Ji-Xiang Fu, Li-Sheng Tseng, Shing-Tung Yau
We present a class of smooth supersymmetric heterotic solutions with a non-compact Eguchi-Hanson space. The non-compact geometry is embedded as the base of a six-dimensional non-Kahler manifold with a non-trivial torus fiber. We solve the non-linear anomaly equation in this background exactly. We also define a new charge that detects the non-Kahlerity of our solutions.
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We construct new examples of torsional heterotic backgrounds using duality with orientifold flux compactifications. We explain how duality provides a perturbative solution to the type I/heterotic string Bianchi identity. The choice of connection used in the Bianchi identity plays an important role in the construction. We propose the existence of a much larger landscape of compact torsional geometries using string duality. Finally, we present some quantum exact metrics that correspond to NS5-branes placed on an elliptic space. These metrics describe how torus isometries are broken by NS flux.
GLSM resolutions of torsional heterotic Z2xZ2 orbifolds
该研究利用规范线性西格玛模型（GLSM），探究带离散挠的杂化Z₂×Z₂轨形的消解过程，发现带挠消解需在例外循环上引入NS5-膜，且可通过带扭曲振子激发的胀开模式重构接近无挠标准嵌入的丛。
We show that a class of torsional compactifications of the heterotic string are dual to conventional Kahler heterotic string compactifications. This observation follows from the recently proposed analogue of the c-map for the heterotic string.
We construct solutions to the Strominger system on a class of noncompact Calabi-Yau 3-folds. These spaces include $\mathbb{C}^3$ and resolved conifold $\mathcal{O}(-1,-1)$ as special examples.
New heterotic torsional geometries are constructed as orbifolds of T^2 bundles over K3. The discrete symmetries considered can be freely-acting or have fixed points and/or fixed curves. We give explicit constructions when the base K3 is Kummer or algebraic. The orbifold geometries can preserve N=1,2 supersymmetry in four dimensions or be non-supersymmetric.
We characterize the geometric moduli of non-Kaehler manifolds with torsion. Heterotic supersymmetric flux compactifications require that the six-dimensional internal manifold be balanced, the gauge bundle be hermitian Yang-Mills, and also the anomaly cancellation be satisfied. We perform the linearized variation of these constraints to derive the defining equations for the local moduli. We explicitly determine the metric deformations of the smooth flux solution corresponding to a torus bundle over K3.
We complete the duality cycle by constructing the geometric transition duals in the type IIB, type I and heterotic theories. We show that in the type IIB theory the background on the closed string side is a Kahler deformed conifold, as expected, even though the mirror type IIA backgrounds are non-Kahler (both before and after the transition). On the other hand, the Type I and heterotic backgrounds are non-Kahler. Therefore, on the heterotic side these backgrounds give rise to new torsional manifolds that have not been studied before. We show the consistency of these backgrounds by verifying th…
We study the superpotential for the heterotic string compactified on non-Kahler complex manifolds. We show that many of the geometrical properties of these manifolds can be understood from the proposed superpotential. In particular we give an estimate of the radial modulus of these manifolds. We also show, how the torsional constraints can be obtained from this superpotential.
Heterotic string compactifications can be conveniently described in the language of (2,0) gauged linear sigma models (GLSMs). Such models allow for Fayet-Iliopoulos (FI)-terms, which can be interpreted as Kahler parameters and axions on the target space geometry. We show that field dependent non-gauge invariant FI-terms lead to a Green-Schwarz-like mechanism on the worldsheet which can be used to cancel worldsheet anomalies. However, given that these FI-terms are constrained by quantization conditions due to worldsheet gauge instantons, the anomaly conditions turn out to be still rather constr…
Gravity duals for little string theories --- which give rise to four-dimensional theories that undergo permanent confinement in the infrared --- have not been studied in great detail. We address this question in the framework of heterotic SO(32) and E_8 x E_8 string theory, constructing these backgrounds by wrapping heterotic five-branes on calibrated two-cycles of non-Kahler resolved conifolds. Related to deformations of the underlying little string theories, we find numerous analytic solutions preserving N = 1 supersymmetry in four-dimensions. These theories all have non-abelian global symme…
We study new compactifications of the SO(32) heterotic string theory on compact complex non-Kahler manifolds. These manifolds have many interesting features like fewer moduli, torsional constraints, vanishing Euler character and vanishing first Chern class, which make the four-dimensional theory phenomenologically attractive. We take a particular compact example studied earlier and determine various geometrical properties of it. In particular we calculate the warp factor and study the sigma model description of strings propagating on these backgrounds. The anomaly cancellation condition and en…
We review our present understanding of heterotic compactifications on non-Kahler complex manifolds with torsion. Most of these manifolds can be obtained by duality chasing a consistent F-theory compactification in the presence of fluxes. We show that the duality map generically leads to non-Kahler spaces on the heterotic side, although under some special conditions we recover Kahler compactifications. The dynamics of the heterotic theory is governed by a new superpotential and minimizing this superpotential reproduces all the torsional constraints. This superpotential also fixes most of the mo…
We calculate near-horizon solutions for four-dimensional 4-charge and five-dimensional 3-charge black holes in heterotic string theory from the part of the ten-dimensional tree-level effective action which is connected to gravitational Chern-Simons term by supersymmetry. We obtain that the entropies of large black holes exactly match the alpha'-exact statistical entropies obtained from microstate counting (D=4) and AdS/CFT correspondence (D=5). Especially interesting is that we obtain agreement for both BPS and non-BPS black holes, contrary to the case of R^2-truncated (four-derivative) action…
Gauged linear sigma models with (0,2) supersymmetry allow a larger choice of couplings than models with (2,2) supersymmetry. We use this freedom to find a fully linear construction of torsional heterotic compactifications, including models with branes. As a non-compact example, we describe a family of metrics which correspond to deformations of the heterotic conifold by turning on H-flux. We then describe compact models which are gauge-invariant only at the quantum level. Our construction gives a generalization of symplectic reduction. The resulting spaces are non-Kahler analogues of familiar …
We report on some recent progress in understanding heterotic flux compactifications, from a worldsheet perspective mainly. We consider local models consisting in torus fibration over warped Eguchi-Hanson space and non-Kähler resolved conifold geometries. We analyze the supergravity solutions and define a double-scaling limit of the resolved singularities, defined such that the geometry is smooth and weakly coupled. We show that, remarkably, the heterotic solutions admit solvable worldsheet CFT descriptions in this limit. This allows in particular to understand the important role of worldsheet …
In the same spirit as done for N=2 and N=4 supersymmetric non-linear $\si$ models in 2 space-time dimensions by Zumino and Alvarez- Gaumé and Freedman, we analyse the (2,0) and (4,0) heterotic geometry in holomorphic coordinates. We study the properties of the torsion tensor and give the conditions under which (2,0) geometry is conformally equivalent to a (2,2) one. Using additional isometries, we show that it is difficult to equip a manifold with a closed torsion tensor, but for the real 4 dimensional case where we exhibit new examples. We show that, contrarily to Callan, Harvey and Strominge…
We construct a (1,2) heterotic string with gauge symmetry and determine its particle spectrum. This theory has a local N=1 worldsheet supersymmetry for left movers and a local N=2 worldsheet supersymmetry for right movers and describes particles in either two or three space-time dimensions. We show that fermionizing the bosons of the compactified N=1 space leads to a particle spectrum which has nonabelian gauge symmetry. The fermionic formulation of the theory corresponds to a dimensional reduction of self dual Yang Mills. We also give a worldsheet action for the theory and calculate the one-l…
We consider the possibility that the unification of the electroweak interactions and the strong force arises from string theory, at energies significantly lower than the string scale. As a tool, an effective grand unified field theory in six dimensions is derived from an anisotropic orbifold compactification of the heterotic string. It is explicitly shown that all anomalies cancel in the model, though anomalous Abelian gauge symmetries are present locally at the boundary singularities. In the supersymmetric vacuum additional interactions arise from higher-dimensional operators. We develop meth…
We study a particular N = 1 confining gauge theory with fundamental flavors realised as seven branes in the background of wrapped five branes on a rigid two-cycle of a non-trivial global geometry. In parts of the moduli space, the five branes form bound states with the seven branes. We show that in this regime the local supergravity solution is surprisingly tractable, even though the background topology is non-trivial. New effects such as dipole deformations may be studied in detail, including the full backreactions. Performing the dipole deformations in other ways leads to different warped lo…
The fate of discrete torsion on resolved heterotic Z2xZ2 orbifolds using (0,2) GLSMs
本文利用(0,2) GLSM研究异质Z2xZ2轨形解析化后离散挠率的去向，构造了新的T6/Z2xZ2解析化GLSM，并揭示有挠率情形下需通过对数超场依赖FI项消除混合规范反常，表明例外环上NS5-膜产生H-通量。
The existence of multiple anomalous U(1)s is demonstrated explicitly in a blow-up version of a heterotic Z_3 orbifold. Another blow-up of the same orbifold supports further evidence for the type-I/heterotic duality in four dimensions. It has a single anomalous U(1) which does not factorize universally. As multiple anomalous U(1)s as well as non-universal factorization have never been established on heterotic orbifolds explicitely, these findings might appear contradictory at first sight. Possible inconsistencies are avoided by reinterpreting a charged twisted state as a second non-universal lo…
We explore a ``fertile patch'' of the heterotic landscape based on a Z_6-II orbifold with SO(10) and E_6 local GUT structures. We search for models allowing for the exact MSSM spectrum. Our result is that of order 100 out of a total 3\times 10^4 inequivalent models satisfy this requirement.
In this note we propose a generalization of heterotic/F-theory duality. We introduce a set of non-compact building blocks which we glue together to reach compact examples of generalized duality pairs. The F-theory building blocks consist of non-compact elliptically fibered Calabi-Yau fourfolds which also admit a K3 fibration. The compact elliptic model obtained by gluing need not have a globally defined K3 fibration. By replacing the K3 fiber of each F-theory building block with a T^2, we reach building blocks in a heterotic dual vacuum which includes a position dependent dilaton and three-for…
We compute the partition function for the exotic instanton system corresponding to D-instantons on D7 branes in Type I' theory. We exploit the BRST structure of the moduli action and its deformation by RR background to fully localize the integration. The resulting prepotential describes non-perturbative corrections to the quartic couplings of the gauge field F living on the D7's. The results match perfectly those obtained in the dual heterotic theory from a protected 1-loop computation, thus providing a non-trivial test of the duality itself.
By fibering the duality between the $E_{8}\times E_{8}$ heterotic string on $T^{3}$ and M-theory on K3, we study heterotic duals of M-theory compactified on $G_{2}$ orbifolds of the form $T^{7}/\mathbb{Z}_{2}^{3}$. While the heterotic compactification space is straightforward, the description of the gauge bundle is subtle, involving the physics of point-like instantons on orbifold singularities. By comparing the gauge groups of the dual theories, we deduce behavior of a "half-$G_{2}$" limit, which is the M-theory analog of the stable degeneration limit of F-theory. The heterotic backgrounds ex…
We consider heterotic strings on a warped Eguchi-Hanson space with five-brane and line bundle gauge fluxes. The heterotic string admits an exact CFT description in terms of an asymmetrically gauged SU(2)xSL(2,R) WZW model, in a specific double scaling limit in which the blow-up radius and the string scale are sent to zero simultaneously. This allows us to compute the perturbative 6D spectra for these models in two independent fashions: i) Within the supergravity approximation we employ a representation dependent index; ii) In the double scaling limit we determine all marginal vertex operators …
We consider the heterotic string on Calabi-Yau manifolds admitting a Strominger-Yau-Zaslow fibration. Upon reducing the system in the $T^3$-directions, the Hermitian Yang-Mills conditions can then be reinterpreted as a complex flat connection on $\mathbb{R}^3$ satisfying a certain co-closure condition. We give a number of abelian and non-abelian examples, and also compute the back-reaction on the geometry through the non-trivial $α'$-corrected heterotic Bianchi identity, which includes an important correction to the equations for the complex flat connection. These are all new local solutions t…
We address non-perturbative effects and duality symmetries in $N=2$ heterotic string theories in four dimensions. Specifically, we consider how each of the four lines of enhanced gauge symmetries in the perturbative moduli space of $N=2$ $T_2$ compactifications is split into 2 lines where monopoles and dyons become massless. This amounts to considering non-perturbative effects originating from enhanced gauge symmetries at the microscopic string level. We show that the perturbative and non-perturbative monodromies consistently lead to the results of Seiberg-Witten upon identication of a consist…
We consider heterotic string theory on Eguchi-Hanson space, as a local model of a resolved A_1 singularity in a six-dimensional flux compactification, with an Abelian gauge bundle turned on and non-zero torsion. We show that in a suitable double scaling limit, that isolates the physics near the non-vanishing two-cycle, a worldsheet conformal field theory description can be found. It contains a heterotic coset whose target space is conformal to Eguchi-Hanson. Starting from the blow-down limit of the singularity, it can be viewed as a dynamical deformation of the near-horizon fivebrane backgroun…
A new two dimensional $\mathcal{N}=(0,2)$ Supersymmetric Non-Linear Sigma Model describes the dynamics of internal moduli of the BPS semi-local vortex string supported in four dimensional $\mathcal{N}=2$ SQED. While the core of these strings is very similar to Abrikosov-Nielsen-Olesen vortices, they are defined with a characteristic size modulus, much like the instanton lump size. This entails that the constituting fields of the vortex do not decay exponentially, as one goes far away from the core of the string, but as a rational function. The appearance of an extra scale in the problem also a…
We propose a description of G-fluxes that induce chirality in 4-dimensional F-theory GUT spectra that is intrinsic to F-theory and does not rely on Heterotic/F-theory duality. Using this, we describe how to globally extend fluxes that have been constructed in a semi-local setting and obtain an F-theoretic formula for computing the chiral spectrum that they induce. Chirality computations agree with those from the semi-local Higgs bundle analysis for matter fields that are charged under the GUT-group, and hence with the standard Heterotic formulae where applicable. Finally, the relation of G-flu…
The constraints imposed on heterotic compactifications by global consistency and phenomenology seem to be very finely balanced. We show that weakening these constraints, as was proposed in some recent works, is likely to lead to frivolous results. In particular, we construct an infinite set of such frivolous models having precisely the massless spectrum of the MSSM and other quasi-realistic features. Only one model in this infinite collection (the one constructed in arXiv:hep-th/0512149) is globally consistent and supersymmetric. The others might be interpreted as being anomalous, or as non-su…
We prove that any compact complex homogeneous space with vanishing first Chern class after an appropriate deformation of the complex structure admits a homogeneous Calabi-Yau with torsion structure, provided that it also has an invariant volume form. A description of such spaces among the homogeneous C-spaces is given as well as many examples and a classification in the 3-dimensional case. We calculate the cohomology ring of some of the examples and show that in dimension 14 there are infinitely many simply-connected spaces with the same Hodge numbers and torsional Chern classes admitting such…
We study a heterotic two-dimensional N=(0,2) gauged non-linear sigma-model whose target space is a weighted complex projective space. We consider the case with N positively and N^~=N_F - N negatively charged fields. This model is believed to give a description of the low-energy physics of a non-Abelian semi-local vortex in a four-dimensional N=2 supersymmetric U(N) gauge theory with N_F > N matter hypermultiplets. The supersymmetry in the latter theory is broken down to N=1 by a mass term for the adjoint fields. We solve the model in the large-N approximation and explore a two-dimensional subs…
Instantons on Eguchi-Hanson spaces provide explicit examples of stable bundles on non-compact four dimensional C^2/Z_n orbifold resolutions with non-Abelian structure groups. With this at hand, we can consider compactifications of ten dimensional SO(32) supergravity (arising as the low energy limit of the heterotic string) on the resolved spaces in the presence of non-Abelian bundles. We provide explicit examples in the resolved C^2/Z_3 case, and give a complete classification of all possible effective six dimensional models where the instantons are combined with Abelian gauge fluxes in order …
We continue our exploration of local Double Field Theory (DFT) in terms of symplectic graded manifolds carrying compatible derivations and study the case of heterotic DFT. We start by developing in detail the differential graded manifold that captures heterotic Generalized Geometry which leads to new observations on the generalized metric and its twists. We then give a symplectic pre-NQ-manifold that captures the symmetries and the geometry of local heterotic DFT. We derive a weakened form of the section condition, which arises algebraically from consistency of the symmetry Lie 2-algebra and i…
We argue that $(0,1)$ heterotic string models with 4 non-heterotic spacetime dimensions may provide an instability of the vacuum to gravitational polarization in globally strong gravitational fields. This instability would be triggered during gravitational collapse by a non-local quantum switching process, and lead to the formation of regions of a high temperature broken symmetry phase which would be the equivalent in these string models of conventional black holes.
Local descriptions of the heterotic SU(3) moduli space
本文研究异质弦理论中Hull-Strominger系统模空间的局部结构，通过形变复形和上同调同构证明其预期维数为零，并提供Dolbeault型定理连接Čech上同调。
We continue our study of geometric transitions in type II and heterotic theories. In type IIB theory we discuss an F-theory setup which clarifies many of our earlier assumptions and allows us to study gravity duals of N = 1 gauge theories with arbitrary global symmetry group G. We also point out the subtle differences between global and local metrics, and show that in many cases the global descriptions are far more complicated than discussed earlier. We determine the full global description in type I/heterotic theory. In type IIA, our analysis gives rise to a local non-Kahler metric whose glob…
We desribe in detail a Z_6 orbifold compactification of the heterotic E_8 x E_8 string which leads to the (supersymmetric) standard model gauge group and matter content. The quarks and leptons appear as three 16-plets of SO(10), two of which are localized at fixed points with local SO(10) symmetry. The model has supersymmetric vacua without exotics at low energies and is consistent with gauge coupling unification. Supersymmetry can be broken via gaugino condensation in the hidden sector. The model has large vacuum degeneracy. Certain vacua with approximate B-L symmetry have attractive phenomen…
We construct a 6D supergravity theory which emerges as intermediate step in the compactification of the heterotic string to the supersymmetric standard model in four dimensions. The theory has N=2 supersymmetry and a gravitational sector with one tensor and two hypermultiplets in addition to the supergravity multiplet. Compactification to four dimensions occurs on a T^2/Z_2 orbifold which has two inequivalent pairs of fixed points with unbroken SU(5) and SU(2)xSU(4) symmetry, respectively. All gauge, gravitational and mixed anomalies are cancelled by the Green-Schwarz mechanism. The model has …
We determine threshold corrections to the gauge couplings in local models of N=2 smooth heterotic compactifications with torsion, given by the direct product of a warped Eguchi-Hanson space and a two-torus, together with a line bundle. Using the worldsheet CFT description previously found and by suitably regularising the infinite target space volume divergence, we show that threshold corrections to the various gauge factors are governed by the non-holomorphic completion of the Appell-Lerch sum. While its holomorphic Mock-modular component captures the contribution of states that localise on th…
The study of curved D-brane geometries in type II strings implies a general relation between local singularities $\cx W$ of Calabi-Yau manifolds and gravity free supersymmetric QFT's. The minimal supersymmetric case is described by F-theory compactifications on $\cx W$ and can be used as a starting point to define minimal supersymmetric heterotic string compactifications on compact Calabi-Yau manifolds with holomorphic, stable gauge backgrounds. The geometric construction generalizes to non-perturbative vacua with five-branes and provides a framework to study non-perturbative dynamics of the h…
We reconsider the ingredients of moduli stabilization in heterotic M-theory. On this line we close a gap in the literature deriving the Kaehler potential dependence on vector bundle moduli and charged matter. Crucial in this derivation is our superspace formulation of 5d heterotic M-theory taking into account the Bianchi identities modified by brane terms. Likewise, we obtain the Fayet-Iliopolous terms due to brane localised anomalous U(1)'s. After assembling perturbative and non-perturbative contributions to the superpotential, we study supersymmetric (adS) vacua. It is found that the susy co…
Non-Supersymmetric String-String Dualities via Enriques Surfaces
通过恩里克斯曲面实现非超对称的弦-弦对偶性，利用K3表面的商结构构造轨道理论，并与非超对称异弦非对称轨道理论建立双模关系。
We perform the first systematic analysis of particle spectra obtained from heterotic string compactifications on non-Abelian toroidal orbifolds. After developing a new technique to compute the particle spectrum in the case of standard embedding based on higher dimensional supersymmetry, we compute the Hodge numbers for all recently classified 331 non-Abelian orbifold geometries which yield N=1 supersymmetry for heterotic compactifications. Surprisingly, most Hodge numbers follow the empiric pattern h^(1,1) - h^(2,1) = 0 mod 6, which might be related to the number of three standard model genera…
In a previous paper, referred to as a "Mini-Landscape" search, we explored a "fertile patch" of the heterotic landscape based on a Z6-II orbifold with SO(10) and E6 local GUT structures. In the present paper we extend this analysis. We find many models with the minimal supersymmetric standard model spectra and an exact R parity. In all of these models, the vector-like exotics decouple along D flat directions. We present two "benchmark" models which satisfy many of the constraints of a realistic supersymmetric model, including non-trivial Yukawa matrices for 3 families of quarks and leptons and…
We compute the new supersymmetric index of a large class of N=2 heterotic compactifications with torsion, corresponding to principal two-torus bundles over warped K3 surfaces with H-flux. Starting from a UV description as a (0,2) gauged linear sigma-model with torsion, we use supersymmetric localization techniques to provide an explicit expression of the index as a sum over the Jeffrey-Kirwan residues of the one-loop determinant. We finally propose a geometrical formula that gives the new supersymmetric index in terms of bundle data, regardless of any particular choice of underlying two-dimens…
Heterotic solitons on four-manifolds
本文研究了四维流形上的 Heterotic 孤子，证明了特定闭 Einstein-Weyl 结构模空间的同构性及其形变受阻，并发现了首个非局部同构于超对称背景的 Heterotic 紧致化背景实例。
The heterotic--string models in the free fermionic formulation gave rise to some of the most realistic string models to date, which possess N=1 spacetime supersymmetry. Lack of evidence for supersymmetry at the LHC instigated recent interest in non-supersymmetric heterotic-string vacua. We explore what may be learned in this context from the quasi--realistic free fermionic models. We show that constructions with a low number of families give rise to proliferation of a priori tachyon producing sectors, compared to the non--realistic examples, which typically may contain only one such sector. Th…
We study the moduli-dependent prefactor of M5-instanton corrections to the superpotential in four-dimensional F-theory compactifications. In light of the M-theory and type IIb limits and also heterotic duality, we propose that the explicit moduli dependence of the prefactor can be computed by a study of zero modes localized at intersections between the instanton and seven-branes. We present an instanton prefactor in an E_6 F-theory GUT which does not admit a heterotic dual and show that it vanishes if and only if a point of E_8 enhancement is present in the instanton worldvolume. More generica…
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