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1.
2.
This article explores possible embeddings of the Standard Model gauge group and its matter representations into F‐theory. To this end we construct elliptic fibrations with gauge group as suitable restrictions of a ‐fibration with rank‐two Mordell‐Weil group. We analyse the five inequivalent toric enhancements to gauge group along two independent divisors W3 and W2 in the base. For each of the resulting smooth fibrations, the representation spectrum generically consists of a bifundamental , three types of representations and five types of representations (plus conjugates), in addition to charged singlet states. The precise spectrum of zero‐modes in these representations depends on the 3‐form background. We analyse the geometrically realised Yukawa couplings among all these states and find complete agreement with field theoretic expectations based on their U(1) charges. We classify possible identifications of the found representations with the Standard Model field content extended by right‐handed neutrinos and extra singlets. The linear combination of the two abelian gauge group factors orthogonal to hypercharge acts as a selection rule which, depending on the specific model, can forbid dangerous dimension‐four and ‐five proton decay operators.  相似文献   

3.
We discuss the possible realisation in string/M theory of the recently discovered family of four‐dimensional maximal gauged supergravities, and of an analogous family of seven‐dimensional half‐maximal gauged supergravities. We first prove a no‐go theorem that neither class of gaugings can be realised via a compactification that is locally described by ten‐ or eleven‐dimensional supergravity. In the language of Double Field Theory and its M theory analogue, this implies that the section condition must be violated. Introducing the minimal number of additional coordinates possible, we then show that the standard S 3 and S 7 compactifications of ten‐ and eleven‐dimensional supergravity admit a new class of section‐violating generalised frames with a generalised Lie derivative algebra that reproduces the embedding tensor of the and gaugings respectively. The physical meaning, if any, of these constructions is unclear. They highlight a number of the issues that arise when attempting to apply the formalism of Double Field Theory to non‐toroidal backgrounds. Using a naive brane charge quantisation to determine the periodicities of the additional coordinates restricts the gaugings to an infinite discrete set and excludes all the gaugings other than the standard one.  相似文献   

4.
We study asymmetric simple‐current extensions of Gepner models in dimensions with at least eight supercharges in the right‐moving sector. The models obtained in an extensive stochastic computer search belong to a small number of different classes. These classes can be categorized as dimensional reductions, asymmetric orbifolds with , extra gauge enhancement and as coming from the super Higgs‐effect. Models in the latter class are particularly interesting, as they may correspond to non‐geometric flux compactifications.  相似文献   

5.
In this paper we define the analogue of Calabi–Yau geometry for generic , flux backgrounds in type II supergravity and M‐theory. We show that solutions of the Killing spinor equations are in one‐to‐one correspondence with integrable, globally defined structures in generalised geometry. Such “exceptional Calabi–Yau” geometries are determined by two generalised objects that parametrise hyper‐ and vector‐multiplet degrees of freedom and generalise conventional complex, symplectic and hyper‐Kähler geometries. The integrability conditions for both hyper‐ and vector‐multiplet structures are given by the vanishing of moment maps for the “generalised diffeomorphism group” of diffeomorphisms combined with gauge transformations. We give a number of explicit examples and discuss the structure of the moduli spaces of solutions. We then extend our construction to and flux backgrounds preserving eight supercharges, where similar structures appear, and finally discuss the analogous structures in generalised geometry.  相似文献   

6.
We describe this paper as a Sentimental Journey from Hydrodynamics to Supergravity. Beltrami equation in three dimensions that plays a key role in the hydrodynamics of incompressible fluids has an unsuspected relation with minimal supergravity in seven dimensions. We show that just supergravity and no other theory with the same field content but different coefficients in the lagrangian, admits exact two‐brane solutions where Arnold‐Beltrami fluxes in the transverse directions have been switched on. The rich variety of discrete groups that classify the solutions of Beltrami equation, namely the eigenfunctions of the operator on a three‐torus, are by this newly discovered token injected into the brane world. A new quite extensive playing ground opens up for supergravity and for its dual gauge theories in three dimensions, where all classical fields and all quantum composite operators will be assigned to irreducible representations of discrete crystallographic groups Γ.  相似文献   

7.
During inflation, the geometry of spacetime is described by a (quasi‐)de Sitter phase. Inflationary observables are determined by the underlying (softly broken) de Sitter isometry group which acts like a conformal group on : when the fluctuations are on super‐Hubble scales, the correlators of the scalar fields are constrained by conformal invariance. Heavy fields with mass m larger than the Hubble rate H correspond to operators with imaginary dimensions in the dual Euclidean three‐dimensional conformal field theory. By making use of the dS/CFT correspondence we show that, besides the Boltzmann suppression expected from the thermal properties of de Sitter space, the generic effect of heavy fields in the inflationary correlators of the light fields is to introduce power‐law suppressed corrections of the form . This can be seen, for instance, at the level of the four‐point correlator for which we provide the correction due to a massive scalar field exchange.  相似文献   

8.
The scale invariant gravity theory coupled to conformally invariant matter is investigated. We show that in the non‐supersymmetric case the conformally coupled scalars belong to an manifold, while in the supersymmetric case the scalar manifold becomes isomorphic to the Kählerian space =. In both cases when the underlying scale symmetry is preserved the vacuum corresponds to de Sitter space. Once the scale symmetry is broken by quantum effects, a transition to flat space becomes possible. We argue that the scale violating terms are induced by anomalies related to a symmetry. The anomaly is resolved via the gauging of a Peccei‐Quinn axion shift symmetry. The theory describes an inflationary transition from de Sitter to flat Minkowski space, very similar to the Starobinsky inflationary model. The extension to metastable de Sitter superstring vacua is also investigated. The scalar manifold is extended to a much richer manifold, but it contains always as a sub‐manifold. In superstrings the metastability is induced by axions that cure the anomalies in chiral (or even ) supersymmetric vacua via a Green‐Schwarz/Peccei‐Quinn mechanism generalized to four dimensions. We present some typical superstring models and discuss the possible stabilization of the no‐scale modulus.  相似文献   

9.
We study heterotic backgrounds with non‐trivial H‐flux and non‐vanishing expectation values of fermionic bilinears, often referred to as gaugino condensates. The gaugini appear in the low energy action via the gauge‐invariant three‐form bilinear . For Calabi‐Yau compactifications to four dimensions, the gaugino condensate corresponds to an internal three‐form that must be a singlet of the holonomy group. This condition does not hold anymore when an internal H‐flux is turned on and effects are included. In this paper we study flux compactifications to three and four‐dimensions on G‐structure manifolds. We derive the generic conditions for supersymmetric solutions. We use integrability conditions and Lichnerowicz type arguments to derive a set of constraints whose solution, together with supersymmetry, is sufficient for finding backgrounds with gaugino condensate.  相似文献   

10.
In this paper we consider the ‐graded parity generalizing the ordinary (or Z2‐graded ) parity. Using the ‐graded parity operator, we discuss the minimal bosonization of the N=2 SUSY with ‐graded parity. The lowest energy level is shown to be infinitely degenerate. In order to avoid the infinite degeneracy of the ground state we introduce the paraboson algebra to obtain the para‐supersymmetry. Finally, we discuss the hidden SUSY with Z3‐graded parity.  相似文献   

11.
A new class is introduced of M2‐branes solutions of d=11 supergravity that include internal fluxes obeying Englert equation in 7‐dimensions. A simple criterion for the existence of Killing spinors in such backgrounds is established. Englert equation is viewed as the generalization to d=7 of Beltrami equation defined in d=3 and it is treated accordingly. All 2‐brane solutions of minimal d=7 supergracity can be uplifted to d=11 and have supersymmetry. It is shown that the simple group PSL(2, 7) is crystallographic in d=7 having an integral action on the A7 root lattice. By means of this point‐group and of the T7 torus obtained quotiening with the A7 root lattice we were able to construct new M2 branes with Englert fluxes and . In particular we exhibit here an solution depending on 4‐parameters and admitting a large non abelian discrete symmetry, namely . The dual field theories have the same symmetries and have complicated non linear interactions.  相似文献   

12.
We first review the Coset Space Dimensional Reduction (CSDR) programme and present the best model constructed so far based on the , 10‐dimensional E8 gauge theory reduced over the nearly‐Kähler manifold with the additional use of the Wilson flux mechanism. Then we present the corresponding programme in the case that the extra dimensions are considered to be fuzzy coset spaces and the best model that has been constructed in this framework too. In both cases the best model appears to be the trinification GUT .  相似文献   

13.
The standard model has for massless quarks and leptons “miraculously” no triangle anomalies due to the fact that the sum of all possible traces — where and are the generators of one, of two or of three of the groups and U (1) — over the representations of one family of the left handed fermions and anti‐fermions (and separately of the right handed fermions and anti‐fermions), contributing to the triangle currents, is equal to zero. 1 - 4 It is demonstrated in this paper that this cancellation of the standard model triangle anomaly follows straightforwardly if the and are the subgroups of the orthogonal group , as it is in the spin‐charge‐family theory. 5 - 22 We comment on the anomaly cancellation, which works if handedness and charges are related “by hand”.  相似文献   

14.
We derive the analog of the large N Gross‐Taylor holomorphic string expansion for the refinement of q‐deformed Yang‐Mills theory on a compact oriented Riemann surface. The derivation combines Schur‐Weyl duality for quantum groups with the Etingof‐Kirillov theory of generalized quantum characters which are related to Macdonald polynomials. In the unrefined limit we reproduce the chiral expansion of q‐deformed Yang‐Mills theory derived by de Haro, Ramgoolam and Torrielli. In the classical limit , the expansion defines a new β‐deformation of Hurwitz theory wherein the refined partition function is a generating function for certain parameterized Euler characters, which reduce in the unrefined limit to the orbifold Euler characteristics of Hurwitz spaces of holomorphic maps. We discuss the geometrical meaning of our expansions in relation to quantum spectral curves and β‐ensembles of matrix models arising in refined topological string theory.  相似文献   

15.
We compute the Hodge numbers for the quotients of complete intersection Calabi‐Yau three‐folds by groups of orders divisible by 4. We make use of the polynomial deformation method and the counting of invariant Kähler classes. The quotients studied here have been obtained in the automated classification of V. Braun. Although the computer search found the freely acting groups, the Hodge numbers of the quotients were not calculated. The freely acting groups, G, that arise in the classification are either or contain , , or as a subgroup. The Hodge numbers for the quotients for which the group G contains or have been computed previously. This paper deals with the remaining cases, for which or . We also compute the Hodge numbers for 99 of the 166 CICY's which have quotients.  相似文献   

16.
We study ‐dimensional half‐maximal flux backgrounds using exceptional field theory. We define the relevant generalised structures and also find the integrability conditions which give warped half‐maximal MinkowskiD and AdSD vacua. We then show how to obtain consistent truncations of type II / 11‐dimensional SUGRA which break half the supersymmetry. Such truncations can be defined on backgrounds admitting exceptional generalised structures, where , and N is the number of vector multiplets obtained in the lower‐dimensional theory. Our procedure yields the most general embedding tensors satisfying the linear constraint of half‐maximal gauged SUGRA. We use this to prove that all half‐maximal warped AdSD and MinkowskiD vacua of type II / 11‐dimensional SUGRA admit a consistent truncation keeping only the gravitational supermultiplet. We also show to obtain heterotic double field theory from exceptional field theory and comment on the M‐theory / heterotic duality. In five dimensions, we find a new SO(5, N ) double field theory with a ‐dimensional extended space. Its section condition has one solution corresponding to 10‐dimensional supergravity and another yielding six‐dimensional SUGRA.  相似文献   

17.
We review and develop the general properties of algebras focusing on the gauge structure of the associated field theories. Motivated by the homotopy Lie algebra of closed string field theory and the work of Roytenberg and Weinstein describing the Courant bracket in this language we investigate the structure of general gauge invariant perturbative field theories. We sketch such formulations for non‐abelian gauge theories, Einstein gravity, and for double field theory. We find that there is an algebra for the gauge structure and a larger one for the full interacting field theory. Theories where the gauge structure is a strict Lie algebra often require the full algebra for the interacting theory. The analysis suggests that algebras provide a classification of perturbative gauge invariant classical field theories.  相似文献   

18.
We investigate global symmetries for 6D SCFTs and LSTs having a single “unpaired” tensor, that is, a tensor with no associated gauge symmetry. We verify that for every such theory built from F‐theory whose tensor has Dirac self‐pairing equal to −1, the global symmetry algebra is a subalgebra of . This result is new if the F‐theory presentation of the theory involves a one‐parameter family of nodal or cuspidal rational curves (i.e., Kodaira types I1 or II) rather than elliptic curves (Kodaira type I0). For such theories, this condition on the global symmetry algebra appears to fully capture the constraints on coupling these theories to others in the context of multi‐tensor theories. We also study the analogous problem for theories whose tensor has Dirac self‐pairing equal to −2 and find that the global symmetry algebra is a subalgebra of . However, in this case there are additional constraints on F‐theory constructions for coupling these theories to others.  相似文献   

19.
In this work, we expand the hidden ‐Lorentz superalgebra underlying supergravity, reaching a (hidden) Maxwell superalgebra. The latter can be viewed as an extension involving cosmological constant of the superalgebra underlying supergravity in flat spacetime. We write the Maurer‐Cartan equations in this context and we find some interesting extensions of the antisymmetric 3‐form A (3) appearing in the Free Differential Algebra in Minkowski space. The structure of Free Differential Algebras is obtained by considering the zero curvature equations. We write the parametrization of A (3) in terms of 1‐forms and we rend the topological features of its extensions manifest. We interestingly find out that the structure of these extensions, and consequently the structure of the corresponding boundary contribution , strongly depends on the form of the extra fermionic generator appearing in the hidden Maxwell superalgebra. The model we develop in this work is defined in an enlarged superspace with respect to the ordinary one, and the extra bosonic and fermionic 1‐forms required for the closure of the hidden Maxwell superalgebra must be considered as physical fields in this enlarged superspace.  相似文献   

20.
In this paper we address the general problem of including inflationary models exhibiting Starobinsky‐like potentials into (symmetric) supergravities. This is done by gauging suitable abelian isometries of the hypermultiplet sector and then truncating the resulting theory to a single scalar field. By using the characteristic properties of the global symmetry groups of the supergravities we are able to make a general statement on the possible α‐attractor models which can obtained upon truncation. We find that in symmetric models group theoretical constraints restrict the allowed values of the parameter α to be . This confirms and generalizes results recently obtained in the literature. Our analysis heavily relies on the mathematical structure of symmetric supergravities, in particular on the so called c‐map connection between Quaternionic Kähler manifolds starting from Special Kähler ones. A general statement on the possible consistent truncations of the gauged models, leading to Starobinsky‐like potentials, requires the essential help of Tits Satake universality classes. The paper is mathematically self‐contained and aims at presenting the involved mathematical structures to a public not only of physicists but also of mathematicians. To this end the main mathematical structures and the general gauging procedure of supergravities is reviewed in some detail.  相似文献   

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