D-brane Superpotentials Research Articles

Mirror symmetry, D-brane superpotentials and Ooguri–Vafa invariants of Calabi–Yau manifolds**Supported by Y4JT01VJ01 and NSFC(11475178)

The D-brane superpotential is very important in the low energy effective theory. As the generating function of all disk instantons from the worldsheet point of view, it plays a crucial role in deriving some important properties of the compact Calabi–Yau manifolds. By using the generalized GKZ hypergeometric system, we will calculate the D-brane superpotentials of two non-Fermat type compact Calabi–Yau hypersurfaces in toric varieties, respectively. Then according to the mirror symmetry, we obtain the A-model superpotentials and the Ooguri–Vafa invariants for the mirror Calabi–Yau manifolds.

D-brane superpotentials and Ooguri-Vafa invariants of compact Calabi-Yau threefolds**Supported by NSFC (11075204, 11475178)

We calculate the D-brane superpotentials for two compact Calabi-Yau manifolds X14(1,1,2,3,7) and X8(1,1,1,2,3) which are of non-Fermat type in the type II string theory. By constructing the open mirror symmetry, we also compute the Ooguri-Vafa invariants, which are related to the open Gromov-Witten invariants.

Type II/F-theory superpotentials and Ooguri–Vafa invariants of compact Calabi–Yau threefolds with three deformations

We calculate the D-brane superpotentials for two Calabi–Yau manifolds with three deformations by the generalized hypergeometric GKZ systems, which give rise to the flux superpotentials 𝒲 GVW of the dual F-theory compactification on the relevant Calabi–Yau fourfolds in the weak decoupling limit. We also compute the Ooguri–Vafa invariants from A-model expansion with mirror symmetry, which are related to the open Gromov–Witten invariants.

Off-shell D-brane/F-theory effective superpotentials and Ooguri–Vafa invariants of several compact Calabi–Yau manifolds

According to the open-closed mirror symmetry and generalized Gel'fand–Kapranov–Zelerinski (GKZ) system, we calculate off-shell D-brane/F-theory effective superpotentials of four compact Calabi–Yau (CY) manifolds by integrating special periods of subsystem, and extracting the open Ooguri–Vafa invariants from the expansion of off-shell D-brane superpotentials.

On the arithmetic of D-brane superpotentials: lines and conics on the mirror quintic

Irrational invariants from D-brane superpotentials are pursued on the mirror quintic, systematically according to the degree of a representative curve. Lines are completely understood: the contribution from isolated lines vanishes. All other lines can be deformed holomorphically to the van Geemen lines, whose superpotential is determined via the associated inhomogeneous Picard-Fuchs equation. Substantial progress is made for conics: the families found by Mustat¸ya contain conics reducible to isolated lines, hence they have a vanishing superpotential. The search for all conics invariant under a residual Z2 symmetry reduces to an algebraic problem at the limit of our computational capabilities. The main results are of arithmetic flavor: the extension of the moduli space by the algebraic cycle splits in the large complex structure limit into groups each governed by an algebraic number field. The expansion coefficients ofthe superpotential around large volume remain irrational. The integrality of those coefficients is revealed by a new, arithmetic twist of the di-logarithm: the D-logarithm. There are several options for attempting to explain how these invariants could arise from the A-model perspective. A successful spacetime interpretation will require spaces of BPS states to carry number theoretic structures, such as an action of the Galois group.

The geometry of D-brane superpotentials

The disk partition function of the open topological string computes the spacetime superpotential for D-branes wrapping cycles of a compact Calabi-Yau threefold. We use string duality to show that when appropriately formulated, the problem admits a natural geometrization in terms of a non-compact Calabi-Yau fourfold without D-branes. The duality relates the D-brane superpotential to a flux superpotential on the fourfold. This sheds light on several features of superpotential computations appearing in the literature, in particular on the observation that Calabi-Yau fourfold geometry enters the problem. In one of our examples, we show that the geometry of fourfolds also reproduces the D-brane superpotentials obtained from matrix factorization methods.

Type II/F-theory superpotentials with several deformations and $ \mathcal{N} = 1 $ mirror symmetry

We present a detailed study of D-brane superpotentials depending on several open and closed-string deformations. The relative cohomology group associated with the brane defines a generalized hypergeometric GKZ system which determines the off-shell superpotential and its analytic properties under deformation. Explicit expressions for the N=1 superpotential for families of type II/F-theory compactifications are obtained for a list of multi-parameter examples. Using the Hodge theoretic approach to open-string mirror symmetry, we obtain new predictions for integral disc invariants in the A model instanton expansion. We study the behavior of the brane vacua under extremal transitions between different Calabi-Yau spaces and observe that the web of Calabi-Yau vacua remains connected for a particular class of branes.

D-brane superpotentials: Geometric and worldsheet approaches

From the worldsheet perspective, the superpotential on a D-brane wrapping internal cycles of a Calabi–Yau manifold is given as a generating functional for disk correlation functions. On the other hand, from the geometric point of view, D-brane superpotentials are captured by certain chain integrals. In this work, we explicitly show for branes wrapping internal two-cycles how these two different approaches are related. More specifically, from the worldsheet point of view, D-branes at the Landau–Ginzburg point have a convenient description in terms of matrix factorizations. We use a formula derived by Kapustin and Li to explicitly evaluate disk correlators for families of D2-branes. On the geometry side, we then construct a three-chain whose period gives rise to the effective superpotential and show that the two expressions coincide. Finally, as an explicit example, we choose a particular compact Calabi–Yau hypersurface and compute the effective D2-brane superpotential in different branches of the open moduli space, in both geometric and worldsheet approaches.

Calculations for mirror symmetry with D-branes

We study normal functions capturing D-brane superpotentials on several one- and two-parameter Calabi-Yau hypersurfaces and complete intersections in weighted pro- jective space. We calculate in the B-model and interpret the results using mirror symmetry in the large volume regime, albeit without identifying the precise A-model geometry in all cases. We identify new classes of extensions of Picard-Fuchs equations, as well as a novel type of topology changing phase transition involving quantum D-branes. A 4-d domain wall which is obtained in one region of closed string moduli space from wrapping a four-chain interpolating between two Lagrangian submanifolds is, for other values of the parameters, represented by a disk ending on a single Lagrangian.

D-brane superpotentials and RG flows on the quintic

The behaviour of D2-branes on the quintic under complex structure deformations is analysed by combining Landau-Ginzburg techniques with methods from conformal field theory. It is shown that the boundary renormalisation group flow induced by the bulk deformations is realised as a gradient flow of the effective space time superpotential which is calculated explicitly to all orders in the boundary coupling constant.

CERN LHC signatures of resonantCPviolation in a minimal supersymmetric Higgs sector

We study deformations of Landau-Ginzburg D-branes corresponding to obstructed rational curves on Calabi-Yau threefolds. We determine D-brane moduli spaces and D-brane superpotentials by evaluating higher products up to homotopy in the Landau-Ginzburg orbifold category. For concreteness we work out the details for lines on a perturbed Fermat quintic. In this case we show that our results reproduce the local analytic structure of the Hilbert scheme of curves on the threefold.

String field theory and brane superpotentials

I discuss tree-level amplitudes in cubic topological string field theory, showing that a certain family of gauge conditions leads to an A-infty algebra of tree-level string products which define a potential describing the dynamics of physical states. Upon using results of modern deformation theory, I show that the string moduli space admits two equivalent descriptions, one given in standard Maurer-Cartan fashion and another given in terms of a `homotopy Maurer-Cartan problem', which describes the critical set of the potential. By applying this construction to the topological A and B models, I obtain an intrinsic formulation of `D-brane superpotentials' in terms of string field theory data. This gives a prescription for computing such quantities to all orders, and proves the equivalence of this formulation with the fundamental description in terms of string field moduli. In particular, it clarifies the relation between the Chern-Simons/holomorphic Chern-Simons actions and the superpotential for A/B-type branes.