Fibrewise T-Duality for D-Branes on Elliptic Calabi-Yau

13 Pages·0.17 MB·English

by  Bjorn Andreas

hep-th/0101129
HUB-EP-01/03
Fibrewise T-Duality
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for D-Branes on Elliptic Calabi-Yau
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Bj¨orn Andreas1, Gottfried Curio2, Daniel Herna´ndez Ruip´erez3 and Shing-Tung Yau1
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1 1Department of Mathematics, Harvard University, Cambridge, MA 02138, USA
v
9 2Humboldt-Universita¨t zu Berlin, Institut fu¨r Physik, D-10115 Berlin, Germany
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3Departamento de Matema´ticas, Universidad de Salamanca, 37008, Salamanca, Spain
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h Fibrewise T-duality (Fourier-Mukai transform) for D-branes on an elliptic Calabi-Yau
t
- X is shown to require naturally an appropriate twisting of the operation respectively a
p
e twisted charge. The fibrewise T-duality is furthermore expressed through known mon-
h
: odromies in the context of Kontsevich’s interpretation of mirror symmetry.
v
i
X
r
a
This paper constitutes a shortened and simplified version of [1] (where details and
references are given) intended to present the essential results to an audience of physicists.
The last years saw an intense study on BPS D-branes, with their associated bundles,
in type II string theories on a Calabi-Yau manifold. For the class of elliptic Calabi-Yau’s
not only the description of bundles on the whole Calabi-Yau, related to D6-branes, can be
made explicit via spectral covers (cf. [2]), but it allows also for a version of the Fourier-
Mukai (FM) transform of a bundle V on non-toroidal spaces which in contrast to earlier
transforms in such cases (say on K3) sticks completely to the idea of using the duality on
a torus by building a fibrewise FM transform. This ’fibrewise T-duality’ operates on the
spectrum of D-branes and has also a close connection with the spectral cover construction.
We compute the cohomological invariants of a bundle and its fibrewise dual and show
how the adiabatic character1of the operation is confirmed only by using an appropriately
twisted charge. Alternatively a version of Mukai’s f-map, using the usual Mukai vector as
charge but a slightly twisted operation, has to be used.
Also, in a version of the mirror correspondence, which brings supersymmetric D-
branes on both sides into the play, bundles, or better sheaves (think of them as bundles
supported on holomorphic subvarieties), on a Calabi-Yau X in type IIA are related with
special Lagrangian submanifolds (with an U(1) bundle over them) in the mirror Calabi-
Yau Y with cohomological invariants relating Heven(X) and H3(Y). Furthermore, the
cohomological effect of a line bundle twist (or more general operations related to FM
transforms) is related with a type IIB monodromy resp. (using the mirror identification)
with a type IIA monodromy (around the locus in moduli space where the divisorvanishes).
The fibrewise FM transform is given its place in Kontsevich’s general association of FM
transforms with monodromies.
ConsideranSU(n)bundle V ofc (V) = 0overanellipticallyfibered X orequivalently
1
nD6-branes wrapped overX withinduced D2-andD0-brane charges(D2i-charges meaning
here for now just ch (V)). The T-duality on the T2 fibre maps (subscripts indicate that
i
fibre F or base B is contained (resp. contains) the wrapped world-volume)
D6 → D˜4
B
D4 → D˜6 , D4 → D˜2
B F B
(1)
˜ ˜
D2 → D4 , D2 → D0
B F˜ F
D0 → D˜2
F˜
The operation of fibrewise T-duality can be described on the cohomological data repre-
senting V and its FM dual V˜, i.e. we will mirror the mentioned D-brane relations as a map
1 I.e., decomposingthe cohomology intobase andfibre parts, the operationoffibrewise duality
2
takes the form of an adiabatic extension of the same operation on the (cohomology of a) T ,
fulfilling the expectations from the interpretation as T-duality on D-branes.
1
˜
between ch(V) and ch(V) (this will be made precise). Assume the following decomposition
H0(X) = C
H2(X) = Cσ ⊕H2(B)
  (2)
H4(X) = H2(B)σ ⊕C
H6(X) = C 
 
Then essentially the six entries of the Chern character vector in our decomposition are
pairwise interchanged and the transformation has the block-diagonal form
0 ∗
∗ 0
 
0 ∗
Q = Q˜ (3)
 ∗ 0 
 
 0 ∗
 
 ∗ 0
 
So the cohomological effect of the fibrewise FM transform (T-duality) is given by adiabatic
extension of the effect of T-duality on the fibre where the matrix is [3]
0 1
A = (4)
(cid:18)−1 0(cid:19)
So one would like to see as actual form of the transformation matrix
0 1 0 0 0 0
−1 0 0 0 0 0
 
0 0 0 1 0 0
Q = Q˜ (5)
 0 0 −1 0 0 0
 
 0 0 0 0 0 1
 
 0 0 0 0 −1 0
 
thereby confirming the ’adiabatic’ interpretation of the Fourier-Mukai transform as (with
entries now consisting of 3×3 blocks corresponding to H∗(X) = σH∗(B)⊕π∗H∗(B))
0 1
3 3 (6)
(cid:18)−1 0 (cid:19)
3 3
The Fourier-Mukai transform
The spectral cover construction2 describes the bundle V from a line bundle L on
the spectral cover surface C. For the description of the FM transform we will instead of
2 An SU(n) bundle V on X of c1(V) = 0 decomposes on the typical fibre E (where V is
assumed to be semistable) as a sum of line bundles of degree zero each of which corresponds
(a reference point p chosen) to a point Q on E. When globalized this leads to a hypersurface
i
i : C ֒→ X, a ramified n-fold ’spectral cover’ of B, whose equation can still be twisted so that
C = nσ +η in cohomology. As fibrewise V = p∗P and a twist by a line bundle L over C leaves
∗
the fibrewise isomorphism class unchanged this generalizes to V = p∗(pCL ⊗ P) where p and
pC are the projections on the first and second factor of X ×B C. The condition c1(V) = 0
translates to the fixing π∗(c1(L)) = −π∗(c1(C) − c1)/2) of c1(L) in H1,1(C) up to a class in
ker π∗ : H1,1(C) → H1,1(B).
2
˜
working on X × C work on X × X
B B
X × X˜ −p→2 X˜
B
p1 π2 (7)
 
 
X −π→1 B
y y
where X˜ is the compactified relative Jacobian of X. X˜ parameterizes torsion-free rank 1
and degree zero sheaves of the fibres of X → B and is actually isomorphic with X (see [4]
or [5]). We will identify X˜ and X. V is constructed from the Poincar´e sheaf P via
V = p (p∗(i L)⊗P) (8)
1∗ 2 ∗
with P = O(∆−σ×X˜ −X×σ˜ −c (B)) normalized to be trivial along σ×X˜ and X×σ˜.
1
Let us determine the FM-transform. For this we make use of the fact that the repre-
sentation of V by the (C,L) data already looks in itself like a FM transform. So the FM
transform of V is practically the double transform of i L. But the inverse transform of
∗
FM is not precisely FM itself, but rather a slightly twisted version of the first transform,
as we will recall. Only this twisted version brings us back from V to i L, so if we start on
∗
V, making the original FM transform we get i L times the inverse twist.
∗
So for V = p (p∗(i L)⊗P) the ’fibrewise dual’ is given in spectral cover data by
1∗ 2 ∗
V˜ = R1p (p∗(V)⊗P∗) = i L⊗π∗K (9)
2∗ 1 ∗ 2 B
One wants to show that there exists a matrix M in the block diagonal form (5) relating V
to V˜ or V to i L (the latter option is simply the inverse process as V is the dual of i L).
∗ ∗
Before going on let us recall the FM transform for elliptic fibrations (cf. [4] , [3] , [5]).
We define the Fourier-Mukai functors Si, i = 0,1 by associating with every sheaf V on X
the sheaf Si(F) on X (X and X˜ are identified; P the Poincar´e sheaf on the fibre product)
Si(V) = Rip (p∗(V)⊗P) (10)
1∗ 2
It can be also described as (cf. [2] ) P = I∗ ⊗ p∗O(−σ) ⊗ p∗O(−σ) ⊗ q∗K with q =
1 2 B
π.p = π.p and I = O(∆)∗ the ideal sheaf of the diagonal immersion δ : X → X × X.
1 2 B
We can also define the inverse Fourier-Mukai functors Sˆi, i = 0,1
Sˆi(V) = Rip (p∗(V)⊗P∗ ⊗q∗K−1) (11)
2∗ 1 B
Therelationshipbetween these functors isstatedbest ifweconsider theassociatedfunctors
between the derived categories of complexes of coherent sheaves bounded from above.
S : D−(X) → D−(X); S(G) = Rp (p∗(G)⊗P)
1∗ 2
(12)
Sˆ : D−(X) → D−(X); Sˆ(G) = Rp (p∗(G)⊗P∗ ⊗q∗K−1)
2∗ 1 B
3
Note that one obtains an invertibility result
ˆ ˆ
S(S(G)) = G[−1], S(S(G)) = G[−1]. (13)
The K3 case
Consider first the two-dimensional case of X = K3 and assume the decomposition
H0(X) = C
H2(X) = Cσ ⊕C (14)
H4(X) = C
 
The class of the spectral curve C on which i L is supported is C = nσ +kF.
∗
The Chern characters of i L can be obtained using Grothendieck-Riemann-Roch for
∗
the embedding i : C → X˜, giving with ch(i L)Td(X˜) = i (ch(L)Td(C)) that
∗ ∗
ch(i L) = (0,C,n) (15)
∗
further ch(V) (V is the only FM transform of i L) is given by
∗
ch(V) = (n,0,−k) (16)
Let us introduce3 the new functor T(·) = S(·) ⊗ π∗K−1/2 = S(·) ⊗ O(F) (and similarly
B
Ti(·) = Si(·)⊗π∗K−1/2 = Si(·)⊗O(F)) so that T0(i L) = V ⊗O(F). We get
B ∗
ch(T0(i L)) = ch(V)(1+F) = M ·ch(i L) (17)
∗ ∗
where we reach the matrix we wanted
0 1 0 0
−1 0 0 0
M =  .
0 0 0 1
 0 0 −1 0
 
The Calabi-Yau threefold case
Consider a sheaf V and write its Chern character as ch(V) = (n,xσ +S,ση +aF,s)
(η,S ∈ p∗H2(B)), then, using the decompostion of the cohomology, we find
2
n 0
x −n
   
S η + 1xc
ch(V) = , ch(Sˆ(V)) = 2 1
η   −1nc −S 
   2 1 
a  s+ 1ηc σ + 1 xc2σ 
   2 1 12 1 
 s  −1nσc2 −a− 1σc S +xσc2
   6 1 2 1 1
3 The functor T was introduced [4] because its inverse transform in the sense of (13) is the
∗
“natural” one: T is the FM transform w.r.t. the sheaf P = P ⊗q O(F) and its inverse functor
∗
Tˆ is the FM transform w.r.t. the dual sheaf P . S does not have this property as its inverse
∗ ∗ −1
transform is not the FM transform with respect to P , but this twisted by q K . So we just
B
divided this up in two parts of K−1/2, distributed among the original and the inverse transform.
B
4
If we multiply ch(Sˆ(V)) by the Todd class Td(N) = 1− 1c + 1 c2,4 we get
2 1 12 1
x
−n
 
η
ch(Sˆ(V))·Td(N) =
 −S 
 
s− 1 xσc2
 12 1
 −a+xσc2 
 1 
When V is fibrewise of degree 0 we have x = 0 and then
x 0 1 0 0 0 0 n
−n −1 0 0 0 0 0 x
     
η 0 0 0 1 0 0 S
= · (18)
−S  0 0 −1 0 0 0 η 
     
 s   0 0 0 0 0 1 a
     
−a  0 0 0 0 −1 0  s 
     
that is
Td(N)·ch(Sˆ(V)) = M ·ch(V). (19)
So for fibrewise degree 0, semistable bundles V we have Sˆ0(V) = 0 and Sˆ1(V) = i L and5
∗
Td(N)·ch(i L) = Td(N)·ch(Sˆ1(V)) = −M ·ch(V).
∗
The interpretation6 is that here one must either twist the operation a little bit and stick to
the usual Mukai-vector or one keeps the operation and makes the twist in the usual Mukai-
vector more ’relative’. Explicitly, similarly to the usual twist in the Mukai-vector, here in
the fibrewise situation a twist by Td(N) with the normal bundle N = (j∗TX)/TB =
j∗T = j∗(TX/π∗TB) plays a rpole (where j : B ֒→ X). For a complex of sheaves G on
X/B
X you twist the standard definition of ch to define a “charge” (where |i+1| = (−1)i+1)
Q(G) = (−1)ich(Gi)( Td(N))|i+1| (20)
X p
4 We always confuse the normal bundle N with its pull-back π∗N to X
5 There is an analogous equation to (19) for the direct FM transform S(V): if x = 0 then
−1 0
Td(N )·ch(S(V)) = M·ch(V). Forfibrewisedegree0andsemistablebundlesV wehaveS (V) =
1 −1 −1 1
0 and with S (V) = i∗L, we have Td(N )·ch(i∗L) = Td(N )·ch(S (V)) = −M ·ch(V).
6 In the three-fold case the sole use of the T-functor from the K3-case (for the map between
theChernclassesofthebundle anditsdual)isinsufficienttoexhibit(5)astransformationmatrix.
Rather one needs (just as one has in the usual full, not fibrewise, FM transform actually to take
the Mukai vector with its twist by Td(X) and not just ch(V)) the relevant twisted charge.
p
5
Then
Q(Sˆ(V)) = −ch(Sˆ1(V))( Td(N)) , Q(V) = ch(V)( Td(N))−1 (21)
p p
and thus
Q(Sˆ(V)) = M ·Q(V) (22)
Since Sˆ0(V) = 0 and Sˆ1(V) = i L we have Sˆ(V) = i L[−1] as complexes and (21) gives
∗ ∗
Q(i L[−1]) = M ·Q(V) = M ·Q(S(i L))
∗ ∗
Alternatively the cohomology effect of the FM transform can be described by a so-called
f-map. Fora fibrewise FM transform wecan alsointroduce7 a relativeversion f : H(X) →
r
H(X). But modifying f to
r
f(x) = p (p∗(x)·Z) (23)
1∗ 2
with Z = p∗Td(X)·ch(P)· p∗Td(X), the effective charge of V transforms to
2 1
p p
f(Q(S(V))) = M ·Q(V) (24)
when x = 0. If V is also fibrewise semistable so that S0(V) = 0 and S1(V) = i L, then
∗
−f(Q(i L)) = −f(Q(S1(V))) = f(Q(S(V))) = M ·Q(V)
∗
Fibrewise T-duality on D-branes at the sheaf level
Note that we can also describe the fibrewise T-duality maps in (1) at the sheaf level.
Let us consider the skyscraper sheaf C(x) at a point x of X. Its FM transform S0(C(x))
is a torsion-free rank one sheaf L on the fibre of X over π(x) 8 as we expect from (1) and
x
thus D0 → D2 . The topological invariants are n = x = a = S = η = 0,s = 1 and
F
ch (S0(C(x))) = 0, i = 0,1,3, ch (S0(C(x))) = F (25)
i 2
If we start with F = O ; proceeding as in (3.16) of [5] we have
σ
S0(O ) = O , S1(O ) = 0
σ X σ
(26)
S0(O ) = 0, S1(O ) = O ⊗π∗K
X X σ B
7 It is defined by fr(x) = p1∗(p∗2(x)·Zr) where Zr = p∗2Td(TX/B)·ch(P)· p∗1Td(TX/B).
Then Td(T )·ch(S(V)) = f ch(V)· Td(T ) .pThe effect of f on the peffective charge
X/B r X/B r
given pby the Mukai vector Q(V) =(cid:0)ch(V)·pTd(X) is d(cid:1)escribed by f (Q(V)) = Q(S(V)).
r
8 With the identification X ≃ X˜ the poipnt x corresponds precisely to L (see [4] or [5] )
x
6
Then O transforms to the structure sheaf of X and O transforms to a line bundle on
σ X
σ as we expect from (1) since D4 ↔ D6. We have as before the transformations at the
B
cohomology level (n = 0,x = 1,S = 0,η = 1c ,a = 0,s = 1σc2) and get
2 1 6 1
ch (S0(O )) = 1, ch (S0(O )) = 0, i = 1,2,3 (27)
0 σ i σ
Finally, let us consider a sheaf F on B; by (26) we have S0(O ⊗π∗F) = π∗F,S1(O ⊗
σ σ
π∗F) = 0,S0(π∗F) = 0S1(π∗F) = O ⊗ π∗F ⊗ π∗K Then, a sheaf O ⊗ π∗F = j F
σ B σ ∗
(j : B→X is the section) supported on a curve C˜ in B embedded in X via j transforms
to a sheaf on the elliptic surface supported on the inverse image of C˜ in X and vice versa.
So we have relations (1) for fibrewise T-duality on D-branes at the sheaf level (cf. [6] )
˜
D4 → D6
B
D2 → D˜4 (28)
B F˜
D0 → D˜2
F˜
On the monodromy interpretation of the FM transform
We finally ask whether the Fourier-Mukai transform will be related to a monodromy
matrix in the sense of Kontsevich’s proposal. We are interested first in an explicit
map between the topological invariants of the characteristic classes of the Chan-Paton
sheaf V and the brane charges. Consider the BPS charge lattice which can be identi-
fied with the middle cohomology lattice of the mirror manifold H3(Xˆ,Z) and consider
(we think of two-parameter models) the central charge associated to the integral vector
n = (n ,n1,n2,n ,n1,n2) which is
6 4 4 0 2 2
Z(n) = n Π +n1Π +n2Π +n Π +n1Π +n2Π . (29)
6 1 4 2 4 3 0 4 2 5 2 6
One the other side, one has in the large volume limit of X the lattice of microscopic D-
brane charges (which is identified with the K-theory lattice K(X)). Here one considers the
effective charge Q of a D-brane state η given by the Mukai vector Q = ch(η) Td(X) ∈
Heven(X) with the associated central charge p
t3 t2
Z(t) = Q0 − Q2 +tQ4 −Q6. (30)
Z 6 2
Z
where t is the generic K¨ahler class and one gets
r 1 r 1
Z(Q) = t3 − ch (V)t2 +(ch (V)+ c (X))t−(ch (V)+ ch (V)ch (V)) (31)
1 2 2 3 1 2
6 2 24 24
The comparison of Z(n) and Z(t) leads then to a map between the low energy charges
n and the topological invariants of the K-theory class η.
7
The monodromy transformations S correspond to automorphisms M(D)
D
[V] → [V ⊗O (L)], [V] → [V ⊗O (H)] (32)
X X
and we see that the linear transformations acting on n corresponding to D are
M(D) = S−1 (33)
D
Let us consider now a second type of monodromy transformation proposed by Kont-
sevich. He proposed that the monodromy T about the conifold locus of the mirror corre-
sponds to the automorphism of the derived category whose effect on cohomology is
S : γ → γ − γ ∧Td(T ) ·1 (34)
X X
(cid:16)Z (cid:17)
(1 the standard generator of H0(X,Q)) leading to a change in the topological invariants
X
ch (V)c (X)
1 2
ch(V) → ch(V)−( +ch (V)) (35)
3
12
Using the expression of the prepotential in the large radius limit F = 1(t·J)3 − c2(X)(t·
6 24
J)+.... where (t·J) = t Ja and using the period vector Π we find the expression valid
a
for both models P4 P[8] and P4 [12]
1,1,2,2,2 1,1,2,2,6
ch (V) = n1(J −2J )+n2J
1 4 1 2 4 2
(36)
c (X)
ch (V) = −(n1(J −2J )+n2J ) 2 −n
3 4 1 2 4 2 12 0
leading to the universal shift
n → n +n (37)
6 6 0
comparing this to the monodromy we find for the linear transformation acting on n
S = T−1 (38)
WenowaskwhethertheFourier-Mukaitransformwillberelatedtoamonodromymatrixin
the sense of Kontsevich’s proposal. As the corresponding matrices are by mirror symmetry
identifiable already on the bundle side this comes down essentially to the question whether
the matrix M in (5) , is generated by S ,S ,T. Recall that the transformations in the
H L
Kontsevich association can themselves be considered as FM transforms. This time we will
not work on the fibre product but rather on the ordinary product.
X ×X˜ −q→2 X˜
q1 π2 (39)
 
 
X −π→1 B
y y
8
The Fourier-Mukai functors Si are (E ∈ D(X ×X) in the derived category)
Si(V) = Rip (p∗(V)⊗E) (40)
E 1∗ 2
We can also define the full FM transformation at the derived category level S (G) =
E
Rp (p∗(G)⊗E). For example the twist transformation G→G⊗L considered in (32) comes
1∗ 2
fromE = O ⊗q∗(L)where∆isthediagonalofX×X. Theotheroperation(corresponding
∆ 2
to the ’conifold monodromy’) considered before corresponds to an E whose cohomology is
the ideal sheaf I of the diagonal of X×X. This recovers the “gamma shift” (34)
∆
ch(S (G)) = ch (S (G))−ch(G) = ch(G)∧Td(T ) −ch(G) (41)
I∆ 0 OX×X (cid:16)Z X (cid:17)
Our fibrewise FM transform is specified by using E = j P where j : X × X→X×X
∗ B
is the natural embedding9, so it is build up out of the divisors (resp. their associated line
bundles) σ and π∗K on the one hand and by O on the other.
B ∆
ˆ
We have the M matrix equation for the inverse FM transform. This is S = S with
j∗Q
Q = P∗ ⊗q∗(K−1) = I ⊗p∗O(σ)⊗p∗O(σ)⊗q∗K−2 where I is the ideal of the diagonal
B 1 2 B
of X × X. Then Sˆ = (⊗q∗K−2)◦(⊗O(σ))◦S ◦(⊗O(σ)) or
B B j∗I
Sˆ = S ◦S ◦S ◦S
O∆(2c1) O∆(σ) j∗I O∆(σ)
Having written our fibrewise FM transform Sˆ as a composition of three divisor FM trans-
forms and one FM transform S we expect the following relation for its monodromy,
j∗I
expressed as a product of divisor monodromies and the monodromy around the diagonal
of the fibre product of ideal sheaf I
S = S ·S2 ·S ·S (42)
V σ c1 I σ
Let us make this more explicit by considering the model given by the elliptic fibration
P4 [18], extensively studied in the context of mirror symmetry [7] and in the context
1,1,1,6,9
of D-branes on elliptic Calabi-Yau [8]. Among the degree 18 hypersurfaces is10
z18 +z18 +z18 +z3 +z2 = 0 (43)
1 2 3 4 5
9 FM on X ×B X with respect to P is FM on X ×X with respect to j∗P
10 Atz1 = z2 = z3 = 0theambientspacehasasingularlinewhichintersectsX inasinglepoint.
2
The blow up of this line gives an exceptional divisor E = P in X. A second divisor L (defined by
2
the first order polynomials) is given by the elliptic surface over a line in P and together with E
generates H4(X,Z). The elliptic fibration structure is induced by the linear system |L| generated
by z1,z2,z3 mapping X to P2. The section of the fibration is given by B2 = E. The homology
2
class of the elliptic fibre in H2(X) will be denoted by h = L . Further intersection relations are
2 2 3 3
given by E ·L = 1, E ·L = −3, L = 0, E = 9. Working in the E,L basis (cf. [8]) the
generic K¨ahler class is given by J = t1E+t2L with t1,t2 coordinates on the K¨ahler moduli space.
2 ∗
In the degree 18 model again one has σ = E = B = P and π K = 3L; note that H = 3L+E
B
−3
with a corresponding multiplicative relation S = S ·S for the matrices.
E H L
9