II型弦论中的开放弦多分支与Kähler势研究

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JHEP09(2016)062

Here Π

= exp

(Π

) is again a homotopic special Lagrangian deformation of Π

, and ζ

is a basis of quantised harmonic one-forms on it such that

∧ρ

= l

. Again, these

functions can be expressed in terms of chain integrals as [22]

Im (CΩ) =

Im (CΩ) ∧

= −g

α i

Im(CF

) + g

α Λ i

Im(CX

) (2.21)

where Σ

is the three-chain swept by the two-cycle Poincar´e dual to ζ

, and

is the

extension of this one-form to the four-chain C

. More explicitly we have that

α i

∧

and g

α Λ i

∧

. (2.22)

3 The scalar potential from Minkowski three-forms

The space of background and D-brane deformations described in the last section will be

subject to a scalar potential in certain type IIA compactiﬁcations. In particular, both

K¨ahler and complex structure moduli will develop an F-term scalar potential when NSNS

and RR background ﬂuxes are present [25, 34]. In the absence of open string deformations,

this potential can been reproduced by applying the usual 4d supergravity expression or

by direct dimensional reduction. The latter method involves integrating out the degrees

of freedom associated to three-form ﬁelds in R

1,3

which give a non-vanishing contribution

to the potential [25, 34]. In fact, as shown in [21] one may describe the full F-term

scalar potential purely in terms of contributions coming from Minkowski three-forms if one

performs the dimensional reduction in the democratic formulation of type IIA supergravity.

In the following we will adopt this latter approach, as it allows to incorporate the D-brane

moduli into the computation and derive a scalar potential for open and closed string modes

simultaneously [22, 23].

For simplicity let us consider a Calabi-Yau orientifold compactiﬁcation where only RR

background ﬂuxes are present, ignoring for now the presence of localised sources. Working

in the democratic formulation we have the RR p-form potentials C

with p = 1, 3, 5, 7, 9

that can be arranged in the polyforms

C = C

+ C

or A = C ∧ e

−B

(3.1)

respectively dubbed C and A-basis for the RR potentials in [35]. The corresponding gauge

invariant ﬁeld strengths are given by [35, 36]

G = dC − H ∧ C +

G ∧ e



dA +



∧ e

(3.2)

with

G a formal sum of harmonic (p + 1)-forms of M

to be thought as the background

value for the internal RR ﬂuxes deﬁned in the A-basis. This basis is particularly adequate

to apply Dirac quantisation, since a D(p − 1)-brane will couple to the potential A

via its

Chern-Simons action, and so the standard reasoning gives the quantisation condition [37]

p+1

∈ Z (3.3)

– 8 –

JHEP09(2016)062

for any cycle π

p+1

in the internal space not intersecting a localised source like a background

D-brane. In the A-basis a source wrapping a cycle Π

enters the Bianchi identity as



−B

∧ G



= d



dA +



δ(Π

) ∧ e

−σF

(3.4)

with δ(Π

) the delta-function with support on Π

and indices transverse to it, while F

is the quantised worldvolume ﬂux threading Π

. In the absence of localised sources the

r.h.s. of (3.4) vanishes and the A

are globally well-deﬁned, so they do not contribute to

the integral in (3.3) which becomes a quantisation condition for the ﬂuxes

p+1

. We can

then deﬁne ﬂux quanta in terms of the integer cohomology of M

, namely as

m = l

, m

˜π

, e

(3.5)

where ˜π

∈ H

−

, Z) and π

∈ H

, Z). These deﬁnitions need to be generalised

if we take into account the eﬀect of localised sources in the above Bianchi identity, as we

discuss in the next section. Finally, as we have doubled the p-form degrees of freedom we

need to impose the Hodge duality relations

= (−)

10−2n

. (3.6)

which can be done either by hand or by adding a series of Lagrange multipliers to the 10d

supergravity action, as done in appendix B.

To proceed we deﬁne a set of Minkowski four-form ﬁeld strengths arising from the

dimensional reduction of the 10d RR ﬁeld strengths

= F

+ . . . G

= F

∧ ω

+ . . . G

4 a

∧ ˜ω

+ . . . G

∧ ω

+ . . . (3.7)

Here the four-forms (F

, F

4 a

) have their indices in R

1,3

and (ω

, ˜ω

, ω

) are har-

monic forms of M

In particular l

−2

are the harmonic representatives of H

−

, Z)

deﬁned in the last section and l

−4

˜ω

is a dual basis of harmonic forms in H

, Z) in

the sense that

∧ ˜ω

= l

, a, b, ∈ {1, . . . , h

1,1

−

}. (3.10)

Finally l

−6

= dvol

√

represents the unique harmonic six-form with unit integral

over M

. Following [21], such Minkowski four-forms enter the 4d eﬀective action as

−

−K

32κ

1,3

∧ ∗F

+ g

∧ ∗F

−3φ

4 a

∧ ∗

4 b

−3φ

∧ ∗

4κ

1,3

+ F

4,a

˜ρ

˜ρ (3.11)

In terms of the dimensional reduction of the RR potential to Minkowski three-forms

= c

+ . . . C

= c

∧ ω

+ . . . C

3 a

∧ ˜ω

+ . . . C

∧ ω

+ . . . (3.8)

We have that

= dc

− db

∧ c

4 a

= d

3 a

− K

abc

∧ c

= d

− db

∧

3 a

(3.9)

– 9 –

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