String compactification, QCD axion and axion-photon-photon coupling
物理学咬文嚼字之四十四 Uncertaintyof the Uncertainty Principle(下)
物理学咬文嚼字之四十四
（ 下） Ｕ ｎ ｃ ｅ ｒ ｔ ａ ｉ ｎ ｔ ｆ ｔ ｈ ｅ Ｕ ｎ ｃ ｅ ｒ ｔ ａ ｉ ｎ ｔ ｒ ｉ ｎ ｃ ｉ ｌ ｅ ｙｏ ｙＰ ｐ
曹则贤
（ ） 中国科学院物理研究所 北京 １ ０ ０ １ ９ ０
经典不确定性 所谓的 “ 不能够无限精确地确定一个物理量 ” 给 欧洲的物理学家们 带 来 了 惶 恐 ， 但是经典世界里存 在ｅ ｏ ｓ ｉ ｔ ｉ ｏ ｎ ｉ ｎ ｘ ａ ｃ ｔ ｐ ｇ的 说 法 似 乎 也 是 毫 无 根 据 的． 确定公路上一辆汽 车 的 位 置 ， 精确到毫米量级都是 没必要的 、 原则 上 不 可 能 的 （ 用汽车的哪个部位标 ） ， 定它 的 位 置 呢 ， 部 位 又 如 何 定 位 呢？ 遑论无限精 确． 观察过程中光子 对 车 子 的 位 置 当 然 也 是 有 扰 动 ） 但这丝毫 不 影 响 确 定 车 子 （ 这 的， ｌ ｏ ｃ ａ ｔ ｉ ｎ ｔ ｈ ｅ ｃ ａ ｒ ｇ 个物理事件 ． 一个有必要问的问题是 ， 远小于物体尺 度的位置精度有 意 义 吗 ？ 笔 者 以 为 ， 科学定律可能 比如描述种群数量演 本身就有可容忍的 不 确 定 性 ， 化的ｌ 本身就是处理大数目的群体演 ｏ ｉ ｓ ｔ ｉ ｃ方 程 ， ｇ 而无限精确地 描 述 物 理 世 界 的 想 法 只 是 理 想 化的 ． 化的愿望 ， 数学的理 想 不 可 以 用 来 取 代 或 否 定 现 实 — — 一只足球对运动员来说等价于数学家的 的世界 —
１ ７） 确定 、 一定以及肯定 ” 的吗 ？Ｋ 是“ ａ ｒ ｌ Ｐ ｏ ｅ ｒ继续 ｐ ｐ ２ ０］ ， 对经典决定性加以批判 ［ 他引入了 同 样 属 于 经 典
物理的相对论来佐 证 ． 他认为狭义相对论就堵死了 如 图 ８ 所 示， 在Ｓ 通往决定论的路 ． １ 点能够测到的 物理事件在 Ｓ 而能影响到Ｓ １ 点的 未 来 光 锥 内 ； ２ 点 上事件的物理事件在 它 的 过 去 光 锥 靠 Ｓ １ 未来一侧 的部分 ， 显然前者 大 于 后 者 ． 也 就 是 说， 在此时空点 上我们不可能接近 影 响 未 来 某 时 空 点 的 全 部 事 件 ， 当然也就不可能测量之以作为初始条件 ．
物理学名词
1/4波片quarter-wave plateCG矢量耦合系数Clebsch-Gordan vector coupling coefficient; 简称“CG[矢耦]系数”。
X射线摄谱仪X-ray spectrographX射线衍射X-ray diffractionX射线衍射仪X-ray diffractometer[玻耳兹曼]H定理[Boltzmann] H-theorem[玻耳兹曼]H函数[Boltzmann] H-function[彻]体力body force[冲]击波shock wave[冲]击波前shock front[狄拉克]δ函数[Dirac] δ-function[第二类]拉格朗日方程Lagrange equation[电]极化强度[electric] polarization[反射]镜mirror[光]谱线spectral line[光]谱仪spectrometer[光]照度illuminance[光学]测角计[optical] goniometer[核]同质异能素[nuclear] isomer[化学]平衡常量[chemical] equilibrium constant[基]元电荷elementary charge[激光]散斑speckle[吉布斯]相律[Gibbs] phase rule[可]变形体deformable body[克劳修斯-]克拉珀龙方程[Clausius-] Clapeyron equation[量子]态[quantum] state[麦克斯韦-]玻耳兹曼分布[Maxwell-]Boltzmann distribution[麦克斯韦-]玻耳兹曼统计法[Maxwell-]Boltzmann statistics[普适]气体常量[universal] gas constant[气]泡室bubble chamber[热]对流[heat] convection[热力学]过程[thermodynamic] process[热力学]力[thermodynamic] force[热力学]流[thermodynamic] flux[热力学]循环[thermodynamic] cycle[事件]间隔interval of events[微观粒子]全同性原理identity principle [of microparticles][物]态参量state parameter, state property[相]互作用interaction[相]互作用绘景interaction picture[相]互作用能interaction energy[旋光]糖量计saccharimeter[指]北极north pole, N pole[指]南极south pole, S pole[主]光轴[principal] optical axis[转动]瞬心instantaneous centre [of rotation][转动]瞬轴instantaneous axis [of rotation]t 分布student's t distributiont 检验student's t testＫ俘获K-captureＳ矩阵S-matrixＷＫＢ近似WKB approximationＸ射线X-rayΓ空间Γ-spaceα粒子α-particleα射线α-rayα衰变α-decayβ射线β-rayβ衰变β-decayγ矩阵γ-matrixγ射线γ-rayγ衰变γ-decayλ相变λ-transitionμ空间μ-spaceχ 分布chi square distributionχ 检验chi square test阿贝不变量Abbe invariant阿贝成象原理Abbe principle of image formation阿贝折射计Abbe refractometer阿贝正弦条件Abbe sine condition阿伏伽德罗常量Avogadro constant阿伏伽德罗定律Avogadro law阿基米德原理Archimedes principle阿特伍德机Atwood machine艾里斑Airy disk爱因斯坦-斯莫卢霍夫斯基理论Einstein-Smoluchowski theory 爱因斯坦场方程Einstein field equation爱因斯坦等效原理Einstein equivalence principle爱因斯坦关系Einstein relation爱因斯坦求和约定Einstein summation convention爱因斯坦同步Einstein synchronization爱因斯坦系数Einstein coefficient安[培]匝数ampere-turns安培[分子电流]假说Ampere hypothesis安培定律Ampere law安培环路定理Ampere circuital theorem安培计ammeter安培力Ampere force安培天平Ampere balance昂萨格倒易关系Onsager reciprocal relation凹面光栅concave grating凹面镜concave mirror凹透镜concave lens奥温电桥Owen bridge巴比涅补偿器Babinet compensator巴耳末系Balmer series白光white light摆pendulum板极plate伴线satellite line半波片halfwave plate半波损失half-wave loss半波天线half-wave antenna半导体semiconductor半导体激光器semiconductor laser半衰期half life period半透[明]膜semi-transparent film半影penumbra半周期带half-period zone傍轴近似paraxial approximation傍轴区paraxial region傍轴条件paraxial condition薄膜干涉film interference薄膜光学film optics薄透镜thin lens保守力conservative force保守系conservative system饱和saturation饱和磁化强度saturation magnetization本底background本体瞬心迹polhode本影umbra本征函数eigenfunction本征频率eigenfrequency本征矢[量] eigenvector本征振荡eigen oscillation本征振动eigenvibration本征值eigenvalue本征值方程eigenvalue equation比长仪comparator比荷specific charge; 又称“荷质比(charge-mass ratio)”。
格点量子色动力学进展.pptx
The fDs puzzle?
Experimental results
Comparison
Independent determination of Vcs
More comparisons…
Is is a puzzle?
只有HPQCD的结果偏离最严重 Vcs需要单独确定 目前还不是很严重的问题…
理论方面的探讨
相关的唯象理论工作
Shallow bound state of two D mesons (S.L. Zhu et al, ) PRD77,034003 Tetra-quark resonance above threshold (X.-H Liu et al, PRD77, 094005) Threshold enhancement (J.L. Rosner, ) PRD76,114002
The Berlin wall & lattice QCD c o s t
m m
z
a
z 4, 7
Berlin wall crushing…
More physical…
Pion mass 700MeV below 300MeV where cPT starts to work…
Matrix elements
B_K 的计算 B ˆB KKBK K((8 0|/Q 3 ))L fSK s (L 232M )|(K K 2)0,2/Q 9L 1S L 2(s(43s)d()L)(sJd3)L
Renormalization dilemma
Continuum perturbation done in the MS-bar Naive Dimensional Regularization scheme (MS-bar NDR-scheme)
格格格点点点QCD在在在K物物物理理理中中中的的的新新 ...
K物理中的常规物理量: BK
Short distance 贡献占主导 ⇒ OPE ⇒ Wilson coeff. C(µ) × operator Q∆S=2(µ)
He∆ffS=2 占间接CP 破坏 K 贡献的绝大部份
K
=
exp(i φ
) sin(φ
)
⎡⎢⎢⎢⎢⎣
Im[⟨K
0 He∆ffS=2 ∆MK
格点QCD——超级计算机上的虚拟实验室
格点QCD：超级计算机上的虚拟实验室
格点QCD简介
格点QCD 40年
最早由诺贝奖获得者 Kenneth G. Wilson 在 1974 年提出 计算机上的第一个数值计算由 M. Creutz 在 1979 年实现 QCD 超级计算机 1983 – 2011
未来几年内，QCD 计算机进入Eflops时代，每秒进行1018浮点运算
格子上的QCD
QCD 格点离散化 夸克场位于格点上, ψ(x), xµ = nµa 胶子场由格点之间的链接来表示 Uµ(x ) = eiagAµ(x)
1 mπ
L a
计算机只能模拟有限的自由度 ⇒ 格距a不是无穷小，格子长度L不是无 穷大 欧氏时空路径积分: 闵氏时间被欧氏时间替代 x0 → −it ⇒ e−iHx0 → e−Ht = e−S[ψ,ψ¯,A] 但哈密顿量 H 在闵氏和欧氏时空是一致的
=++
FLAG average for = + RBC/UKQCD 15A RBC/UKQCD 13 FNAL/MILC 12I JLQCD 12 JLQCD 11 RBC/UKQCD 10 RBC/UKQCD 07
FLAG average for = ETM 10D (stat. err. only) ETM 09A QCDSF 07 (stat. err. only) RBC 06 JLQCD 05 JLQCD 05
《粒子物理学教学讲义》6.4－节重夸克和重夸克偶素
确定了J/Ψ粒子子的夸克组分——正反粲夸克对，我们就可以解
释它的窄宽度了：
质量最轻的含粲D介子子质量为1870MeV，两个D介子子的阈能
为3740MeV，所以J/Ψ不能衰变到D介子子末态，它的强衰变末
态只能是由u,d,s夸克组成的轻强子子。
J/Ψ的强子子衰变的费曼图可以画作，
u d
c
d
c
c
d
c
d
它和J/Ψ粒子子类似, 也只能通过把b 和 b 湮没掉的Zweig禁 戒过程做强衰变，衰变概率被大大大大地压低，表现出突出的“重 质量，窄宽度”性质． 同样ϒ (2S), ϒ(3S)也具有同样的性质。ϒ (4S)以及更高高质量的 态则可以衰变到两个B介子子。因而而具有很大大的宽度。
17
对前3个ϒ共振态, ϒ→ 2B 的衰变道没有打开,他们所有的强衰 变道都是OZI禁戒过程而受到压低, 衰变宽度很窄。 但对下面更高质量的ϒ高激发态, 均可以衰变到两个B介子,因 而具有很大的几十个MeV的宽度
16
ϒ粒子子所有的强衰变道都是OZI 禁戒过程而而受到压低，
分支支比比最大大的衰变道也是二二级电磁衰变的 e+e− 道，µ +µ − 道 和 τ +τ − 道．
含底夸克b的最轻的粒子子是 B+(bu) 和 B0 (bd) 粒子子，其质量为
m = (5279)MeV，因此ϒ粒子子不可能衰变为一一对正反B介子子。
14
如果粲夸克存在，那么可以将u，d，s的SU(3) 味道对称性扩 充到u,d,s,c的SU(4) 对称性，则应该存在含有若干（反）粲夸 克的重子子和介子子多重态，
1976年发现了含粲介子子态（ D0 = cu , D+ = cd ）， 1977年又又发现了含粲奇异介子子 Ds+ = cs ，
弦论与宇宙学
弦论与宇宙学弦论（String Theory）是近年来物理学领域的一项重要研究课题。
它被认为是统一量子力学和爱因斯坦的广义相对论的理论框架。
而宇宙学（Cosmology）则是研究整个宇宙的起源、演化和结构等问题的学科。
本文将探讨弦论与宇宙学之间的关系以及对宇宙起源和演化的启示。
一、弦论基础弦论是一种基本粒子描述的理论，将基本粒子看作是一维的弦状结构，而不再是零维点状结构。
这样一来，物质和力场都可以通过弦的振动模式来描述。
弦论不仅包含了量子力学的规则，还包含了引力的效应，因此具备统一量子力学和引力理论的潜力。
二、弦论与宇宙起源弦论对于解释宇宙的起源提供了一种可能性。
宇宙学家普遍接受的宇宙起源理论是大爆炸理论，即宇宙在一个初始的高温高密度点上产生，然后逐渐膨胀。
然而，大爆炸之前的宇宙是什么样子的呢？弦论提出了一个有趣的观点，即在大爆炸之前宇宙是一个高度弯曲的时空，弦理论则给出了描绘这样一个宇宙的方法。
三、宇宙学观测与弦论验证弦论的一个重要特征是它预测了一些现象，这些现象在目前的宇宙学观测中还没有被观测到。
然而，科学家们认为这些现象可能在未来的观测中被证实，从而验证弦论的有效性。
例如，弦论预测了额外的维度存在，这些维度的大小可能相当微小，因此很难直接观测到。
但是，一些宇宙学实验可能会通过探测引力微弱的效应来间接证实这些额外维度的存在。
四、弦论与黑洞黑洞是宇宙中的一种极端天体，强大的引力使得物质被压缩到极限，形成一个密度无穷大、引力无法逃脱的区域。
弦论对于黑洞的研究也取得了重要的进展。
根据弦论的描述，黑洞并不是完全无法逃离的，而是通过黑洞边界的一些量子效应，信息可能得以逃离黑洞，这与传统的黑洞观点有所不同。
五、宇宙学的开放问题和弦论的应用宇宙学仍然存在许多未解之谜，如宇宙膨胀加速的原因、宇宙中的暗物质和暗能量是什么等。
弦论具有潜在解决这些问题的能力。
例如，弦论中的超弦可以被认为是暗物质的候选者之一，因为它们与普通物质相互作用很弱，难以被观测到。
基本粒子质量与耦合常数
基本粒子质量与耦合常数基本粒子质量和耦合常数是粒子物理学中两个重要的物理量，它们对于描述基本粒子的性质和相互作用至关重要。
本文将探讨基本粒子质量和耦合常数的概念、影响因素以及其在粒子物理学中的重要性。
一、基本粒子质量基本粒子质量是指基本粒子所具有的质量特征。
在粒子物理学中，基本粒子包括了夸克、轻子、强子以及弱子等多种类型。
每种基本粒子都具有自己特定的质量值。
基本粒子的质量是由赋予它们的标准模型希格斯场决定的。
标准模型是物理学家们用来描述基本粒子及其相互作用的理论框架。
希格斯场通过与基本粒子相互作用，赋予了它们质量。
这也是为什么一些基本粒子如夸克、轻子具有质量，而光子等其他基本粒子没有质量的原因。
基本粒子的质量并不是固定不变的，而是可以受到物理过程的影响而发生变化。
例如，夸克在高能强子对撞实验中可以生成更重的夸克，这就涉及了质量产生和转化的过程。
二、耦合常数耦合常数是描述基本粒子之间相互作用强度的量。
在标准模型中，有多种类型的相互作用，包括强相互作用、电磁相互作用以及弱相互作用等。
强相互作用的耦合常数称为强耦合常数，通常用α_s表示。
电磁相互作用的耦合常数称为电磁耦合常数，常用α_e表示。
弱相互作用的耦合常数称为弱耦合常数，一般用α_w表示。
耦合常数的数值大小与相互作用的强度成正比。
例如，强耦合常数越大，说明相互作用越强；弱耦合常数越小，说明相互作用越弱。
三、基本粒子质量与耦合常数的关系基本粒子质量和耦合常数之间存在一定的关系。
根据标准模型理论计算和实验测量结果，基本粒子的质量和耦合常数可以互相影响。
在标准模型中，基本粒子的质量与希格斯场的耦合强度有关。
希格斯场的耦合强度越大，基本粒子的质量也越大。
因此，希格斯场的耦合常数与基本粒子质量的大小密切相关。
另外，基本粒子的质量和耦合常数还与相互作用类型有关。
不同类型的相互作用会带来不同的质量与耦合常数的影响。
例如，强子相互作用对基本粒子质量的影响较大，而轻子相互作用对基本粒子质量的影响较小。
THE THEORY OF SUPERSTRING WITH FLUX ON NON-K¨AHLER MANIFOLDS AND THE COMPLEX MONGE-AMP`ERE EQUATION
j.differential geometry78(2008)369-428THE THEORY OF SUPERSTRING WITH FLUX ON NON-K¨AHLER MANIFOLDS AND THE COMPLEXMONGE-AMP`ERE EQUATIONJi-Xiang Fu&Shing-Tung YauAbstractThe purpose of this paper is to solve a problem posed by Stro-minger in constructing smooth models of superstring theory withflux.These are given by non-K¨a hler manifolds with torsion.1.IntroductionThe purpose of this paper is twofold.Thefirst purpose is to solve an old problem posed by Strominger in constructing smooth models of superstring theory withflux.These are given by non-K¨a hler mani-folds with torsion.To achieve this,we solve a nonlinear Monge-Amp`e re equation which is more complicated than the equation in the Calabi con-jecture.The estimate of the volume form gives extra complication,for example.The second purpose is to point out the connection of the newly constructed geometry based on Strominger’s equations in realizing the proposal of M.Reid[19]on connecting one Calabi-Yau manifold to an-other one with different topology.In Reid’s proposal,the construction of Clemens-Friedman(see[9])is needed where a Calabi-Yau manifold is deformed to complex manifolds diffeomorphic to connected sums of S3×S3.These are non-K¨a hler manifolds.There is a rich class of non-K¨a hler complex manifolds for dimensions greater than two.It is therefore important to construct canonical ge-ometry on such manifolds.Since for non-K¨a hler geometry,the complex structure is not quite compatible with the Riemannian metric,it has been difficult tofind a reasonable class of Hermitian metric that ex-hibits rich geometry.We believe that metrics motivated by theoretic physics should have good properties.This is especially true for those metrics which admit parallel spinors.The work of Strominger provided such a candidate.In this paper,we provide a smooth solution to the Strominger system.This has been an important open problem through the past twenty years.Our method is based on a priori estimates whichReceived05/24/2007.369370J.-X.FU &S.-T.YAUcan be generalized to elliptic fibration over general Calabi-Yau mani-folds.However,in this paper,for the sake of importance in string the-ory,we shall restrict ourselves to complex three-dimensional manifolds.The structure of the equations for higher-dimensional Calabi-Yau man-ifolds is a little bit different.They are also more relevant to algebraic geometry and hence will be treated in a later occasion.The physical context of the solutions is discussed in a companion paper [3]written jointly with K.Becker,M.Becker,and L.-S.Tseng.Acknowledgement.The authors would like to thank K.Becker,M.Becker,and L.-S.Tseng for useful discussions.J.-X.Fu would also like to thank J.Li and X.-P.Zhu for useful discussions.J.-X.Fu is supported in part by NSFC grant 10471026.S.-T.Yau is supported in part by NSF grants DMS-0244462,DMS-0354737and DMS-0306600.2.Motivation from string theoryIn the original proposal for compactification of superstring [5],Can-delas,Horowitz,Strominger,and Witten constructed the metric prod-uct of a maximal symmetric four-dimensional spacetime M with a six-dimensional Calabi-Yau vacuum X as the ten-dimensional spacetime;they identified the Yang-Mills connection with the SU (3)connection of the Calabi-Yau metric and set the dilaton to be a constant.Adapting the second author’s suggestion of using Uhlenbeck-Yau’s theorem [22]on constructing Hermitian-Yang-Mills connections over stable bundles,Witten [23]and later Horava-Witten [13]proposed to use higher rank bundles for strong coupled heterotic string theory so that the gauge groups can be SU (4)or SU (5).At around the same time,Strominger [20]analyzed heterotic super-string background with spacetime supersymmetry and non-zero torsion by allowing a scalar “warp factor”for the spacetime metric.He consid-ered a ten-dimensional spacetime that is a warped product of a maximal symmetric four-dimensional spacetime M and an internal space X ;the metric on M ×X takes the form g 0=e 2D (y ) g μν(x )00g ij (y ),x ∈M,y ∈X ;the connection on an auxiliary bundle is Hermitian-Yang-Mills connec-tion over X :F ∧ω2=0,F 2,0=F 0,2=0.Here ωis the Hermitian form ω=√−12g i ¯j dz i ∧d ¯z j defined on the internal space X .In this system,the physical relevant quantities are h =−√−1(¯∂−∂)ω,φ=−12log Ω +φ0,THE THEORY OF SUPERSTRING 371and g 0ij=e 2φ0 Ω −1g ij ,for a constant φ0.In order for the ansatz to provide a supersymmetric configuration,one introduces a Majorana-Weyl spinor so thatδψM = M −18h MNP γNP =0,δλ=γM ∂M φ −112h MNP γMNP =0,δχ=γMN F MN =0,where ψM is the gravitino,λis the dilatino,χis the gluino,φis the dilaton and h is the Kalb-Ramond field strength obeyingdh =α 2(tr F ∧F −tr R ∧R ),where α is positive.Strominger [20]showed that in order to achieve spacetime supersymmetry,the internal six manifold X must be a com-plex manifold with a non-vanishing holomorphic three-form Ω;and the anomaly cancellation demands that the Hermitian form ωobey 1√−1∂¯∂ω=α 4(tr R ∧R −tr F ∧F )and supersymmetry requires 2d ∗ω=√−1(¯∂−∂)log Ω ω.Accordingly,he proposed the system(2.1)F H ∧ω2=0;(2.2)F 2,0H =F 0,2H =0;(2.3)√−1∂¯∂ω=α 4(tr R ∧R −tr F H ∧F H );(2.4)d ∗ω=√−1(¯∂−∂)ln Ω ω.This system gives a solution of a superstring theory with flux that allows a non-trivial dilaton field and a Yang-Mills field.(It turns out D (y )=φand is the dilaton field.)Here ωis the Hermitian form and R is the curvature tensor of the Hermitian metric ω;H is the Hermitian metric and F is its curvature of a vector bundle E ;tr is the trace of the endomorphism bundle of either E or T X .In [17],Li and Yau observed the following:1The curvature F of the vector bundle E in ref.[20]is real,i.e.,c 1(E )=F 2π.But we are used to taking the curvature F such that c 1(E )=√−12πF .So this equationcorrects eq.(2.18)of ref.[20]by a minus sign.2See eq.(56)of ref.[21],which corrects eq.(2.30)of ref.[20]by a minus sign.372J.-X.FU &S.-T.YAULemma 1.Equation (2.4)is equivalent to(2.5)d ( Ω ωω2)=0.In fact,Li and Yau gave the first irreducible non-singular solution of the supersymmetric system of Strominger for U (4)and U (5)prin-ciple bundle.They obtained their solutions by perturbing around the Calabi-Yau vacuum coupled with the sum of tangent bundle and triv-ial line bundles.In this paper,we consider the solution on complex manifolds which do not admit K¨a hler structures.Study of non-K¨a hler manifolds should be useful to understand the speculation of M.Reid that all Calabi-Yau manifolds can be deformed to each other through conifold transition.An example of non-K¨a hler manifolds X is given by T 2-bundles over Calabi-Yau varieties [2,4,10,12,14].Since we demand that the internal six manifold X is a complex manifold with a non-vanishing holomorphic three form Ω,we consider the T 2−bundle (X,ω,Ω)over a complex surface (S,ωS ,ΩS )with a non-vanishing holomorphic 2-form ΩS .According to the classification of complex surfaces by Enriques and Kodaira,such complex surfaces must be finite quotients of K3sur-face,complex torus (K¨a hler),and Kodaira surface (non-K¨a hler).If (X,ω,Ω)satisfies Strominger’s equation (2.4),Lemma 1shows thatd ( Ω ωω2)=0.Let ω= Ω 12ωω.Then dω 2=0,i.e.,ω is a balanced metric [18].The balanced metric was studied extensively by Michelsohn.She proved that the balanced condition is preserved under proper holomorphic submersions.Note that Alessandrini and Bassanelli [1]proved that this condition is also preserved under mod-ifications of complex manifolds.Hence if a holomorphic submersion πfrom a balanced manifold X to a complex surface S is proper,S is also balanced (actually π∗ω 2is the balanced metric on S ,see Proposition1.9in [18]).When the dimension of complex manifold is two,the condi-tions of being balanced and K¨a hler coincide.Hence there is no solution to Strominger’s equation (1.4)on T 2bundles over Kodaira surface and we consider T 2-bundles over K 3surface and complex torus only.On the other hand,duality from M -theory suggests that there is no supersymmetric solution when the base manifold is a complex torus (see[3]).This class of three manifolds includes the Iwasawa manifold.But the solution to Strominger’s system should exist when the base is K 3surface.In this paper we prove the existence of solutions to Strominger’s system on such torus bundles over K 3surfaces.3.Statement of main resultLet (S,ωS ,ΩS )be a K3surface or a complex torus with a K¨a hler form ωS and a non-vanishing holomorphic (2,0)-form ΩS .Let ω1and ω2be anti-self-dual (1,1)-forms such that ω12πand ω22πrepresent integralTHE THEORY OF SUPERSTRING 373cohomology ing these two forms,Goldstein and Prokushkin[10]constructed a non-K¨a hler manifold X such that π:X →S is aholomorphic T 2-fibration over S with a Hermitian form ω0=π∗ωS +√−12θ∧¯θand a holomorphic (3,0)-form Ω=ΩS ∧θ(for the definition of θ,see section 3).Note that (ω0,Ω)satisfies equation (2.5).(Sethipointed out that in papers [6]and [2]similar ansatz was discussed.However the major problem of solving equations was not addressed in the literature.)Let u be any smooth function on S and let ωu =π∗(e u ωS )+√−12θ∧¯θ.Then (ωu ,Ω)also satisfies equation (2.5)(see [10]or Lemma 12),i.e.,ωu is conformal balanced.The stability concept can be defined on a vector bundle over a complex manifold using the Gauduchon metric [16],and hence for complex manifolds with balanced metrics.Note that the stability concept of the vector bundle depends only on the conformal class of metric.Let V →X be a stable bundle over X with degree zero with respect to the metric ωu .(Such bundles can be obtained by pulling back stable bundles over a K 3surface or a complex torus,see Lemma 16.)According to Li-Yau’s theorem [16],there is a Hermitian-Yang-Mills metric H on V ,which is unique up to positive constants.The curvature F H of the Hermitian metric H satisfies equation (2.1)and (2.2).So (V,F H ,X,ωu )satisfies Strominger’s equations (2.1),(2.2)and (2.4).Therefore we only need to consider equation (2.3).As ω1and ω2are harmonic,¯∂ω1=¯∂ω2=0.According to ¯∂-Poincar´e Lemma,we can write ω1and ω2locally asω1=¯∂ξ=¯∂(ξ1dz 1+ξ2dz 2)and ω2=¯∂ζ=¯∂(ζ1dz 1+ζ2dz 2),where (z 1,z 2)is a local coordinate on S .Let B = ξ1+√−1ζ1ξ2+√−1ζ2.We can use B to compute tr R 0∧R 0of the metric ω0(see Proposition8)and tr R u ∧R u of the metric ωu (see Lemma 14).Then we reduce equation (2.3)to √−1∂¯∂e u ∧ωS −α 2∂¯∂(e −u tr(¯∂B ∧∂B ∗·g −1))−α 2∂¯∂u ∧∂¯∂u (3.1)=α 4tr R S ∧R S −α 4tr F H ∧F H −12( ω1 2ωS + ω2 2ωS )ω2S 2!,where g =(g i ¯j )is the Ricci-flat metric on S associated to the K¨a hler form ωS and g −1is the inverse matrix of g ;R S is the curvature of g .374J.-X.FU &S.-T.YAUTaking wedge product with ωu and integrating both sides of the above equation over X ,we obtain(3.2)α X {tr R S ∧R S −tr F H ∧F H }∧ωu −2 X( ω1 2ωS + ω2 2ωS )ω2S 2!∧ωu =0.When S =T 4,R S =0.We obtain immediatelyProposition 2.There is no solution of Strominger’s system on the torus bundle X over T 4if the metric has the form e u ωS +√−12θ∧¯θ.This situation is different if the base is a K 3surface.If E is a stable bundle over S with degree 0with respect to the metric ωS ,then V =π∗E is also a stable bundle with degree 0over X with respect to the Hermitian metric ωu .In this case,equation (3.1)on X can be considered as an equation on S .Integrating equation (3.1)over S ,(3.3)α S {tr R S ∧R S −tr F H ∧F H }=2 S ( ω1 2ωS + ω2 2ωS )ω2S 2!.As S tr R S ∧R S =8π2c 2(V )=8π2×24,and S tr F H ∧F H =8π2×(c 2(E )−12c 21(E ))≥0,we can rewrite equation (3.3)as (3.4)α 24− c 2(E )−12c 21(E ) = Sω12π 2ωS + ω22π 2ωS ω2S 2!.For a compact,oriented,simply connected four-manifold S ,the Poincar´e duality gives rise to a pairingQ :H 2(S ;Z )×H 2(S ;Z )→Zdefined byQ (β,γ)=S β∧γ.We shall denote Q (β,β)by Q (β).Then for an integral anti-self-dual (1,1)-form ω12π,the intersection number Q (ω12π)can be expressed as − Sω12π 2ω2S 2!.On the other hand,the intersection form on K 3surface isgiven by [7]3 0110⊕2(−E 8),where E 8=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝20−1020−1−102−1−1−12−1−12−1−12−1−12−1−12⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.THE THEORY OF SUPERSTRING375Hence Q(ω12π)∈{−2,−4,−6,...}.We shall use the following convention for vector bundles over a com-pact oriented four-manifold:κ(E)=c2(E)for SU(r)bundle E,=c2(E)−12c21(E)for U(r)bundle E,=−12p1(E)for SO(r)bundle E.Then(3.4)implies(3.5)α (24−κ(E))+Qω12π+Qω22π=0,which means that there is a smooth functionμsuch that(3.6)α4tr R S∧R S−α4tr F H∧F H−12( ω1 2+ ω2 2)ω2S2!=−μω2S2!andSμω22!=0.Inserting(3.6)into(3.1),we obtain the followingequation:(3.7)√−1∂¯∂e u∧ωS−α2∂¯∂(e−u tr(¯∂B∧∂B∗·g−1))−α2∂¯∂u∧∂¯∂u+μω2S2!=0,where tr(¯∂B∧∂B∗·g−1)is a smooth well-defined(1,1)-form on S.In particular,whenω2=nω1,n∈Z,tr(¯∂B∧∂B∗·g−1)=√−11+n24ω1 2ωSωS(see Proposition11).Hence if we set f=1+n24 ω1 2ωS,we can rewriteequation(3.7)as the standard complex Monge-Amp`e re equation:(3.8)Δ(e u−α2fe−u)+4αdet u i¯jdet g i¯j+μ=0,where u i¯j denotes∂2u∂z i∂¯z j and =2g i¯j∂2∂z i∂¯z j.We shall solve equation(3.7)by the continuity method[24].Our main theorem isTheorem3.The equation(3.7)has a smooth solution u such thatω =e uωS−α2√−1e−u tr(¯∂B∧∂B∗·g−1)+α√−1∂¯∂udefines a Hermitian metric on S.Our solution u satisfiesSe−4u14=A 1.Actually we can provethat inf u≥−ln(C1A)(see Proposition21)where A must be very small (see Proposition22)and our solution u must be very big.376J.-X.FU&S.-T.YAUTheorem 4.Let S be a K3surface with a Ricci-flat metricωS.Letω1andω2be anti-self-dual(1,1)-forms on S such thatω12π,ω22π∈H2(S,Z).Let X be a T2-bundle over S constructed byω1andω2.Let E be a stable bundle over S with degree0.Supposeω1,ω2andκ(E) satisfy condition(3.5).Then there exist a smooth function u on S and a Hermitian-Yang-Mills metric H on E such that(V=π∗E,π∗F H,X,ωu) is a solution of Strominger’s system.Since it is easy tofind(ω1,ω2,κ(E))which satisfies condition(3.5), this theorem providesfirst examples of solutions to Strominger’s system on non-K¨a hler manifolds.4.Geometric modelIn this section,we take the geometric model of Goldstein and Pro-kushkin for complex non-K¨a hler manifolds with an SU(3)structure[10]. We summarize their results as follows:Theorem5.[10]Let(S,ωS,ΩS)be a Calabi-Yau2-fold with a non-vanishing holomorphic(2,0)−formΩS.Letω1andω2be anti-self-dual(1,1)-forms on S such thatω12π∈H2(S,Z)andω22π∈H2(S,Z).Thenthere is a Hermitian3-fold X such thatπ:X→S is a holomorphic T2-fibration over S and the following holds:1.For any real1-formsα1andα2defined on some open subset of Sthat satisfy dα1=ω1and dα2=ω2,there are local coordinates x and y on X such that dx+idy is a holomorphic form on T2-fibers and a metric on X has the following form:(4.1)g0=π∗g+(dx+π∗α1)2+(dy+π∗α2)2,where g is a Calabi-Yau metric on S corresponding to the K¨a hler formωS.2.X admits a nowhere vanishing holomorphic(3,0)-form with unitlength:Ω=((dx+π∗α1)+i(dy+π∗α2))∧π∗ΩS.3.If eitherω1orω2represents a non-trivial cohomological class thenX admits no K¨a hler metric.4.X is a balanced manifold.The Hermitian form(4.2)ω0=π∗ωS+(dx+π∗α1)∧(dy+π∗α2)corresponding to the metric(4.1)is balanced,i.e.,dω20=0.5.Furthermore,for any smooth function u on S,the Hermitian met-ricωu=π∗(e uωS)+(dx+π∗α1)∧(dy+π∗α2)is conformal balanced.Actually(ωu,Ω)satisfies equation(2.5).THE THEORY OF SUPERSTRING 377Goldstein and Prokushkin also studied the cohomology of this non-K¨a hler manifold X :h 1,0(X )=h 1,0(S ),h 0,1(X )=h 0,1(S )+1;In particular,h 0,1(X )=h 1,0(X )+1.Moreover,b 1(X )=b 1(S )+1,when ω2=nω1,b 1(X )=b 1(S ),when ω2=nω1;b 2(X )=b 2(S )−1,when ω2=nω1,b 2(X )=b 2(S )−2,when ω2=nω1andχ(X )=0.The above topological results can be explained as follows.Let L 1be a holomorphic line bundle over S with the first Chern class c 1(L 1)=[−ω12π].Then we can choose a Hermitian metric h 1on L 1such that its curvature is √−1ω1.Let S 1={v ∈L 1|h 1(v,v )=1}which is a circle bundle over S .Locally we write ω1=dα1U for some real 1-form α1U on some open subset U on S .Such α1U define a connection on S 1,i.e.,there is a section ξU on S 1such that ξU =√−1α1U ⊗ξU .The section ξU defines a local coordinate x U on fibers of S 1|U ,i.e.,we can describe the circle S 1by e √−1x U ξU .If we write ω1=dα1V onanother open set V of S ,then there is another section ξV such that (4.3) ξV =√−1α1V ⊗ξVand this section ξV defines another coordinate x V on fiber of S 1|V .On U ∩V ,d (α1U −α1V )=0and there is a function f UV such that (4.4)d f UV =α1U −α1V .On the other hand,on U ∩V ,there is also a function g UV on U ∩V such that ξV =e √−1g UV ξU .We computeξV = (e √−1g UV ξU )=(√−1dg UV +√−1α1U )⊗(e √−1g UV ξU )=(√−1dg UV +√−1α1U )⊗ξV .Comparing the above equality with (4.3),we get(4.5)−dg UV =α1U −α1V .378J.-X.FU &S.-T.YAUSo combining (4.4),we find(4.6)g UV =f UV +c UV ,where c UV is some constant on U ∩V .On U ∩V ,from eix U ξU =e ix V ξV =e √−1x V e √−1g UV ξU ,we obtain(4.7)x U =x V +g UV +2kπ=x V +f UV +c UV +2kπ.(4.4)and (4.7)imply (4.8)dx U −dx V =d f UV =−α1U +α1V .So dx U +α1U is a globally defined 1-form on X .We denote it by dx +α1.We construct another line bundle L 2with the first Chern class [−ω22π].Similarly,we write locally ω2=dα2,and define a coordinate y on fibers such that dy +α2is a well-defined 1-form on the circle bundle S 1of L 2.On X ,ω1=d (dx +α1)and ω2=d (dy +α2),and so [ω1]=[ω2]=0∈H 2(X,R ).When ω2=nω1,d (n (dx +α1)−(dy +α2))=0.So[n (dx +α1)−(dy +α2)]∈H 1(X,R ).Finally we define θ=dx +α1+√−1(dy +α2).Then θis a (1,0)-form on X ,see [10]or the next section.Because d ¯θ=ω1−√−1ω2is a (1,1)-form on X ,its (0,2)-component ¯∂¯θ=0.So [¯θ]∈H 0,1¯∂(X )∼=H 1(X,O ).5.The calculation of tr R ∧RIn order to calculate the curvature R and tr R ∧R ,we express the Hermitian metric (4.1)in terms of a basis of holomorphic (1,0)vector fields.Hence we need to write down the complex structure of X .Let {U,z j =x j +√−1y j ,j =1,2}be a local coordinate in S .The horizontallifts of vector fields ∂∂x j and ∂∂y j ,which are in the kernel of dx +π∗α1and dy +π∗α2,are X j =∂∂x j −α1 ∂∂x j ∂∂x −α2 ∂∂x j ∂∂yfor j =1,2,Y j =∂∂y j −α1 ∂∂y j ∂∂x −α2 ∂∂y j ∂∂yfor j =1,2.The complex structure ˜Ion X is defined as ˜IXj =Y j ,˜IY j =−X j ,for j =1,2,˜I ∂∂x =∂∂y ,˜I ∂∂y =−∂∂x .THE THEORY OF SUPERSTRING 379LetU j=X j −√−1˜IX j =X j −√−1Y j ,U 0=∂∂x −√−1˜I ∂∂x =∂∂x −√−1∂∂y .Then {U j ,U 0}is the basis of the (1,0)vector fields on X .The metric (4.1)takes the following Hermitian form:(5.1) (g i ¯j )001as U 1and U 2are in the kernel of dx +π∗α1and dy +π∗α2.Let (5.2)θ=dx +√−1dy +π∗(α1+√−1α2).It’s easy to check that {π∗dz j ,θ}annihilates the {U j ,U 0}and is the basis of (0,1)-forms on X .So {π∗dz j ,θ}are (1,0)-forms on X .Certainly π∗dz j are holomorphic (1,0)-forms and θis not.We need to construct another holomorphic (1,0)-form on X .Because ω1and ω2are harmonic forms on S ,∂ω1=∂ω2=0.By ¯∂-Poincar´e Lemma,locally we can find (1,0)-forms ξ=ξ1dz 1+ξ2dz 2and ζ=ζ1dz 1+ζ2dz 2on S ,where ξi and ζj are smooth complex functions on some open set of S ,such that ω1=∂ξand ω2=∂ζ.Let θ0=θ−π∗(ξ+√−1ζ)=(dx +√−1dy )+π∗(α1+√−1α2)−π∗(ξ+√−1ζ).We claim that θ0is a holomorphic (1,0)-form.By our construction,θ0is the (1,0)-form.But dθ=d (dx +√−1dy +π∗(α1+√−1α2))=π∗(ω1+√−1ω2)is a (1,1)-form on X .So(5.3)∂θ=0and ∂θ=dθ=π∗(ω1+iω2).Thus,∂θ0=∂θ−∂π∗(ξ+√−1ζ)=π∗(ω1+√−1ω2)−π∗(ω1+√−1ω2)=0.So θ0is a holomorphic (1,0)-form and {π∗dz j ,θ0}forms a basis of holo-morphic (1,0)-forms on X .Letϕj =ξj +√−1ζj for j =1,2and˜U j =U j +ϕj U 0for j =1,2.Then {˜Uj ,U 0}is dual to {π∗dz j ,θ0}because U j is in the kernel of θ.It’s the basis of holomorphic (1,0)-vector fields.The metric g 0then380J.-X.FU&S.-T.YAU becomes the following Hermitian matrix:(5.4)H X=⎛⎝g1¯1+|ϕ1|2g1¯2+ϕ1ϕ2ϕ1g2¯1+ϕ2ϕ1g2¯2+|ϕ2|2ϕ2ϕ1ϕ21⎞⎠=g+B·B∗BB∗1,where g is the Calabi-Yau metric on S and B=(ϕ1,ϕ2)t.According to Strominger’s explanation in[20],when the manifold is not K¨a hler,we should take the curvature of Hermitian connection on the holomorphic tangent bundle T ing the metric(5.4),we computethe curvature to beR=∂(∂H X·H−1X)=R1¯1R1¯2R2¯1R2¯2,whereR1¯1=R S+∂B∧(∂B∗·g−1)+B·∂(∂B∗·g−1),R1¯2=−R S B+(∂g·g−1)∧∂B−∂B∧(∂B∗·g−1)B−B∂(∂B∗·g−1)B+B(∂B∗·g−1)∧∂B+∂∂B, R2¯1=∂(∂B∗·g−1),R2¯2=−∂(∂B∗·g−1)B+(∂B∗·g−1)∧∂B,and R S is the curvature of Calabi-Yau metric g on S.It is easy to check that tr(∂B∧(∂B∗·g−1)+B·∂(∂B∗·g−1))−∂(∂B∗·g−1)B+(∂B∗·g−1)∧∂B=0.So tr R=π∗tr R S.Proposition6([11]).The Ricci forms of the Hermitian connections on X and S have the relation tr R=π∗tr R S.Remark7.In the above calculation,we don’t use the condition that the metric g on S is Calabi-Yau.Proposition8.(5.5)tr R∧R=π∗(tr R S∧R S+2tr∂∂(∂B∧∂B∗·g−1)).Proof.Fix any point p∈S,we pick B such that B(p)=0.Otherwise, B(p)=0and we simply replace B by B−B(p).Hence in the calculation of tr R∧R at p,all terms containing the factor B will vanish.Thus tr R∧R=tr R S∧R S+2tr R S∧∂B∧(∂B∗·g−1)+2tr∂g·g−1∧∂B∧¯∂(∂B∗·g−1)+2tr∂∂B∧∂(∂B∗·g−1). Proposition8follows from the next two lemmas.q.e.d.THE THEORY OF SUPERSTRING381 Lemma9.tr∂∂(∂B∧∂B∗·g−1)=tr R S∧∂B∧(∂B∗·g−1)+tr∂g·g−1∧∂B∧¯∂(∂B∗·g−1)+tr∂∂B∧∂(∂B∗·g−1).Proof.tr∂∂(∂B∧∂B∗·g−1)=−tr∂(∂B∧∂(∂B∗·g−1))=tr∂∂B∧∂(∂B∗·g−1)+tr∂B∧∂(∂B∗∧∂g−1)=tr∂∂B∧∂(∂B∗·g−1)−tr∂B∧∂(∂B∗·g−1∧∂g·g−1)=tr∂∂B∧∂(∂B∗·g−1)−tr∂B∧∂(∂B∗·g−1)∧∂g·g−1+tr∂B∧(∂B∗·g−1)∧¯∂(∂g·g−1)=tr∂∂B∧∂(∂B∗·g−1)−tr∂B∧∂(∂B∗·g−1)∧∂g·g−1+tr∂B∧(∂B∗·g−1)∧R S=tr(∂∂B∧∂(∂B∗·g−1))+tr(R S∧∂B∧∂B∗·g−1)+tr(∂g·g−1∧∂B∧∂(∂B∗·g−1).q.e.d. Lemma10.tr(¯∂B∧∂B∗·g−1)is a well-defined(1,1)-form on S.Proof.We take local coordinates(U,z i)and(W,w j)on S such that U∩W=∅.Let J=(∂w i∂z j)and(ω1+√−1ω2)|U=∂(ϕ1dz1+ϕ2dz2)=∂ϕ1∧dz1+¯∂ϕ2∧dz2,(ω1+√−1ω2)|W=∂(γ1dw1+γ2dw2)=∂γ1∧dw1+¯∂γ2∧dw2.Then on U∩W,¯∂γ1¯∂γ2∧dw1dw2=¯∂ϕ1¯∂ϕ2∧dz1dz2.So¯∂ϕ1¯∂ϕ2=¯∂γ1¯∂γ2J.(5.6)On the other hand,we have(5.7)g(z)=J t g(w)J,382J.-X.FU &S.-T.YAUwhere g (z )=(g i ¯j (z ))and g (w )=(g i ¯j (w )).Then on U ∩W ,using (5.6),(5.7),we havetr ¯∂γ1¯∂γ2∧ ∂¯γ1∂¯γ2 ·g −1(w )=tr(J t )−1 ¯∂ϕ1¯∂ϕ2 ∧ ∂ϕ1∂ϕ2 ¯J −1·¯J ·g −1(z )·J t =tr ¯∂ϕ1¯∂ϕ2 ∧ ∂¯ϕ1∂¯ϕ2 ·g −1(z ),which proves that tr(¯∂B∧∂B ∗·g −1)is a well-defined (1,1)-form on S .q.e.d.Although tr(¯∂B∧∂B ∗·g −1)is a well-defined (1,1)-form on S ,we can not express it by ω1and ω2.But in some particular cases,we can.Proposition 11.When ω2=nω1,n ∈Z ,(5.8)tr(¯∂B ∧∂B ∗·g −1)=√−14(1+n 2) ω1 2ωS ωS ,where g is the given Calabi-Yau metric on S and ωS is the corresponding K¨a hler form.Proof.We recall that locally,ω1=¯∂ξ,ξ=ξ1dz 1+ξ2dz 2,ω2=¯∂ζ,ζ=ζ1dz 1+ζ2dz 2,ϕj =ξj +√−1ζj ,for j =1,2,B = ϕ1ϕ2,B ∗= ¯ϕ1¯ϕ2 .When ω2=nω1,we take ζ=nξ.Then ¯∂ζj =n ¯∂ξj ,¯∂B= ¯∂ϕ1¯∂ϕ2 =(1+n √−1) ¯∂ξ1¯∂ξ2 and∂B ∗= ∂¯ϕ1∂¯ϕ2 =(1−n √−1) ∂¯ξ1∂¯ξ2 .THE THEORY OF SUPERSTRING 383Using above equalities,we find(5.9)tr(¯∂B ∧∂B ∗·g −1)=(1+n 2)tr ¯∂ξ1¯∂ξ2 ∧ ∂¯ξ1∂¯ξ2 ·g −1=1+n 2det g tr ∂ξ1∂¯z i d ¯z i ∂ξ2∂¯z i d ¯z i ∧ ∂ξ1∂¯z j dz j ∂ξ2∂¯z j dz j · g 2¯2−g 1¯2−g 2¯1g 1¯1 =1+n 2det g tr ∂ξ1∂¯z i ∂ξ2∂¯z i∧ ∂ξ1∂¯z j ∂ξ2∂¯z j · g 2¯2−g 1¯2−g 2¯1g 1¯1 d ¯z i ∧dz j .In order to get the global formula,we need to calculate ω1.As ω1is real,(5.10)∂ξi ∂¯z j =−∂ξj ∂¯z i for i,j =1,2.Since ω1is anti-self-dual,i.e.,ω1∧ωS =0,we have(5.11)g 1¯1∂ξ2∂¯z 2+g 2¯2∂ξ1∂¯z 1−g 1¯2∂ξ2∂¯z 1−g 2¯1∂ξ1∂¯z 2=0.Because(5.12)ω1∧ω1=−ω1∧∗ω1=−ω1∗¯ω1=− ω1 2ωS ω2S 2!,locally we also have (5.13)1det(g ) ∂ξ1∂¯z 1∂ξ2∂¯z 2−∂ξ1∂¯z 2∂ξ2∂¯z 1=18ω1 2ωS .Now using above (5.10),(5.11)and (5.13),we calculate the component of d ¯z 1∧dz 1in (5.9)to be(5.14)1+n 2det(g ) g 2¯2∂ξ1∂¯z 1∂ξ1∂¯z 1−g 2¯1∂ξ1∂¯z 1∂ξ2∂¯z 1−g 1¯2∂ξ2∂¯z 1∂ξ1∂¯z 1−g 1¯1∂ξ2∂¯z 1∂ξ2∂¯z 1 =1+n 2det(g ) ∂ξ1∂¯z 1 g 1¯1∂ξ2∂¯z 2+g 2¯2∂ξ1∂¯z 1 −g 2¯2 ∂ξ1∂¯z 1 2−g 1¯1∂ξ2∂¯z 1∂ξ1∂¯z 2 =1+n 2det(g )g 1¯1 ∂ξ1∂¯z 1∂ξ2∂¯z 2−∂ξ2∂¯z 1∂ξ1∂¯z 2=1+n 28ω1 2ωS g 1¯1.Similarly,the components of d ¯z 2∧dz 1,d ¯z 1∧dz 2and d ¯z 2∧dz 2in (5.9)are 1+n 28 ω1 2ωS g 1¯2,1+n 28 ω1 2ωS g 2¯1and 1+n 28ω1 2ωS g 2¯2respectively.384J.-X.FU &S.-T.YAUSo we obtaintr(¯∂A ∧∂A ∗·g −1)=√−11+n 24ω1 2ωS ωS .q.e.d.6.Reduction of Strominger’s systemConsider a 3-dimensional Hermitian manifold (X,ω0,Ω)as described in the section 2.Let ωS be the Calabi-Yau metric on S .Let θ=dx +α1+√−1(dy +α2),then the Hermitian form ω0in (4.2)is ω0=π∗ωS +√−12θ∧¯θ.Because Ω ω=1,and ω1and ω2are anti-self-dual,we use (5.3)to computed ( Ω ω0ω20)(6.1)=d (π∗ω2S +√−1π∗ωS ∧θ∧¯θ)=√−1π∗ωS ∧dθ∧¯θ−√−1π∗ωS ∧θ∧d ¯θ=√−1π∗ωS ∧((ω1+√−1ω2)∧¯θ−(ω1−√−1ω2)∧θ)=0.According to Lemma 1,(ω0,Ω)is the solution of equation (2.4).Let u be any smooth function on S and let (6.2)ωu =π∗(e u ωS )+√−12θ∧¯θ.Then Ω 2ωu =ω30ω3u =1e2u andΩ ωu ω2u =ω20+(e u −1)ω2S .Using (6.1),we obtaind ( Ω ωu ω2u)=dω20+d (e u −1)∧ω2S =0because e u is a function on S .Hence we have proven the followingLemma 12([10]).The metric (6.2)defined on X satisfies equation (2.5)and so satisfies equation (2.4).Let V be a stable vector bundle over X with degree 0with respec-tive to the metric ωu .According to Li-Yau’s theorem [16],there is a Hermitian-Yang-Mills metric H on V ,which is unique up to constant.THE THEORY OF SUPERSTRING 385Then (V,H,X,ωu )satisfies equation (2.1),(2.2)and (2.4)of the Stro-minge’s system.Hence to look for a solution to Strominger’s system,we need only to consider equation (2.3):(6.3)√−1∂¯∂ωu =α 4(tr R u ∧R u −tr F H ∧F H ),where R u is the curvature of Hermitian connection of metric ωu on the holomorphic tangent bundle T X .Define the Laplacian operator with respective to the metric ωS asψω2S 2!=√−1∂¯∂ψ∧ωS .Lemma 13.√−1∂¯∂ωu = e u ·ω2S 2!+12( ω1 2ωS + ω2 2ωS )ω2S 2!ing (5.3)and (5.12),we compute √−1∂∂ωu =√−1∂∂(e u ωS +√−12θ∧θ)=√−1∂¯∂e u ∧ωS −12¯∂θ∧∂¯θ= e u ·ω2S 2!+12( ω1 2ωS + ω2 2ωS )ω2S 2!.q.e.d.Lemma 14.tr R u ∧R u =π∗tr R S ∧R S +2√−1π∗(∂¯∂u ∧∂¯∂u )+2π∗(∂¯∂(e −u ρ)),where ρ=−√−1tr(¯∂B∧∂B ∗·g −1).Proof.In the proof of Proposition 8we don’t use the condition thatωS is K¨a hler.So if we replace metric g by e u g ,we can still obtain:(6.4)tr R u ∧R u =π∗(tr R u S∧R u S +2√−1∂¯∂(e −u ρ)),here R u S denotes the curvature of Hermitian connection of the metric e u g on holomorphic tangent bundle T S .SoR u S=¯∂∂u ·I +R S and(6.5)tr R u S∧R u S =tr R S ∧R S +2∂¯∂u ∧∂¯∂u,here we use the fact that tr R S =0because the Hermitian metric g is the Calabi-Yau metric on S .Inserting (6.5)into (6.4),we have proven the lemma.q.e.d.From Lemma 13and 14,we can rewrite equation (6.3)as(6.6)√−1∂¯∂e u ∧ωS −√−1α 2∂¯∂(e −u ρ)−α 2∂¯∂u ∧∂¯∂u =α 4tr R S ∧R S −α 4tr F H ∧F H −12( ω1 2+ ω2 2ωS )ω2S 2!.。
弦理论(原稿)(1)
量子遇上引力20世纪发展出了两套现代物理学的基础，量子力学和广义相对论。
广义相对论描述宏观物体的运动，量子力学描述微观粒子的运动，它们在各自的领域都取得了巨大的成功，然而，当量子力学和广义相对论结合时，却出现了问题。
20世纪20年代，狄拉克将量子力学与狭义相对论结合，创立了量子场论，这是后来标准模型的雏形。
量子场论将粒子看做场的激发，粒子的概率波其实就是弥漫在空间中的场，粒子在某一位置某一时刻的出现就是场在那一时刻那一位置的激发。
利用这一点，理论物理学家创立了描述微观下电磁相互作用的量子电动力学（QED），为了使QED有定量计算的意义，物理学家理查德.费曼提出了费曼图，就是将粒子的运动轨迹和相互作用情况画成一张图，当发生电磁相互作用时，激发出的光子的路径与原粒子的路径的交点称为顶点，利用费曼图，可以计算顶点处的量子数，这就使QED能够进行定量计算。
然而，由于量子力学的多重历史性（即在出现相同结果时，过程不一定相同），需要将所有的历史（过程）叠加，然而，历史的个数是无穷个，因此，最后计算的结果必然是无穷大。
然而，无穷大是没有意义的，于是，费曼又发明了重整化方法。
费曼发现，由于QED的耦合常数（即理论里相互作用的强度，耦合可以理解为相互作用）小于1，这就使得每一阶的计算（即每一张费曼图的计算）越往后对结果的影响越来越小，因此，只用计算前几阶的结果就能得到相当精确的结果，这就是重整化方法，重整化使QED中的无穷大消除了。
然而，当物理学家将引力用上述方法与量子力学结合时，在尺度较大时，还能够成功，然而，当研究的尺度小到被称为普朗克长度时，理论中的耦合常数突然大于1，也就是说，重整化不再适用，量子力学与广义相对论在普朗克尺度下的结合是无穷大，显然，现代物理学的两大基石-----量子力学和广义相对论的统一失败了。
可是，这又有什么关系呢？广义相对论和量子力学研究的对象看起来完全不同，广义相对论研究大质量的物体，量子力学研究小尺度的物体，然而，在我们的宇宙中，恰恰有既是沉的，又是小的东西，比如黑洞的奇点，以及宇宙大爆炸之初，量子力学和广义相对论结合的失败是我们无法探求宇宙中的极端的物质，更关键的是，宇宙为什么要有两套法则？400年的物理研究使得物理学家相信，宇宙必然由一套法则来支配，因此，也必然有一个理论可以解释所有的自然现象。
溴到硼酸酯
Materials Chemistry C
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Cite this: J. Mater. Chem. C, 2015, 3, 861
However, these oligouorene functionalized oligomers may suffer from the unwanted long wavelength emission under long-term device operation, similar to polyuorene-based macromolecules.34–36
Received 26th September 2014 Accepted 17th November 2014 DOI: 10.1039/c4tc02173h /MaterialsC
Starburst 4,40,400-tris(carbazol-9-yl)triphenylamine-based deep-blue fluorescent emitters with tunable oligophenyl length for solution-processed undoped organic lightemitting diodes†
Introduction
Since 1987, organic light-emitting diodes (OLEDs) have attracted tremendous attention due to their potential applications in at-panel displays and solid-state lightings.1–10 To achieve fullcolor displays, three primary RGB luminescent materials with excellent stability, efficiency and color purity are required. With respect to green and red counterparts, it seems to be a big challenge to develop blue emitters, especially deep-blue ones that have a Commission Internationale de l'Eclairage (CIE) y coordinate value of <0.10, because the intrinsic wide bandgap would inevitably result in inefficient charge injection to an emitting layer (EML).11 Therefore, great efforts should be paid to the design of deep-blue emitters to push forward the commercialization of full-color OLEDs.