• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Seiberg-Witten/Whitham Equations and Instanton Corrections in N=2 Supersymmetric Yang-Mills Theory?

    2018-06-11 12:21:14JiaLiangDai戴佳亮andEnGuiFan范恩貴
    Communications in Theoretical Physics 2018年5期

    Jia-Liang Dai(戴佳亮) and En-Gui Fan(范恩貴)

    School of Mathematical Science,Fudan University,Shanghai 200433,China

    1 Introduction

    The solution of exact low energy effective action in N=2 supersymmetric Yang-Mills theory with SU(2)gauge group was obtained by Seiberg and Witten in Ref.[1]and their work have been generalized to the other higher rank gauge group SU(n),SO(n)and Sp(n)without or with matter hypermultiplets in the fundamental representation as well as to the exceptional group.The key analysis in Ref.[1]was that the quantum moduli space of N=2 supersymmetric gauge theories coupled with or without hypermultiplets could be naturally identi fi ed with the moduli space of certain hyperelliptic curves or Riemann surfaces.More specifically,with the help of these hyperelliptic curves we can describe the low energy effective action by a single holomorphic function F called prepotential and the exact solution for the prepotential is completely determined from the period integrals of a meromorphic differential on the hyperelliptic curves.[2?4]In general,the expression of prepotential F in the weakcoupling region may be divided into three parts:classical part Fclass,perturbative part Fpertwhich arises only from one-loop effects,and a sum of n-instanton part Fnin[5?6]

    it is well known that various kinds of methods have been developed to derive the prepotential from the Seiberg-Witten curves such as hypergeometric functions,[3]Picard-Fuchs equations[7?12]and the renormalization group type equations,[13]however,their complexity increases rapidly as the rank of the gauge group is large,even without matter hypermultiplets.In addition,another important observation about prepotential was noticed by Nekrasov[14]who introduced a partition function which is the generating function of the integral of equivariant cohomology class on the moduli space of framed instantons and showed[15]that the logarithm of this partition function is the instanton part of the Seiberg-Witten prepotential in N=2 supersymmetric four-dimensional gauge theory with gauge group SU(n).Meanwhile,Nakajima and Yoshioka[16]independently proved Nekrasov’s conjecture using the blowup techniques and Hilbert scheme which relates the Seiberg-Witten curves,prepotential and partition function together.These give a framework for understanding the instantons in gauge theory,integrable systems and representation theory of in finite-dimensional algebras in an intricate way.

    On the other hand,it was soon discovered that there exists a deep connection between Seiberg-Witten gauge theory and integrable system.[17?21]Roughly speaking,the Seiberg-Witten solution for the N=2 supersymmetric Yang-Mills theory is equivalent to a homogeneous solution of the Whitham hierarchy as well as the prepotential F corresponds to the logarithm of the Toda’s quasiclassical tau function.In the theory of Whitham hierarchy,there is a new family of variables introduced into the prepotential known as Whitham slow times Tnwhile the Whitham equations parameterized by the slow times characterize the deformations of the Seiberg-Witten curves.Around this time,the RG equations in Seiberg-Witten theory were fi rst derived by Gorsky,Marshakov,Mironov,and Morozov in Ref.[13]with the aid of Whitham hierarchy.Furthermore,based on their work Takasaki pointed out that the deformations by T1are interpreted as the renormalization group flows while the other Whitham deformations may be viewed as generalized RG flows.[22?23]More importantly,it is of great significance to remark that the second derivative of the prepotential F with respect to the slow time Tnleads to the appearance of the Riemann Theta-function.In this sense,after appropriately rescaling the gauge invariant parameters,the T1can be naturally regarded as the dynamical scale Λ which explicitly occurs in the Seiberg-Witten theory and the main RG equations required in this paper are given by[24?25]

    here the symbol ΘE(0|τ)is the Riemann’s theta function associated to the hyperelliptic curve C

    where E=[α;β]Tstands for an even half integer characteristic and in the case of pure SU(n)gauge theory it will be in the form of E=[0,...,0;1/2,...,1/2]T.From the above discussion,there is no difficulty in seeing that one can calculate any desired order instanton correction terms in SU(n)supersymmetric gauge theory by comparing the expansion coefficients of powers of Λ on both sides of Eq.(2)if we insert into the semiclassical expression of the prepotential F.Therefore,one of the most fundamental results of the Seiberg-Witten/Whitham equations is that they provide us a precise description of general recursion relations for any order instanton correction coefficient Fnin terms of the lower order instanton correction terms.It then follows that from this point of view,in principle,we could obtain arbitrary higher order correction coefficient Fnwithout solving the explicit expressions for the Seiberg-Witten periods a,aDwhich is the major feature of this article.

    The paper is organized as follows. In Sec.2 we briefly deduce the instanton correction coefficients using the Seiberg-Witten/Whitham equations in the case of SU(2)supersymmetric gauge theory as an illustrative example.In Sec.3 we generalize this approach to the general SU(n)situation and mainly we compute one-and two-instanton correction coefficients in detail,moreover,our results are in agreement with those in Ref.[5].Section 4 contains some conclusions and discussions and Appendix supplies us with relevant calculations and specific proofs we needed in Sec.3.

    2 SU(2)Case

    Let us first discuss the simplest case of non-abelian SU(2)supersymmetric gauge theory.For the moment due to the instanton effect,it is sufficient to consider the prepotential in SU(2)case without hypermultiplets and the general form of the prepotential F is[3]

    here the first term in Eq.(4)is one-loop expression for the prepotential which does not receive higher order perturbative corrections and the coefficients Fkin second term are constants.Notice that searching for the exact low energy effective action solution is equivalent to evaluating Fkfor all the k,fortunately,the Seiberg-Witten/Whitham equations give us an effective and standard procedure to determine these coefficients.Firstly one has

    here we set F0=6 which makes the constant coefficient term to be zero and this will be re fl ected in our choice for the normalization of F if we are rescaling Λ appropriately.We apply Eq.(2)to receive

    Substituting τ into the Riemann’s theta function(3)we find

    and the derivative of the Riemann’s theta function is

    obviously,the Seiberg-Witten/Whitham equations in SU(2)case turns out to be extremely simple

    matching the coefficient of(Λ/a)4nterm on both sides of Eq.(9),we then obtain the recursion relations for Fnas follows

    here ΘjandeΘjare defined as in Eqs.(7),(8)respectively.Now if rescaling the renormalization parameter Λ?→Λ/2 we can reexpress the prepotential F more explicitly

    here the instanton corrections coefficients are F1=1/25,F2=5/214,F3=3/218and so on.[3]

    Alternatively one can consider the physics near the N=1 singularities where generically corresponding to N?1 massless magnetic monopoles and the basic idea now is to evaluate the dual variables aD,kobtained from the akafter an S duality transformation.As shown in Ref.[26],the general expression of the strong coupling expansion of the prepotential at such singularity is given by

    in this case the variable a and coupling coefficient τDcan be formally calculated as above

    by analogy with Eq.(6)we have

    However,it should note here that from the point of view of the dual transformation,the characteristic of the theta function near the N=1 points must be replaced byα=(1/2,...,1/2)andβ=(0,...,0),at the same time,the Riemann’s theta function becomes Θ = ∑∞n=?∞exp(iπ(n+1/2)2τD),consequently it reads

    We assume FD0= ?1 and that is just the outcome of normalization of the FDkif we are rescaling Λ appropriately.In this way,the prepotential FDtakes a more familiar form as that in[3]

    here=iaD/Λ and taking advantage of these relations,one could derive the strong coupling expansion coefficients FDnrecursively through comparing the coefficient of the term(aD/Λ)n+1/8on both sides of the Seiberg-Witten/Whitham equation(2).

    3 SU(n)Case

    In this section we mainly consider the general SU(n)non-abelian supersymmetric Yang-Mills theory without doubt that the associated formulae and computations are much more complicated.To illustrate this detailedly,let us first recall that the general form of the prepotential F in the pure SU(n)supersymmetric gauge theory consists of three parts:the classical prepotential,the perturbative one-loop effects,and the k-instanton corrections,[5]namely

    in this formalism,the derivatives of the prepotential F follow directly that

    In view of the restriction to the constrain hyperplanewe should view the prepotential F(a1,...,aN)as a function of all the independent variables akexcept aNwhich results in ?i(ak?aN)= δki+1(here?imeans the partial derivative with variable ai,1≤i≤N?1).Thus we can put the RG equations in the following precise way

    hererespectively.Now the need to pay attention to is that because of the constraint conditionthe dual variables aD,iare

    and after a simple calculation we get

    which leads to a fairly explicit expression for the coupling coefficients τij= ?aD,i/?aj? ?aD,i/?aN(ij)

    in particular,the coupling coefficients τiiare

    From the previous argument we are aware of that in the general SU(n)non-abelian case,the most crucial ingredient in our analysis is the expression of Θ function.Therefore under the substitution of Eqs.(22),(23)into the de fi nition of Θ,it turns out to be

    where

    herecomes from the aNkterms in Eqs.(22)and(23).We already eliminate the e?(3/2)Nαfactor in Eq.(24)by appropriately rescaling Λ and as explained above this will be reflected in our choice for the normalization of the Fk.In addition,Θ(0)=1.Then without more efforts one is able to write down the formula for the derivative of the Θ with respect to the period matrix τijin the following form although the expressions ofare rather more involved.However,it is not difficult to prove that the indice l starts from 1,since when l=0 all integers nimust be equal to zero and from Eq.(26)we easily conclude that the coefficientsvanish.

    Now let us insert the formulae of ΘE(0|τ),?τijΘE(0|τ)obtained above into Eq.(2)

    which provides an exact expression for the instanton corrections coefficients

    here β =2N and comparing the coefficients of powers of Λ makes it possible to compute the Fkin a purely algebraic combinatorial way

    We remark here that Eq.(29)is a fundamental recursion relation for us to derive the exact n-instanton coefficient Fnby starting just from the coefficient F1through the Whitham hierarchy method in pure SU(n)supersymmetric gauge theory.As a matter of fact,the above results can even be extended to the more sophisticated situations of the massless or massive hypermultiplets included.Generally speaking,it is essentially no difficulty in applying this approach to the massless or massive hypermultiplets cases by repeating these relations(29)recursively,and after finitely many steps one is capable of getting all the correct instanton coefficients.However,in each case when k is larger,the functionsbecome much more complicated and of course the procedure of computation turns into cumbersome.Thereby for the purpose of the concrete evaluation of the Fk,one has to resort to the help of symbolic computation.In this article we basically compute 1-instanton and 2-instanton correction coefficients to illustrate the calculational procedure and then compare them with the results in paper.[5]

    3.1 1-Instanton Correction

    To derive the 1-instanton correction coefficient F1we consider k=1 in Eq.(29)which reads

    obviously from the definition ofwe have α=2,that is

    and we observe that the solutions of Eq.(31)are divided into two cases

    In the first case(a),under the condition ni=1 for some i,the expression A(a;n1,...,nN?1)turns out to be

    then analogous to Eq.(33)in case(b)we have

    clearly,notice that when ni= ?1 and nj= ?ni=1 we find A(?i)=A(i),A(?i,j)=A(i,?j)respectively.Then recalling the coefficients

    after a straightforward calculation,Eq.(30)becomes

    here we introduce the notationfor convenience and make use of(see the proof in Appendix A),the 1-instanton correction part of F can be written as

    now if defining the functionwe can rewrite Eq.(37)in the form ofwhich is the same as the expression in Ref.[5].

    3.2 2-instanton Correction

    In the following section we mainly compute the 2-instant correction coefficient and we are thus led,on account of the recursion relation(29),to the F2

    evidently,the calculation of F2is identical to the sum of three terms in Eq.(38)respectively.

    (a)term

    Taking into account of the definition oftogether with Eq.(35),we simplify the first term as

    here the accurate expression of F1,i(see Appendix B)is

    now utilizing Eq.(55)below,we may rewrite Eq.(39)as

    (b)term

    In order to obtain the explicit expression ofwe note that from Eq.(26)there are two conditions make contributions to the coefficients of Λ4N:(i) α =4,m=0;(ii) α =2,m=1,k=1.For the first condition α =4,or equivalentlywe find that the corresponding solutions are separated into two cases(we mainly consider N≥5)

    thus according to the definition of Eq.(25)one obtains as well as

    due to the symmetry between indices l and m,the functions of(ij)are now given by

    here the summation is for l,m and theare

    here we sum over for j,l,m.Now using these consequences and the expressions forogether with Eqs.(43),(44)above,we calculate

    Furthermore substituting the expressionsinto Eq.(47),it is immediate to see that

    and a tedious algebraic calculation of the mixed derivatives of the F1shows that

    the detailed proof of Eq.(49)can be consulted in Appendix B.

    (c) F1Θ(1)term

    Finally we want to compute the F1Θ(1)term and to begin with

    Hence,let us combineand we have

    Now to proceed further it is necessary to calculate the bibjblbmterms in Eqs.(41)and(49)explicitly,actually we find

    and from Appendix C,we conclude that

    The above analysis enables us to derive the 2-instanton correction coefficient F2in terms of variables aijand bk.Indeed,putting all these Eqs.(41),(47)–(49)and(51)–(53)together,we therefore arrive at a more compact expression as follows

    In particular,using the key relationit follows that

    and as a consequence,various identities can now be deduced from the above Eq.(55)which will play an important role to help us simplify the expression of F2.For instance,by differentiating Eq.(55)with respect to aiand multiplyingon both sides of the result equation,then summing over for i one finds that

    moreover,the similar process gives rise to

    analogously let us take derivative with respect to aion both sides of Eq.(57),multiplyon the result equation and sum over for i,through a direct calculation it yields

    hence inserting Eqs.(56)and(58)into(54)with the aid of Eq.(57),the computation of F2now is straightforward,and one finally obtains

    As a final comment,it is worth mentioning here that taking advantage of the notation Sk(x),the evaluation of 2-instanton correction coefficient F2can also be expanded in a more familiar form

    We point out here that the above expression is precisely the same as derived by E.D’Hoker,D.H.Phong and I.M.Krichever in Ref.[5].

    4 Conclusion

    In this paper,we primarily describe how to obtain arbitrary order instanton corrections coefficients of the effective prepotential F in N=2 pure SU(n)supersymmetric Yang-Mills theory from Whitham hierarchy and Seiberg-Witten/Whitham equations.The most important feature of this method is that there is no necessary to know the exact expressions of the Seiberg-Witten periods as functions of the moduli parameters.It is natural to generalize this idea to the other classical gauge group theory with or without hypermultiplets,which allows us to calculate the instanton corrections terms in various different Seiberg-Witten curves within a unified framework.We emphasize here that if one wants to get the recursion relations of instanton corrections coefficients from Seiberg-Witten/Whitham equations with massive or massless hypermultiplets,the number of hypermultiplets must be an even integer and the massive hypermultiplets must come up in degenerated pairs as shown in Ref.[10].Therefore it is essential to modify the formalism of the Whitham hierarchy and RG equations in order to extend our approach to the generic cases of unpaired and arbitrary masses.This would be interesting to further study.

    Appendix A

    For presenting the proof,it is convenient to define polynomial fand the first basic result is trivialBelow we will mostly focus on

    here(aN)means omitting the term fi(aN)in the products of f1(aN)··fi(aN)··fN?1(aN).Next for simplicity it is useful to introduce polynomialobviously we find gj(ak)=0 for jk,k≤N ?1,which provides

    Now let us make some general considerations on the following polynomial G(x)

    in fact,we notice that the degree of G(x)is N?2 but the polynomial has N?1 roots ai,i=1,...,N?1 which tells us that the polynomial G(x)is identically equal to zero,in other words,T(x)≡ (?1)(N?2)(N?3)/2∏N?1j

    on the other hand,from the definition of fi(x)we have

    then inserting Eqs.(A4)and(A5)into Eq.(A1),one obtains

    that is complete the proof.

    Appendix B

    Here we will exhibit some elementary identities about bi,which are applicable for our purpose and the corresponding proofs are straightforward

    According to these equations we can calculate the explicit form of F1,ij,indeed a simple and direct calculation shows that

    Then we give some details about how to evaluate the second derivative of Eq.(A8)with respect to the variable ajwhich can be seen as follows

    proceeding as before one finds(ij)

    Now if taking into account of Eq.(A7),we obtain a certain number of identities about the terms in Eq.(A9),these are

    Finally,let us substitute Eq.(A11)into(A9)together with Eq.(A10),we get

    Appendix C

    To begin with it is well known that from

    we have

    here the termsvanishing due to the antisymmetry of the indices i,j foror l,m for

    Analogously with the help of the equation

    it is enough to present that

    the termsvanishing because of the antisymmetry of the indices l,i or m,j for aliamj.

    Similarly from the identity

    we also have

    which gives rise to

    as explained above the first termsvanishing since the antisymmetry of the indices i,j or l,m for almaij.Now combining Eqs.(A14),(A16)with(A19)one can easily verify that

    [1]N.Seiberg and E.Witten,Nucl.Phys.B 426(1994)19.

    [2]A.Gorsky,I.Krichever,A.Marshakov,et al.,Phys.Lett.B 355(1995)466.

    [3]A.Klemm,W.Lerche,and S.Theisen,Int.J.Mod.Phys.A 11(1996)1929.

    [4]H.Itoyama and A.Morozov,Nucl.Phys.B 477(1996)855.

    [5]E.D’Hoker,D.H.Phong,and I.M.Krichever,Nucl.Phys.B 489(1997)179.

    [6]E.D’Hoker,D.H.Phong,and I.M.Krichever,Nucl.Phys.B 489(1997)211.

    [7]K.Ito and N.Sasakura,Nucl.Phys.B 484(1997)141.

    [8]J.M.Isidro,A.Mukherjee,J.P.Nunes,and H.J.Schnitzer,Nucl.Phys.B 492(1997)647.

    [9]M.Alishahiha,Phys.Lett.B 398(1997)100.

    [10]J.M.Isidro,A.Mukherjee,J.P.Nunes,and H.J.Schnitzer,Nucl.Phys.B 502(1997)363.

    [11]Y.Ohta,J.Math.Phys.40(1999)6292.

    [12]J.M.Isidro,arXiv:hep-th/0011253.

    [13]A.Gorsky,A.Marshakov,A.Mironov,and A.Morozov,Nucl.Phys.B 527(1998)690.

    [14]N.Nekrasov,Adv.Theor.Math.Phys.7(2004)831.

    [15]N.Nekrasov and A.Okounkov,arXiv:hep-th/0306238.

    [16]H.Nakajima and K.Yoshioka,Invent.Math.162(2005)313.

    [17]E.Martinec and N.Warner,Nucl.Phys.B 459(1995)97.

    [18]T.Nakatsu and K.Takasaki,Mod.Phys.Lett.A 11(1996)157.

    [19]E.D’Hoker and D.H.Phong,arXiv:hep-th/9903068.

    [20]A.Marshakov,Seiberg-Witten Theory and Integrable Systems,World Scientific,Singapore(1999).

    [21]A.Marshakov and N.Nekrasov,arXiv:hep-th/0612019.

    [22]K.Takasaki,Int.J.Mod.Phys.A 15(2000)3635.

    [23]K.Takasaki,Prog.Theor.Phys.Suppl.135(1999)53.

    [24]J.D.Edelstein and J.Mas,arXiv:hep-th/9902161.

    [25]J.D.Edelstein,M.G.Reino,and J.Mas,Nucl.Phys.B 561(1999)273.

    [26]J.D.Edelstein and J.Mas,Phys.Lett.B 452(1999)69.

    欧美高清成人免费视频www| 免费av中文字幕在线| 99久久综合免费| 国产69精品久久久久777片| 成人综合一区亚洲| 交换朋友夫妻互换小说| 精品久久久久久久久av| 最近2019中文字幕mv第一页| 精品一品国产午夜福利视频| 肉色欧美久久久久久久蜜桃| 国产一区二区在线观看日韩| 亚洲自偷自拍三级| 99久久中文字幕三级久久日本| 日韩制服骚丝袜av| 九九久久精品国产亚洲av麻豆| 一本色道久久久久久精品综合| 亚洲婷婷狠狠爱综合网| 嫩草影院入口| 视频中文字幕在线观看| 国产淫片久久久久久久久| 精品亚洲成国产av| 国产免费福利视频在线观看| 最近最新中文字幕免费大全7| 嫩草影院入口| 国产 一区 欧美 日韩| av福利片在线观看| 亚洲欧美成人综合另类久久久| 99九九线精品视频在线观看视频| 国产无遮挡羞羞视频在线观看| 欧美老熟妇乱子伦牲交| 亚洲精品久久久久久婷婷小说| 有码 亚洲区| 日本欧美国产在线视频| 国产淫语在线视频| av在线蜜桃| 亚洲一级一片aⅴ在线观看| 一区二区三区免费毛片| 涩涩av久久男人的天堂| 亚洲欧美精品自产自拍| 久久人人爽人人爽人人片va| 亚洲精品日本国产第一区| 久久久久久久亚洲中文字幕| av在线蜜桃| 中文精品一卡2卡3卡4更新| 毛片女人毛片| 人妻 亚洲 视频| 国内少妇人妻偷人精品xxx网站| 啦啦啦在线观看免费高清www| 亚洲欧美清纯卡通| 男女无遮挡免费网站观看| 大话2 男鬼变身卡| 免费av中文字幕在线| 亚洲欧美一区二区三区国产| 中文资源天堂在线| 哪个播放器可以免费观看大片| 成人18禁高潮啪啪吃奶动态图 | 久久久成人免费电影| 啦啦啦视频在线资源免费观看| 婷婷色av中文字幕| 国产高清不卡午夜福利| av又黄又爽大尺度在线免费看| 亚洲精品国产av成人精品| 尤物成人国产欧美一区二区三区| 寂寞人妻少妇视频99o| 又粗又硬又长又爽又黄的视频| 大香蕉97超碰在线| tube8黄色片| 一级毛片黄色毛片免费观看视频| 国产精品免费大片| 少妇人妻精品综合一区二区| 欧美激情国产日韩精品一区| 哪个播放器可以免费观看大片| 国产黄片视频在线免费观看| 三级国产精品片| 国产毛片在线视频| 男女下面进入的视频免费午夜| 免费观看无遮挡的男女| 最近最新中文字幕大全电影3| 在线天堂最新版资源| 国产一区亚洲一区在线观看| 在现免费观看毛片| 精品视频人人做人人爽| 亚洲精品,欧美精品| 成人漫画全彩无遮挡| 成人二区视频| 欧美日韩综合久久久久久| 七月丁香在线播放| 熟女人妻精品中文字幕| 十八禁网站网址无遮挡 | 深爱激情五月婷婷| 一级av片app| 看免费成人av毛片| 久久久久视频综合| 亚洲精品成人av观看孕妇| 国产又色又爽无遮挡免| 国产精品福利在线免费观看| 黄色怎么调成土黄色| 亚洲av不卡在线观看| 人人妻人人澡人人爽人人夜夜| 久久99热这里只有精品18| 丰满少妇做爰视频| 80岁老熟妇乱子伦牲交| 在线观看免费日韩欧美大片 | 久久ye,这里只有精品| 免费在线观看成人毛片| 久久久久久久久久久免费av| 国产国拍精品亚洲av在线观看| 精品人妻熟女av久视频| 成人高潮视频无遮挡免费网站| 男女下面进入的视频免费午夜| 一区二区三区免费毛片| 国产午夜精品一二区理论片| 最新中文字幕久久久久| 18禁裸乳无遮挡免费网站照片| 一级爰片在线观看| 只有这里有精品99| 插阴视频在线观看视频| 精品视频人人做人人爽| 五月伊人婷婷丁香| 欧美日韩国产mv在线观看视频 | 国产精品女同一区二区软件| 亚洲av在线观看美女高潮| 肉色欧美久久久久久久蜜桃| 欧美一区二区亚洲| 精品亚洲成国产av| 亚洲一区二区三区欧美精品| 一二三四中文在线观看免费高清| 国产免费又黄又爽又色| 国产午夜精品久久久久久一区二区三区| av在线播放精品| 中文字幕人妻熟人妻熟丝袜美| 永久网站在线| 成人漫画全彩无遮挡| 欧美极品一区二区三区四区| 边亲边吃奶的免费视频| 国产成人免费无遮挡视频| 老女人水多毛片| 亚洲av欧美aⅴ国产| 激情 狠狠 欧美| 国产一级毛片在线| 天堂中文最新版在线下载| 男女边吃奶边做爰视频| 国产爽快片一区二区三区| 九九在线视频观看精品| 一级a做视频免费观看| 国产乱来视频区| 看免费成人av毛片| 18禁裸乳无遮挡动漫免费视频| 水蜜桃什么品种好| 久久久久久久久久久免费av| 国产亚洲最大av| 亚洲欧洲国产日韩| av女优亚洲男人天堂| 亚洲精品一二三| 成人毛片a级毛片在线播放| 一区在线观看完整版| 久久久国产一区二区| 中国国产av一级| 日本爱情动作片www.在线观看| 亚洲av免费高清在线观看| 内地一区二区视频在线| 亚洲av中文字字幕乱码综合| 欧美最新免费一区二区三区| 日韩制服骚丝袜av| 免费观看无遮挡的男女| 欧美丝袜亚洲另类| 欧美 日韩 精品 国产| h视频一区二区三区| 成人毛片a级毛片在线播放| 久久人人爽av亚洲精品天堂 | 国产精品人妻久久久影院| 狂野欧美白嫩少妇大欣赏| 日本黄色日本黄色录像| 精品久久久久久久久av| 精品亚洲乱码少妇综合久久| 在线观看av片永久免费下载| 国产av国产精品国产| 老女人水多毛片| 午夜福利在线在线| 久久国内精品自在自线图片| 国产欧美亚洲国产| 狂野欧美激情性xxxx在线观看| 国产欧美日韩一区二区三区在线 | 夫妻性生交免费视频一级片| 舔av片在线| 国产乱人偷精品视频| 成人毛片a级毛片在线播放| 日本爱情动作片www.在线观看| 观看免费一级毛片| 日产精品乱码卡一卡2卡三| 美女内射精品一级片tv| 国产成人一区二区在线| 亚洲av福利一区| 天堂俺去俺来也www色官网| 成人亚洲精品一区在线观看 | 一个人免费看片子| 国产精品国产三级专区第一集| 精品久久久久久久末码| 91精品一卡2卡3卡4卡| 又大又黄又爽视频免费| 建设人人有责人人尽责人人享有的 | 亚洲精品视频女| 国精品久久久久久国模美| 中文字幕久久专区| 日韩制服骚丝袜av| 国产一级毛片在线| 久久精品人妻少妇| 高清欧美精品videossex| 欧美日本视频| 在线观看三级黄色| 久久久久久久国产电影| 欧美亚洲 丝袜 人妻 在线| 嫩草影院入口| 一个人看视频在线观看www免费| 一本色道久久久久久精品综合| 日本黄色片子视频| 人妻夜夜爽99麻豆av| 久久97久久精品| 成年女人在线观看亚洲视频| 色婷婷av一区二区三区视频| 韩国高清视频一区二区三区| 国产爱豆传媒在线观看| 春色校园在线视频观看| 国产欧美亚洲国产| 国产精品精品国产色婷婷| 观看免费一级毛片| 成人国产麻豆网| 最近的中文字幕免费完整| 成人免费观看视频高清| 欧美一区二区亚洲| 日韩一区二区视频免费看| 亚洲av成人精品一二三区| 日韩 亚洲 欧美在线| 亚洲色图av天堂| 久久久精品94久久精品| 久久久久久久亚洲中文字幕| 日韩制服骚丝袜av| 国产精品一二三区在线看| 国产精品久久久久久久电影| 视频中文字幕在线观看| 欧美变态另类bdsm刘玥| www.色视频.com| 美女高潮的动态| 亚洲电影在线观看av| 超碰av人人做人人爽久久| 精品亚洲成a人片在线观看 | 国产色爽女视频免费观看| 街头女战士在线观看网站| 老司机影院成人| 五月开心婷婷网| 亚洲国产欧美人成| 成人综合一区亚洲| 一本—道久久a久久精品蜜桃钙片| 亚洲av综合色区一区| 成人毛片60女人毛片免费| 日日摸夜夜添夜夜爱| 久久久精品免费免费高清| www.色视频.com| 亚洲av综合色区一区| 国产免费一级a男人的天堂| 国产成人a∨麻豆精品| 国产精品不卡视频一区二区| 免费久久久久久久精品成人欧美视频 | 久久精品国产亚洲av涩爱| 大片电影免费在线观看免费| 在线观看一区二区三区激情| 日韩精品有码人妻一区| 久热久热在线精品观看| 22中文网久久字幕| 久久久久网色| 99精国产麻豆久久婷婷| 嫩草影院入口| 成人免费观看视频高清| 亚洲欧美日韩卡通动漫| 人人妻人人爽人人添夜夜欢视频 | 精品少妇久久久久久888优播| 精品视频人人做人人爽| 国产男人的电影天堂91| 免费黄频网站在线观看国产| 看非洲黑人一级黄片| 麻豆成人午夜福利视频| 国产有黄有色有爽视频| 99久久综合免费| 人人妻人人爽人人添夜夜欢视频 | 日本爱情动作片www.在线观看| 国产伦精品一区二区三区四那| 99久久中文字幕三级久久日本| 女的被弄到高潮叫床怎么办| 哪个播放器可以免费观看大片| 最近的中文字幕免费完整| 亚洲第一区二区三区不卡| 亚洲aⅴ乱码一区二区在线播放| 午夜免费男女啪啪视频观看| 久久久精品94久久精品| 久热这里只有精品99| 国产精品人妻久久久影院| 老熟女久久久| 亚洲欧洲国产日韩| 久久久久久人妻| 亚洲欧美日韩无卡精品| 免费看av在线观看网站| 欧美精品一区二区免费开放| 热re99久久精品国产66热6| 国内少妇人妻偷人精品xxx网站| 一区二区三区四区激情视频| 国产在线男女| 一级毛片电影观看| 精品一区二区三卡| 中国三级夫妇交换| 国产大屁股一区二区在线视频| 国产一区二区在线观看日韩| 纯流量卡能插随身wifi吗| 日产精品乱码卡一卡2卡三| 国产精品久久久久久精品电影小说 | 欧美少妇被猛烈插入视频| 亚洲av成人精品一区久久| 日本wwww免费看| 日韩一区二区三区影片| 日产精品乱码卡一卡2卡三| 精品人妻熟女av久视频| 22中文网久久字幕| 狂野欧美白嫩少妇大欣赏| 三级国产精品欧美在线观看| av不卡在线播放| 亚洲成人av在线免费| 亚洲欧美日韩卡通动漫| 国产精品一区二区三区四区免费观看| 久久人人爽av亚洲精品天堂 | 91精品国产国语对白视频| 欧美激情国产日韩精品一区| 精品少妇久久久久久888优播| 久久亚洲国产成人精品v| 国产精品国产三级国产av玫瑰| 狂野欧美激情性bbbbbb| 亚洲精品一二三| 日本色播在线视频| 久久精品久久精品一区二区三区| 亚洲国产精品国产精品| 亚洲欧美一区二区三区黑人 | 国产成人精品婷婷| 国产成人a区在线观看| www.色视频.com| 欧美成人午夜免费资源| 亚洲伊人久久精品综合| 99久久人妻综合| 少妇丰满av| 国产精品蜜桃在线观看| 又大又黄又爽视频免费| 久久精品国产a三级三级三级| 日韩伦理黄色片| 一级av片app| 日本与韩国留学比较| 国产中年淑女户外野战色| 日韩人妻高清精品专区| 亚洲精品第二区| 亚洲精品乱码久久久久久按摩| 水蜜桃什么品种好| 这个男人来自地球电影免费观看 | 2021少妇久久久久久久久久久| 国产有黄有色有爽视频| 男人和女人高潮做爰伦理| 日本免费在线观看一区| 日韩欧美精品免费久久| 日本-黄色视频高清免费观看| 夜夜骑夜夜射夜夜干| 国产男女超爽视频在线观看| 性色avwww在线观看| 99视频精品全部免费 在线| 欧美xxxx黑人xx丫x性爽| 三级经典国产精品| 欧美成人一区二区免费高清观看| 亚洲电影在线观看av| 久久国内精品自在自线图片| 国产高潮美女av| a 毛片基地| 中文字幕av成人在线电影| 国产黄色视频一区二区在线观看| 男女啪啪激烈高潮av片| 免费在线观看成人毛片| 成人黄色视频免费在线看| 蜜桃亚洲精品一区二区三区| 亚洲欧美成人综合另类久久久| 蜜桃久久精品国产亚洲av| 最新中文字幕久久久久| 中文字幕人妻熟人妻熟丝袜美| av免费在线看不卡| 精品一区二区三卡| av一本久久久久| 在线播放无遮挡| 国产精品女同一区二区软件| 国产欧美亚洲国产| videossex国产| 精品国产露脸久久av麻豆| 日韩免费高清中文字幕av| 国产有黄有色有爽视频| 在现免费观看毛片| 成人毛片a级毛片在线播放| 国产美女午夜福利| 蜜臀久久99精品久久宅男| 国产精品国产三级国产专区5o| 街头女战士在线观看网站| 亚洲精品456在线播放app| 久久99蜜桃精品久久| 性色av一级| 91aial.com中文字幕在线观看| 国产v大片淫在线免费观看| 日本色播在线视频| 亚洲av在线观看美女高潮| 七月丁香在线播放| 成人一区二区视频在线观看| 国产精品三级大全| kizo精华| 久久99热这里只频精品6学生| 国产在线免费精品| 丰满少妇做爰视频| 人妻夜夜爽99麻豆av| 国产美女午夜福利| 精品人妻视频免费看| 亚洲国产成人一精品久久久| 一级毛片 在线播放| 国产色婷婷99| av一本久久久久| 亚洲性久久影院| 久久97久久精品| 国产男女超爽视频在线观看| 女性被躁到高潮视频| 亚洲精品乱码久久久久久按摩| 美女高潮的动态| 精品国产乱码久久久久久小说| 丝瓜视频免费看黄片| 亚洲综合色惰| 美女中出高潮动态图| 亚洲精品国产色婷婷电影| 在线精品无人区一区二区三 | 精品熟女少妇av免费看| 乱系列少妇在线播放| 亚洲精品久久久久久婷婷小说| 男人狂女人下面高潮的视频| 搡女人真爽免费视频火全软件| av黄色大香蕉| 国产亚洲欧美精品永久| 日本欧美视频一区| 日韩av免费高清视频| 亚洲精品一二三| 久久综合国产亚洲精品| 一个人看的www免费观看视频| 啦啦啦视频在线资源免费观看| 99国产精品免费福利视频| 搡老乐熟女国产| 一个人免费看片子| 美女国产视频在线观看| 国产精品一二三区在线看| 亚洲av日韩在线播放| 国产成人免费无遮挡视频| 日韩国内少妇激情av| 丰满迷人的少妇在线观看| 99热这里只有是精品50| 男女下面进入的视频免费午夜| 91精品国产九色| 精品一区二区免费观看| 国产免费又黄又爽又色| 欧美另类一区| 国产成人精品福利久久| 春色校园在线视频观看| 日日啪夜夜撸| 熟妇人妻不卡中文字幕| 在线观看三级黄色| 亚洲性久久影院| 久久国产亚洲av麻豆专区| 亚洲第一区二区三区不卡| 亚洲欧美日韩另类电影网站 | 国产人妻一区二区三区在| 国产黄频视频在线观看| 免费黄网站久久成人精品| www.色视频.com| 亚洲国产毛片av蜜桃av| 国产精品一区二区在线不卡| 精品酒店卫生间| 日本欧美国产在线视频| 久久久午夜欧美精品| 香蕉精品网在线| 久久久成人免费电影| 国产淫语在线视频| 亚洲精品久久午夜乱码| 日本免费在线观看一区| 国产精品国产三级国产专区5o| 成人黄色视频免费在线看| 国产日韩欧美亚洲二区| 国产亚洲最大av| 五月开心婷婷网| 黄片wwwwww| 国产日韩欧美亚洲二区| 亚洲精品国产色婷婷电影| 色综合色国产| 日日摸夜夜添夜夜爱| 制服丝袜香蕉在线| 妹子高潮喷水视频| 国产精品秋霞免费鲁丝片| 欧美一级a爱片免费观看看| 狂野欧美激情性xxxx在线观看| 伦理电影大哥的女人| av不卡在线播放| 在线观看免费日韩欧美大片 | 婷婷色av中文字幕| 亚洲精品乱久久久久久| 人妻一区二区av| 能在线免费看毛片的网站| 男女啪啪激烈高潮av片| 亚洲精品乱码久久久v下载方式| 99久久中文字幕三级久久日本| 另类亚洲欧美激情| 毛片一级片免费看久久久久| 中文天堂在线官网| 女的被弄到高潮叫床怎么办| 久久久国产一区二区| 一个人看视频在线观看www免费| 亚洲欧美清纯卡通| 亚洲欧美精品专区久久| 少妇精品久久久久久久| 欧美日韩视频高清一区二区三区二| 国产男女超爽视频在线观看| 一级黄片播放器| 精品国产三级普通话版| 国产av一区二区精品久久 | 高清视频免费观看一区二区| 能在线免费看毛片的网站| 婷婷色麻豆天堂久久| 国产深夜福利视频在线观看| 国产精品三级大全| 国产白丝娇喘喷水9色精品| 在线免费十八禁| 一区在线观看完整版| 免费少妇av软件| 黄色怎么调成土黄色| 亚洲欧美一区二区三区黑人 | 伦理电影大哥的女人| 国产精品久久久久久精品电影小说 | 国产 精品1| 99热国产这里只有精品6| h视频一区二区三区| 91久久精品国产一区二区成人| 日韩欧美 国产精品| 精品99又大又爽又粗少妇毛片| 精品久久国产蜜桃| 色综合色国产| 少妇人妻精品综合一区二区| 99热网站在线观看| 中国美白少妇内射xxxbb| 亚洲综合精品二区| 自拍偷自拍亚洲精品老妇| 极品少妇高潮喷水抽搐| 噜噜噜噜噜久久久久久91| 99精国产麻豆久久婷婷| 日本黄色片子视频| av又黄又爽大尺度在线免费看| 亚洲在久久综合| 亚洲精品日本国产第一区| 3wmmmm亚洲av在线观看| a级一级毛片免费在线观看| 99久久综合免费| 汤姆久久久久久久影院中文字幕| 最近最新中文字幕大全电影3| 少妇裸体淫交视频免费看高清| 亚洲电影在线观看av| 国产一级毛片在线| 九色成人免费人妻av| 网址你懂的国产日韩在线| 久久精品夜色国产| 一级毛片 在线播放| 亚洲久久久国产精品| 99久国产av精品国产电影| 精品久久久久久久久亚洲| 大又大粗又爽又黄少妇毛片口| 欧美少妇被猛烈插入视频| 欧美激情国产日韩精品一区| 午夜福利影视在线免费观看| 亚洲熟女精品中文字幕| 国产成人精品一,二区| 伦理电影免费视频| 久久久久国产网址| 一本一本综合久久| 国内揄拍国产精品人妻在线| 亚洲人成网站在线观看播放| 亚洲精品国产av成人精品| 免费观看a级毛片全部| 99久久精品一区二区三区| 97热精品久久久久久| 国产精品无大码| 免费人成在线观看视频色| 少妇猛男粗大的猛烈进出视频| freevideosex欧美| 亚洲人成网站高清观看| 91久久精品国产一区二区成人| 久久久午夜欧美精品| 99热这里只有是精品50| 91精品国产国语对白视频| 久久久午夜欧美精品| 亚洲人成网站高清观看| 一二三四中文在线观看免费高清| freevideosex欧美| 免费少妇av软件| 嫩草影院入口| 欧美日韩在线观看h| 日韩一区二区视频免费看| 性色avwww在线观看| 五月伊人婷婷丁香| 国产精品国产av在线观看| av不卡在线播放| 搡女人真爽免费视频火全软件| 日日摸夜夜添夜夜添av毛片| 精品一区二区三区视频在线| 久久99热这里只有精品18| 内射极品少妇av片p| 久久精品夜色国产| 国产欧美另类精品又又久久亚洲欧美|