Enhanced Gauge Group Research Articles

Eight-dimensional non-geometric heterotic strings and enhanced gauge groups

Eight-dimensional non-geometric heterotic strings and enhanced gauge groups

Symmetry enhancements in 7d heterotic strings

We use a moduli space exploration algorithm to produce a complete list of maximally enhanced gauge groups that are realized in the heterotic string in 7d, encompassing the usual Narain component, and five other components with rank reduction realized via nontrivial holonomy triples. Using lattice embedding techniques we find an explicit match with the mechanism of singularity freezing in M-theory on K3. The complete global data for each gauge group is explicitly given.

Unbroken $E7 \times E7$ nongeometric heterotic strings, stable degenerations and enhanced gauge groups in F-theory duals

Eight-dimensional non-geometric heterotic strings with gauge algebra $\mathfrak{e}_8\mathfrak{e}_7$ were constructed by Malmendier and Morrison as heterotic duals of F-theory on K3 surfaces with $\Lambda^{1,1}\oplus E_8\oplus E_7$ lattice polarization. Clingher, Malmendier and Shaska extended these constructions to eight-dimensional non-geometric heterotic strings with gauge algebra $\mathfrak{e}_7\mathfrak{e}_7$ as heterotic duals of F-theory on $\Lambda^{1,1}\oplus E_7\oplus E_7$ lattice polarized K3 surfaces. In this study, we analyze the points in the moduli of non-geometric heterotic strings with gauge algebra $\mathfrak{e}_7\mathfrak{e}_7$, at which the non-Abelian gauge groups on the F-theory side are maximally enhanced. The gauge groups on the heterotic side do not allow for the perturbative interpretation at these points. We show that these theories can be described as deformations of the stable degenerations, as a result of coincident 7-branes on the F-theory side. From the heterotic viewpoint, this effect corresponds to the insertion of 5-branes. These effects can be used to understand nonperturbative aspects of nongeometric heterotic strings. Additionally, we build a family of elliptic Calabi-Yau 3-folds by fibering elliptic K3 surfaces, which belong to the F-theory side of the moduli of non-geometric heterotic strings with gauge algebra $\mathfrak{e}_7\mathfrak{e}_7$, over $\mathbb{P}^1$. We find that highly enhanced gauge symmetries arise on F-theory on the built elliptic Calabi-Yau 3-folds.

Exploring the landscape of heterotic strings on Td

Compactifications of the heterotic string on Td are the simplest, yet rich enough playgrounds to uncover swampland ideas: the U(1)d+16 left-moving gauge symmetry gets enhanced at special points in moduli space only to certain groups. We state criteria, based on lattice embedding techniques, to establish whether a gauge group is realized or not. For generic d, we further show how to obtain the moduli that lead to a given gauge group by modifying the method of deleting nodes in the extended Dynkin diagram of the Narain lattice II1,17. More general algorithms to explore the moduli space are also developed. For d = 1 and 2 we list all the maximally enhanced gauge groups, moduli, and other relevant information about the embedding in IId,d+16. In agreement with the duality between heterotic on T2 and F-theory on K3, all possible gauge groups on T2 match all possible ADE types of singular fibers of elliptic K3 surfaces. We also present a simple method to transform the moduli under the duality group, and we build the map that relates the charge lattices and moduli of the compactification of the E8 × E8 and Spin(32)/ℤ2 heterotic theories.

Nongeometric heterotic strings and dual F-theory with enhanced gauge groups

Eight-dimensional nongeometric heterotic strings were constructed as duals of F-theory on Λ1,1 ⊕ E8 ⊕ E7 lattice polarized K3 surfaces by Malmendier and Morrison. We study the structure of the moduli space of this construction. There are special points in this space at which the ranks of the non-Abelian gauge groups on the 7-branes in F-theory are enhanced to 18. We demonstrate that the enhanced rank-18 non-Abelian gauge groups arise as a consequence of the coincident 7-branes, which deform stable degenerations on the F-theory side. This observation suggests that the non-geometric heterotic strings include nonperturbative effects of the coincident 7-branes on the F-theory side. The gauge groups that arise at these special points in the moduli space do not allow for perturbative descriptions on the heterotic side.We also construct a family of elliptically fibered Calabi-Yau 3-folds by fibering K3 surfaces with enhanced singularities over ℙ1. Highly enhanced gauge groups arise in F-theory compactifications on the resulting Calabi-Yau 3-folds.

A new twist on heterotic string compactifications

A rich pattern of gauge symmetries is found in the moduli space of heterotic string toroidal compactifications, at fixed points of the T-duality transformations. We analyze this pattern for generic tori, and scrutinize in full detail compactifications on a circle, where we find all the maximal gauge symmetry groups and the points where they arise. We present figures of two-dimensional slices of the 17-dimensional moduli space of Wilson lines and circle radii, showing the rich pattern of points and curves of symmetry enhancement. We then study the target space realization of the duality symmetry. Although the global continuous duality symmetries of dimensionally reduced heterotic supergravity are completely broken by the structure constants of the maximally enhanced gauge groups, the low energy effective action can be written in a manifestly duality covariant form using heterotic double field theory. As a byproduct, we show that a unique deformation of the generalized diffeomorphisms accounts for both SO(32) and E8 × E8 heterotic effective field theories, which can thus be considered two different backgrounds of the same double field theory even before compactification. Finally we discuss the spontaneous gauge symmetry breaking and Higgs mechanism that occurs when slightly perturbing the background fields, both from the string and the field theory perspectives.

The bosonic string on string-size tori from double field theory

We construct the effective action for toroidal compactifications of bosonic string theory from generalized Scherk-Schwarz reductions of double field theory. The enhanced gauge symmetry arising at special points in moduli space is incorporated into this framework by promoting the O(k, k) duality group of k-tori compactifications to O(n, n), n being the dimension of the enhanced gauge group, which allows to account for the full massless sector of the theory. We show that the effective action reproduces the right masses of scalar and vector fields when moving sligthly away from the points of maximal symmetry enhancement. The neighborhood of the enhancement points in moduli space can be neatly explored by spontaneous symmetry breaking. We generically discuss toroidal com-pactifications of arbitrary dimensions and maximally enhanced gauge groups, and then inspect more closely the example of T2 at the SU(3)L × SU(3)R point, which is the simplest setup containing all the non-trivialities of the generic case. We show that the entire moduli space can be described in a unified way by considering compactifications on higher dimensional tori.

Matter from Toric Geometry and its Search at the LHC

Toric geometry is applied for construction the enhanced gauge groups in F-theory compactified on elliptic Calabi-Yau fourfolds. The Hodge numbers calculated from the polyhedra for the chain H = SU (1), ... ,SU (5), SO(10), E6, E7 determine the number of tensor multiplets, vector multiplets and hypermultiplets of soli- tonic states that appear from singularities of elliptic fibration. Due to duality between the compactification of E8timesE8 heterotic string and the type IIA string compactification on a Calabi-Yau manifold there is a natural sequence of E-group embeddings which gives the matter content of Minimal Supersymmetric Standard Model and the possibility of searching for supersymmetry at the LHC.

Enhanced gauge groups inN=4topological amplitudes and Lorentzian Borcherds algebras

We continue our study of algebraic properties of $\mathcal{N}=4$ topological amplitudes in heterotic string theory compactified on ${\mathbb{T}}^{2}$, initiated in arXiv:1102.1821. In this work we evaluate a particular one-loop amplitude for any enhanced gauge group $\mathfrak{h}\ensuremath{\subset}{\mathfrak{e}}_{8}\ensuremath{\bigoplus}{\mathfrak{e}}_{8}$, i.e. for arbitrary choice of Wilson line moduli. We show that a certain analytic part of the result has an infinite product representation, where the product is taken over the positive roots of a Lorentzian Kac-Moody algebra ${\mathfrak{g}}^{++}$. The latter is obtained through double extension of the complement $\mathfrak{g}=({\mathfrak{e}}_{8}\ensuremath{\bigoplus}{\mathfrak{e}}_{8})/\mathfrak{h}$. The infinite product is automorphic with respect to a finite index subgroup of the full $T$-duality group $SO(2,18;\mathbb{Z})$ and, through the philosophy of Borcherds-Gritsenko-Nikulin, this defines the denominator formula of a generalized Kac-Moody algebra $\mathcal{G}({\mathfrak{g}}^{++})$, which is an 'automorphic correction' of ${\mathfrak{g}}^{++}$. We explicitly give the root multiplicities of $\mathcal{G}({\mathfrak{g}}^{++})$ for a number of examples.

D-brane deconstructions in IIB orientifolds

With model building applications in mind, we collect and develop basic techniques to analyze the landscape of D7-branes in type IIB compact Calabi-Yau orientifolds, in three different pictures: F-theory, the D7 worldvolume theory and D9-anti-D9 tachyon condensation. A significant complication is that consistent D7-branes in the presence of O7− planes are generically singular, with singularities locally modeled by the Whitney Umbrella. This invalidates the standard formulae for charges, moduli space and flux lattice dimensions. We infer the correct formulae by comparison to F-theory and derive them independently and more generally from the tachyon picture, and relate these numbers to the closed string massless spectrum of the orientifold compactification in an interesting way. We furthermore give concrete recipes to explicitly and systematically construct nontrivial D-brane worldvolume flux vacua in arbitrary Calabi-Yau orientifolds, illustrate how to read off D-brane flux content, enhanced gauge groups and charged matter spectra from tachyon matrices, and demonstrate how brane recombination in general leads to flux creation, as required by charge conservation and by equivalence of geometric and gauge theory moduli spaces.

M theory fivebrane and confining phase of N = 1 SO ( Nc) gauge theories

The moduli space of vacua for the confining phase of N = 1 SO ( N c ) supersymmetric gauge theories in four dimensions is analyzed by studying the M theory fivebrane. The type IIA brane configuration consists of a single NS5 brane, multiple copies of NS'5 branes, D4 branes between them, and D6 branes intersecting D4 branes. We construct M theory fivebrane configuration corresponding to the superpotential perturbation where the power of adjoint field is connected to the number of NS'5 branes. At a singular point in the moduli space where mutually local dyons become massless, the quadratic degeneracy of the N = 2 SO ( N c ) hyperelliptic curve determines whether this singular point gives a N = 1 vacua or not. The comparison of the meson vevs in M theory fivebrane configuration with field theory results turns out that the effective superpotential by the “integrating in” method with enhanced gauge group SU (2) is exact.

Confining phase of N=1 Sp(Nc) gauge theories via M theory fivebrane

The moduli space of vacua for the confining phase of N=1 $Sp(N_c)$ supersymmetric gauge theories in four dimensions is studied by M theory fivebrane. We construct M theory fivebrane configuration corresponding to the perturbation of superpotential in which the power of adjoint field is related to the number of NS'5 branes in type IIA brane configuration. We interpret the dyon vacuum expectation values in field theory results as the brane geometry and the comparison with meson vevs will turn out that the low energy effective superpotential with enhanced gauge group SU(2) is exact.

Compactifications of F-theory on Calabi-Yau threefolds at constant coupling

Generalizing the work of Sen, we analyze special points in the moduli space of the compactification of the F-theory on elliptically fibered Calabi-Yau threefolds where the coupling remains constant. These contain points where they can be realized as orbifolds of six torus $T^6$ by $Z_m \times Z_n (m, n=2, 3, 4, 6)$. At various types of intersection points of singularities, we find that the enhancement of gauge symmetries arises from the intersection of two kinds of singularities. We also argue that when we take the Hirzebruch surface as a base for the Calabi- Yau threefold, the condition for constant coupling corresponds to the case where the point like instantons coalesce, giving rise to enhanced gauge group of $Sp(k)$.

Monopole condensation and confining phase of N = 1 gauge theories via M-theory fivebrane

The fivebrane of M-theory is used in order to study the moduli space of vacua of confining phase N = 1 supersymmetric gauge theories in four dimensions. The supersymmetric vacua correspond to the condensation of massless monopoles and confinement of photons. The monopole and meson vacuum expectation values are computed using the fivebrane configuration. The comparison of the fivebrane computation and the field theory analysis shows that at vacua with a classically enhanced gauge group SU( r) the effective superpotential obtained by the “integrating in” method is exact for r = 2 but is not exact for r > 2. The fivebrane configuration corresponding to N = 1 gauge theories with Landau-Ginzburg type superpotentials is studied. N = 1 non-trivial fixed points are analyzed using the brane geometry.

Toric geometry and enhanced gauge symmetry of F-theory/heterotic vacua

We study F-theory compactified on elliptic Calabi-Yau threefolds that are realized as hypersurfaces in toric varieties. The enhanced gauge group as well as the number of massless tensor multiplets has a very simple description in terms of toric geometry. We find a large number of examples where the gauge group is not a subgroup of E 8 × E 8, but rather, is much bigger (with rank as high as 296). The largest of these groups is the group recently found by Aspinwall and Gross. Our algorithm can also be applied to elliptic fourfolds, for which the groups can become very large indeed (with rank as high as 121328). We present the gauge content for two of the fourfolds recently studied by Klemm et al.

Symmetries of 2-D gravity

Two-dimensional gravity in the light-cone gauge was shown to exhibit an underlying sl(2, R) current algebra. It is the purpose of this note to offer a possible explanation about the origin of this important algebra. The essential point is that two-dimensional gravity is governed by a topological field theory. The gauge group is sl(2, R) and it is this enhanced gauge group that yields Polyakov's current algebra.

Duality symmetries in N = 2 heterotic superstring

We review the derivation and the basic properties of the perturbative prepotential in N=2 compactifications of the heterotic superstring. We discuss the structure of the perturbative monodromy group and the embedding of rigidly supersymmetric monodromies associated with enhanced gauge groups, at both perturbative and non-perturbative level.

Duality symmetries in N = 2 heterotic superstring

We review the derivation and the basic properties of the perturbative prepotential in N = 2 compactifications of the heterotic superstring. We discuss the structure of the perturbative monodromy group and the embedding of rigidly supersymmetric monodromies associated with enhanced gauge groups, at both perturbative and nonperturbative level.

Enhanced gauge symmetries and K3 surfaces

String-string duality dictates that type IIA strings compactified on a K3 surface acquire non-abelian gauge groups for certain values of the K3 moduli. We argue that, contrary to expectation, the theories for which such enhanced gauge symmetries appear are not orbifolds in the string sense. For a specific example we show that a theory with enhanced gauge symmetry and an orbifold theory have the same classical K3 surface as a target space but the value of the “ B-field” differs. This raises the possibility that the conformal field theory associated to a string theory with an enhanced gauge group is badly behaved in some way.

Affine Kac-Moody algebras, CHL strings and the classification of tops

Candelas and Font introduced the notion of a `top' as half of a three dimensional reflexive polytope and noticed that Dynkin diagrams of enhanced gauge groups in string theory can be read off from them. We classify all tops satisfying a generalized definition as a lattice polytope with one facet containing the origin and the other facets at distance one from the origin. These objects torically encode the local geometry of a degeneration of an elliptic fibration. We give a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top (and more generally to any elliptic fibration structure) in a precise way that involves the lengths of simple roots and the coefficients of null roots. Tops related to twisted Kac-Moody algebras can be used to construct string compactifications with reduced rank of the gauge group.