BPS equations and non-trivial compactifications

The effective D-brane superpotential determines the vacuum structure and is the generating function of the Ooguri-Vafa invariants. It is of great significance in both physics and mathematics. In this work, the superpotential of the system with three D-branes is calculated by mirror symmetry, GKZ-system (hypergeometric system of Gel'fand, Kapranov, and Zelevinski) method, and the type Ⅱ/F-theory duality. On compact Calabi-Yau manifold, the superpotenial and geometric invariants of D-brane system in the three phases are studied. The calculations show that the superpotentials in the three phases are different. This implies that the enhanced gauge symmetry in low energy theory and geometric singularity may lead to the phase transition.

We describe how local toric singularities, including the Toric Lego construction, can be embedded in compact Calabi-Yau manifolds. We study in detail the addition of D-branes, including non-compact flavor branes as typically used in semi-realistic model building. The global geometry provides constraints on allowable local models. As an illustration of our discussion we focus on D3 and D7-branes on (the partially resolved) (dP 0)3 singularity, its embedding in a specific Calabi-Yau manifold as a hypersurface in a toric variety, the related type IIB orientifold compactification, as well as the corresponding F-theory uplift. Our techniques generalize naturally to complete intersections, and to a large class of F-theory backgrounds with singularities.

The symplectic structure of N=2 supergravity and its central extension

Motivated by studies on 4d black holes and q-deformed 2d Yang–Mills theory, and borrowing ideas from compact geometry of the blowing up of affine ADE singularities, we build a class of local Calabi–Yau threefolds (CY 3) extending the local 2-torus model O ( m ) ⊕ O ( − m ) → T 2 considered in [C. Gomez, S. Montanez, A comment on quantum distribution functions and the OSV conjecture, hep-th/0608162] to test OSV conjecture. We first study toric realizations of T 2 and then build a toric representation of X 3 using intersections of local Calabi–Yau threefolds O ( m ) ⊕ O ( − m − 2 ) → P 1 . We develop the 2d N = 2 linear σ-model for this class of toric CY 3s. Then we use these local backgrounds to study partition function of 4d black holes in type IIA string theory and the underlying q-deformed 2d quiver gauge theories. We also make comments on 4d black holes obtained from D-branes wrapping cycles in O ( m ) ⊕ O ( − m − 2 ) → B k with m = ( m 1 , … , m k ) a k-dim integer vector and B k a compact complex one dimension base consisting of the intersection of k 2-spheres S i 2 with generic intersection matrix I i j . We give as well the explicit expression of the q-deformed path integral measure of the partition function of the 2d quiver gauge theory in terms of I i j . A comment on the link between our analysis and the construction of [N. Caporaso, M. Cirafici, L. Griguolo, S. Pasquetti, D. Seminara, R.J. Szabo, Topological strings, two-dimensional Yang–Mills theory and Chern–Simons theory on torus bundles, hep-th/0609129] is also given.

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.

D7-brane moduli are stabilized by worldvolume fluxes, which contribute to the D3-brane tadpole. We calculate this contribution in the Type IIB limit of F-theory compactifications on Calabi-Yau four-folds with a weak Fano base, and are able to prove a no-go theorem for vast swathes of the landscape of compactifications. When the genus of the curve dual to the D7 worldvolume fluxes is fixed and the number of moduli grows, we find that the D3 charge sourced by the fluxes grows faster than 7/16 of the number of moduli, which supports the Tadpole Conjecture of ref. [1]. Our lower bound for the induced D3 charge decreases when the genus of the curves dual to the stabilizing fluxes increase, and does not allow to rule out a sliver of flux configurations dual to high-genus high-degree curves. However, we argue that most of these fluxes have very high curvature, which is likely to be above the string scale except on extremely large (and experimentally ruled out) compactification manifolds.

In the classification of Riemannian holonomy groups, the exceptional holonomy groups are G 2 in 7 dimensions, and Spin(7) in 8 dimensions.We outline the construction of the first known examples of compact 7-and 8-manifolds with holonomy G 2 and Spin(7).In the case of G 2 , we first choose a finite group of automorphisms of the torus T 7 and a flat -invariant G 2 -structure on T 7 , so that T 7 / is an orbifold.Then we resolve the singularities of T 7 / to get a compact 7-manifold M .Finally we use analysis, and an understanding of Calabi-Yau metrics, to construct a family of metrics with holonomy G 2 on M , which converge to the singular metric on T 7 /.

The mixed Yukawa couplings of the 27 and 27* superfields to other superfields are studied in a model stemming from a superstring model compactified on a generic Calabi-Yau manifold. The coupling to the standard E6-algebra-valued flux-loop operator, as well as an analogous E6-invariant operator, is argued to be related to certain Yukawa couplings. A vacuum expectation value of the latter operator is shown to give masses to all E6-singlet scalar superfields, except for those stemming from the supergravity multiplet.

We compactify four-dimensional N=1 gauged supergravity theories on a circle including fluxes for shift-symmetric scalars. Four-dimensional Taub-NUT gravitational instantons universally correct the three-dimensional superpotential in the absence of fluxes. In the presence of fluxes these Taub-NUT instanton contributions are no longer gauge-invariant. Invariance can be restored by gauge instantons on top of Taub-NUT instantons. We establish the embedding of this scenario into M-theory. Circle fluxes and gaugings arise from a restricted class of M-theory four-form fluxes on a resolved Calabi-Yau fourfold. The M5-brane on the base of the elliptic fourfold dualizes into the universal Taub-NUT instanton. In the presence of fluxes this M5-brane is anomalous. We argue that anomaly free contributions arise from involved M5-brane geometries dual to gauge-instantons on top of Taub-NUT instantons. Adding a four-dimensional superpotential to the gravitational instanton corrections leads to three-dimensional Anti-de Sitter vacua at stabilized compactification radius. We comment on the possibility to uplift these M-theory vacua, and to tunnel to four-dimensional F-theory vacua.

It is found the most general local form of the 11-dimensional supergravity backgrounds which, by reduction along one isometry, gives rise to IIA supergravity solutions with a RR field and a nontrivial dilaton, and for which the condition ${F}^{(1,1)}=0$ holds.This condition is stronger than the usual condition ${F}^{ab}{J}_{ab}=0$, required by supersymmetry. It is shown that these D6 wrapped backgrounds arise from the direct sum of the flat Minkowski metric with certain ${G}_{2}$ holonomy metrics admitting an $U(1)$ action, with a local form found by Apostolov and Salamon. Indeed, the strong supersymmetry condition is equivalent to the statement that there is a new isometry on the ${G}_{2}$ manifold, which commutes with the old one; therefore these metrics are inherently toric. An example that is asymptotically Calabi-Yau metric is presented. We find other ${G}_{2}$ metrics which give rise to half-flat $SU(3)$ structures. All these examples possess an $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ isometry subgroup. Supergravity solutions without fluxes corresponding to these ${G}_{2}$ metrics are constructed. The presence of a ${T}^{3}$ subgroup of isometries permits to apply the $\ensuremath{\gamma}$-deformation technique in order to generate new supergravity solutions with fluxes.

The complete Type IIB supergravity solutions with 16 supersymmetries are obtained on the manifold AdS4 ? S2 ? S2 ? ? with SO(2, 3) ? SO(3) ? SO(3) symmetry in terms of two holomorphic functions on a Riemann surface ?, which generally has a boundary. This is achieved by reducing the BPS equations using the above symmetry requirements, proving that all solutions of the BPS equations solve the full Type IIB supergravity field equations, mapping the BPS equations onto a new integrable system akin to the Liouville and Sine-Gordon theories, and mapping this integrable system to a linear equation which can be solved exactly. Amongst the infinite class of solutions, a non-singular Janus solution is identified which provides the AdS/CFT dual of the maximally supersymmetric Yang-Mills interface theory discovered recently. The construction of general classes of globally non-singular solutions, including fully back-reacted AdS5 ? S5 and supersymmetric Janus doped with D5 and/or NS5 branes, is deferred to a companion paper [1].

Exactly Solvable String Compactifications on Manifolds of Su(N) Holonomy

In this short note we are concerned with the Kähler-Einstein metrics near cone type log canonical singularities. By two different approaches, we construct a Kähler-Einstein metric with negative scalar curvature in a neighborhood of the cone over a Calabi-Yau manifold which is complete towards the vertex. This provides a local model for the further study of the global Kähler-Einstein metrics on singular varieties. In the first approach, we show that the singularity is uniformized by a complex ball and hence the induced metric from the Bergman metric of the ball is the desired one. In the second approach, we obtain a Kähler-Einstein metric with the desired properties by Calabi Ansatz. At last, we show that the obtained metrics are the same.

We study the presence of discrete flavor symmetries in D-brane models of particle physics. By analyzing the compact extra dimensions of these models one can determine when such symmetries exist both in the context of intersecting and magnetized D-brane constructions. Our approach allows to distinguish between approximate and exact discrete symmetries, and it can be applied to compactification manifolds with continuous isometries or to manifolds that only contain discrete isometries, like Calabi-Yau three-folds. We analyze in detail the class of rigid D-branes models based on a Z_2 x Z'_2 toroidal orientifold, for which the flavor symmetry group is either the dihedral group D_4 or tensor products of it. We construct explicit Pati-Salam examples in which families transform in non-Abelian representations of the flavor symmetry group, constraining Yukawa couplings beyond the effect of massive U(1) D-brane symmetries.

We study the modular symmetry in heterotic string theory on Calabi-Yau threefolds. In particular, we examine whether moduli-dependent holomorphic Yukawa couplings are described by modular forms in the context of heterotic string theory with standard embedding. We find that SL(2, ℤ) modular symmetry emerges in asymptotic regions of the Calabi-Yau moduli space. The instanton-corrected holomorphic Yukawa couplings are then given by modular forms under SL(2, ℤ) or its congruence subgroups such as Γ0(3) and Γ0(4). In addition to the modular symmetry, it turns out that another coupling selection rule controls the structure of holomorphic Yukawa couplings. Furthermore, the coexistence of both the positive and negative modular weights for matter fields leads to a hierarchical structure of matter field Kähler metric. Thus, these holomorphic modular forms and the matter field Kähler metric play an important role in realizing a hierarchical structure of physical Yukawa couplings.