonalwith respect to g yields g =τtg τ . Let S be an n n-
xx 2 yy 2 xx ×
matrix such g =St S . Then we can introduce new coordinates(x′,y′) defined
xx xx xx
by the following equation
x′ x S 0 x
=S = xx .
(cid:18)y′(cid:19) (cid:18)y(cid:19) (cid:18) 0 Sxxτ2−1(cid:19)(cid:18)y(cid:19)
With respect to these new coordinates the metric and the complex structure have
thestandardformdiscussedabove. InthenewcoordinatestheD-braneisdescribed
by the gluing matrixR˜ :=SRS−1. Note that T is invariant under such coordinate
transformations. So the mirror D-brane is described by R˜′ = R˜T. However, this
descriptionisintermsofcertainunknowncoordinates(x′′,yˇ′)onY. Recallingthat
yˇis dual to y, a reasonable guess for the definition of these new coordinates is
x′′ x S 0 x
=Sˇ = xx .
(cid:18)yˇ′(cid:19) (cid:18)yˇ(cid:19) (cid:18) 0 S−tτt(cid:19)(cid:18)yˇ(cid:19)
xx 2
4 CHRISTIANVANENCKEVORT
In the coordinates (x′′,yˇ′)=(x′,yˇ′), the metric on Y is the standard metric, so in
the coordinates (x,yˇ) the metric is
g 0
gˇ=SˇtSˇ= xx .
(cid:18) 0 g−1(cid:19)
yy
Similarly we find the complex structure
0 I 0 g−1τt
Jˇ=Sˇ−1 n Sˇ= xx 2 .
(cid:18)−In 0(cid:19) (cid:18)−τ2−tgxx 0 (cid:19)
Comparing to (2.1), we see τˇ = τ−tg . Together with gˇ = g and gˇ = g−1,
2 2 xx xx xx yˇyˇ yy
this defines the mirror map on the background fields. To compare this with the
mirror map in e.g. [Enc00], let us define k =g τ =τ−tg . One can easily check
yy 2 2 xx
that the K¨ahler form can be written as an antisymmetric block matrix
0 kt
.
(cid:18) k 0(cid:19)
−
So k parametrises the K¨ahler structure. In this notation the mirror map on the
background fields is given by kˇ = τ and τˇ = k, which matches the result in
2 2
[Enc00].
2.4. D-brane charges. To D-branes one can associate so-called charges. These
are easiest to define for B-branes. For a vector bundle E on a torus Y the charge
vector is given by the Chern class ch(E). For more general Calabi-Yau manifolds
one usually uses the so-called Mukai vector v(E) := ch(E) td(Y). Of course for
a torus td(Y)= 1, so v(E) =ch(E). Using these charges, wpe can define the Euler
characteristic of a pair (E,F) of B-branes
χ(E,F):= ch(F) ch(Eˇ) td(Y)= v(F) v(Eˇ).
Z ∪ ∪ Z ∪
Y Y
According to the Riemann-Roch theorem we have
(2.2) χ(E,F)= ( 1)ihi(E,F),
−
Xi
wherehi(E,F):=dimExti(E,F). Forthepurposesofthisarticlethesedefinitions
suffice. However, it has been suggested in the literature that one should really use
K-theory instead of cohomology classes.
For A-branes the definition is quite simple in the case of Lagrangian branes. In
this case the charge vector is given by rPD(L), where PD(L) is the Poincar´e dual
of the Lagrangian submanifold L and r is the rank of the flat bundle on L. For
more general coisotropic submanifolds Z X with a bundle F defined on Z, we
⊂
can define the charge as follows
c(Z,F):=i (ch(F)),
∗
where i : Z X is the embedding. For Lagrangian submanifolds this reproduces
→
the definition formulated above. The natural guess for the definition of the Euler
characteristic for a pair of A-branes (Z ,F ) (i=1,2) is
i i
χ((Z ,F ),(Z ,F )):= c(Z ,F ) c(Z ,Fˇ ).
1 1 2 2 2 2 1 1
Z ∪
X
Here Fˇ is the dual bundle of F on Z . In the simplest case of two Lagrangian
1 1 1
submanifoldsL withrankr flatvectorbundlesF ,thisEulercharacteristicreduces
i i i
to
χ((L ,F ),(L ,F )):=r r PD(L ) PD(L )=r r L L ,
1 1 2 2 1 2 2 1 1 2 1 2
Z ∪ ·
X
NOTE ON MIRROR SYMMETRY AND COISOTROPIC D-BRANES ON TORI 5
where L L counts the intersection points of L and L with sign. Recall that
1 2 1 2
a basis of·the complex Hom∗((L ,F ),(L ,F )) in the Fukaya category is given by
1 1 2 2
theintersectionpoints(tensoredwithlinearmapsbetweenthecorrespondingfibres
of F and F ). Because the Euler characteristic of a complex is equal to the Euler
1 2
characteristic of its cohomology,the analog of (2.2) holds in this simple case. It is
unclear if a similar result is valid for coisotropic branes, because for that case we
do not have a definition of the space of morphisms.
The mirror map on charge vectors is expected to be given by fibrewise Fourier-
MukaitransformFM:A∗ A∗ ofdifferentialforms,whereA∗ denotesthespace
Y → X X
of differential forms on X and similarly for A∗. This map is defined as follows
Y
FM(α)= αe−hdyˇ,dyi.
ZTyˇn
Let µ be the mirror map of D-branes, mapping holomorphic D-branes on Y to
objects of the Fukaya category on X. Then one expects
FM(ch(E))=c(µ(E)).
The charges depend on the support of the D-brane and the curvature of the con-
nection. This is also the information containedin the gluing matrix. In both cases
we have a description of how mirror symmetry should act. In the sequel we will
analyse mirror symmetry using both descriptions and compare the results.
3. Mirror symmetry for D-branes
3.1. Linebundles. LetusfirstconsideralinebundleLonY anditsmirrorobject
on X. The first Chern class of this line bundle can be written as
c (L)= 1 dx,Adx + 1 dyˇ,Bdyˇ + Cdx,dyˇ .
1 2h i 2h i h i
Because c (L) should be integral, we see that the entries of the n n matrices A,
1
×
B and C haveto be integers. Note that the matrices A and B can be chosento be
antisymmetric. To make contact with the usual notation in physics, the curvature
matrixF oftheline bundleLcan,intermsofthe matricesA,B andC,be written
as F = A Ct .
−C B
Anot(cid:0)her requ(cid:1)irement is that c (L) should be a (1,1)-form. Recall that complex
1
coordinates on Y are defined by z = τx+y, where for simplicity we assume that
τ = iτ is purely imaginary. A simple computation shows that y = (z+z¯)/2 and
2
x=(2iτ )−1(z z¯). So the (0,2)-part of c (L) is given by
2 1
−
c (L)(0,2) = 1 dz¯,τ−tAτ−1dz¯ + 1 dz¯,Bdz¯ i dz¯,τ−tCdz¯ .
1 −8h 2 2 i 8h i− 4h 2 i
So for c (L) to be a (1,1)-form τ−tC should be symmetric and the antisymmetric
1 2
matrices τ−tAτ−1 and B should be equal.
2 2
For the SYZ-fibration that we fixed above, mirror symmetry is given by the
T-duality matrix T = In 0 . As discussed above a very general description of
0 −In
T-duality can be given(cid:0)in term(cid:1)s of the gluing matrices. For simplicity we will first
assumethatthe metricis the standardmetric givenby the matrixg =I . Inthat
2n
case the gluing matrix for the line bundle L is given by R=(I +F)−1(I F).
2n 2n
−
The gluing matrix of the dual A-brane is then given by R˜ = RT. For a geometric
interpretationofthe B-braneintermsofaLagrangiansubmanifoldweneedthatR˜
is a symmetric matrix. Using the antisymmetry of F and the symmetry of T this
condition can be written as (I F)T(I F)=(I +F)T(I +F). A small
2n 2n 2n 2n
− −
calculationwith block matrices shows that condition is met when A and B vanish.
So we find that the geometric case is exactly the case investigated in [Enc00].
6 CHRISTIANVANENCKEVORT
In this case R˜ can easily be calculated. We can write R˜ = α γ . If we write
γt β
the defining equation for R˜ as (I +F)R˜ = (I F)T, it is(cid:0) a m(cid:1)atter of some
2n 2n
−
careful computations with block matrices to find
I 2Ct(I +CCt)−1C 2Ct(I +CCt)−1
R˜ = n− n n
(cid:18) 2(In+CCt)−1C In+2CCt(In+CCt)−1(cid:19)
−
One can verify that the eigenspace with eigenvalue 1 is ( x ) and that its or-
{ Cx }
thogonal complement −Cty is the eigenspace with eigenvalue 1. This is in
y −
complete agreement w(cid:8)it(cid:0)h [Enc(cid:1)0(cid:9)0], where the mirror brane was found to be given
by the equation y =Cx+α for some shift vector α Rn, which we cannot find in
∈
this way.
Itisinterestingtocheckwhatthislookslikewhenthemetricisnotthestandard
metric. As we did above, we will use a coordinate transform given by a block
diagonal matrix S = diag(S ,S τ−1) to transform to the standard metric and
xx xx 2
the standard complex structure (note however, that we are starting on Y and
transforming to X instead of the other way around). We will denote the matrices
onthe B-sidewithrespecttothe new orthogonalcoordinateswithprimes. We find
R′ =(I +F′)−1(I F′). AgainR′ willonlybesymmetricwhenF′ =S−tFS−1
2n 2n
has the form F′ = −0 C′t , where C′ = S−tτtCS−1. Because T′ = T, we
−C′ 0 xx 2 xx
can use the result ab(cid:0)ove to fi(cid:1)nd R˜′. However, this result is still with respect to
transformed coordinates. The transformation to the original coordinates on X is
given by S˜=diag(S ,S−tτt). So the eigenspace of R˜ for the eigenvalue 1 can be
xx xx 2
written as
ER˜ = S˜−1 x′ x′ Rn = Sx−x1x′ x′ Rn = x x Rn .
1 (cid:8) (cid:0)C′x′(cid:1)| ∈ (cid:9) (cid:8)(cid:0)CSx−x1x′(cid:1)| ∈ (cid:9) {(cid:0)Cx(cid:1)| ∈ }
So we see that the mirror brane does not depend on the metric.
The more complicated case is when R˜ is not symmetric. We will first describe
this case in terms of charges. Using an orthogonal basis transformation we can
write obtain a basis compatible with the decomposition R2n = kerA kerB
⊕ ⊕
(kerA)⊥ (kerB)⊥. Notethatorthogonalandperpendiculariswithrespecttothe
standard⊕metric on R2n (and not the one inducing the metric on the torus). The
corresponding coordinates will be denoted with (x ,yˇ ,x ,yˇ ).
0 0 1 1
Using these coordinates we can write A = 00A0′ (w.r.t. x0, x1), B = 00B0′
(w.r.t. yˇ , yˇ ) and C = C00 C01 (w.r.t. yˇ , yˇ(cid:0)and(cid:1)x , x ). In this notati(cid:0)on th(cid:1)e
0 1 C10 C11 0 1 0 1
matrix F can be written(cid:0)as (cid:1)
0 Ct 0 Ct
00 10
F =−C00 0 −C01 0 
0 Ct A′ Ct
 01 11
 C10 0 C11 B′
− − 
The charge on the B-side is given by the Chern character
ch(L)=e12hdv,Fdvi,
where v = (x ,yˇ ,x ,yˇ ). On the A-side the D-brane is given by a coisotropic
0 0 1 1
submanifoldZ X. OnthesubmanifoldZ aC∞ vectorbundleE withconnection
⊂
is defined. The charge of such a D-brane should be defined as i ch(E), where
∗
∇
i : Z X is the inclusion. A fibrewise Fourier-Mukai transform on differential
→
forms should mapa representativeofthe chargecohomologyclass on the B-side to
NOTE ON MIRROR SYMMETRY AND COISOTROPIC D-BRANES ON TORI 7
one representing the charge on the A-side, i.e.,
FM(ch(L))= e21hdv,Fdvie−hdyˇ0,dy0i−hdyˇ1,dy1i
ZTyˇn
= e21hdx1,A′dx1i+21hdy1,B′dy1i+hC00dx0+C01dx1−y0,dyˇ0i
ZTyˇn
ehC10dx0+C11dx1−dy1,dyˇ1i.
×
Writing Tn as Tn−p Tp and integrating over Tp, we obtain using (A.2)
yˇ yˇ0 × yˇ1 yˇ0
FM(ch(L))=e12hdx1,A′dx1idp(C00dx0+C01dx1 y0)
−
e21hdy1,B′dy1i+hC10dx0+C11dx1−dy1,dyˇ1i.
×ZTyˇn1−p
Using (A.1) we find
(3.1) FM(ch(L))=dp(C x +C x y )
00 0 01 1 0
−
e21hdx1,A′dx1ie12hC10dx0+C11dx1−dy1,(B′)−1(C10dx0+C11dx1−dy1)i det(B′).
×
p
This is a very nice formula, but the interpretation is a bit complicated. The easy
case is when A = B = 0, which we already discussed in terms of gluing matrices
above. Inthatcaseweonlykeepx ,y andC . Droppingtheindices0,weobtain
0 0 00
the following formula
FM(ch(L))=dn(Cx y),
−
which is the Poincar´e dual of the Lagrangian submanifold given by the equation
y =Cx. So this is in full agreement with previous results.
Inthe generalcase we expect this to be of the formPD(Z)ch(F), where Z X
⊂
is a coisotropic submanifold of X and F is a vector bundle defined on Z. If we
define
Z := (x ,C x +C x ,x ,y ) x Rp, x ,y Rn−p ,
0 00 0 01 1 1 1 0 1 1
{ | ∈ ∈ }
then the first factor in (3.1) can be identified with the Poincar´e dual PD(Z) of Z.
The second line of (3.1) should therefore be interpreted as ch(F). The degree 0
part of the Chern character is the rank of the vector bundle. So we see that F has
rank √B′. The 2-form
1 dx ,Ct (B′)−1C dx + 1 dx ,(A′+Ct (B′)−1C )dx
2h 0 10 10 0i 2h 1 11 11 1i
+ 1 dy ,(B′)−1dy dx ,Ct dy dx ,Ct (B′)−1dy + dx ,Ct C dx
2h 1 1i−h 0 10 1i−h 1 11 1i h 0 10 11 1i
is not integral on Zp Zp Zn−p Zn−p. However,there exists an integral 1(n
× × × 2 −
p) 1(n p)-matrix B′′ such that
× 2 −
0 B′′
B′ = .
(cid:18) (B′′)t 0 (cid:19)
−
Using this matrix we can define the lattice ΛB′′ :=Zp Zp Zn−p B′′Z21(n−p)
Z12(n−p). Theabove2-formisintegralonthislattice. S×owec×anfind×alinebundle×L
onthe torusY˜ =R2n/ΛB′′ withfirstChernclassgivenbytheabove2-form. Using
the isogeny i : Y˜ Y, we can define the vector bundle F˜ := i L. The restriction
∗
of F˜ to Z is then→the vector bundle on Z that we are looking for.
Checking if this matches the description in terms of gluing matrices is rather
complicated. In the coordinates discussed above mirror symmetry is given by the
matrixT =diag(I , I ,I , I ). The gluingmatrixR˜ ofthe mirrorD-brane
p p n−p n−p
− −
is given by the equation (I +F)R˜ =(I F)T. Recallthat the gluing matrices
2n 2n
−
areorthogonalmatrices,sotheeigenspacesfordifferenteigenvaluesareorthogonal.
It is again easiest to use a suitable coordinate transform so that we only have to
8 CHRISTIANVANENCKEVORT
deal with the standard metric and the standard complex structure. Then one can
check that the orthogonal complement of the eigenspace for eigenvalue 1 of R˜ is
−
indeed given by Z. Further checks get increasingly complicated.
3.2. Higher rank bundles. To describe higher rank bundles on the B-side and
their mirrors on the A-side, we use the classification of semi-homogeneous vector
bundlesontori(seeChapter4in[Enc00]). Suchvectorbundlescanbeconstructed
outoflinebundlesontoriusingisogeniesandtensorproductwithflatlinebundles.
We will discuss these two possibilities in turn.
3.2.1. Tensor product with flat vector bundle. The simplest possibility is tensor
productwithaflatvectorbundle. Flatvectorbundlescorrespondtorepresentations
of the fundamental group. For the torus Tn Tˇn on the B-side, we have π (Tn
1
Tˇn)=Zn Zn. We can always use a holom×orphic bundle isomorphism to find a×n
equivalent×bundleF suchthattherepresentationistrivialonthe secondZn-factor.
Such a representation also induces a vector bundle on the base torus Tn. This
bundle on the base torus can be pulled back to a vector bundle F˜ on the mirror
torus Tn Tn. One has
× µ(E F)=µ(E) F˜.
⊗ ⊗
3.2.2. Isogenies. Anotherwaytoconstructvectorbundlesofrankgreaterthanone,
istouseisogenies. Forageneralisogenyitisnotclearwhattheeffectonthemirror
will be, but there are two special cases which can be described. These two cases
are when the isogeny is compatible with the SYZ-fibration.
The easiest of these two special cases is an isogeny i : T˜n Tn on the base
→
torus alone. This simply commutes with the mirror map, so we have
µ(i (L))=i (µ(L)).
∗ ∗
Here we slightly abuse notation by using i for both isogenies T˜n Tn Tn Tn
and T˜n Tˇn Tn Tˇn. × → ×
Thes×econd→caseis×anisogenyonthefibresalone. Leti:T˜n Tn beanisogeny.
Thisinducesanisogenyˇı:Tˇn Tˇ˜n. Letusalsowriteiandˇıfo→rthecorresponding
maps Tn T˜n Tn Tn an→d Tn Tˇn Tn Tˇ˜n, that are the identity on the
× → × × → ×
first factor. The mirror map intertwinesˇı∗ and i as follows
∗
µ(ˇı∗(L))=i (µ(L)).
∗
The isogeny that we used above, can be interpreted in this way. Similarly, the
mirror map should also intertwineˇı and i∗.
∗
4. Conclusions and outlook
Wehavegivenapreliminarydescriptionofthemirrorobjectsofsemi-homogeneous
vector bundles. These include D-branes that are coisotropic, but not Lagrangian.
We have also investigated some of their properties and the results fit in nicely
with the expectations from physics in particular from mirror symmetry. Semi-
homogeneous vector bundles are conjectured to generate in a certain sense the full
derived category. So we may hope that this provides us with a fairly complete
understanding of the mirror map on the level of objects.
To do these calculations we restricted the class of background fields that we
considered. In [vE] we will discuss an extensionto more generalbackgroundfields.
ThisshouldalsoclarifytheroleoftheB-field. Inthatarticlewewillalsogivesome
more details on things that were left rather sketchy in this article.
Another restriction is that we described the mirror map only on objects. How-
ever, it should be a functor of D-brane categories, so it should also define a map
on morphisms. Here we have made less progress, because it remains difficult to
NOTE ON MIRROR SYMMETRY AND COISOTROPIC D-BRANES ON TORI 9
define morphisms in the Fukaya category, especially when non-Lagrangianobjects
are involved.
Appendix A. Gaussian integrals with differential forms on a torus
Let T2m be an even dimensional torus and let A be a nondegenerate anti-
symmetric 2m 2m-matrix. We can choose coordinates y ,...,y , such that
1 2m
A = 0 B .×This allows us to write the Gaussian integral, that we want to
−Bt 0
calcul(cid:0)ate, as f(cid:1)ollows
1 1
e2hdy,Adyi = e i,jdyi∧dyjBi,j−m = ( dyi dyjBi,j−m)m.
ZT2m ZT2m P ZT2m m! Xi,j ∧
Here the sum is over i = 1,...,m and j = m+1,...,2m. Because the integral is
only nonvanishing when all dy and dy occur exactly once, we find
i j
ZT2me21hdy,Adyi =ZT2mj1,X...,jmdy1∧dyj1 ∧···∧dym∧dyjmB1,j1−m...Bm,jm−m
= sgn(σ)dy dy dy dy B ...B
1 m+1 m 2m 1,σ(1) m,σ(m)
ZT2mσX∈Sm ∧ ∧···∧ ∧
=det(B)vol(T2m)= det(A)vol(T2m).
p
More general integrals can be computed by ‘completing the square’.
e21hdy,Adyi+ha,dyi= e21hdy−A−1a,A(y−A−1a)i−12hA−1a,ai
(A.1) ZT2m ZT2m
=e21ha,A−1ai det(A).
p
Here a is a vector of 1-forms vanishing on T2m, so dy,a = a,dy . We also
h i −h i
use the antisymmetry of A to rewrite A−1a,Ady = a,dy . The shift over
−h i h i
A−1a does not affect the Gaussian integral, because we have 2m different 1-forms
dy (A−1a) . So wedge products of more than 2m factors vanish automatically.
i i
−
The integral picks out the term with only dy , because we are in dimension 2m.
i
Another simple, but useful formula for integrals of differential forms on a torus
is
(A.2) eha,dyi =a a .
1 m
Z ∧···∧
Tm
Here a is a vector of 1-forms vanishing on Tm.
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10 CHRISTIANVANENCKEVORT
JohannesGutenbergUniversita¨t, Mainz,Germany
E-mail address: [email protected]