On (2+1) Dimensional Topologically Massive Non-linear Electrodynamics

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Junctions of three quantum wires

2 tion on the nature of the ﬁxed points for a range of the parameter g , and that diﬀerent methods that we describe in the paper are needed to shed light onto the nature of the strong coupling ﬁxed points. In Sec. VI we establish the mapping of the Y-junction model onto the DHM and deduce its consequences for the phase diagram of the Yjunction model. In Sec. VII we introduce the study of the system by looking at the boundary conditions satisﬁed by the bosonic ﬁelds at the junction. In Sec. VIII we discuss the twisted structure of the Hilbert space that is manifest in the compactiﬁcation of the bosons so as to take proper care of the fermionic statistics of the electrons in the wires. In Sec. IX we introduce what we refer to as the method of delayed evaluation of boundary conditions (DEBC), which is less reliant on the apparatus of BCFT. In Sec. X we apply all the results from the previous sections to analyze the RG ﬁxed points in the junction problem. We also show in this section that asymmetries in the couplings between the three wires are irrelevant at low energies and temperatures, so the simple Z3 symmetric model of the system may indeed be a good description of realistic junctions. In Sec. XI we use simple arguments based on energy conservation to obtain constraints on the conductance tensor. We show that all the ﬁxed points that we are able to understand in this paper correspond to physical situations where no energy is transfered to neutral (non-charge carrying) modes in the quantum wires. In Sec. XII we discuss how the conductance tensors are modiﬁed if the Luttinger liquid wires are connected to Fermi liquid leads far from the junction. Depending on details of the potential experiment, this may be a better model. In Sec. XIII we present and discuss open problems. In Appendix A we give results on the non-interacting version of our model. In Appendix B we review boundary conformal ﬁeld theory. In Appendix C we review the boundary conformal ﬁeld theory of standard free bosons. In Appendix D we review the boundary conformal ﬁeld theory approach to calculating the conductance. In Appendix E we discuss the related problem of a Y-junction of quantum Hall edge states.

一些随机矩阵的数值范围

Numerical range of some doubly stochastic matricesqKristin A.Camenga a ,Patrick X.Rault b ,Daniel J.Rossi b ,Tsvetanka Sendova c ,Ilya M.Spitkovsky d ,⇑aDepartment of Mathematics and Computer Science,Houghton College,Houghton,NY 14744,USA bDepartment of Mathematics,State University of New York at Geneseo,Geneseo,NY 14454,USA cDepartment of Mathematics and Computer Science,Bennett College,Greensboro,NC 27401,USA dDepartment of Mathematics,College of William and Mary,Williamsburg,VA 23187,USAa r t i c l e i n f o Keywords:Numerical rangeDoubly stochastic matricesa b s t r a c tA classiﬁcation of all possible shapes is given for numerical ranges of 4Â4doubly stochas-tic matrices.The tests determining the shape are also provided,along with illustrating examples.Ó2013Elsevier Inc.All rights reserved.1.IntroductionWith F standing for either the ﬁeld R of real or C of complex numbers,we denote by F n Âm the linear space (algebra,if m ¼n )of the n Âm matrices with entries in F .We also use the standard abbreviation F n :¼F n Â1,and supply C n with the innerproduct h x ;y i ¼P n j ¼1x j y j and the respective norm k x k :¼h x ;x i1=2.The numerical range of A 2C n Ânby deﬁnition isW ðA Þ¼fh Ax ;x i :k x k ¼1g :This is a compact convex subset of C containing the spectrum r ðA Þof A ;see e.g.[1]or [2,Chapter 1]for these and other fundamental properties of W ðA Þ.We will repeatedly be using the following:(i)W ðA Þis unitarily invariant:W ðU T AU Þ¼W ðA Þfor any unitary U 2C n Ân .(ii)If A is unitarily reducible,that is,unitarily similar to the direct sum A 1ÈA 2,then W ðA Þis the convex hull ofW ðA 1Þ[W ðA 2Þ;that is W ðA Þ¼conv f W ðA 1Þ;W ðA 2Þg .From these properties it follows in particular that W ðA Þ¼conv r ðA Þwhenever A is normal.On the other hand,W ðA Þis an ellipse with the foci at the eigenvalues of A if A 2C 2Â2and is not normal (the Elliptical Range Theorem).The case n ¼3has also been treated fully,see [3]and its English translation [4]for the classiﬁcation,and [5–7]for the pertinent tests.However,starting with n ¼4,the shape of W ðA Þis known only for some special classes of matrices.Here we concentrate our attention on doubly stochastic matrices,that is,matrices A 2R n Ân for which the entries are non-negative while the row and column sums are all equal to one:Xn j ¼1a ij ¼1for i ¼1;...;n ;and Xn i ¼1a ij ¼1for j ¼1;...;n :ð1:1Þ0096-3003/$-see front matter Ó2013Elsevier Inc.All rights reserved./10.1016/j.amc.2013.06.011qThe work on this paper was begun during the REUF workshop at the American Institute of Mathematics in July 2011,supported by the NSF.⇑Corresponding author.E-mail addresses:kristin.camenga@ (K.A.Camenga),rault@ (P.X.Rault),tsendova@ (T.Sendova),ilya@math. ,imspitkovsky@ (I.M.Spitkovsky).Observe that (1.1)can be interpreted as e ¼½1;...;1 T being an eigenvector of both A and A T ,corresponding to the eigenvalue one.Thus,doubly stochastic matrices are unitarily reducible to the direct sum of ð1Þwith some A 12R ðn À1ÞÂðn À1Þ:U T AU ¼ð1ÞÈA 1;ð1:2Þwhere U is an orthogonal matrix with the ﬁrst column equal 1ﬃﬃp e .Consequently,W ðA Þin this case is the convex hull of W ðA 1Þwith the point 1.Moreover,1is a corner point of W ðA Þ[8].If A (equivalently,A 1)is normal,then W ðA Þis the convex hull of r ðA Þ¼r ðA 1Þ[f 1g .Thus we are only interested in the case of non-normal A 1.This,combined with the Elliptical Range Theorem,immediately implies a complete description of numerical ranges for 3Â3doubly stochastic matrices,normal or otherwise;see [9]for details.Our aim in this paper is to realize a similar plan for n =4.According to [4],for a non-normal A 1there are a priori three possibilities:(i)W ðA 1Þis the convex hull of a point and an ellipse (with the point lying either inside or outside the ellipse),(ii)the boundary of W ðA 1Þcontains a ﬂat portion,with the rest of it lying on a 4th degree algebraic curve,or (iii)W ðA 1Þhas an ovular shape,bounded by a 6th degree algebraic curve.Since W ðA 1Þalso is symmetric with respect to the real axis (as A 1has real entries),the ellipse in case (i)has one of its axes on R ,and the ﬂat portion in case (ii)is vertical.In the forthcoming sections we provide criteria to distinguish between the three possibilities,show that each of them actually materializes,and describe the respective shapes of each W ðA Þ.To simplify the terminology,we label the shape of W ðA Þby the degree of a non-ﬂat portion of its boundary.2.Numerical range of 2nd degreeIn this section we characterize the case when the non-ﬂat portion of @W ðA Þis an elliptical arc.Theorem 1.Let A be a 4Â4doubly stochastic matrix.Then @W ðA Þconsists of elliptical arcs and line segments if and only ifl :¼tr A À1þtr A 3Àtr ðA T A 2Þtr ðA T A ÞÀtr A 2ð2:1Þis an eigenvalue of A (multiple,ifl ¼1).If,in addition,tr A À1À3l >0;ðtr A À1À3l Þ2Àtr ðA T A Þþ1þ2ðdet A Þ=l þl 2>0;ð2:2Þthen W ðA Þalso has corner points at l and 1,and thus four ﬂat portions on the boundary.Otherwise,1is the only corner point of W ðA Þ,and @W ðA Þconsists of two ﬂat portions and one elliptical arc.Proof.Due to (1.2),the right hand side of (2.1)coincides with tr A 1þtr A 31Àtr ðA T 1A 21Þtr ðA T 1A 1ÞÀtr A 21.From [7,Theorem 5],it then follows thatW ðA 1Þis the convex hull of an ellipse and a point if and only if l is an eigenvalue of A 1.This proves the ﬁrst statement of the theorem.Further,recall from [5]that if the above condition on l is satisﬁed,then W ðA 1Þ¼conv f E ;l g ,where E is an ellipse withfoci at two other eigenvalues k 1;k 2of A 1and the minor axis d ¼tr A T 1A 1Àj k 1j 2Àj k 2j 2Àl 2.So,W ðA Þ¼conv f E ;l ;1g has two corner points (at 1and l )if l lies to the left of E ,and only one corner point (at 1)otherwise.Now observe that l lies to the left of E if and only if it is located in the exterior of E ,that is,ðj k 1Àl j þj k 2Àl jÞ2Àj k 1Àk 2j 2>d ;ð2:3Þand to the left of its center ðk 1þk 2Þ=2.The second condition is equivalent to3l <k 1þk 2þl ;where the right hand side is of course tr A À1.Collecting all the terms on the same side of the inequality,we arrive at the ﬁrst inequality in (2.2).It remains to show the equivalence of its second inequality and (2.3).To this end,we consider separately the case of complex conjugate and real k j .Case 1.k 1¼k 2¼:a þbi .Inequality (2.3)then can be rewritten as4ða Àl Þ2þb 2Àð2b Þ2>tr ðA T 1A 1ÞÀ2ða 2þb 2ÞÀl 2:Simplifying the left side and right side individually yieldsðtr A 1À3l Þ2>tr ðA T 1A 1ÞÀ2ðdet A 1Þ=l Àl 2:Putting this in terms of A ,we arrive at the desired second inequality in (2.2).Case 2.All eigenvalues of A 1are real.Recalling that l lies to the left of the eigenvalues,inequality (2.3)becomesðk 1þk 2À2l Þ2Àðk 1Àk 2Þ2>tr ðA T 1A 1ÞÀk 21Àk 22Àl 2:K.A.Camenga et al./Applied Mathematics and Computation 221(2013)40–4741Working with the left hand side gives4l2þ4k1k2À4ðk1þk2Þl>trðA T1A1ÞÀk21Àk22Àl2:Rewriting this and collecting terms yields5l2þ2k1k2À4ðtr A1ÀlÞlþðk1þk2Þ2ÀtrðA T1A1Þ>0:After putting this in terms of A and factoring gives5l2þ2ðdet AÞ=lþðtr AÀ1ÀlÞðtr AÀ1À5lÞÀtrðA T AÞþ1>0:Noting thatðtr AÀ1À3lÞÀ2lðÞðtr AÀ1À3lÞþ2lðÞ¼ðtr AÀ1À3lÞ2À4l2completes the proof.hThe following examples demonstrate that each case considered by Theorem1actually occurs. Example1.Consider the doubly stochastic matrixA¼011510111917537019B BB@1C CC A:Using equation(2.1),we compute l¼À1=3.The characteristic polynomial of A is1=128Àð11xÞ=192Àð193x2Þ=384Àð43x3Þ=96þx4;which evaluated atÀ1=3is0.In addition,we compute the formulas in inequalities(2.2)astr AÀ1À3l¼43=96andðtr AÀ1À3lÞ2ÀtrðA T AÞþ1þ2ðdet AÞ=lþl2¼À779=9216:Since the latter is negative,Theorem1states that@WðAÞhas twoﬂat portions and one elliptical arc.Indeed,A reduces uni-tarily toð1ÞÈA1for some3Â3matrix A1,and in Fig.1we give WðAÞ,and the horizontal ellipse it contains,WðA1Þ. Example2.Next,we give a similar example where the elliptical arc is part of a vertical ellipse:A¼0131451213012161418143851213240124B BB@1C CC A:By coincidence,we again compute l¼À1=3,and though the characteristic polynomial is now Numerical range of A with twoﬂat portions and an elliptical arc on the boundary,and the ellipse it contains.The eigenvalues of A are indicated 42K.A.Camenga et al./Applied Mathematics and Computation221(2013)40–47Example3.Next,we give an example of a doubly stochastic matrix,A,where@WðAÞconsists of fourﬂat portions and two elliptical arcs.Let a be the second largest root of the four real roots of the polynomial À28995327þ108880894xþ143424512x2À778960896x3þ603979776x4and let b be the second smallest of the four real roots ofÀ5072þ17953xþ310274x2À1013248x3þ786432x4.WithA¼0a13Àa1011b11191Àb2Àb255Àa0À509þbþa 0B BB@1C CC A;the characteristic polynomial evaluated at l is identically0.The two formulas in inequality(2.2)come out to two positive numbers,roughly0:65and0:31respectively.Hence WðAÞis the convex hull of an ellipse and two points outside of the el-lipse.Fig.3shows WðAÞ,and that the ellipse is vertical.stly,we give an example of a doubly stochastic matrix,A,where WðAÞis the convex hull of a horizontal ellipse and two points.Let a be the second smallest root of the polynomialÀ301228513þ853379010xþ4421068800x2À17180393472x3þ14495514624x4and let b be the second largest root of the polynomial À10016À19031xþ1045714x2À3505152x3þ3145728x4.Just as in the previous examples,the characteristic polynomial of the matrixA¼0a13Àa1011b10951214275512Àb23Àb403512Àa0À6971536þbþaB BB@1C CC Arange A with twoﬂat portions and an elliptical arc on the boundary,and the contained vertical ellipse.The eigenvaluesvanishes at l .In this case the two formulas in inequality (2.2)come out to two positive numbers of about 0:77and 0:21respectively.Hence W ðA Þis the convex hull of an ellipse and two points outside of the ellipse and has four ﬂat portions.Fig.4shows W ðA Þ,and that in this case the ellipse is horizontal.3.Numerical range of 4th degreeBy Birkhoff’s Theorem,the doubly stochastic matrices form a convex polytope in R n Ân ,its vertices being the n !permuta-tion matrices.So,it is not surprising that the set of all doubly stochastic matrices A with the ﬁxed symmetric part H ¼ðA þA T Þ=2also forms a polytope.The latter is stated more rigorously in the following lemma.Lemma 2.Consider A 2R n Ân with given symmetric part H:A ¼H þK ;where K ¼ðA ÀA T Þ=2:ð3:1ÞThen A is doubly stochastic if and only if H is doubly stochastic,the row (and thus,column)sums of K are zero,and its upper diag-onal entries satisfyj k ij j 6h ij ;16i <j 6n :ð3:2ÞProof.Directly from (1.1)it follows that if A is doubly stochastic,then its symmetric part H is doubly stochastic as well.On the other hand,if H is doubly stochastic,then for A given by (3.1)to be doubly stochastic it is necessary and sufﬁcient that K has zero row/column sums and h ij Æk ij P 0.The latter condition is equivalent to (3.2).hNote that antisymmetric matrices with zero row/column sums form a ðn À1Þðn À2Þ-dimensional subspace of R n Ân .So,(3.2)can be rewritten as a system of n ðn À1Þ=2linear inequalities in ðn À1Þðn À2Þ=2variables.In particular,for n ¼4we haveK ¼x 001À10000À10110À102666437775þy 010À1À10100À1011À12666437775þz 010À1À1001001À102666437775;ð3:3Þand (3.2)takes the formj x j 6h 13;j y j 6h 23;j z j 6h 24;j x þy j 6h 34;j y þz j 6h 12;j x þy þz j 6h 14:ð3:4ÞFor the following theorem,recall our notation e for the vector with all the entries equal to one.Theorem 3.Let A 2R 4Â4be a doubly stochastic matrix.Then @W ðA Þconsists of 4th degree algebraic arcs and line segments if andonly if in its representation (3.1),H ¼k I þ1Àk 4ee T Æff Tð3:5Þfor some f 2R 4orthogonal to e,and k satisfyingÀ1ð1þ4min fÆf 2i gÞ6k 61þ4min i –j fÆf i f j g ;ð3:6ÞNumerical range of A with four ﬂat portions and elliptical arcs on the boundary,and the numerical range of the unitary reduction A 1of a horizontal ellipse and l .The eigenvalues are indicated by the points.while K is given by(3.3),satisﬁes(3.4),and has no eigenvectors lying in f f;e g?.The upper signs in(3.5)and(3.6)correspond to the case when@WðAÞhas threeﬂat portions,one of them being vertical and lying to the left of WðAÞ.Alternatively,if(3.5)and(3.6) hold with the lower signs,@WðAÞhas only twoﬂat portions,with1being their common endpoint.Proof.In order for@WðAÞto contain algebraic arc(s)of4th degree,the same should be true for@WðA1Þ,with A1as in(1.2). According to[5,Section3],this happens if and only if A1is unitarily irreducible and its real part H1has a multiple maximal or minimal eigenvalue k.Equivalently,k is a multiple eigenvalue of H from(3.1),which is either minimal or second in value after one.Along with the double stochasticity of H(see Lemma2),this implies(3.5)and(3.6).Conditions(3.3)and(3.4)on K follow from the description of all doubly stochastic matrices withﬁxed symmetric part (Lemma2again).Finally,A1is unitarily irreducible if and only if e is the only common eigenvector of H and K,that is,if K has no eigenvectors orthogonal to e and f.Suppose now that all the conditions(3.3)–(3.6)are satisﬁed.Then A1is unitarily irreducible and,according to[5],its numerical range has aﬂat portion on its boundary lying on the vertical line x¼k.Just as A reduces toð1ÞÈA1,we reduce H to ð1ÞÈH1.If k is the minimal eigenvalue of H1(that is,(3.5)and(3.6)hold with the upper signs),then this vertical portion is to the left of WðA1Þand thus of WðAÞ.Therefore,it is part of@WðAÞ,along with two otherﬂat portions,originating at1.On the other hand,for(3.5)and(3.6)holding with the lower signs,the vertical portion of@WðA1Þis to the right of WðA1Þand is thus absorbed by the interior of WðAÞ.hTo demonstrate that such matrices actually exist,we give examples of numerical ranges withﬂat portions on the bound-ary to illustrate the above theorem.Example5.Consider the following matrix.A¼20212012 03021220B BB@1C CC A:Eq.(3.1)becomes A¼HþK,where H¼ðAþA TÞ=2and K¼ðAÀA TÞ=2.H has three eigenspaces:theﬁrst generated by e, with eigenvalue1,the second generated by½À3;1;1;1 T with eigenvalue k2¼1,and the third a2-dimensional space gener-ated by½0;À1;0;1 T and½0;À1;1;0 T corresponding to the double eigenvalue k3¼À25¼puting HÀk3IÀ1Àk34ee T yieldsa symmetric positive semi deﬁnite matrixM¼920À320À320À320À320120120120À320120120120À320120120120B BB@1C CC Aof rank1,and the vector f from its representation M¼ff T is determined up to a sign and can be found via dividing any of theK.A.Camenga et al./Applied Mathematics and Computation221(2013)40–4745A ¼05121413111151216161411110B B B B B B @1C C C C CC A :The Hermitian part,H ,again has three eigenspaces:the ﬁrst generated by e ,with eigenvalue 1,the second generated by½À3;1;1;1 T with eigenvalue k 2¼À13,and the third a 2-dimensional space generated again by ½0;À1;0;1 T and ½0;À1;1;0 Tand now corresponding to the double eigenvalue k 3¼À1¼k puting H Àk 3I À1Àk 3ee T yields a symmetric negative semi deﬁnite matrix M of rank 1.Similarly to how this was done in Example 5,M ¼Àff T ,where f ¼14ﬃﬃ3p ½À3;1;1;1 T ,and thus H ¼k 3I þ1Àk 34ee T Àff T .Furthermore,note that inequality (3.6)is satisﬁed:À112 À112 1112.We conclude that when we reduce A to ð1ÞÈA 1,withA 1¼À1À1À10À1À100À1B @1C A ;@W ðA 1Þhas a vertical ﬂat portion on the right.The convex hull of W ðA 1Þand f 1g ,which is the numerical range of A ,does not have a vertical ﬂat portion.These numerical ranges are both illustrated in Fig.6.4.Numerical range of 6th degreeAll numerical ranges which do not fall into one of the above categories will have an ovular shaped boundary curve of de-gree 6.Example 7.In this example we will show that the following matrix does not satisfy the requirements of Theorems 1and 3.A ¼3101103103103101531015121333110B B B @1C C C A :We ﬁrst compute l ¼1=55in (2.1).The characteristic polynomial of A evaluated at l is nonzero,and indeed the ﬁrst of the inequalities in (2.2)does not hold.Theorem 1then implies that the numerical range of A is not the convex hull of an ellipse and two points.Next,H ¼12ðA þA T Þ¼31131171171131110B B B @1C C C A46K.A.Camenga et al./Applied Mathematics and Computation 221(2013)40–47A 1¼À111011À10À10B @1C A tells us that this ovular shape actually is W ðA 1Þ.Fig.7shows the numerical ranges of both A and A 1.References[1]K.E.Gustafson,D.K.M.Rao,Numerical Range,The Field of Values of Linear Operators and Matrices,Springer,New York,1997.[2]R.A.Horn,C.R.Johnson,Topics in Matrix Analysis,Cambridge University Press,Cambridge,1991.[3]R.Kippenhahn,Über den Wertevorrat einer matrix,Math.Nachr.6(1951)193–228.[4]R.Kippenhahn,On the numerical range of a matrix,Linear Multilinear Algebra 56(1–2)(2008)185–225(translated from the German by Paul F.Zachlin and Michiel E.Hochstenbach).[5]D.Keeler,L.Rodman,I.Spitkovsky,The numerical range of 3Â3matrices,Linear Algebra Appl.252(1997)115–139.[6]L.Rodman,I.M.Spitkovsky,3Â3matrices with a ﬂat portion on the boundary of the numerical range,Linear Algebra Appl.397(2005)193–207.[7]P.Rault,T.Sendova,I.M.Spitkovsky,3-by-3Matrices with elliptical numerical range revisited,Electron.Linear Algebra 26(2013)158–167.[8]C.R.Johnson,An inclusion region for the ﬁeld of values of a doubly stochastic matrix based on its graph,Aequationes Math.17(2–3)(1978)305–310.[9]P.Nylen,T.Y.Tam,Numerical range of a doubly stochastic matrix,Linear Algebra Appl.153(1991)161–176.。

史树中凸分析参考答案

史树中凸分析参考答案
什么是史树？
史树（Genealogy）实际上就是构成谱系的一系列个体（人类或非
人类）和家庭关系之间的联系。

它可以用来帮助我们了解和追踪某一
地区或特定家庭的历史、非人类物种的发展，也可以帮助我们研究某
些种族的起源。

史树也可以用来研究社会、法律、军事以及宗教等方
面的发展。

什么是凸分析？
凸分析（Convex Analysis）是一种统计分析方法，该方法旨在解
决多元数据的可视化以及分析凸形函数的构成组成和寻找最优解等问题。

此外，凸分析还可以应用于人类和非人类家谱关系（社会关系）、多项式函数以及信号处理等方面，旨在发现家谱中的凹凸变化或构成
团体等。

史树中凸分析参考答案
史树中凸分析的具体用途主要有以下几点：
1、可以用来分析两个特定家庭之间的关系，比如血缘关系、配偶
构成的最短距离等。

2、可以用来查明社会地位的差异，以及在家谱内人物彼此之间互
动以及家庭利益的分配。

3、可以用来研究跨越整个家族的形态，并分析家谱的发展过程，
以及家谱群体的构成。

4、可以用来研究多样化的家庭价值观以及家谱构成的对应结构，
以及家谱社会系统之间的关联。

5、可以用来研究社会发展，探讨多种变迁以及传承问题。

此外，史树中凸分析还可以作为更广泛的研究的重要参考工具，
比如比较社会结构和历史现象，理解物种适应以及物种进化的过程等。

单连通区域上全纯自同构的拓扑共轭分类

单连通区域上全纯自同构的拓扑共轭分类(摘要)本文主要介绍了单连通区域上全纯自同构的拓扑共轭分类.我们说两个变换f:X→X和g:Y→Y是拓扑共轭的,如果存在一个同胚h:X→Y使得h∘f=g∘h,这里∘是映射的复合.单连通区域主要包括:复平面,扩充复平面,单位圆盘和上半平面.关于复平面上全纯自同构的拓扑共轭分类问题,Budnitska得到了更一般的结论,即有限维向量空间上仿射算子的拓扑共轭分类.具体地讲,如果仿射算子f(x)=Ax+b有一个不动点,那么f拓扑共轭于它的线性部分A;如果仿射算子f:U→U无不动点,我们证明f拓扑共轭于一个仿射算子g:U→U,这里U是g 的不变子空间V和W的正交直和,g在V上的限制g|V是一个拥有如下形式的V的标准正交基底的仿射算子(x1,x2,...,x n)→(x1+1,x2,...,x n−1,εx n),ε=±1,它是由f唯一确定的,g在W上的限制g|W是一个通过幂零Jordan矩阵给出的W的标准正交基底的线性算子,是由f唯一确定的.对于扩充复平面上的分式线性变换的拓扑共轭分类问题, Rybalkina和S ergeichuk已经给出了相应的结果.我们在此整理并总结了复平面和扩充复平面上全纯自同构的拓扑共轭分类,关于单位圆盘和上半平面上全纯自同构的拓扑共轭之前没有得到完全的分类,本文将就这一问题给出答案.我们通过旋转理论及构造同胚的方法,证明了:上半平面上（单位圆盘）所有无不动点的全纯自同构之间都是拓扑共轭的;两个有不动点的全纯自同构f和g是拓扑共轭的当且仅当ρ(f)=±ρ(g),mod Z;无不动点的全纯自同构与有不动点的全纯自同构之间是不拓扑共轭的.关键词:拓扑共轭,全纯自同构,单位圆盘,上半平面,复平面,扩充复平面.Topologically conjugate classiﬁcations of the holomorphic isomorphisms on simple connected domains(Abstract)In this paper,we are interested in the topologically conjugate classiﬁcations of the holo-morphic isomorphisms on simple connected domains.We said that two transformations f: X→X and g:Y→Y are topologically conjugate if there exists a homeomorphism h:X→Y such that h∘f=g∘h,where∘is the composition of mappings.Simple connected do-mains includes:complex plane,extended complex plane,unit disk and upper half plane. For the problem of topologically conjugate classiﬁcations of the holomorphic isomorphisms on complex plane,Budnitska gave more general result.He obtained the topologically conjugate classiﬁcations of aﬃne operators onﬁnite dimensional vector space.Speciﬁcally speaking if the aﬃne operator f(x)=Ax+b has aﬁxed point,then f is topologically conjugate to its linear part A.If the aﬃne operator f:U→U has noﬁxed point,f is topologically conjugate to an aﬃne operator g:U→U,where U is an orthogonal direct sum of g-invariant subapaces V and W,the restriction g|V is an aﬃne operator with the form(x1,x2,...,x n)→(x1+1,x2,...,x n−1,εx n) under the orthogonal basis of V,ε=±1,and it is uniquely determined by f.The restriction g|W is a linear operator with a nilpotent Jordon form under the orthogonal basis of W,and it is uniquely determined by f.For the problem of topologically conjugate classiﬁcations of the holomorphic isomorphisms on extended complex plane,Rybalkina and Sergeichuk have given the answer.It is not known of the completely classiﬁcations for the holomorphic isomorphisms on unit disk or upper half plane before.Now we will solve the problem.By rotation theory and some constructions of homeomorphisms,we prove that all holomorphic automorphisms on upper half plane(or unit disk)having noﬁxed points are topologically conjugate;two holo-morphic automorphisms f and g havingﬁxed points are topologically conjugate if and only if ρ(f)=±ρ(g),mod Z;a holomorphic automorphism with noﬁxed points and a holomorphic automorphism withﬁxed points are not topologically conjugate.Key Words:topological conjugacy,holomorphic automorphism,unit disk,upper half plane, complex plane,extended complex plane.目录摘要 (i)Abstract (ii)第1章引言 (1)第2章复平面上全纯自同构的拓扑共轭分类 (5)2.1有不动点的仿射算子 (6)2.2无不动点的仿射算子 (8)第3章扩充复平面上全纯自同构的拓扑共轭分类 (13)3.1线性算子的拓扑分类 (13)3.2定理3.1的证明 (16)3.3推论3.2的证明 (17)第4章单位圆盘与上半平面上全纯自同构的拓扑共轭分类 (18)4.1单位圆盘与上半平面上的全纯自同构 (18)4.2上半平面上全纯自同构的拓扑共轭分类 (20)第5章结论 (23)参考文献 (24)作者简介及在学期间所取得的科研成果 (26)致谢 (27)第1章引言拓扑动力系统是指一个偶对(X,f),这里X是一个完备度量空间,f:X→X是X上的一个连续映射.我们关心空间中点在迭代作用f n下轨道的渐进性态,这里f n= f∘f∘···∘f,∘是映射复合.其中,拓扑共轭分类问题是该系统领域中的核心问题.定义1.1设X,Y是两个完备度量空间,f:X→X和g:Y→Y分别是X和Y上的连续（全纯,线性）自映射.如果存在（全纯,线性）同胚映射h:X→Y使得h∘f=g∘h.则称f和g是拓扑（全纯,线性）共轭的.一般而言,完全的拓扑共轭分类很难获得,但对于一些特殊的系统,我们已知如下结果:Poincare给出了圆周上旋转映射的拓扑共轭分类[1];Walters考察了n维环面T n和紧交换拓扑群上仿射变换的拓扑共轭分类[2,3];Robbin和Kuiper研究了有限维向量空间上线性自同态的拓扑共轭分类[4,5].本文我们将给出单连通区域上全纯自同构的拓扑共轭分类.记C为复平面,ˆC C∪{∞}为扩充复平面,H为上半平面,D为单位开圆盘.进而用Aut(C),Aut(ˆC),Aut(H)和Aut(D)分别表示复平面,扩充复平面,上半平面和单位开圆盘上的所有全纯自同构构成的集合.本文我们将给出在复平面,扩充复平面,单位圆盘和上半平面上全纯自同构的拓扑共轭分类.对于复平面上全纯自同构的拓扑共轭分类问题,Budnitska[6]给出了更一般的形式,即有限维向量空间上仿射算子的拓扑共轭分类问题.一个仿射算子f:V→V是一个形如f(x)=Ax+b的映射,这里A:V→V是一个线性算子并且b∈V.为了简便我们通常令V=F n,其中F=C或R,那么f:F n→F n有如下形式:f(x)=Ax+b,A∈M n×n(F), b∈F n.这里M n×n(F)是数域F上的n×n阶矩阵.两个仿射算子f,g:F n→F n是共轭的,如果存在一个双射h:F n→F n使得g=h−1f h,(1)我们说(a)如果h在(1)中是一个线性算子,那么它是线性共轭.(b)如果h是一个仿射算子,那么它是仿射共轭.(c)如果h是一个双射,那么它是双正则共轭,这意味着h和h−1有如下形式(x1,...,x n)→(ϕ1(x1,...,x n),...ϕn(x1,...,x n))(2)在这里所有的ϕi都是F上的多项式.(d)如果h是一个同胚,那么它是拓扑共轭,这意味着h和h−1是连续的双射.线性共轭蕴含着仿射共轭蕴含着双正则共轭蕴含着拓扑共轭.下面简单介绍一下(a)−(d)仿射算子共轭分类的结果.在(a)中一个线性共轭y=Ax+b的变换相当于一个在F n上的基底的变换,形式为(A,b)→(S−1AS,S−1b),S∈F n×n是非奇异(3)关于这些变换的仿射算子的规范形式是很容易构造的:如果F=C,我们令A是Jordan标准形式,然后通过(3)中的变换来化简b,因此S随着Jordan矩阵A变换,它的形式在[7]中给出.在(b)中一个仿射共轭的变换相当于一个在F n上的基底的仿射变换.我们说仿射算子x→Ax+b是非奇异的如果它的矩阵A是非奇异的.Blanc[8]证明了非奇异的仿射算子x→Ax+b和x→Cx+d在特征值为0的代数闭域上是仿射共轭的,当且仅当它们的矩阵A和C是相似的,即对于一些非奇异的S有S−1AS=C.在(c)中Blanc[8]也给出了特征值为0的代数闭域K上非奇异仿射算子的双正则共轭分类.两个K上有不动点的非奇异仿射算子是双正则共轭的,当且仅当它们的矩阵是相似的.一个无不动点的非奇异仿射算子f:K n→K n双正则共轭于一个对角的仿射算子.(x1,x2,...,x n)→(x1+1,λ2x2,...,λn x n)(4)在这里1,λ2,...,λn∈K∖0是矩阵的特征值,仿射算子(4)是由f唯一确定的.在(d)中R2上的仿射算子的拓扑共轭分类由E phramowitsch[9]给出.在本文中我们将给出R n和C n上的仿射算子的分类.对于扩充复平面上全纯自同构的拓扑共轭分类问题,我们主要考虑Mobius变换[10]. Mobius变换是一个在扩充复平面上的如下形式的线性分式变换:f(z)=az+bcz+d,ad−bc 0,a,b,c,d∈C(5)Mobius变换的理论基础可参见[11]和[12].a,b,c,d同时乘以任意一个非零的数,f不变,变换(5)可以写成矩阵的形式M f:=1√ad−bc⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝a bc d⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(6)它的行列式为1并且是由f唯一确定的至多差一个-1的乘子.变换的复合相当于矩阵的乘积M f g=M f M g(7)矩阵A的迹记做tr A.两个Mobius变换f和g是共轭的,如果存在一个Mobius变换h使得g=h−1f h:两个Mobius变换f和g是拓扑共轭的,如果存在一个同胚h:ˆC→ˆC,使得g=h−1f h.如果两个Mobius变换是共轭的,那么它们是拓扑共轭,因为每一个Mobius变换都是一个同胚.共轭的标准形式很容易从[11]中获得,不同的Mobius变换f和g是共轭的当且仅当tr M f=±tr M g,因此共轭的Mobius变换是由相似矩阵给出的至多差一个-1的乘子.我们给出变换矩阵M h使得M g是M h的Jordan形式.由于det M f=det M g=1,M g=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝λ001/λ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(λ ±1,0)或者M g=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝λ10λ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(λ=±1)(8)矩阵M g是由f唯一确定的至多差一个-1的乘子和(8)中第一个矩阵对角元素λ和1/λ的交换.因此f共轭于z→λ2z,或者z→(1/λ2)z,或者z→z+1,我们从[11]中获得了规范形式.每个Mobius变换恰好共轭于如下形式mµ(z)=µz(µ 0,1)或者m1(z)=z+1(9)在这里µ是由1/µ确定的.µ1=µ和µ1=1/µ被叫做f的乘子.它们是由Riemann面上的全纯映射决定的[13];对于不同的Mobius变换f它们能通过下面的公式计算µi=f′(z i z i ∞,i=1,2µi=limz→∞1f′(z),z i=∞,i=1,2这里z1和z2是f的不动点.第2章复平面上全纯自同构的拓扑共轭分类本章我们将介绍复平面上全纯自同构的拓扑共轭分类,Budnitska[6]给出两种不同形式的仿射算子:（1）有不动点并且没有以1的根为特征值的仿射算子.有不动点的仿射算子的拓扑共轭分类问题是所有线性算子的拓扑共轭分类问题.实际上每一个线性算子x→Ax可以看作是一个有不动点x=0的仿射算子x→Ax+0.相反的,如果仿射算子是拓扑共轭的,那么每一个有不动点的x→Ax+b都能被它的线性部分x→Ax代替,因为有下面的引理2.1我们证明它们是拓扑共轭的.Kuiper和Robbin[5,4]给出了在R上的没有以1的根为特征值的线性算子的标准形式,在定理2.2中我们将标准扩展到C上，给出在R和C上的没有以1的根为特征值的线性算子的标准形式.为了简便我们不考虑以1的根为特征值的线性算子;像这种算子的拓扑分类问题已经由Kuiper和Robbin[5,4],Cappell和Shaneson [14,15,16,17,18],Hsiang和Pardon[19],Madsen和Rothenberg[20],和Schultz[21]给出.（2）无不动点的仿射算子.在定理2.3中我们证明了在F=C或R上的无不动点的仿射算子恰好共轭于一个如下形式的仿射算子x→(I k⊕J0)x+[1,0,...,0]T,如果当F=R时,x→(I k⊕[−1]⊕J0)x+[1,0,...,0]T,在这里k≥1并且J0是一个幂零的Jordan矩阵由f唯一确定.对任意一个F∈{C,R}上的方阵A,都有一个非奇异的矩阵A*和一个F上的幂零矩阵A0,使得A相似于A*⊕A0(10)在下面的定理中我们总结了仿射算子拓扑共轭的条件.定理2.1令f(X)=Ax+b和g(X)=Cx+d是F=C或R上的仿射算子.（1）假设f和g有不动点,那么f和g是拓扑共轭的,当且仅当x→Ax和x→Cx是拓扑共轭的.（2）假设f有一个不动点,g无不动点,那么f和g不拓扑共轭.（3）假设f和g无不动点:如果F=C,那么f和g是拓扑共轭的,当且仅当A0相似于B0.如果F=R,那么f和g是拓扑共轭的,当且仅当A*和C*的行列式有相同的迹并且A0相似于B0.2.1有不动点的仿射算子在这一部分我们将给出一个有不动点的仿射算子f(X)=Ax+b拓扑共轭的标准形式,这里矩阵A没有相同的特征值.我们可以仅仅考虑线性算子,因为在下面的引理中我们把有不动点仿射算子的分类问题化简为线性算子的分类问题.引理2.1一个C或R上的仿射算子f(X)=Ax+b拓扑共轭于它的线性部分f lin(X)= Ax,当且仅当f有一个不动点.如果p是f的一个不动点,那么f lin=h−1f h,h(x):=x+p.证明.如果f(p)=p,那么Ap+b=p并且(h−1f h)(x)=(h−1f)(x+p)=h−1(A(x+p)+b)= h−1(Ax+(p−b)+b)=h−1(Ax+p)=Ax=f lin(x).相反的,如果f和f lin是拓扑共轭的,那么f和f lin有相同的不动点,因为f lin(0)=0,f也有一个不动点.对任意的λ∈C,我们记J n(λ):=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣λ 01λ···... ............ 0···1λ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦对于每一个n×n复矩阵A=[a kl+b kl i],a kl,b kl∈R,我们记¯A=[akl−b kl i](11)并且通过A的实化A R来表示:通过用块⎡⎢⎢⎢⎢⎢⎢⎣a kl−b klb kl a kl⎤⎥⎥⎥⎥⎥⎥⎦(12)来代替每一个元素a kl+b kl i,由A得到了一个2n×2n的实矩阵.每一个在F∈{C,R}的方阵A相似于A0⊕A01⊕A1⊕A1∞,(13)在这里所有A0（A01A1,A1∞)的特征值λ都满足λ=0(0<|λ|<1,|λ|=1,|λ|>1).A0和(10)中的一样,A01⊕A1⊕A1∞和(10)中的A*相似.定理2.2(a)令f(x)=Ax和g(x)=Bx是F=C或R上没有以1的根为特征值的线性算子,令A0,...,A1∞和B0,...,B1∞像(13)中的一样,分别对应A和B的分解.（i）如果F=R,那么f和g是拓扑共轭的,当且仅当A0相似于B0,size A01=size B01,这里size是矩阵维数的意思,det(A01B01)>0,A1相似于B1，size A1∞=size B1∞，det(A1∞B1∞)>0,(14)（ii）如果F=C,那么f和g是拓扑共轭的,当且仅当A0相似于B0,size A01=size B01,A1⊕¯A1相似于B1⊕¯B1,size A1∞=size B1∞,(15)(b)每一个F=C或R上没有以1的根为特征值的线性算子拓扑共轭于一个线性算子,它的矩阵是一个由直和项的置换唯一确定的直和.（i）当F=R直和项的数字为J k(0),[1/2],J k(λ)R,[2](16)在这里λ是一个系数为1的复数,是由¯λ的置换确定的,显然它没有相同的特征值.至多一个1×1的直和项[-1/2]至多一个1×1的直和项[-2]（ii）当F=CJ k(0),[1/2],J k(λ),[2](17)在这里λ是一个系数为1的复数,是由¯λ的置换确定的,显然它没有相同的特征值.证明.（a）结论（i）已经由Kuiper和Robbin[5,4]证明,下面我们来证明（ii）关于加法的阿贝尔交换群V=C n可看做是C上的n维矢量空间V C和R上的2n维矢量空间V R.进一步的,我们可以将V C看做一个有标准正交基底的酉空间e1=[1,0,...,0]T,e2=[0,1,...,0]T,...,e n=[0,0,...,1]T,(18)将V R看做一个有标准正交基底的欧式空间e1,ie1,e2,ie2,...,e n,ie n(19)对每一个v=(α1+β1i)e1+...+(αn+βn i)e n∈V,αkβk∈R它在V C和V R的长度是相同的:+β21+...+α2n+β2n)1/2|v|=(α21因此映射h:V→V是V C的一个同胚当且仅当h是V R的一个同胚.(20)每一个线性算子f:V C→C定义一个线性算子f R:V R→R两个线性算子f,g:V C→C是拓扑共轭的当且仅当f R,g R:V R→R是拓扑共轭的.(21)令f(x)=Ax和g(x)=Bx是V C上没有相同特征值的线性算子,显然A和B是它们的有正交基底的矩阵.考虑f和g在V R上的线性算子f R和g R,我们发现f R和g R的基底是A和B 的实化A R和B R.因为S−1AS=A0⊕A01⊕A1⊕A1∞对于一些非奇异的S,我们有(S R)−1A R S R=A R0⊕A R01⊕A R1⊕A R1∞类似的B R相似于B R0⊕B R01⊕B R1⊕B R1∞通过(21)和定理2.2（a）的结论（i）f和g是拓扑共轭的当且仅当f R和g R是拓扑共轭的当且仅当A R 0相似于B R,size A R01=size B R01,det(A R01B R01)>0,A R 1相似于B R1,size A R1∞=size B R1∞,det(A R1∞B R1∞)>0,(22)对于每一个复矩阵M,它的实化M R相似于M⊕¯M,因为⎡⎢⎢⎢⎢⎢⎢⎣11−i i⎤⎥⎥⎥⎥⎥⎥⎦−1⎡⎢⎢⎢⎢⎢⎢⎣a−bb a⎤⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎣11−i i⎤⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎣a+bi00a−bi⎤⎥⎥⎥⎥⎥⎥⎦因为A0的Jordan标准形式是一个幂零的Jordan矩阵,¯A0相似于A0.因此条件A R相似于B R0,等价于条件A0⊕¯A0相似于B0⊕¯B0等价于条件A0相似于B0.条件size A R01=B R01等价于条件size A01=B01.条件det(A R01B R01)>0保持不变,因为det(A R01B R01)=det(A01B01)R=det(A01B01⊕A01B01)>0类似的我们考虑(22)中剩下的三个条件,得到(22)等价于(15),这就证明了（ii）结论.2.2无不动点的仿射算子在这一部分,我们将证明下面的定理,它给出了拓扑共轭的标准,并且给出了无不动点的仿射算子的拓扑共轭的标准形式.定理2.3(a)令f (x )=Ax +b 和g (x )=Cx +d 是F =C 或R 上的无不动点的仿射算子,令A *,A 0和C *,C 0像（10）中那样,分别对应A 和C 的分解.（b ）每一个F =C 或R 上的无不动点的仿射算子f 拓扑共轭于一个如下形式的仿射算子x →(I k ⊕J 0)x +[1,0,...,0]T(23)或者当F =R 时x →(I k ⊕[−1]⊕J 0)x +[1,0,...,0]T(24)在这里k ≥1并且J 0是一个由f 唯一确定的幂零Jordan 矩阵.我们通过对(A ,b )给出一个仿射算子f (x )=Ax +b ，写成f =(A ,b ).对于两个仿射算子f :F m →F m 和g :F n →F n ,我们通过(f ⊕g )⎡⎢⎢⎢⎢⎢⎢⎣x y ⎤⎥⎥⎥⎥⎥⎥⎦:=⎡⎢⎢⎢⎢⎢⎢⎣f (x )g (y )⎤⎥⎥⎥⎥⎥⎥⎦定义f ⊕g :F m +n →F m +n 因此(A ,b )⊕(C ,d )=(⎡⎢⎢⎢⎢⎢⎢⎣A 00C ⎤⎥⎥⎥⎥⎥⎥⎦,⎡⎢⎢⎢⎢⎢⎢⎣b d ⎤⎥⎥⎥⎥⎥⎥⎦)当f 和g 在F 上拓扑共轭时,我们记做f ∼F g 进一步我们有f ∼F f ′和g ∼F g ′=⇒f ⊕g ∼F f ′⊕g′(25)在这一部分,我们将继续通过拓扑共轭变换把F =C 或R 上的无不动点的仿射算子化简为(23)或者(24)的形式.第一步:将y =Ax +b 化简为如下的形式:p ⨁︁i =1(J m i (1),a i )⊕r ⨁︁i =p +1(J m i (1),a i )⊕(J 0,s )⊕(B ,c )(26)在这里J 0是A 0的Jordan 标准形式,1和0不是B 的特征值,每一个a 1,...,a p 都有一个非零的第一坐标,每一个a p +1,...,a r 都有一个零的第一坐标.我们是通过F 上的(3)中的线性共轭的变换来化简的.第二步:把(26)化简成如下的形式:p⨁︁i=1(J mi(1),a i)⊕r⨁︁i=p+1(J mi(1),0)⊕(J0,0)⊕(B,0)(27)在这里每一个a i都有一个非零的第一坐标,我们是通过(25)来进行化简的,共轭是(J m(1),a)∼F(J m(1),0),(J0,s)∼F(J0,0),(B,c)∼F(B,0)(28)在这里a的第一坐标是0,共轭(28)通过引理2.1保持不变,因为(J0,s)和(B,c)有不动点.记p≥1因为(27)是一个有不动点为0的线性算子,但是f没有不动点.第三步:把(27)化简成如下的形式:p⨁︁i=1(J mi(1),e1)⊕(C,0)⊕(J0,0)(29)在这里e1=[1,0,...,0]T和C:=⨁︀ri=p+1J mi(1)⨁︀B是非奇异的.我们用的共轭是(J m(1),a)∼F(J m(1),e1)(30)在这里a的第一坐标是0,a被下面的形式代替a=b[1,a2,...,a n]T,b 0,这个共轭是线性的,它保持不变因为(S J m(1)S−1,S e1)=(J m(1)a)S=b ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1 0a21······...a3a21···... ...............a n...a3a21⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦第四步:把(29)化简成如下的形式:p⨁︁i=1(I mi,e1)⊕(C,0)⊕(J0,0)(31)我们用的共轭是(J m(1),e1)∼F(I m,e1)(32)第五步:把(31)化简成如下的形式:(I 1,[1])⊕(D ,0)⊕(J 0,0)(33)在这里D :=I ⊕C 是非奇异的.我们用的共轭是p ⨁︁i =1(I m i ,e 1)∼F (I p ,[1,...,1]T )⊕(I q ,0)∼F (I 1,[1])⊕(J q +p −1,0)最后的共轭保持不变,因为(I 2,[1,1]T )∼F (I 2,e 1),它有如下的形式(S −1I 2S ,S −1⎡⎢⎢⎢⎢⎢⎢⎣11⎤⎥⎥⎥⎥⎥⎥⎦)=(I 2,e 1),S :=⎡⎢⎢⎢⎢⎢⎢⎣1011⎤⎥⎥⎥⎥⎥⎥⎦第六步:我们把（33）化简到（23）,（24）的形式,在这里我们分两方面来考虑.当F =R ,我们有共轭f ∼Fg ,f :=(I 1,[1])⊕(εF ,0),g :=(I 1,[1])⊕(εI m ,0)(34)实际上,g =h −1f h 对于映射h :R m +1→R m +1是由h :⎡⎢⎢⎢⎢⎢⎢⎣x y ⎤⎥⎥⎥⎥⎥⎥⎦→⎡⎢⎢⎢⎢⎢⎢⎣x εF x y ⎤⎥⎥⎥⎥⎥⎥⎦,x ∈R ,y ∈R m 决定的.因为hg ⎡⎢⎢⎢⎢⎢⎢⎣x y ⎤⎥⎥⎥⎥⎥⎥⎦=h ⎡⎢⎢⎢⎢⎢⎢⎣x +1εy ⎤⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎣x +1ε2F x +1y ⎤⎥⎥⎥⎥⎥⎥⎦=f ⎡⎢⎢⎢⎢⎢⎢⎣x εF x y ⎤⎥⎥⎥⎥⎥⎥⎦=f h ⎡⎢⎢⎢⎢⎢⎢⎣x y ⎤⎥⎥⎥⎥⎥⎥⎦映射h 是连续的,因为级数F x=exG=1+xG +(xG )22!+(xG )33!+...(35)收敛半径是无穷的,这里G 是实矩阵使得F =e G.逆映射h :⎡⎢⎢⎢⎢⎢⎢⎣x y ⎤⎥⎥⎥⎥⎥⎥⎦→⎡⎢⎢⎢⎢⎢⎢⎣x εF −x y ⎤⎥⎥⎥⎥⎥⎥⎦,x ∈R ,y ∈R m 也是连续的.这就证明了(34).应用线性共轭(3)到(33)的变换,我们化简D 到P ⊕(−Q )的形式,这里P 是一个非奇异的实的没有负的实特征值的p ×p 矩阵,Q 是一个非奇异的实的有正的实特征值的q ×q 矩阵.这里线性算子(33)有如下形式(I 1,[1]⊕(P ,0)⊕(−Q ,0)⊕(J 0,0);它拓扑共轭于(I 1,[1]⊕(I p ,0)⊕(−I q ,0)⊕(J 0,0)(36)在(34)中令ε=1,F=−I2,我们得到(I1,[1]⊕(−I2,0)∼R(I3,e1)应用这个共轭多次,我们把(36)化简为(23),(24)的形式,我们证明了在R上的无不动点的仿射算子拓扑共轭于(23)或(24).当F=C,让我们证明f∼C g,f:=(I1,[1])⊕(D,0),g:=(I1,[1])⊕(I m,0)(37)在这里D是一个非奇异的m×m复矩阵,实际上,g=h−1f h,这里h:C m+1→C m+1是由h:⎡⎢⎢⎢⎢⎢⎢⎣xy⎤⎥⎥⎥⎥⎥⎥⎦→⎡⎢⎢⎢⎢⎢⎢⎣xD x y⎤⎥⎥⎥⎥⎥⎥⎦,x∈C,y∈C m决定的.映射h是一个同胚因为D x被(35)中的F:=D代替了.这就证明了(37),用它我们把(33)化简为(23),我们证明了在C上的无不动点的仿射算子拓扑共轭于(23).第3章扩充复平面上全纯自同构的拓扑共轭分类本章我们将介绍扩充复平面上全纯自同构的拓扑共轭分类,关于Mobius变换的拓扑共轭分类由Rybalkina在[10]中给出.定理3.1对于任意的非奇异的Mobius变换f,g:ˆC→ˆC,下面的四条是等价的.（i）f和g是拓扑共轭的;（ii）M f的迹和M g的迹不属于[-2;2],或者trM f=±trM g;（iii）如果λ是M f的任意特征值,λ′是M g的任意特征值,那么|λ|,|λ′| 1,或者λ=±λ′,或者λ=±¯λ′;（iv）如果µ是f的任意乘子,ν是g的任意乘子,那么|µ|,|ν| 1,或者µ=ν,或者µ=¯ν;（9）中的标准形式被用到下面的定义中:一个非奇异的Mobius变换被叫做双曲如果它解析共轭于z→µz同时1 µ∈R;斜驶如果它解析共轭于z→µz同时µ R并且|µ| 1;椭圆如果它解析共轭于z→µz同时|µ|=1并且µ 1;抛物如果它解析共轭于z→z+1.一个Mobius变换的拓扑共轭的标准形式很容易从定理3.1中（i）到（iv）的等价形式中获得:推论3.1（a）每一个双曲或斜驶的Mobius变换拓扑共轭于z→2z.（b）每一个椭圆的Mobius变换拓扑共轭于z→µz,它是由µ通过¯µ的替换唯一确定的.（c）每一个抛物的Mobius变换拓扑共轭于z→z+1.两个线性算子A,B:C2→C2是拓扑共轭的,如果存在一个同胚h:C2→C2使得B=h−1Ah.Mobius变换的线性算子x→M f x(x∈C2)是由-1的乘子决定的,我们得到了一个ˆC上的Mobius变换和C2上的行列式值为1的线性算子的一一对应.推论3.2下面的两个条件对于Mobius变换f和g是等价的（i）f和g是拓扑共轭的;（ii）在C2上的线性算子x→M f x拓扑共轭于x→M g x或−x→M g x.3.1线性算子的拓扑分类在本小节我们将给出一些关于线性算子的拓扑分类[5,4].对于每一个复方阵A=[a i j],我们定义矩阵¯A=[¯a i j]是矩阵A的复共轭,利用矩阵的直和构造一个A的分解S−1AS=A0⊕A01⊕A1⊕A1∞(S是非奇异的矩阵）(38)在这里所有A0的特征值λ满足λ=0分别的A01,A1,A1∞的特征值λ满足0<|λ|<1,|λ|= 1,|λ|>1.两个线性算子f,g:R n→R n是共轭的,如果存在一个同胚h:R n→R n使得g=h−1f h.算子f具有周期性,如果对于一些自然数k有f k是恒等映射.Kuiper和Robbin[5]证明了如果假设两个周期的线性算子是拓扑共轭的当且仅当它们是线性共轭的.定理3.2令f(x)=Ax和g(x)=Bx是V=R m或C m上的线性算子,令A0,A01,A1,A1∞和B0,B01,B1,B1∞像（38）中一样,分别对应A和B的分解.（i）如果V=R m同时m≤5,那么f和g是拓扑共轭的,当且仅当A0相似于B0,size A01=size B01,det(A01B01)>0,A1相似于B1,size A1∞=size B1∞,det(A1∞B1∞)>0,（ii）如果V=C m同时m≤2,那么f和g是拓扑共轭的,当且仅当A0相似于B0,size A01=size B01,A1⊕¯A1相似于B1⊕¯B1,size A1∞=size B1∞,(39)推论3.3令f(x)=Ax和g(x)=Bx是C2上的两个非奇异的线性算子,矩阵A和B的行列式为1并且是对角矩阵.令λ和λ′是A和B的特征值,那么f和g是拓扑共轭的,当且仅当|λ|,|λ′| 1,或者λ=λ′或者λ=¯λ′.证明.线性算子的共轭意味着它们拓扑共轭,因此我们假设f(x)=Ax和g(x)=Bx的矩阵由它们的Jordan形式给出:A=⎡⎢⎢⎢⎢⎢⎢⎣λ00λ−1⎤⎥⎥⎥⎥⎥⎥⎦,⎡⎢⎢⎢⎢⎢⎢⎣λ′00λ′−1⎤⎥⎥⎥⎥⎥⎥⎦,λ,λ′ ±1,0下面我们分四种情况来讨论:第一种:|λ| 1并且|λ′| 1.那么在（38）中A01⊕A1∞=[λ]⊕[λ−1],B01⊕B1∞=[λ′]⊕[λ′−1]通过定理3.2的（ii）f和g是拓扑共轭的.第二种:|λ|=|λ′|=1,那么A1⊕¯A1=[λ]⊕[¯λ],B1⊕¯B1=[λ′]⊕[¯λ′]通过定理3.2的（ii）f和g是拓扑共轭的,当且仅当λ=λ′或者λ=¯λ′第三种:|λ|=1并且|λ′| 1,那么A1⊕¯A1=[λ]⊕[¯λ],B01⊕B1∞=[λ′]⊕[λ′−1]通过定理3.2的（ii）f和g不是拓扑共轭的.第四种:|λ| 1并且|λ′|=1,那么f和g不是拓扑共轭的.引理3.1Mobius变换f(z)=az和g(z)=bz是拓扑共轭的当且仅当|a|,|b| 1或a=b或a=¯b(40)证明.⇐=.假设f和g满足(40)如果f和g满足|a|,|b|<1或|a|,|b|>1或a=b或a=¯b(41)那么通过定理3.2的（ii）在C上的线性映射z→az和z→bz通过同胚η:C→C是拓扑共轭的,因此f和g通过同胚h:ˆC→ˆC是拓扑共轭的,定义成如下形式h(z):=η(z)如果z∈C并且h(∞):=∞.如果f和g不满足(41)（但是满足(40)）,那么|a|<1并且|b|>1或者|a|>1并且|b|<1.假设|a|<1并且|b|>1.那么|1/b|<1并且通过(41)f拓扑共轭于g−1z=(1/b)z,通过ˆC上的同胚z→1/z拓扑共轭于g.=⇒.令ˆC上的Mobius变换f(z)=az和g(z)=bz是拓扑共轭的,那么存在着一个同胚h:ˆC→ˆC,使得hg(z)=f h(z)z∈ˆC.(42)因为h把所有g的不动点转换成f的不动点,他们的不动点是0和∞,下面我们分两种情形来讨论.第一种情形:h(∞)=∞并且h(0)=0.通过(42)线性算子z→az和z→bz在C上是拓扑共轭的,通过h到C上的限制同胚.定理3.2的（ii）确定|a|,|b|<1或|a|,|b|>1或a=b或a=¯b(43)第二种情形:h(∞)=0并且h(0)=∞.Mobius变换f−1(z)=(1/a)z和g(z)=bz是拓扑共轭的,通过同胚h1:=ϕh在这里ϕ(z):=1/z.因为h1(∞)=∞,我们有(43),在这里a被1/a代替.上面的两种情形a和b都满足(40).3.2定理3.1的证明我们记f和g是两个非恒等映射的Mobius变换,记λ和λ′分别为M f和M g的任一特征值,记n(f)和n(g)为f和g的不动点个数.(44)另外,注意到任意非恒等映射的Mobius变换的不动点个数为1或2.（i）⇐⇒（iv）.假设（i）存在,那么n(f)=n(g).因为f和g不是恒等的,下面我们分两种情形来讨论.第一种情形:n(f)=n(g)=1,通过(9)f和g共轭于m1(z)=z+1,它的乘子是1,这就确定了（iv）.第二种情形:n(f)=n(g)=2,令µ,ν 0,1是f和g的乘子,通过(9)f和g共轭于mµ(z)=µz和mν(z)=νz,他们的不动点是0或者∞.引理3.1确定了（iv）.因此推出（i）=⇒（iv）,相反的讨论我们得出（i）⇐=（iv）.（iii）⇐⇒（iv）.通过(9)和(6),如果µ是f的乘子,那么f共轭于mµ(z)=µz(µ 0,1)或者m1(z)=z+1(µ=1),因此M f相似于M mµ=±1√µ⎡⎢⎢⎢⎢⎢⎢⎣µ001⎤⎥⎥⎥⎥⎥⎥⎦=±⎡⎢⎢⎢⎢⎢⎢⎣√µ001/√µ⎤⎥⎥⎥⎥⎥⎥⎦(µ 0,1)或者M mµ=±⎡⎢⎢⎢⎢⎢⎢⎣1101⎤⎥⎥⎥⎥⎥⎥⎦(µ=1)因此矩阵M f有一个特征值λ等于√µ或−√µ(µ 0)(45)类似的,如果ν是g的乘子,那么M g有一个特征值λ′等于√νor−√ν(ν 0),这就证明了（iii）和（iv）是等价的.（ii）⇐⇒（iii）.让我们证明|λ|=1⇐⇒trM f=±trM mµ∈[−2;2].(46)等式trM f=±trM mµ成立是因为M f和M mµ是相似的.如果|λ|=1,那么通过(8)trM mµ=λ+λ−1=λ+¯λ∈[−2;2].如果|λ| 1,那么λ−1=¯λ并且trM mµ=λ+λ−1 [−2;2],这就证明了(46).下面我们分三种情形来讨论.第一种情形:|λ| 1并且|λ′| 1,那么（iii ）存在,并且通过(46),（ii)也存在.第二种情形:|λ|=1并且|λ′| 1,或者|λ| 1并且|λ′|=1,那么（ii ）和（iii ）都不存在.第三种情形:|λ|=|λ′|=1.条件trM f =±trM g 等价于trM m µ=±trM m ν等价于λ+¯λ=±(λ′+¯λ′)等价于λ=±λ′或者λ=±¯λ′.3.3推论3.2的证明对于任意的Mobius 变换f 和g 我们将分以下四种情形分别讨论.第一种情形:n (f ) n (g ),那么推论3.2的断言（i ）不存在,让我们证明（ii ）也不存在.假设n (f )<n (g ).如果n (g )=∞,那么g 是恒等的,n (f )∈1,2,并且（ii ）是不存在的.假设n (g )<∞,那么n (f )=1并且n (g )=2.通过（9）和（45）,f 共轭于m 1(z )=z +1并且g 共轭于m µ(z )=λ2z .线性算子x →M f x 和x →M g x 共轭于x →±M m 1x 和x →±M m µx ,在这里M m 1=⎡⎢⎢⎢⎢⎢⎢⎣1101⎤⎥⎥⎥⎥⎥⎥⎦,M m µ=⎡⎢⎢⎢⎢⎢⎢⎣λ00λ−1⎤⎥⎥⎥⎥⎥⎥⎦(λ 0,1)矢量[0,0]T 是线性算子x →±M m µx 的唯一不动点.所有的矢量[a ,0]T (a ∈C )是x →±M m 1x 的不动点.因此断言（ii ）不存在.第二种情形:n (f )=n (g )=1,通过（9）f 和g 共轭于z =z +1,通过（8）矩阵M f 和M g 相似于⎡⎢⎢⎢⎢⎢⎢⎣1101⎤⎥⎥⎥⎥⎥⎥⎦或者⎡⎢⎢⎢⎢⎢⎢⎣−110−1⎤⎥⎥⎥⎥⎥⎥⎦因此M f 相似于M g 或者−M g ,因此x →M f x 共轭于x →M g x ,因此它们拓扑共轭.第三种情形:n (f )=n (g )=2,通过（9）和（45）f 共轭于z →λ2z 并且g 共轭于z →λ′2z ,在这里λ和λ′是M f 和M g 的特征值.M f 和M g 的Jordan 形式是±J f 和±J g ,在这里J f :=⎡⎢⎢⎢⎢⎢⎢⎣λ00λ−1⎤⎥⎥⎥⎥⎥⎥⎦,J g :=⎡⎢⎢⎢⎢⎢⎢⎣λ′00λ′−1⎤⎥⎥⎥⎥⎥⎥⎦,λ,λ′ {0,±1}.通过定理3.1（iii ）,f 和g 共轭当且仅当|λ|,|λ′| 1,或者λ=±λ′,或者λ=±¯λ′,当且仅当线性算子x →J f x 和x →±J g x 拓扑共轭,当且仅当线性算子x →M f x 和x →±M g x 拓扑共轭.第四种情形:n (f )=n (g )>2，断言（i ）和（ii ）存在因为f 和g 是恒等映射并且M f =±M g =±I 2,在这里I 2是恒等矩阵.第4章单位圆盘与上半平面上全纯自同构的拓扑共轭分类4.1单位圆盘与上半平面上的全纯自同构众所周知,存在从单位圆盘到上半平面的解析双射,所以Aut(H)和Aut(D)是全纯同构的.那么当我们明确了上半平面上的全纯自同构的拓扑共轭分类时,也就可以给出单位圆盘上全纯自同构的拓扑共轭分类.现在我们从Aut(H)入手,上半平面上的全纯自同构必具有如下形式f(z)=az+bcz+d,这里a,b,c,d∈R且ad−bc=1.记PSL(2,R) SL(2,R)/{I2,−I2}为实数域上的2阶射影特殊线性群,这里I2表示2阶单位矩阵.注意到Aut(H) PSL(2,R),那么我们可以根据Jordon标准型对上半平面上的全纯自同构进行分类,而且具有相同Jordon标准型的全纯自同构是全纯共轭的（这个全纯共轭是由相似变换诱导的）.Type I.若⎛⎜⎜⎜⎜⎜⎜⎝a bc d⎞⎟⎟⎟⎟⎟⎟⎠≃⎛⎜⎜⎜⎜⎜⎜⎝λ100λ2⎞⎟⎟⎟⎟⎟⎟⎠这里λ1 λ2.则f(z)=az+bcz+d全纯共轭于某个g(z)=λz,λ>1.Type II.若⎛⎜⎜⎜⎜⎜⎜⎝a bc d⎞⎟⎟⎟⎟⎟⎟⎠≃⎛⎜⎜⎜⎜⎜⎜⎝1101⎞⎟⎟⎟⎟⎟⎟⎠或⎛⎜⎜⎜⎜⎜⎜⎝−110−1⎞⎟⎟⎟⎟⎟⎟⎠.则f(z)=az+bcz+d全纯共轭于g(z)=z+1.Type III.若⎛⎜⎜⎜⎜⎜⎜⎝a bc d⎞⎟⎟⎟⎟⎟⎟⎠≃⎛⎜⎜⎜⎜⎜⎜⎝cosθ−sinθsinθcosθ⎞⎟⎟⎟⎟⎟⎟⎠这里θ∈[0,2π).则f(z)=az+bcz+d全纯共轭于某个g(z),g(z)=cosθz−sinθsinθz+cosθ,这里θ∈[−π2,π2 ).上半平面上的全纯自同构必为Type I―III中的一种.另外,注意到Type I和Type II的全纯自同构是没有不动点的,称之为NFP-型全纯自同构;Type III的全纯自同构必有不动点,称之为EFP-型全纯自同构.由于具有不动点这一性质是拓扑共轭不变性,所以NFP-型全纯自同构与EFP-型全纯自同构之间是不拓扑共轭的.另外,通过上半平面到单位圆盘的解析双射,Type III型的全纯自同构对应了（全纯共轭）单位圆盘上的旋转映射.引理4.1令f(z):H→H为上半平面上的全纯自同构f(z)=cosθz−sinθsinθz+cosθ,令g(ω):D→D为单位圆盘上的全纯自同构g(ω)=e i2θω,这里θ∈[−π2,π2).则f与g全纯共轭(拓扑共轭).证明.定义映射h:H→D为,对任意的z∈C,h(z)=z−i z+i.则h是一个全纯同构(同胚).往证下面图表是交换的.H f−→Hh↓↓hD g−→D令ω=h(z)=z−iz+i ,则z=h−1(ω)=ω+11−ωi.那么f h−1(ω)=cosθω+11−ωi−sinθsinθω+11−ωi+cosθ=ωi cosθ+i cosθ−sinθ+ωsinθωi sinθ+i sinθ+cosθ−ωcosθ,进而h f h−1(ω)=ωi cosθ+i cosθ−sinθ+ωsinθωi sinθ+i sinθ+cosθ−ωcosθ−i ωi cosθ+i cosθ−sinθ+ωsinθωi sinθ+i sinθ+cosθ−ωcosθ+i=ωi cosθ+i cosθ−sinθ+ωsinθ+ωsinθ+sinθ−i cosθ+iωcosθωi cosθ+i cosθ−sinθ+ωsinθ−ωsinθ−sinθ+i cosθ−iωcosθ=2ωi cosθ+2ωsinθ2i cosθ−2sinθ=i cosθ+sinθi cosθ−sinθω=(i cosθ+sinθ)(i cosθ+sinθ)−cosθ·cosθ−sinθ·sinθω=−(−cosθ·cosθ+2i sinθcosθ+sinθ·sinθ)ω=(cos2θ−i sin2θ)ω=e i2θω=g(ω).至此,我们可以称Type III的全纯自同构为上半平面上的旋转,若其对应θ∈[−π2,π2 ),则称θπ为这个旋转的旋转数,记做ρ(f).4.2上半平面上全纯自同构的拓扑共轭分类现在我们将进一步讨论Type I-III型全纯自同构,进而给出上半平面上全纯自同构拓扑共轭分类.如不作特殊说明,本节中的映射均指上半平面上的连续自映射.首先对于Type I的全纯自同构,我们可以证明它们都是拓扑共轭的.命题4.1设g(z)=2z.那么对每个f(z)=az,a>1,f(z)拓扑共轭于g(z).证明.对任意的a>1,令α=log2a−1.定义映射h:H→H为h(z)=z|z|α,对任意z∈H.则h是一个同胚.一方面,h∘g(z)=h(2z)=2z|2z|α=2α+1z|z|α=az|z|α,另一方面,f∘h(z)=f(z|z|α)=az|z|α,故h∘g(z)=f∘h(z),因此f拓扑共轭于g.。

Topological strings in generalized complex space

a rXiv:h ep-th/063145v24J u n26PUTP-2188ITEP-TH-104/05Topological strings in generalized complex space V asily Pestun ∗Physics Department,Princeton University,Princeton NJ 08544,USA e-mail:pestun@ March 2006Abstract A two-dimensional topological sigma-model on a generalized Calabi-Yau target space X is deﬁned.The model is constructed in Batalin-Vilkovisky formalism using only a generalized complex structure J and a pure spinor ρon X .In the present construction the algebra of Q -transformations automatically closes oﬀ-shell,the model transparently depends only on J ,the algebra of observables and correlation functions for topologically trivial maps in genus zero are easily deﬁned.The extended moduli space appears naturally.The familiar action of the twisted N =2CFT can be recovered after a gauge ﬁxing.In the open case,we consider an example of generalized deformation of complex structure by a holomorphic Poisson bivector βand recover holomorphic noncommutative Kontsevich ∗-product.1IntroductionThe topological [1,2]A/B-sigma-models [3]were introduced by Witten as twists of the N =2two-dimensional supersymmetric conformal ﬁeld theory of maps from the world-sheet Σinto the target space X .Topological strings [4]are obtained by coupling topologi-cal sigma-models to topological 2D gravity on Σ,so that in the path integral one integrates over the moduli space of complex structures on Σ.See [5]for references and an extensive review of the progress in the subject.The topological A-model can be deﬁned for any almost Kahler manifold X .An almost Kahler structure is a pair (ω,J )of a symplectic structure ωand a compatible almost complex structure J .In the Batalin-Vilkovisky approach,the almost complex structure J is used for the gauge ﬁxing of the A-model,so the correlation functions depend only on the symplectic structure ω.The observables t of the A-model are identiﬁed with the de Rham cohomology classes t ∈H ∗(X,C ).The A-model free energy F g (t )is the generating function of genus g Gromov-Witten invariants of X [6,7].2V asily PestunOn a Kahler manifold,the space of physical observables φ(x )i 1...i p ¯i 1...¯i q χi 1...χi p χ¯i 1...χ¯i q of the A -model [3]is conveniently graded by a pair (p,q )according to the Hodge decompo-sition of the de Rham cohomology H k (X,C )=⊕p +q =k H p,q (X,C ).The cohomology class of symplectic structure ωcan be deformed by an element of H 2(X,C ),which corresponds to an observable of degree p +q =2.Physically,such observables give rise to ghost number zero deformations of the action [3]for the A -model.The deformations corresponding to the observables of degree p +q =2will be called geometrical deformations.The hermi-tian observables of type (1,1)are deformations of the Kahler structure.If we include the B-ﬁeld then the hermitian condition can be dropped.In other words,the real B -ﬁeld and the real symplectic structure ωcan be combined together into a complex two-form ω+iB .If H 2,0(X,C )is trivial,then deformations of type (1,1)are the only geometrical defor-mations of the A -model.However,if H 2,0(X,C )is not trivial,then there are also observ-ables of type (2,0)and (0,2).Let us call the moduli space generated by the observables of degree (1,1)—the ordinary geometric moduli space,for the degrees (2,0),(1,1),(0,2)—the geometric moduli space,and for all (p,q )—the extended moduli space following [3].Now consider the topological B-model on a Calabi-Yau manifold X .The B-model cou-ples to a complex structure J on X .Its observables φi 1...i p ¯j 1...¯j q (x )θi 1...θi p η¯j 1...η¯j q [3]of type (−p,q )are identiﬁed with the Dolbeault cohomology classes H q (Λp T X 1,0).1The ordinarygeometric moduli space of complex structures is generated by Beltrami diﬀerentials µi ¯j .They deform the Dolbeault diﬀerential ¯∂¯j →¯∂¯j +µi ¯j ∂i.The genus g free energy F g of the B-model does not have such a clear geometrical description as in the case of the A -model.For the recent mathematical progress in deﬁnition of the B-model see [8,9]and [10,11].In genus zero the extended moduli space of the B-model was studied in [12,13,14].Again,the observables of type (−1,1)correspond to ordinary geometrical deformationsof complex structure generated by Beltrami diﬀerentials µi ¯j .If H 0(Λ2T X 1,0)or H 0,2(X )are nontrivial,then there exist generalized deformations of complex structure generatedby a holomorphic Poisson bivector βij or by a closed two-form B ¯i ¯j .ordinary geometric moduli (p,q )=(1,1)⊂(Hitchin’s generalized )geometric moduli p +q =2⊂extended moduli all (p,q )(1.1)The geometrical meaning of these deformations was unclear in [3,4].The missing no-tions appeared in the studies of the homological mirror symmetry [15]and the generalized complex geometry [16].2In the framework of the Kontsevich homological mirror symmetry conjecture [15],we need to consider not only closed strings,but also open strings together with all possible branes.Physically,branes are the boundary conditions for the open strings.Mathemati-cally,branes are described by certain derived categories.For example,in the case of the B-model,branes could be holomorphic vector bundles supported on holomorphic subman-ifolds.The full category of B-branes is the bounded derived category D b (X )of coherentTopological strings in generalized complex space3 sheaves on X.Physically,the worldsheet path integral deﬁnes the structure of algebra on the space of open string states A.For example,the product A⊗A→A can be deﬁned in terms of the open string three-point function.The worldsheet action in the path integral can be deformed by closed string observables.By deﬁnition,this deforms the open string algebra.Therefore,deformations of the open string algebra(modulo some symmetry)are described by the closed string states[17,18,19,20,21,8,9].Mathematically,these deformations of the category of branes are computed by the Hochschild cohomology HH(D b(X))[22,23,24,21,20,25,14].In[24]it was argued,that computation of HH(D b(X))can be reduced to the computation of the Hochschild coho-mology of the single space-ﬁlling brane.The algebra of observables on such a brane is just an algebra of holomorphic functions O(X).Thus,the hard computation of HH(D b(X)) reduces to an easier computation of HH(O(X))and gives the result[24,26]HH(O(X))=⊕H q(Λp(T X1,0)).This is precisely the moduli space of the closed B-model[3].Given this observation, the geometrical nature of(−2,0)and(0,2)deformations of the closed B-model might be interpreted in terms of equivalent deformation of the open B-model algebra.It was claimed in[27,28]that(0,2)deformations by a closed two-form b¯i¯j transform a sheaf structure on a brane into a more general gerby structure.On the other hand,(−2,0) deformations by a holomorphic bivectorβij deform the ordinary product of functions on a brane into the noncommutative Kontsevich∗-product[22,23,21,29,30].See[31]for a review of noncommutativeﬁeld theories,and[32,33,34]for study of mirror symmetry in the context of generalized complex structures.The notion of a generalized complex structure and generalized Calabi-Yau manifold was deﬁned by Hitchin[16,35]and then fully developed in Gualtieri’s thesis[36].There are two ideas behind the notion of generalized complex structure.Theﬁrst,coming from theory of constrained systems,is to generalize structures on the tangent bundle T X to structures on the direct sum of the tangent and the cotangent bundle E=T X⊕T∗X and consider Dirac structure on E[37].The second,coming from string theory,is to extend the diﬀeomorphism group by action of B-ﬁeld.An almost complex structure is a section I of End(T X)such that I2=−1.Similarly, an almost generalized complex structure is a section J of End(T X⊕T∗X)such that J∗=−J,J2=−1.An almost complex structure is integrable if+i-eigenbundle of J (holomorphic subbundle of T X)is involutive under the Lie bracket on T X.Similarly,an almost generalized complex structure is integrable if+i-eigenbundle L of J(holomorphic subbundle of T X⊕T∗X)is involutive under the Courant bracket[37].The generalized complex geometry incorporates symplectic structures and ordinary complex structures as particular cases.Therefore,the topological A-model and the topo-logical B-model could be particular cases of a certain generalized topological model[30, 38,39].This topological sigma-model of maps fromΣto X depends on a generalized complex structure J,so let us call it the topological J-model.See[40]for the analogue of the present construction in the real case and[41]for studies of the current algebra associated with J.4V asily Pestun If J is an ordinary symplectic(complex)structure,let us call it to be of A(B)-type. Gualtieri[36]shows that deformations of a generalized complex structure at the symplectic point(A-type)are parameterized by H2(X,C),while deformations in the complex point (B-type)are parameterized by⊕p+q=2H q(Λp T X1,0).This is the space of p+q=2 deformations of the topological A/B-model[3].Thus it is very natural to suggest that the topological A/B-model is a particular case of the generalized J-model.Then one can also ask the following question.If the Hitchin’s generalized complex geometry corresponds to p+q=2deformations of the A/B-model,then what could be the extended complex geometry that corresponds to arbitrary(p,q)deformations?nn The answer is known under the name BV geometry.Itﬁrst appeared in papers of Batalin and Vilkovisky[42,43],who suggested a powerful generalization of BRST quantization method for the case when gauge symmetries are reducible.A clear geometric interpretation of the BV formalism was given in[44,45,46].A general method to construct a topological sigma model for a BV target manifold was suggested in[47]by Alexandrov,Kontsevich,Schwarz and Zaboronsky(AKSZ).They illustrated the method by example of Chern-Simons theory and the topological A(B)ter AKSZ method was used by Kontsevich[22]toﬁnd the∗-product formula in the context of deformation quantization,see also[48,49,50].This is a summary of the steps to get the J-modelgeneralized CY structure1−→Lie bialgebroid2−→BV geometry3−→J-model.1.A generalized complex structure J deﬁnes decomposition of the bundle E=(T X⊕T∗X)⊗C into the+i-eigenbundle L and its conjugate L∗≃¯L,so E=L⊕¯L[16,36]. Each of the bundles L and¯L has a structure of Lie algebroid[51,37,52,53].The Lie bracket{,}:Γ(L)⊗Γ(L)→Γ(L)is equal to the restriction of the Courant bracket from T X⊕T∗X to L.The Lie bracket on L∗is also the restriction of the Courant bracket. The Lie algebroid diﬀerential¯∂:Γ(Λk L∗)→Γ(Λk+1L∗)is canonically deﬁned by the Lie bracket on L.It satisﬁes¯∂2=0.The pair(L,L∗)has the structure of Lie bialgebroid (L,L∗)[54],which means that the operator¯∂,deﬁned by the Lie bracket on L,satisﬁes the Leibnitz’s rule for the Lie bracket on L∗.2.A BV manifold(M,ω,ρ)is generally deﬁned as a supermanifold M equipped witha nondegenerate symplectic structureωand a measureρsuch that the corresponding odd Laplacian∆squares to zero∆2=0[45,46].Here the Laplacian of a function f is the divergence∆f:=divX f of the vectorﬁeld X f generated by f,where the divergence of a vectorﬁeld X is deﬁned by the formulaρdivX:=d(i Xρ).The Poisson bracket{,} and the solution S of the BV classical master equation{S,S}=0deﬁne the operator Q on functions on M as Qf:={S,f}.The operator Q is a derivation of the Poisson bracket{,}and satisﬁes Q2=0.The space of functions A:=C∞(M)on such a manifold M is a diﬀerential BV algebra[55,56,49,57].This algebra has an odd Lie bracket {,}:A⊗A→A,canonically deﬁned by the odd Poisson structure{f,g}:=∂i fωij∂j g, and the diﬀerential Q:A→A,deﬁned by the solution of the classical master equation Qf:={S,f}.The Poisson bracket in A is generated by the BV Laplacian∆as{a,b}= (−1)|a|(∆(ab)−(∆a)b−(−1)a a(∆b)).Now consider the supermanifold N=ΠL,which is the total space of the bundle L with parity reversed on theﬁbers.In other words,coordinates in theﬁbers are oddTopological strings in generalized complex space5 (fermionic)variables.The space of functions A=C∞(N)is naturally identiﬁed with the space of sections of⊕kΛk L∗.The Lie algebroid diﬀerential¯∂:Γ(Λk L∗)→Γ(Λk+1L∗) is equivalent to the BV diﬀerential Q of degree1on A.The Lie algebroid bracket{,}:Γ(L∗)⊗Γ(L∗)→Γ(L∗)is equivalent to the BV bracket A⊗A→A of degree-1.The generalized CY condition is equivalent to the existence of BV measureρ.It allows us to deﬁne the BV Laplacian∆.In terms of the Lie algebroid one has∆=∂,where∂is the generalized∂-divergenceΓ(Λk L∗)→Γ(Λk−1L∗).Actually,the generators of the Poisson bracket onΠL are in one to one correspondence withﬂat Lie algebroid L∗-connections on the top external powerΛdeg L(L∗)–generalized divergence operators[58,59,56,60].Thus for any generalized CY manifold there is a corresponding BV supermanifold N=ΠL equipped with a homological diﬀerential Q.Assume that the odd Poisson structure on N is invertible,so N is a symplectic manifold.If it is not,then we will take symplectic realization M of N[61,62,63].Since the odd vectorﬁeld Q preserves the symplectic structure,the form i Qωis closed.If it is also exact,then i Qω=dS.The function S is a Hamiltonian function for the vectorﬁeld Q,so Q={S,·}.3.Given the BV target manifold N and the Hamiltonian function S that satisﬁes {S,S}=0,the topological J-model is obtained along the lines of[47,49].The BVﬁelds of the model are maps fromΠTΣto N,where the supermanifoldΠTΣis the total space of the tangent bundle of a Riemann surfaceΣwith parity reversed on theﬁbers.Let X i and p i be local coordinates on N,which are canonically conjugate with respect to the BV odd symplectic structure.Then the BV action of the model is simplyS= ΠTΣp i dX i+S(p i,X i).(1.2)The physical action of the sigma-model is obtained after the gaugeﬁxing of this BV master action.See[40,64,65]on the Poisson sigma-model.In section2we consider in more details the algebraic and geometric structures men-tioned above.In section3we will construct the J-model.In section4we will compute the correlation functions on the boundary of the disk for the J-model obtained by aﬁ-nite deformation of the B-model by a holomorphic Poisson bivectorβij and reproduce the Kontsevich∗-product formula.Section5concludes the paper.2A diﬀerential BV algebra A of a generalized CY manifoldA generalized complex structure[36,16]on a manifold X is a section of J∈End(T X⊕T∗X)such that J∗=−J and J2=−1with a certain integrability condition:the +i-eigenbundle L⊂(T X⊕T∗X)⊗C is involutive with respect to Courant bracket[37]1[X+ξ,Y+η]=[X,Y]+L Xη−L Yξ−6V asily Pestun Then(T X⊕T∗X)⊗C=L⊕¯L,where¯L can be identiﬁed with L∗,since L is maximal isotropic3and L∩¯L=0.The bundle L has a structure of a Lie algebroid[36,16,53,59,52].The Lie algebroid bracket is the restriction of the Courant bracket on L.The anchor map a:L→T X is the restriction of the projection T X⊕T∗X→T X.Generally speaking,the structure of Lie algebroid on a bundle L can be conveniently described by an odd2-nilpotent vectorﬁeld Q of degree one onΠL[47].Let(xµ,ψa)be coordinates inΠL.Functions f∈C∞(ΠL)can be expanded inψaf(x,ψ)= 1∂xµ−1∂ψa(2.3)it corresponds to the Lie algebroid diﬀerential···¯∂→Λk L∗¯∂→Λk+1L∗¯∂→···.(2.4) The cohomology groups of this complex are called the Lie algebroid cohomology groups H k(L). The structure functions eµa and f a bc deﬁne the anchor map L→T X and the Lie algebroid bracket.The notion of Lie algebroid generalizes the notion of Lie algebra and the tangent bundle.For a Lie algebra L one can put eµa=0and f a bc to be the structure constants of L.For the tangent bundle one can put eµa to be the identity map and f a bc=0.For the case when L is the Lie algebroid that describes the+i-eigenbundle of a gener-alized complex structure on X,we will use the supermanifoldΠL to construct the target space of the J-model.(In the case of A-model,ΠL is actually the target space).The algebra A of functions onΠL is(super-)associative,commutative algebra with the diﬀer-ential Q.The diﬀerential Q onΠL is not enough to deﬁne a quantumﬁeld theory in the BV formalism.One also needs a BV bracket on the algebra A=C∞(ΠL)=Γ(Λ•L∗).4Thereis a natural notion to deﬁne simultaneously the compatible diﬀerential and the bracket onΠL–a Lie bialgebroid[53,56,58,52].A Lie bialgebroid is a pair(L,L∗)of Lie algebroidL and its dual L∗,such that the diﬀerential Q:Λk L∗→Λk+1L∗of the Lie algebroid L satisﬁes the Leibnitz’s rule for the Lie algebroid bracket on L∗.This is the case for the pair(L,L∗)associated with generalized complex structure[36,52,53].The Lie bracket {,}L∗on L∗can be extended toΛ•L∗.That equips the algebra A=C∞(ΠL)with the odd Poisson bracket of degree-1.This is the BV bracket.It satisﬁes the Leibnitz’s rule for the Lie algebroid diﬀerential¯∂.To summarize,a Lie bialgebroid structure(L,L∗)Topological strings in generalized complex space7 is equivalent to the structure on A=C∞(ΠL)of a diﬀerential odd Poisson algebra5 (A,¯∂,[,]L∗)[58,56,59].A(diﬀerential)Batalin-Vilkovisky algebra is a special case of a(diﬀerential)odd Pois-son algebra.In a BV algebra[45,46,44]the Poisson bracket is generated by an odd2-nilpotent diﬀerential operator of the second order∆,which is called BV Laplacian[56,58]. It generates BV bracket on A according to the formula{a,b}=(−1)|a|∆(a∧b)−∆a∧b−(−1)|a|a∧∆b for a,b∈A.(2.5)Such an operator∆can be naturally constructed for a generalized CY manifold as follows.The spinors of Spin(T X,T∗X)are identiﬁed with all diﬀerential formsΩ•(X)≡Λ•T∗X.The canonical line bundle U0[36,16]is a pure spinor line bundle U0deﬁned as a subbundle of those spinors of Spin(T X,T∗X),which are annihilated by all sections of L. In the Fock space spin representation,the U0is the vacuum line bundle,the sections of L are lowering operators and the sections of L∗are increasing operators.The canonical line bundle U0deﬁnes the alternative grading onΛ•T∗X as followsΛ•T∗X=U0⊕U1⊕···⊕U2n,where U k=Λk L∗⊗U0.It is explained in[36]that integrability of L is equivalent to the condition d(Γ(U0))⊂d(Γ(U1)).Then[36]one can deﬁne generalized∂and¯∂operators on Λ•(T∗X)in such a way that d=∂+¯∂and¯∂:Γ(U k)→Γ(U k+1)and∂:Γ(U k)→Γ(U k−1).A generalized CY manifold X is deﬁned[16,36]by the condition that on X exists a nowhere-vanishing closed sectionρof the canonical bundle U0–‘pure spinor’.(In the case of CY manifoldρis the holomorphic(3,0)form.In the case of a symplectic manifoldρis e iω,whereωis a symplectic structure.)Usingρ,the operators∂and¯∂on the diﬀerential formsΛ•T∗X can be mapped to the operators∂and¯∂on sections ofΛ•L∗by the formula ¯∂(µ·ρ)=¯∂µ·ρ,∂(µ·ρ)=∂µ·ρforµ∈Γ(Λ•L∗).(2.6)To summarize,a generalized CY structure deﬁnes a diﬀerential BV algebra(A,Q,∆) where A=Γ(Λ•L∗),Q=¯∂,∆=∂[56,53,36,58,55,12].The BV bracket{,}is deﬁned in terms of∆=∂by(2.5).Since∂¯∂+¯∂∂=0,the operator¯∂is a derivation of the bracket{,},and since∂2=0,the operator∂is a derivation of the bracket{,}.To describe quantumﬁeld theory in BV formalism one also needs a measure on the space ofﬁelds A.It is called trace map tr:A→C on the algebra A.It satisﬁestr((∆a)b)=(−1)a tr(a∆b),(2.7)tr((Qa)b)=(−1)a+1tr(a Qb)(2.8)For a generalized CY manifold,the trace map is deﬁned by a sectionρof the canonical line bundle U0.Contracting withρ,one can map an elementµof A=Γ(Λ•L∗)to a diﬀerential form inΩ•(X).This diﬀerential formµ·ρis also a spinor of Spin(T X,T∗X).8V asily Pestun There is a natural Spin(T X,T∗X)invariant bilinear form onΩ•(X)[36,16]with values inΩdim X(X).It is given by the wedge product with a certain sign(a,b)=˜a∧b,(2.9) where˜a=a for deg a=4k,4k+1,and˜a=−a for deg a=4k+2,4k+3.This bilinear form is symmetric in dimension4k,4k+1and antisymmetric otherwise.Using the bilinear form(,)and the canonical pure spinorρwe deﬁnetr a≡ X(a·Ω,Ω).(2.10)ExamplesA generalized complex structure J that corre-sponds to an ordinary complex structure has the following matrix J∈End(T X⊕T∗X)J= −I00I T ,(2.11)with+i-eigenbundle L=T X01⊕T∗X10,so that L∗=T X10⊕T∗X01.We consider the case of Calabi-Yau,then the canonical pure spinor is a nowhere vanishing holomorphic (n,0)formρ.The algebra A=C∞(ΠL)=Γ(Λ•L∗)is the familiar complex[3,12,55]of the observables of the B-modelΛ•L∗=Λ•(T X10⊕T∗X01)=Ω•(Λ•T X10).The Lie algebroid diﬀerential Q=¯∂is the standard Dolbeault diﬀerential¯∂:Ω−p,q→Ω−p,q+1.The BV Laplacian∆is the holomorphic divergence∆=∂·:Ω−p,q→Ω−p+1,q. The BV bracket on A is generated by∆=∂and can be viewed as the Lie bracket on holomorphic polyvectorﬁelds with coeﬃcients in(0,q)-forms.As explained above,the deﬁnition of∆depends on the existence of the pure spinor,or,equivalently,measureρon the BV manifold.For the B-model we take the pure spinor to be the holomorphic(Λp T X10). (n,0)-form.The Lie algebroid cohomology is the Dolbeault cohomology H q¯∂Explicitly,let(x i,x¯i)be complex coordinates on a CY manifold X and let(ψi,ψ¯i)be coordinates in theﬁbers of L=T X10⊕T∗X01.Then the algebra A=C∞(ΠL)is the algebra of functions f(x i,x¯i,ψi,ψ¯i).If we take local coordinates,where the coeﬃcients of the holomorphic(n,0)form are constant functions,then the BV operators Q and∆have the following form6∂Q=¯∂=ψ¯i∂∂ψi6The expression for Q is the same in any coordinates.However,the expression for∆involves holo-morphic(n,0)-formρ.For example,ifρ:=ρdx1∧dx2···∧dx n,then the divergence of the vectorﬁeld µi is deﬁned as∆µ=∂iµi+µi∂i logρ.To simplify formulas,in the text below,we will always assume that the formula for∆is written in the local coordinates whereρis constant.It is easy to recover the general formula,using the deﬁnition of∆by means of the Dolbeault diﬀerential∂and the isomorphism mapρ:Λ•(L∗)≃Λ•(T X∗).Topological strings in generalized complex space 9The trace map on A is familiar from the standard path integral of the B-model [3]tr a = X (a ·Ω)∧Ω.It can be written as a Berezian ρon the superspace ΠLtr a =ΠLρa (x i ,x ¯j ,ψi ,ψ¯j ),(2.14)whereρ=Ωi 1,...,i n ∂∂ψi n Ωj 1,...,j n dx j 1...dx j n dx ¯k 1...dx ¯k n ∂∂ψ¯k n .(2.15)2.The symplectic case (the A-model)∂x i ,(2.18)Λ=ωij ∂∂ψj(2.19)and ∂=∆=δ=[Λ,d ].In Darboux coordinates,where ωis constant,one simply has ∆=ωij ∂∂ψi .In other words ∆is the symplectic conjugate of d .The trace map on the algebra A ≃Γ(Ω(X ))is deﬁned as the integral of the top degree component tr a =Xa for a ∈Ω•(X ).(2.20)10V asily Pestun One can also take a dual point of view and consider the isomorphismΠL≃ΠT∗Xand¯∂=Q={S,·} induced byω.Then in the coordinates(x i,ψi)one has∆=∂∂x jwith S=ωijψiψj.Other types of generalized complex structure in the example of K37In[69]it argued that the relation holds for D-branes.More precisely,let X is a complex manifold equipped with holomorphic(2,0)formσ=b+iω.Then the b-ﬁeld transformation of the A-model(X,ω) is equivalent to theβ-transformation byβ=σ−1of the B-model.In terms of matrices of generalized complex structure for b-transformation of symplectic structureωwe have J= ω−1b−ω−1ω+bω−1b−bω−1 . This is actually aβ-transformation of the complex structure I=ω−1b.(One hasω+bω−1b=0as the condition for the existence of space-ﬁlling coisotropic branes in the A-model[70].)Topological strings in generalized complex space11 A2-plane is spanned by real and imaginary part of the canonical pure spinorΩ.The phys-ical N=(2,2)CFT with K3-surface target space requires existence of two generalized orthogonal complex structures(J,J′)represented by two orthogonal pure spinorsΩ1,Ω2. Each of these spinors spans a2-plane in H•(K3),and these planes must be orthogonal. Therefore,the moduli space of N=(2,2)CFT is locally parameterized by the Grassma-nian Gr2,2(R24)of two orthogonal2-planes in R24.The moduli space of N=(4,4)CFT’s is locally the space of4-planes Gr4(R24).The moduli space of N=(2,2)CFT’sﬁbers over the moduli space of N=(4,4)CFT’s.Theﬁbers S2×S2parametrize decomposition of a4-plane into two orthogonal2-planes[72,73].The topological twisting[3]makes theory to depend only on a half of this N=(2,2) CFT structure.That is,the twisted theory couples only to J and does not couple to J′. Thus,we see that in the case of K3surface,the present construction is in agreement with the traditional considerations[72,73]and[30].3The two-dimensional topological J-modelThe data(X,Q,∆,ρ)associated with a generalized CY manifold X allow us to constructa topological J-model using the AKSZ method[47].3.1The AKSZ construction of sigma-model for Maps(ˆΣ,M)for a targetspace M with a P Q-structureA P Q-manifold is a supermanifold equipped with an odd symplectic structureωand a Hamiltonian vectorﬁeld Q.In[47]a topological sigma-model was constructed for any such target space M.Let us review the key properties of a P Q-target space.The symplectic structure deﬁnes the odd Poisson bracket{·,·}.Sinceωis Q-invariantL Qω=(di Q+i Q d)ω=d(i Qω)=0,(3.1)the one-form i Qωis closed.We consider the case when i Qωis exact,so there is a function S such that i Qω=dS.Such a function on X is called Hamiltonian function for the vector ﬁeld Q.For any function f∈C∞(M)its Lie derivative L Q f can be computed as a bracket with the Hamiltonian function S associated with QL Q f={S,f}.(3.2)The homological property Q2=0of the Q-structure can be written as the BV classical master equation[45,46,44]{S,S}=0.(3.3)A canonical example of a Q-manifold is a tangent bundle with parity reversed on the ﬁbersΠT X.In coordinates(xµ,ψµ)one has Q=ψµ∂µ.As discussed in the previous section,the tangent bundle is an example of a Lie algebroid.The total spaceΠL of any Lie algebroid is a particular case of Q-manifold with the vectorﬁeld Q of degree1.12V asily PestunA canonical example of a P-manifold is a cotangent bundle with parity reversed on the ﬁbersΠT∗X.In coordinates(xµ,πµ)the canonical symplectic form isω(δx,δπ)=δxµδπµ.A P Q-manifold M can be constructed starting from any Q-manifold N as M=ΠT∗N. If(xµ,πµ)are coordinates onΠT∗N,then for any vectorﬁeld Q=vµ∂∂θ1∂∂θ1∂∂σi onˆΣis thestandard de Rham diﬀerential d onΣ.The idea of[47]is to construct the BV structure on the space Maps(ˆΣ,M)using the Q-structure QΣon the worldsheet,the Q-structure Q M on the target space generated by the Hamiltonian function S M,the odd symplectic structure on the target spaceωand the integral on the worldsheet ˆΣ.The space Maps(ˆΣ,M)is the BV phase space of the model.The odd symplectic structureˆωon Maps(ˆΣ,M)is deﬁned by the integral of the pullback of the odd symplectic structureωfrom the target space.For variationsδφ1,δφ2of a mapφ∈Maps(ˆΣ,M)we deﬁne the value ofˆωonδφ1,δφ2to be the integral overˆΣ˜ω(δφ1,δφ2)= ˆΣω(δφ1,δφ2).(3.5) As for the Q-structure on Maps(ˆΣ,M)we can takeQ=QΣ+Q M.(3.6)Any Q-structure generates diﬀeomorphism.There is a natural action of Diff(ˆΣ)and Diff(M)on the space Maps(ˆΣ,M).The Q-structure(3.6)corresponds to composition of two diﬀeomorphisms[47,49].For any such Q there is a corresponding Hamiltonian function S.The function S is the called BV master action[42,43].To write down SΣ,it is convenient to represent the odd symplectic target space M as ΠT∗N for some manifold N.Let xµbe coordinates on N and pµbe coordinates on the。

一维磁性原子链系统中的Majorana费米子态

一维磁性原子链系统中的Majorana费米子态杨双波【摘要】对处于螺旋形磁场及横向均匀磁场的一维磁性原子链模型,在平均场近似下通过自洽地求解Bogoliubov-de-Genes方程我们计算了系统的能谱.我们发现在一定参数值的范围内能谱随螺旋形磁场振幅值演化呈现能量为零的Majorana 费米子态.我们计算了局域态密度发现对Majorana费米子其态密度的峰值出现在链的两端(或中点)位置.我们计算了波函数其空间分布,发现它与局域态密度的结果一致.%For a model of one dimensional magnetic atomic chain in both a helical magnetic field and a transverse uniform magnetic field,we calculate its energy spectrum by solving Bogoliubov-de-Genes equation selfconsistently in the mean field approximation. We find that for a certain parameter setting,energy spectrum evolving with amplitude of helical magnetic field,appears Majorana fermion eigenstates. We calculate local density of states,and find that the local density of states for Majorana fermion shows peaks at the both ends(or at middle)of the magnetic atomic chain. We calculate wave function,and its spatial distribution agrees with local density of states.【期刊名称】《南京师大学报（自然科学版）》【年(卷),期】2017(040)003【总页数】8页(P110-117)【关键词】Majorana费米子;磁性原子链;BdG方程;局域态密度【作者】杨双波【作者单位】南京师范大学物理科学与技术学院,江苏省大规模复杂系统数值模拟重点实验室,江苏南京210023【正文语种】中文【中图分类】O413.1Majorana fermion[1] which is a particle of the same as its own antiparticle,has been attracting great attention. Firstly,because of Majorana fermion being connected with topological phase concept,secondly because of its topological character,it provides a platform of potential application in topological quantum computing and quantum storing[2-4]. So experimentally and theoretically search for physical system of Majorana fermion has been a very hot research topic. The purpose of all these researches is to generate a topological superconductor,so that the Majorana fermion appears as a single excitation at the boundary. Recently,Majorana fermion has been studied for a model of an atomic chain in a helical magnetic field in close proximity to a s-wave superconductor[5-7],and the result shows that at a certain parameter setting,the Majorana fermion is localized at the both end of the magnetic atomic chain. This is a spatially uniform system,after a gauge transformation,the Hamiltonian of the system will become an invariant form for space displacement. In this paper we modified this system by adding a new Zeeman term in the original Hamiltonian,which correspondsto a uniform magnetic field h perpendicular to the atomic chain being applied to the original system. Because of the new Zeeman term,the system is nonuniform spatially,we will study the structure of the Majorana fermion for the system of h≠0.In this paper,we get the system eigenenergies and eigenvectors by numerically solving BdG equation,and then we study the birth and the localization in space of the Majorana fermion by calculating the spatially resolved local density of states and wave function. The structure of the paper is as the following,the Model and theory is in section 1,the result of numerical calculation and discussion is in section 2,the summary of the paper is in section 3.Consider a N-atom atomic chain in a helical magnetic filed or magnetic structure. The magnetic filed at site n is =B0(cosnθ+sinnθ),where θ is the angle made by the magnetic fields at the adjacent sites of the atomic chain,the whole atomic chain is in proximity to the surface of a s-wave superconductor,and in the transverse direction of the atomic chain a uniform magnetic field h is applied. The Hamiltonian of this magnetic atomic chain in mean field approximation is given byH=tx(cn+1α+h.c.)-μcnα+(cnβ+Δ(n)(+h.c.)+h(σz)ααcnα,where tx is the jumping amplitude for electron between two adjacent sites,μ is chemical potential,Δ(n)is the superconductor pairing potential or order parameter at sit n,h is the weak uniform magnetic field for tuning system energy spectrum. or cnα is the operator to creat or annihilate an electron of spin α respectively at site n, is pauli matrix vec tor,and h.c. standfor complex conjugate. By introducing Nambu spinor representationψi=(ci↑,ci↓,,-)T,then Hamiltonian(1)can be written as BdG form,i.e.H=Hijψj,where Hij is the BdG Hamiltonian at site i,which can be written as where Kij=tx(δi+1,j+δi-1,j)-μδij,γij=(h-B0cosiθ)δij. This is a 4N×4N matrix,whose energy eigenvalue εn and eigenfunctionψn(i)=(un(↑,i),un(↓,i),vn(↓,i),vn(↑,i))T, for i=1,2,…,N,is determined by eigenvalue equationand boundary condition. In mean field approximation the order parameter at site i takes the form[8]and the mean number of electron at site i iswhere fn=1/(1+eεn/kBT) is the Fermi distribution,T is temperature in Kelvin. The total number of electron isincluding spin up and spin down electrons. To determine eigenenergy,eigenfunction,order parameter,we need to selfconsistently solve eigenvalue equation(3)with(4)-(6). In this paper,we deal with the case of temperature T=0,then the order parameter and the mean number of electron in site i are given bySpecial case:h=0 and Δ(i)=Δ0,a constant. The Hamiltonian in(1)can be transformed into spatially unform form by a gauge transformation. The topologically nontrivial region of the parameter set is given bywhere Majorana fermion corresponds to εn=0. As |h|≠0,Hamiltonian in(1)is nonuniform in space,and the nontrivial region of parameter set can not be obtained analytically.In this paper we deal with open boundary condition with and without themiddle magnetic domain wall,and at the magnetic domain wall we replace θ by -θ. We have also studied under the periodic boundary condition,and found the result has no significant changes.We first study the character of the Majorana fermion as the order parameter Δ and chemical potential μ are constants,then we study the influence of nonuniform Δ(i)on the result of Majorana fermion by doing selfconsistent calculation. In calculation,we choose the number of site N for the atomic chain according to the angle θ,so that the magnetic field at the both ends of the atomic chain points to the same direction.2.1 Energy Spectrum and Wave FunctionsFor a one-dimensional magnetic atomic chain with magnetic domain wall in the middle,the parameter set is chosen asΔ0=1.0,tx=1.0,μ=2.5,h=0.1,θ=π/2,and length of the chain is chosen asN=81 sites. For every value of B0 in the interval[1.0,4.0],we diagonalize the 4N×4N BdG Hamiltonian matrix(2),we get 4N energy eigenvalues and 4N eigenvectors. In open boundary condition,the energy spectrum is shown in Fig.1. For B0 in the interval[1.486 6,3.996 8],we can see that thereexists eigenstates whose eigenenergy εn=0,and these eigenstates are Majorana fermions. The interval for the existense of Majorana fermion in the case of h=0.1 is very close to the interval[1.476,4.039]calculatedfrom(9)for the h=0 case. For Majorana fer mion at B0=2.1,εn=0,shown in red dot in Fig.1,we calculate its wave functionun(↑,i),un(↓,i),vn(↑,i),vn(↓,i),the result is shown in Fig.2(a-d). We can see that the amplitude of the wave function concentrates on the both ends and themiddle of the atomic chain. In Fig.3,we show wave function for the same magnetic atomic chain without magnetic domain wall in the middle,the amplitude of the wave function concentrates on both ends of the magnetic atomic chain. This is similar to the previous result for the 1-dimensional magnetic atomic chain without uniform magnetic field,h=0.2.2 Local Density of States and Total Density of StatesIn this subsection,we study the space distribution of density ofstates(DOS),which is called local density of states(LDOS)and is defined as ρ(ε,i)is a function of energy and space position,and the total density of states(TDOS)can be written as,i.e.,the arithmetic mean of local density of states,a function of energy only. In numerical calculation,we replace δ by a Lorentz function. Fo r parameter setting h=0.1,tx=1.0,Δ=1.0,μ=2.5,θ=π/2,N=81,and B0=2.1,the local density of states for a Majorana fermion and the mean number of electrons on each site are shown in Fig.4(a-d)for the magnetic atomic chain with magnetic domain wall in the middle. Fig.4(a)shows the local density of states ρ(ε,i)in a 3D-plot;Fig.4(b)shows the local density of states ρ(ε,i)in a 2D contour plot;Fig.4(c)shows the local density of states for Majorana fermion ρ(ε=0,i);Fig.4(d)shows the mean number of electron on each si te of atomic chain<n(i)>. We can see from the Fig.4 that the Majorana fermion is localized at two ends and middle for the magnetic atomic chain with magnetic domain wall in the middle. The mean number of electrons on each site of the atomic chain is around 1.5. In Fig.5(a-d)we show theresult for the same magnetic atomic chain without magnetic domain wall in the middle,then we see density of states for Majorana fermion is peaked only at both ends of the magnetic atomic chain.2.3 The Self-consistent ResultAs the order parameter Δ(i)is space position i dependent,we calculate the energy spectrum and local density of states by selfconsistently solving the eigenvalue equation(3)with equations(7)and(8). For parameter seth=0.1,tx=1.0,μ=2.5,U0=4.0,θ=π/2,N=81,the selfconsistently calculated energy spectrum is shown in Fig.6,the Majorana fermion region can be seen,is still there,but the interval is shorten. The local density of states and mean numbers of electron on each site for Majorana fermion at B0=1.57 are shown in Fig.7(a-d)and Fig.9(a-d). By comparison with Fig.4(a-d),we find the main characters are same,but peak position for selfconsistent result moved inside a little bit. We calculate the selfconsistent wave function for Majorana fermion,and the results are shown in Fig.8(a-d)and Fig.10(a-d). The amplitude is significiently large at both ends for magnetic atomic chain without magnetic domain wall,and significiently large at both ends and middle for a magnetic chain with magnetic domain wall in the middle. This agrees with the result of local density of states.In mean field approximation,and by numerically solving Bogoliubov-de-Genes(BdG)equation,this paper studies the birth,and the localization in space of the Majorana fermion in a one dimensional atomic chain in helical magnetic field,and a uniform magnetic field h which is perpendicular to the atomic chain. Studies find that at a certain parameter setting,theevolution of the energy spectrum with helical magnetic field amplitude B0 appears the zero energy eigenstates,which corresponding to the Majorana fermion. We calculate the local density of states,and find that the local density of states for the Majorana fermion has two peaks on the both end of the magnetic atomic chain. When a magnetic domain wall is applied at the middle of the maqgnetic atomic chain,the Majorana fermion shows peaks at both ends and the middle of the magnetic chain. As the order parameter is a function of space coordinate,we do selfconsistent calculation,and find that by comparing with the result of nonselfconsistent calculation,the energy spectrum and the shape of the local density of states are changed a little bit,but the main character does not change. [1] MAJORANA E. Symmetric theory of electron and positrons[J]. Nuovo Cimento,1937,14(1):171-181.[2] WILCZEK F. Majorana returns[J]. Nat Phys,2009,5(9):614-618.[3] NAYAK C,SIMON S H,STERN A,et al. Non-Abelian anyons and topological quantum computation[J]. Rev Mod Phys,2008,80(3):1 083-1 159.[4] ALICEA J. New directions in the persuit of Majorana fermions in solid state system[J]. Rep Prog Phys,2012,75(7):076501-1-36.[5] NADJ-PERGE S,DROZDOV I K,BERNEVIG B A,et al. Proposal for realizing Majorana fermions in chain of magnetic atoms on a superconductor [J]. Phys Rev B,2013,88(2):020407-1-5(R).[6] PÖYHÖNEN K,WESTSTRÖM A,RÖNTYNEN J,et al. Majorana state in helical shiba chain and ladders[J]. Phys Rev B,2014,89(11):115109-1-7.[7] VAZIFEH M M,FRANZ M. Self-organized topological state with Majorana fermions[J]. Phys Rev Lett,2013,111(20):206802-1-5.[8] SACRAMENTO P D,DUGAEV V K,VIEIRA V R. Magnetic impurities in a superconductors:effect of domainwall and interference[J]. Phys RevB,2007,76(1):014512-1-21.[9] EBISU H,YADA K,KASAI H,et al. Odd frequency pairing in topological superconductivity in a one dimensional magnetic chain[J]. Phys RevB,2015,91(5):054518-1-15.【相关文献】Received data:2016-11-17.Corresponding author:Yang Shuangbo,professor,majored in nonlinear physics and low dimensionalsystem.E-mail:*********************.cndoi:10.3969/j.issn.1001-4616.2017.03.016CLC number:O413.1 Document codeA Article ID1001-4616(2017)03-0110-08。

数学专业英语词汇（T）_数学物理英语词汇

t distribution 学生t分布t number t数t statistic t统计量t test t检验t1topological space t1拓扑空间t2topological space t2拓扑空间t3topological space 分离空间t4topological space 正则拓扑空间t5 topological space 正规空间t6topological space 遗传正规空间table 表table of random numbers 随机数表table of sines 正弦表table of square roots 平方根表table of values 值表tabular 表的tabular value 表值tabulate 制表tabulation 造表tabulator 制表机tacnode 互切点tag 标签tame 驯顺嵌入tame distribution 缓增广义函数tamely imbedded 驯顺嵌入tangency 接触tangent 正切tangent bundle 切丛tangent cone 切线锥面tangent curve 正切曲线tangent function 正切tangent line 切线tangent of an angle 角的正切tangent plane 切平面tangent plane method 切面法tangent surface 切曲面tangent vector 切向量tangent vector field 切向量场tangent vector space 切向量空间tangential approximation 切线逼近tangential component 切线分量tangential curve 正切曲线tangential equation 切线方程tangential stress 切向应力tangents method 切线法tape 纸带tape inscription 纸带记录tariff 税tautology 重言taylor circle 泰勒圆taylor expansion 泰勒展开taylor formula 泰勒公式taylor series 泰勒级数technics 技术technique 技术telegraph equation 电报方程teleparallelism 绝对平行性temperature 温度tempered distribution 缓增广义函数tend 倾向tendency 瞧tension 张力tensor 张量tensor algebra 张量代数tensor analysis 张量分析tensor bundle 张量丛tensor calculus 张量演算法tensor density 张量密度tensor differential equation 张量微分方程tensor field 张量场tensor form 张量形式tensor form of the first kind 第一张量形式tensor function 张量函数tensor of torsion 挠率张量tensor product 张量乘积tensor product functor 张量乘积函子tensor representation 张量表示tensor space 张量空间tensor subspace 张量子空间tensor surface 张量曲面tensorial multiplication 张量乘法term 项term of higher degree 高次项term of higher order 高次项term of series 级数的项terminability 有限性terminable 有限的terminal decision 最后判决terminal edge 终结边terminal point 终点terminal unit 级端设备terminal vertex 悬挂点terminate 终止terminating chain 可终止的链terminating continued fraction 有尽连分数terminating decimal 有尽小数termination 终止terminology 专门名词termwise 逐项的termwise addition 逐项加法termwise differentiation 逐项微分termwise integration 逐项积分ternary 三元的ternary connective 三元联结ternary form 三元形式ternary notation 三进制记数法ternary number system 三进制数系ternary operation 三项运算ternary relation 三项关系ternary representation og numbers 三进制记数法tertiary obstruction 第三障碍tesseral harmonic 田形函数tesseral legendre function 田形函数test 检验test for additivity 加性检验test for uniform convergence 一致收敛检验test function 测试函数test of dispersion 色散检验test of goodness of fit 拟合优度检验test of hypothesis 假设检验test of independence 独立性检验test of linearity 线性检验test of normality 正规性检验test point 测试点test routine 检验程序test statistic 检验统计量tetracyclic coordinates 四圆坐标tetrad 四元组tetragon 四角形tetragonal 正方的tetrahedral 四面角tetrahedral angle 四面角tetrahedral co ordinates 四面坐标tetrahedral group 四面体群tetrahedral surface 四面曲面tetrahedroid 四面体tetrahedron 四面形tetrahedron equation 四面体方程theorem 定理theorem for damping 阻尼定理theorem of alternative 择一定理theorem of identity for power series 幂级数的一致定理theorem of implicit functions 隐函数定理theorem of mean value 平均值定理theorem of principal axes 轴定理theorem of residues 残数定理theorem of riemann roch type 黎曼洛赫型定理theorem on embedding 嵌入定理theorems for limits 极限定理theoretical curve 理论曲线theoretical model 理论模型theory of automata 自动机理论theory of cardinals 基数论theory of complex multiplication 复数乘法论theory of complexity of computations 计算的复杂性理论theory of correlation 相关论theory of differential equations 微分方程论theory of dimensions 维数论theory of elementary divisors 初等因子理论theory of elementary particles 基本粒子论theory of equations 方程论theory of errors 误差论theory of estimation 估计论theory of functions 函数论theory of games 对策论theory of hyperbolic functions 双曲函数论theory of judgment 判断论theory of numbers 数论theory of ordinals 序数论theory of perturbations 摄动理论theory of probability 概率论theory of proportions 比例论theory of relativity 相对论theory of reliability 可靠性理论theory of representations 表示论theory of sets 集论theory of sheaves 层理论theory of singularities 奇点理论theory of testing 检验论theory of time series 时间序列论theory of transversals 横断线论theory of types 类型论thermal 热的thermodynamic 热力学的thermodynamics 热力学theta function 函数theta series 级数thick 厚的thickness 厚度thin 薄的thin set 薄集third boundary condition 第三边界条件third boundary value problem 第三边界值问题third fundamental form 第三基本形式third isomorphism theorem 第三同构定理third proportional 比例第三项third root 立方根thom class 汤姆类thom complex 汤姆复形three body problem 三体问题three dimensional 三维的three dimensional space 三维空间three dimensional torus 三维环面three eighths rule 八分之三法three faced 三面的three figur 三位的three place 三位的three point problem 三点问题three series theorem 三级数定理three sheeted 三叶的three sided 三面的three sigma rule 三规则three termed 三项的three valued 三值的three valued logic 三值逻辑three valued logic calculus 三值逻辑学threshold logic 阈逻辑time interval 时程time lag 时滞time series analysis 时序分析timesharing 分时toeplitz matrix 托普利兹矩阵tolerance 容许tolerance distribution 容许分布tolerance estimation 容许估计tolerance factor 容许因子tolerance level 耐受水平tolerance limit 容许界限tolerance region 容许区域top digit 最高位数字topological 拓扑的topological abelian group 拓扑阿贝耳群topological algebra 拓扑代数topological cell 拓扑胞腔topological circle 拓扑圆topological completeness 拓扑完备性topological complex 拓扑复形topological convergence 拓扑收敛topological dimension 拓扑维topological direct sum 拓扑直和topological dynamics 拓扑动力学topological embedding 拓扑嵌入topological field 拓扑域topological group 拓扑群topological homeomorphism 拓扑同胚topological index 拓扑指数topological invariant 拓扑不变量topological limit 拓扑极限topological linear space 拓扑线性空间topological manifold 拓扑廖topological mapping 拓扑同胚topological pair 拓扑偶topological polyhedron 曲多面体topological product 拓扑积topological residue class ring 拓扑剩余类环topological ring 拓扑环topological simplex 拓扑单形topological skew field 拓扑非交换域topological space 拓扑空间topological sphere 拓扑球面topological structure 拓扑结构topological sum 拓扑和topological type 拓扑型topologically complete set 拓扑完备集topologically complete space 拓扑完备空间topologically equivalent space 拓扑等价空间topologically nilpotent element 拓扑幂零元topologically ringed space 拓扑环式空间topologically solvable group 拓扑可解群topologico differential invariant 拓扑微分不变式topologize 拓扑化topology 拓扑topology of bounded convergence 有界收敛拓扑topology of compact convergence 紧收敛拓扑topology of uniform convergence 一致收敛拓扑toroid 超环面toroidal coordinates 圆环坐标toroidal function 圆环函数torque 转矩torsion 挠率torsion coefficient 挠系数torsion form 挠率形式torsion free group 非挠群torsion group 挠群torsion module 挠模torsion of a curve 曲线的挠率torsion product 挠积torsion subgroup 挠子群torsion tensor 挠率张量torsion vector 挠向量torsionfree connection 非挠联络torsionfree module 无挠模torsionfree ring 无挠环torus 环面torus function 圆环函数torus group 环面群torusknot 环面纽结total 总和total correlation 全相关total curvature 全曲率total degree 全次数total differential 全微分total differential equation 全微分方程total error 全误差total graph 全图total image 全象total inspection 全检查total instability 全不稳定性total inverse image 全逆象total matrix algebra 全阵环total matrix ring 全阵环total order 全序total predicate 全谓词total probability 总概率total probability formula 总概率公式total regression 总回归total relation 通用关系total space 全空间total stability 全稳定性total step iteration 整步迭代法total step method 整步迭代法total stiefel whitney class 全斯蒂费尔惠特尼类total subset 全子集total sum 总和total variation 全变差totally bounded set 准紧集totally bounded space 准紧空间totally differentiable 完全可微分的totally differentiable function 完全可微函数totally disconnected 完全不连通的totally disconnected graph 完全不连通图totally disconnected groupoid 完全不连通广群totally disconnected set 完全不连通集totally disconnected space 完全不连通空间totally geodesic 全测地的totally nonnegative matrix 全非负矩阵totally ordered group 全有序群totally ordered set 线性有序集totally positive 全正的totally positive matrix 全正矩阵totally quasi ordered set 完全拟有序集totally real field 全实域totally reflexive relation 完全自反关系totally regular matrix method 完全正则矩阵法totally singular subspace 全奇异子空间totally symmetric loop 完全对称圈totally symmetric quasigroup 完全对称拟群touch 相切tournament 竞赛图trace 迹trace form 迹型trace function 迹函数trace of dyadic 并向量的迹trace of matrix 矩阵的迹trace of tensor 张量的迹tracing point 追迹点track 轨迹tractrix 曳物线trajectory 轨道transcendence 超越性transcendence basis 超越基transcendence degree 超越次数transcendency 超越性transcendental element 超越元素transcendental equation 超越方程transcendental function 超越函数transcendental integral function 超越整函数transcendental number 超越数transcendental singularity 超越奇点transcendental surface 超越曲面transfer 转移transfer function 转移函数transfinite 超限的transfinite diameter 超限直径transfinite induction 超限归纳法transfinite number 超限序数transfinite ordinal 超限序数transform 变换transformation 变换transformation equation 变换方程transformation factor 变换因子transformation formulas of the coordinates 坐标的变换公式transformation function 变换函数transformation group 变换群transformation of air mass 气团变性transformation of coordinates 坐标的变换transformation of parameter 参数变换transformation of state 状态变换transformation of the variable 变量的更换transformation rules 变换规则transformation theory 变换论transformation to principal axes 轴变换transgression 超渡transient response 瞬态响应transient stability 瞬态稳定性transient state 瞬态transient time 过渡时间transition function 转移函数transition graph 转换图transition matrix 转移矩阵transition probability 转移函数transitive closure 传递闭包transitive graph 传递图transitive group of motions 可迁运动群transitive law 可迁律transitive permutation group 可迁置换群transitive relation 传递关系transitive set 可递集transitivity 可递性transitivity laws 可迁律translatable design 可旋转试验设计translate 转移translation 平移translation curve 平移曲线translation group 平移群translation invariant 平移不变的translation invariant metric 平移不变度量translation number 殆周期translation of axes 坐标轴的平移translation operator 平移算子translation surface 平移曲面translation symmetry 平移对称translation theorem 平移定理transmission channel 传输通道transmission ratio 传输比transport problem 运输问题transportation algorithm 运输算法transportation matrix 运输矩阵transportation network 运输网络transportation problem 运输问题transpose 转置transposed inverse matrix 转置逆矩阵transposed kernel 转置核transposed map 转置映射transposed matrix 转置阵transposition 对换transversal 横截矩阵胚transversal curve 横截曲线transversal field 模截场transversal lines 截线transversality 横截性transversality condition 横截条件transverse axis 横截轴transverse surface 横截曲面trapezium 不规则四边形trapezoid 不规则四边形trapezoid formula 梯形公式trapezoid method 梯形公式traveling salesman problem 转播塞尔斯曼问题tree 树trefoil 三叶形trefoil knot 三叶形纽结trend 瞧trend line 瞧直线triad 三元组trial 试验triangle 三角形triangle axiom 三角形公理triangle condition 三角形公理triangle inequality 三角形公理triangulable 可三角剖分的triangular decomposition 三角分解triangular form 三角型triangular matrix 三角形矩阵triangular number 三角数triangular prism 三棱柱triangular pyramid 四面形triangular surface 三角曲面triangulate 分成三角形triangulation 三角剖分triaxial 三轴的triaxial ellipsoid 三维椭面trichotomy 三分法trident of newton 牛顿三叉线tridiagonal matrix 三对角线矩阵tridimensional 三维的trigammafunction 三函数trigonometric 三角的trigonometric approximation polynomial 三角近似多项式trigonometric equation 三角方程trigonometric function 三角函数trigonometric moment problem 三角矩问题trigonometric polynomial 三角多项式trigonometric series 三角级数trigonometrical interpolation 三角内插法trigonometry 三角学trihedral 三面形的trihedral angle 三面角trihedron 三面体trilateral 三边的trilinear 三线的trilinear coordinates 三线坐标trilinear form 三线性形式trinomial 三项式;三项式的trinomial equation 三项方程triplanar point 三切面重点 ?triple 三元组triple curve 三重曲线triple integral 三重积分triple point 三重点triple product 纯量三重积triple product of vectors 向量三重积triple root 三重根triple series 三重级数triple tangent 三重切线triply orthogonal system 三重正交系triply tangent 三重切线的trirectangular spherical triangle 三直角球面三角形trisecant 三度割线trisect 把...三等分trisection 三等分trisection of an angle 角的三等分trisectrix 三等分角线trivalent map 三价地图trivector 三向量trivial 平凡的trivial character 单位特贞trivial cohomology functor 平凡上同弹子trivial extension 平凡扩张trivial fibre bundle 平凡纤维丛trivial graph 平凡图trivial homogeneous ideal 平凡齐次理想trivial knot 平凡纽结trivial solution 平凡解trivial subset 平凡子集trivial topology 密着拓扑trivial valuation 平凡赋值triviality 平凡性trivialization 平凡化trochoid 摆线trochoidal 余摆线的trochoidal curve 摆线true error 真误差true formula 真公式true proposition 真命题true sign 直符号true value 真值truncated cone 截锥truncated cylinder 截柱truncated distribution 截尾分布truncated pyramid 截棱锥truncated sample 截样本truncated sequence 截序列truncation 舍位truncation error 舍位误差truncation point 舍位点truth 真值truth function 真值函项truth matrix 真值表truth set 真值集合truth symbol 真符号truth table 真值表truthvalue 真值tube 管tubular knot 管状纽结tubular neighborhood 管状邻域tubular surface 管状曲面turbulence 湍流turbulent 湍聊turing computability 图灵机可计算性turing computable 图灵机可计算的turing machine 图录机turn 转向turning point 转向点twice 再次twice differentiable function 二次可微函数twin primes 素数对twisted curve 空间曲线twisted torus 挠环面two address 二地址的two address code 二地址代码two address instruction 二地址指令two body problem 二体问题two decision problem 二判定问题two digit 二位的two dimensional 二维的two dimensional laplace transformation 二重拉普拉斯变换two dimensional normal distribution 二元正态分布two dimensional quadric 二维二次曲面two dimensional vector space 二维向量空间two fold transitive group 双重可迁群two person game 两人对策two person zero sum game 二人零和对策two phase sampling 二相抽样法two place 二位的two point distribution 二点分布two point form 两点式two sample method 二样本法two sample problem 二样本问题two sample test 双样本检验two sheet 双叶的two sided condition 双边条件two sided decomposition 双边分解two sided divisor 双边因子two sided ideal 双边理想two sided inverse 双边逆元two sided module 双边模two sided neighborhood 双侧邻域two sided surface 双侧曲面two sided test 双侧检定two stage sampling 两阶段抽样法two termed expression 二项式two valued logic 二值逻辑two valued measure 二值测度two variable matrix 双变量矩阵two way array 二向分类two way classification 二向分类twopoint boundary value problem 两点边值问题type 型type problem 类型问题typenumber 型数typical mean 典型平均。

Invariants of Legendrian Knots and Coherent Orientations

1
2
J. ETNYRE, L. NG, AND J. SABLOFF
Chekanov’s original DGA can be recovered by setting t = 1, which will force the grading to be reduced modulo 2r, and taking the coeﬃcients modulo 2. Part I of the paper is devoted to this goal. Concurrent with Chekanov’s work on his DGA, Eliashberg and Hofer adapted the ideas of Floer homology to the contact setting. Though we will ﬂesh out a relative version of their “contact homology theory” in Section 7, the story goes roughly as follows: let (M, α) be a contact manifold with a Legendrian submanifold K . Let A be the free associative unital algebra generated by the Reeb chords — i.e. Reeb trajectories that begin and end on K . The generators are graded by something akin to the Maslov index. There is a diﬀerential on A that comes from counting rigid J -holomorphic disks in the symplectization (M × R, d(eτ α)) of M . Here, J is a vertically-invariant almost complex structure compatible with d(eτ α). Using Floer and Hofer’s idea of coherent orientations [8, 12], it is possible to orient all of the moduli spaces of rigid J -holomorphic disks used in the deﬁnition of the diﬀerential. As a result, we may use Z coeﬃcients in the deﬁnition of the algebra A. The second goal of this paper, carried out in Part II, is to prove that Chekanov’s DGA, and our generalization of it, is a combinatorial translation of relative contact homology. Knowing the relation between the combinatorial and geometric versions of contact homology is quite useful. In particular, the lifting of Chekanov’s DGA from Z/2 to Z[t, t−1 ] was accomplished by studying this relationship. Moreover, explicit computations in the framework of Eliashberg and Hofer’s contact homology theory can be diﬃcult, while computations in the combinatorial theory are more straightforward. Thus, our translation between the two theories yields many explicit computations in contact homology. The paper consists of essentially two parts. After recalling several basic ideas from contact geometry in Section 2, we proceed, in Part I, to describe the combinatorial theory. This part is self-contained apart from a few technical proofs that are relegated to an appendix of Part I. In Part II of the paper we discuss Eliashberg and Hofer’s contact homology and coherent orientations. We then prove the combinatorial theory developed in Part I is a faithful translation of this more geometric theory. 2. Basic Notions We begin by describing some basic notions in three-dimensional contact geometry. A contact structure on a 3-manifold M is a completely non-integrable 2-plane ﬁeld ξ . Locally, a contact structure is the kernel of a 1-form α that satisﬁes the following non-degeneracy condition at every point in M : α ∧ dα = 0. In this paper, we will be interested in the standard contact structure ξ0 on R3 , which is deﬁned to be the kernel of the 1-form α0 = dz + x dy. To each contact form α, we may associate a Reeb ﬁeld Xα that satisﬁes dα(Xα , ·) = 0 and α(Xα ) = 1. By Darboux’s theorem, every contact manifold is locally contactomorphic to (R3 , ξ0 ). See [2, chapter 8] for an introduction to the fundamentals of contact geometry. Our primary objects of study are Legendrian knots in R3 , i.e. knots that are everywhere tangent to the standard contact structure ξ0 . In particular, we examine Legendrian isotopy classes of Legendrian knots, in which two knots are deemed equivalent if they are related by an isotopy through Legendrian knots.

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the frame of this model electrical sources are conﬁned for δ ≥
3 2
.
Energy
of the electric ﬁeld generated by the external charge is inﬁnite due to the
divergence at large distance. Moreover, the dipole-like external source gives
The Pagels-Tomboulis model has been also considered as an example of
a ﬁeld theory with the vanishing trace of the energy-momentum tensor. In
case of any (n + 1)-dimensional gauge theory deﬁned by a Lagrangian
a ﬁnite energy ﬁeld conﬁguration. The energy E behaves like
E = c |q| Λ R , 2δ
4δ−4 2δ−3
0 2δ−1 2δ−1 2δ−1
(2)
where c0 is a numerical constant, q is an external charge and R is the dista√nce
Abstract
The (2+1) dimensional non-linear electrodynamics, the so called Pagels–Tomboulis electrodynamics, with the Chern–Simons term is considered. We obtain ”generalized self–dual equation” and ﬁnd the corresponding generalized massive Chern–Simons Lagrangian. Similar results for (2+1) massive dilaton electrodynamics have been obtained.
a) Institute of Physics, Jagiellonian University, Reymonta 4, Krakow, Poland
b) Department of Physics, University of Alberta, Edmonton, AB T6G 2J1, Canada February 1, 2008
L = L(F ),
where F = FµaνF aµν, the trace T = Tµµ has the following form
dL
T
=
4F dF
− (n + 1)L.
(3)
One can easily ﬁnd that for any (n + 1)-dimensional space–time there exists the unique Lagrangian (up to a multiplicative constant) , which gives the vanishing trace of the energy–momentum tensor
L
=
1 − 4 (Fµν F
µν )δ
+
m 4
ǫµνρAµ
Fνρ.
(6)
Here, for simplicity, the dimensional constant Λ has been neglected. This model is the natural generalization of the non–linear electrodynamics (5) considered in [5]. It is well known that the Chern–Simons part of the Lagrangian (6) does not enter explicitly to the expression for the energy. It is due to the fact that this term is metric independent. Thus the energy– momentum tensor remains unchanged in comparison with the pure non-linear electrodynamics case. It has been shown using the ﬁeld equations that in the Maxwell limit i.e. for δ = 1 the gauge ﬁeld from (6) is proportional to the dual strength tensor [8], [9]
of motion read
∂ν
(FσρF σρ)δ−1F νµ
+
m 2δ
ǫµνρFνρ
=
0.
(9)
The solution of the second order equations (9) has the generalized form of the self-dual equation (7)
1 ǫµνρfµ∂ν fρ,
(11)
where the U(1) gauge ﬁeld in the generalized generalized massive Chern–
Aµ
=
δ 2m
(FσλF
σλ
)δ−1
ǫµν
ρFν
ρ.
(10)
It is immediately seen that after diﬀerentiation of both side of generalized self–dual equation and multiplication by ǫαβγ we obtain (9). As in the
∗mslus@phys.ualberta.ca †wereszcz@.pl
1
is the standard ﬁeld tensor, δ is a dimensionless parameter and Λ is a dimen-
sional constant. The gauge ﬁeld is SU(2) type i.e. a = 1, 2, 3. One can check
that
this
Lagrangian
has
positively
deﬁned
energy
for
δ
≥
1 2
.
The Pagels–Tomboulis theory was originally proposed as an eﬀective
model for the low energy (3+1) QCD [1]. In fact, it was shown that in
3
Maxwell case, the generalized self–dual equation emerges as a ﬁeld equation from generalized massive Chern–Simons Lagrangian
Lmass
=
1 4
(fµ
f
µ
)
δ 2δ−1
−
Dδ 2δ −
Aµ
=
1 2m
ǫµν
ρ
F
νρ.
(7)
Of course, using the U(1) gauge transformation Aµ → Aµ + ∂µψ one can generate the whole family gauge equivalent solutions. The solution (7) corresponds to the Lorentz gauge. This self–dual equation can be derived also from the massive Chern–Simons Lagrangian [10]
potential appears in the limit δ → ∞. Highly non-linear gauge models
appear in studying of the (2+1) QCD as well. Corresponding conﬁning force
has been recently obtained in [4].
L
=
−
1 4
(#43;1 4
.
(4)
Only in (3+1)-dimensional space–time such a Lagrangian is a linear function of F . For instance, in (2 + 1) dimension the pertinent model takes the form
arXiv:hep-th/0204104v3 28 Apr 2003
On (2+1)Dimensional Topologically Massive Non-linear Electrodynamics
M. S´lusarczyk a,b) ∗ and A. Wereszczyn´ski a) †
2
This short and incomplete list of applications of the Pagels–Tomboulis model in various areas of theoretical physics shows that the model is very interesting and has rich mathematical structure. Unfortunately, in contradistinction to other non-linear gauge theories (for example the Born–Infeld theory [6]) it has not been considered in the systematic way. In the present paper we focus on the (2+1) Abelian Pagels–Tomboulis model with the additional topological term - the Chern–Simons term