Gauge Bundle Research Articles

Family unification via quasi-Nambu–Goldstone fermions in string theory

Some of the supersymmetric nonlinear sigma models on exceptional groups (E7 or E8) are known to yield almost minimal necessary content of matter fields for phenomenological model building, including three chiral families and a Higgs multiplet. We explore a possible realization of such a family unification in heterotic string theory, where the spontaneously broken symmetry that we focus on is the one associated with the change of global orientation of the spin bundle embedded in the gauge bundle, which cannot be ignored in the presence of a magnetic source of the B field. We show in a simple model that it indeed gives rise to chiral matter fields on the defect similarly to the domain-wall fermions, and compare the chiral zero mode spectrum with the sigma model. The setup is reduced to a system similar to that postulated in the orbifold GUT or the grand gauge-Higgs unification model, where the origin of Z2 identification is accounted for as arising from the bolt of the Atiyah-Hitchin manifold.

Twisted Differential String and Fivebrane Structures

In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structures and Spin-lifting gerbes and their differential refinement to smooth Spin-connections. We show that all these structures can be encoded in terms of nonabelian cohomology, twisted nonabelian cohomology, and differential twisted nonabelian cohomology, extending the differential generalized abelian cohomology as developed by Hopkins and Singer and shown by Freed to formalize the global description of anomaly cancellation problems in higher gauge theories arising in string theory. We demonstrate that the Green-Schwarz mechanism for the H_3-field, as well as its magnetic dual version for the H_7-field define cocycles in differential twisted nonabelian cohomology that may be called, respectively, differential twisted Spin(n)-, String(n)- and Fivebrane(n)-structures on target space, where the twist in each case is provided by the obstruction to lifting the classifying map of the gauge bundle through a higher connected cover of U(n) or O(n). We show that the twisted Bianchi identities in string theory can be captured by the (nonabelian) L-infinity-algebra valued differential form data provided by the differential refinements of these twisted cocycles.

Heterotic-type II duality in the hypermultiplet sector

We revisit the duality between heterotic string theory compactified on K3 x T^2 and type IIA compactified on a Calabi-Yau threefold X in the hypermultiplet sector. We derive an explicit map between the field variables of the respective moduli spaces at the level of the classical effective actions. We determine the parametrization of the K3 moduli space consistent with the Ferrara-Sabharwal form. From the expression of the holomorphic prepotential we are led to conjecture that both X and its mirror must be K3 fibrations in order for the type IIA theory to have an heterotic dual. We then focus on the region of the moduli space where the metric is expressed in terms of a prepotential on both sides of the duality. Applying the duality we derive the heterotic hypermultiplet metric for a gauge bundle which is reduced to 24 point-like instantons. This result is confirmed by using the duality between the heterotic theory on T^3 and M-theory on K3. We finally study the hyper-Kaehler metric on the moduli space of an SU(2) bundle on K3.

The MSSM spectrum from (0,2)-deformations of the heterotic standard embedding

We construct supersymmetric compactifications of E 8×E 8 heterotic string theory which realise exactly the massless spectrum of the Minimal Supersymmetric Standard Model (MSSM) at low energies. The starting point is the standard embedding on a Calabi-Yau threefold which has Hodge numbers (h 1,1, h 2,1) = (1, 4) and fundamental group $ {\mathbb{Z}_{{12}}} $ , which gives an E 6 grand unified theory with three net chiral generations. The gauge sym- metry is then broken to that of the standard model by a combination of discrete Wilson lines and continuous deformation of the gauge bundle. On eight distinct branches of the moduli space, we find stable bundles with appropriate cohomology groups to give exactly the massless spectrum of the MSSM.

6D effective action of heterotic compactification on K3 with nontrivial gauge bundles

We compute the six-dimensional effective action of the heterotic string compactified on K3 for the standard embedding and for a class of backgrounds with line bundles and appropriate Yang-Mills fluxes. We compute the couplings of the charged scalars and the bundle moduli as functions of the geometrical K3 moduli from a Kaluza-Klein analysis. We derive the D-term potential and show that in the flux backgrounds U (1) vector multiplets become massive by a Stückelberg mechanism.

Recent advances in F‐theory compactifications

AbstractWe review recent progress in the construction of F‐theory compactifications on elliptically fibered Calabi‐Yau 4‐folds. Of special importance is the resolution of the singularities in complex codimension one, two and three. These encode 7‐branes along holomorphic divisors carrying non‐abelian gauge groups, localised matter at the intersection of pairs of 7‐branes and Yukawa interactions, respectively. Toric methods admit the explicit construction of a large class of resolved 4‐folds. In particular the matter surfaces and interaction points can be analysed in great detail via resolution by blow‐up. Engineering chiral matter spectra requires non‐trivial gauge bundles described as certain types of G4‐fluxes. The recent construction of such fluxes paves the way for globally defined 3‐generation models e.g. of SU(5) GUT type, possibly including extra U(1) factors.

Classifying A-field and B-field configurations in the presence of D-branes. Part II: Stacks of D-branes

In the paper of Bonora et al. (2008) [3] we have shown, in the context of type II superstring theory, the classification of the allowed B-field and A-field configurations in the presence of anomaly-free D-branes, the mathematical framework being provided by the geometry of gerbes. Here we complete the discussion considering in detail the case of a stack of D-branes, carrying a non-abelian gauge theory, which was just sketched in Bonora et al. (2008) [3]. In this case we have to mix the geometry of abelian gerbes, describing the B-field, with the one of higher-rank bundles, ordinary or twisted. We describe in detail the various cases that arise according to such a classification, as we did for a single D-brane, showing under which hypotheses the A-field turns out to be a connection on a canonical gauge bundle. We also generalize to the non-abelian setting the discussion about “gauge bundles with non-integral Chern classes”, relating them to twisted bundles with connection. Finally, we analyze the geometrical nature of the Wilson loop for each kind of gauge theory on a D-brane or stack of D-branes.

The Atiyah class and complex structure stabilization in heterotic Calabi-Yau compactifications

Holomorphic gauge fields in N=1 supersymmetric heterotic compactifications can constrain the complex structure moduli of a Calabi-Yau manifold. In this paper, the tools necessary to use holomorphic bundles as a mechanism for moduli stabilization are systematically developed. We review the requisite deformation theory -- including the Atiyah class, which determines the deformations of the complex structure for which the gauge bundle becomes non-holomorphic and, hence, non-supersymmetric. In addition, two equivalent approaches to this mechanism of moduli stabilization are presented. The first is an efficient computational algorithm for determining the supersymmetric moduli space, while the second is an F-term potential in the four-dimensional theory associated with vector bundle holomorphy. These three methods are proven to be rigorously equivalent. We present explicit examples in which large numbers of complex structure moduli are stabilized. Finally, higher-order corrections to the moduli space are discussed.

Stabilizing all geometric moduli in heterotic Calabi-Yau vacua

We propose a scenario to stabilize all geometric moduli - that is, the complex structure, Kahler moduli and the dilaton - in smooth heterotic Calabi-Yau compactifications without Neveu-Schwarz three-form flux. This is accomplished using the gauge bundle required in any heterotic compactification, whose perturbative effects on the moduli are combined with non-perturbative corrections. We argue that, for appropriate gauge bundles, all complex structure and a large number of other moduli can be perturbatively stabilized - in the most restrictive case, leaving only one combination of Kahler moduli and the dilaton as a flat direction. At this stage, the remaining moduli space consists of Minkowski vacua. That is, the perturbative superpotential vanishes in the vacuum without the necessity to fine-tune flux. Finally, we incorporate non-perturbative effects such as gaugino condensation and/or instantons. These are strongly constrained by the anomalous U(1) symmetries which arise from the required bundle constructions. We present a specific example, with a consistent choice of non-perturbative effects, where all remaining flat directions are stabilized in an AdS vacuum.

Heterotic orbifold resolutions as (2,0) gauged linear sigma models

AbstractIn this work we attempt to bridge the gap between heterotic orbifold models and Calabi‐Yau compactifications using gauged linear sigma models (GLSMs) with (2,0) worldsheet supersymmetry. We associate a specific GLSM to a heterotic orbifold model with twisted states that have non‐vanishing vacuum expectation values (VEVs): The charges of the GLSM superfields are essentially determined by the shifted momenta of these states. When a twisted state contains an oscillator excitation, a fermionic gauging is introduced on the worldsheet, inducing a non‐Abelian gauge bundle, e.g. the standard embedding. However, irrespectively of whether the twisted states contain oscillators or not, they can be interpreted as blow‐up modes, as their VEVs are correlated with sizes of exceptional cycles in the resolved geometry. We show that the GLSM anomaly cancellation conditions ensure that the Bianchi identities are fulfilled for all possible triangulations of a resolution. By considering marginal deformations of a GLSM in the large volume limit we are able to directly determine its effective four dimensional spectrum. In the cases considered the spectra coincide with those computed via index theorems.

Topological Geon Black Holes in Einstein-Yang-Mills Theory

We construct topological geon quotients of two families of Einstein-Yang-Mills black holes. For Kuenzle's static, spherically symmetric SU(n) black holes with n>2, a geon quotient exists but generically requires promoting charge conjugation into a gauge symmetry. For Kleihaus and Kunz's static, axially symmetric SU(2) black holes a geon quotient exists without gauging charge conjugation, and the parity of the gauge field winding number determines whether the geon gauge bundle is trivial. The geon's gauge bundle structure is expected to have an imprint in the Hawking-Unruh effect for quantum fields that couple to the background gauge field.

Stabilizing the complex structure in heterotic Calabi-Yau vacua

In this paper, we show that the presence of gauge fields in heterotic Calabi-Yau compacitifications causes the stabilisation of some, or all, of the complex structure moduli of the Calabi-Yau manifold while maintaining a Minkowski vacuum. Certain deformations of the Calabi-Yau complex structure, with all other moduli held fixed, can lead to the gauge bundle becoming non-holomorphic and, hence, non-supersymmetric. This leads to an F-term potential which stabilizes the corresponding complex structure moduli. We use 10- and 4-dimensional field theory arguments as well as a derivation based purely on algebraic geometry to show that this picture is indeed correct. An explicit example is presented in which a large subset of complex structure moduli is fixed. We demonstrate that this type of theory can serve as the hidden sector in heterotic vacua and can co-exist with realistic particle physics.

Transitions in the web of heterotic vacua

AbstractWe analyze transitions between heterotic vacua with distinct gauge bundles using two complementary methods – the effective four‐dimensional field theory and the corresponding geometry. From the viewpoint of effective field theory, such transitions occur between flat directions of the potential energy associated with heterotic stability walls. Geometrically, this branch structure corresponds to smooth deformations of the gauge bundle coupled to the chamber structure of Kähler moduli space. We demonstrate how such transitions can change important properties of the effective theory, including the gauge symmetry and the massless spectrum. Geometrically, this study is divided into deformations of the vector bundle which preserve the rank of the gauge bundle and those which change the rank. In the latter case, our results provide explicit solutions to a class of Li‐Yau type deformation problems. Finally, we use the framework of stability walls and their effective theory to study Donaldson‐Thomas invariants on Calabi‐Yau threefolds.

Product preserving gauge bundle functors on all principal bundle homomorphisms

Product preserving gauge bundle functors on all principal bundle homomorphisms

Towards the MSSM from F-theory

We study the MSSM in F-theory. Its group is the commutant to a structure group SU ( 5 ) × U ( 1 ) Y of a gauge bundle in E 8 . The spectrum contains three generations of quarks and leptons plus vectorlike electroweak and colored Higgses. The minimal MSSM Yukawa couplings with matter parity is obtained at the renormalizable level.

Finite energy shifts inSU(n)supersymmetric Yang-Mills theory onT3×Rat weak coupling

We consider a semi-classical treatment, in the regime of weak gauge coupling, of supersymmetric Yang-Mills theory in a space-time of the form T^3xR with SU(n)/Z_n gauge group and a non-trivial gauge bundle. More specifically, we consider the theories obtained as power series expansions around a certain class of normalizable vacua of the classical theory, corresponding to isolated points in the moduli space of flat connections, and the perturbative corrections to the free energy eigenstates and eigenvalues in the weakly interacting theory. The perturbation theory construction of the interacting Hilbert space is complicated by the divergence of the norm of the interacting states. Consequently, the free and interacting Hilbert furnish unitarily inequivalent representation of the algebra of creation and annihilation operators of the quantum theory. We discuss a consistent redefinition of the Hilbert space norm to obtain the interacting Hilbert space and the properties of the interacting representation. In particular, we consider the lowest non-vanishing corrections to the free energy spectrum and discuss the crucial importance of supersymmetry for these corrections to be finite.

The geometric phase and the spin-statistics relation

The exchange phase for two spins is studied here from the point of view of the quantization of a fermion in the framework of Nelson’s stochastic mechanics. This introduces a direction vector attached to a space–time point depicting the spin degrees of freedom. In this formalism, a fermion appears as a scalar particle attached with a magnetic-flux quantum, and a quantum spin can be described in terms of an SU(2) gauge bundle. This helps us to recast the Berry–Robbins formalism where the exchange phase appears as an unfamiliar geometric phase arising out of the ‘exchange rotation’ in a transported spin basis in terms of gauge currents. However, for polarized fermions, the exchange phase is found to be given by the Berry phase.

Flux-induced isometry gauging in heterotic strings

We study the effect of flux-induced isometry gauging of the scalar manifold in N = 2 heterotic string compactification with gauge fluxes. We show that a vanishing theorem by Witten provides the protection mechanism. The other ungauged isometries in hyper-moduli space could also be protected, depending on the gauge bundle structure. We also discuss the related issue in IIB setting.

Automorphic properties of (2, 0) theory on T 6

We consider ADE-type (2, 0) theory on a family of flat six-tori endowed with flat Sp(4) connections coupled to the R-symmetry. Our main objects of interest are the components of the `partition vector' of the theory. These constitute an element of a certain finite dimensional vector space, carrying an irreducible representation of a discrete Heisenberg group related to the 't Hooft fluxes of the theory. Covariance under the SL_6(Z) mapping class group of a six-torus amounts to a certain automorphic transformation law for the partition vector, which we derive. Because of the absence of a Lagrangian formulation of (2, 0) theory, this transformation property is not manifest, and gives useful non-trivial constraints on the partition vector. As an application, we derive a shifted quantization law for the spatial momentum of (2, 0) theory on a space-time of the form R x T5. This quantization law is in agreement with an earlier result based on the relationship between (2, 0) theory and maximally supersymmetric Yang-Mills theory together with certain geometric facts about gauge bundles.

Heterotic resolved conifolds with torsion, from supergravity to CFT

We obtain a family of heterotic supergravity backgrounds describing non-Kahler warped conifolds with three-form flux and an Abelian gauge bundle, preserving N=1 supersymmetry in four dimensions. At large distance from the singularity the usual Ricci-flat conifold is recovered. By performing a Z_2 orbifold of the T^{1,1} base, the conifold singularity can be blown-up to a four-cycle, leading to a completely smooth geometry. Remarkably, the throat regions of the solutions, which can be isolated from the asymptotic Ricci-flat geometry using a double-scaling limit, possess a worldsheet CFT description in terms of heterotic cosets whose target space is the warped resolved orbifoldized conifold. Thus this construction provides exact solutions of the modified Bianchi identity. By solving algebraically these CFTs we compute the exact tree-level heterotic string spectrum and describe worldsheet non-perturbative effects. The holographic dual of these solutions, in particular their confining behavior, and the embedding of these fluxed singularities into heterotic compactifications with torsion are also discussed.