II. Preliminaries
1.7. Fluxes in Type IIB String Theory
supergravity by mass deformations as showed in [48]. More generally, in a conformally flat three dimensional internal space, one can turn on fluxes in a controlled and stable way to break 4D N = 4 to 4D N = 3, 2, 1, 0 as showed in [49–52].
1.7.1. Gauge fluxes on D7-branes
Besides the above bulk three-form fluxes, one can also turn on the gauge fluxes F originating from the world-volume gauge field A of the D-branes in compactifications.
D-branes with curved world-volume Here we would like to introduce relevant aspects for D-branes with curved world-volumes, which is a typical situation in compactification. As we know from DBI action, in absence of fluxes F , a static space-filling Dp-brane with the world-volume R1,3× Σ has energy E from the 4D perspective with
E(Σ) ∝ Vol(Σ), (1.115)
where Vol(Σ) denotes the volume of the cycles wrapped by Dp-brane in the internal space. Hence if one would like to preserve (partial) supersymmetry in the 4D, the Dp-brane should satisfy the BPS condition, which especially implies that Dp-brane shall stay at the lowest energy level. From the above, it means that they have to wrap along the submanifolds whose volumes are the minimums among the same dimensional submanifolds, which are the supersymmetric cycles, or in mathematic terms, calibrated submanifolds. The calibrated submanifolds Σcali can be defined with respect to a calibration Φc, which is r-form with two conditions:
Algebraic condition :Φc|Σcali6 vol(Σcali) :=
q
g|Σcalidσ
Differential condition :dΦc= 0.
(1.116)
In the case of Calabi-Yau three-manifolds, the top-form is unique Ω3 and the (product
of) Kähler form J2 can be calibrations. The corresponding calibrated manifolds with the top-form Ω3 are special Lagrangian submanifolds, and holomorphic cycles for the Kähler
form J2. In type IIA, the supersymmetric cycles wrapped by space-filling D-branes are
only special Lagrangian cycles as it has odd dimensions and wrapped by D6-branes and O6-planes. Whereas in Type IIB, holomorphic curves can be wrapped by D3, D5, D7, D9-branes, as well as Op-plane, and preserve the (partial) supersymmetry. To be more precise, let’s take the Type IIB strings three-fold Calabi-Yau compactification as an example. The calibration with respect to the Kähler form J2 for the BPS space-filling Dp-branes are
(see e.g. [53])
dp−3ξpdetg = 1
(p−3) 2 !
J(p−3)2 . (1.117)
One should also need to take into account the F when the non-trivial background flux F and B2 are turned on. For example, the modified calibration condition for a D7-brane
which wraps a four-cycle S in X3 yields
Z S d4ξpdet(g − iF ) = Z S 1 2e −iθ(J + F )(J + F ), (1.118)
where θ is a real parameter that characterizing the unbroken supersymmetry in terms of linear combination of two supercharges. Noting that
Z S d4ξpdet(g − iF ) := Z S 1 2J ∧ J − 1 2F ∧ F , (1.119)
combining with the fact that in the presence of orientifold plane, then one can obtain that the condition that the D7-branes to be BPS, and necessarily being calibrated as
Z
S
J ∧ F = 0. (1.120)
This should be reflected by the D-term or F-term in the 4D effective world-volume gauge theory of the D7-brane. It turns out the D-term take the job, that is we expect that D-term should read as [53]
D ∼ Z
S
J ∧ F . (1.121)
As for the effective gauge theory, one can still use the dimensional reduction of 10D N = 1 SYM to obtain the low energy SYM theory, as we mentioned in 1.5. However, such dimensional reduction typically needs to perform topological twists for some supercharges in order to persevere certain supersymmetries. In a curved space, there is no guarantee for the existences of covariantly constant spinor, as we have seen from the conditions leading to Calabi-Yau spaces. A topological twist typically changes the spin structures and thus certain supercharges could survive in a curved space [54].
For later purpose, let us focus on D7-branes. The consistent configuration for internal gauge fluxes can be described by a stable holomorphic vector bundles 25 with the identification of the curvature with the gauge field strength. For all the concrete applications in this thesis we will restrict ourselves further to using a line bundles La to characterize gauge informations
for D7-branes. Typically, the gauge information on a D7-brane wrapping a divisor Di shall
be encoded in the Picard group of the line bundle Li, which is isomorphic to the first sheaf cohomology group of O∗D
i: H
1(O∗
Di). The Picard group enjoys a short exact sequence as
0 → J1H1(Di) → H1(O∗Di) → H
1,1
Z (Di) → 0, (1.122)
where the third term is the gauge flux F = dA, given by the first Chern class c1(Li) ∈ HZ1,1(Di)
and the second term J1H1(Di) := HC1(Di)/F1HC1(Di) + HZ1(Di) is a Jacobian, topology of a
tours, which, in the absence of the gauge flux, parametrizes the Wilson line moduli of the gauge field A, i.e. the holonomy of the gauge field A over a non-trivial 1-cycle. Note that if Di is simply connected, i.e. π(Di) = 0 and no non-trivial one-cylces, then the Jacobian J1H1(Di) is
trivial and the gauge dates are uniquely specified by the first Chern class c1(Li) ∈ HZ1,1(Di). In
the sequel, we omit the degrees of freedoms of Wilson lines and focus on the gauge fluxes.
25A finer and more reasonable description shall involve the (coherent) sheaf, which was first suggested by J.
Harvey and G. Moore in [55] for modeling D-branes on large-radius Calabi-Yau manifolds and since then it has been under vigorous development and become a common weapon for string physicists to attack the physics of D-branes, see more details in [56]. Simply put, a sheaf is the mathematical machinery need to make sense when a bundle is no longer a sensible concept such as a vector bundle living only over a submanifold or certain singular space. For our purpose, the holomorphic vector (line) bundles are good enough. Throughout this thesis, unless they are mentioned, we only talk about vector bundles instead of sheaves for characterizing the non-trivial gauge backgrounds of D-branes.