Phases of Quantum Gravity in AdS3 and Linear Dilaton Backgrounds

a r X i v :h e p -t h /0503121v 1 15 M a r 2005Phases of Quantum Gravity in

AdS 3and Linear Dilaton Backgrounds A.Giveon 1,D.Kutasov 2,E.Rabinovici 1and A.Sever 11Racah Institute of Physics,The Hebrew University,Jerusalem,91904,Israel 2EFI and Department of Physics,University of Chicago,5640S.Ellis Ave.,Chicago,IL 60637,USA We show that string theory in AdS 3has two distinct phases depending on the radius of curvature R AdS =√C)invariant vacuum of the spacetime conformal ?eld theory is normalizable,the high energy density of states is given by the Cardy formula with c e?=c ,and generic high energy states look like large BTZ black holes.For k <1,the SL (2,

2

2).The states responsible

for this growth are two dimensional black holes for k >1,and highly excited perturbative strings living in the linear dilaton throat for k <1.The change of behavior at k =1in the two cases is an example of a string/black hole transition.The entropies of black holes and strings coincide at k =1.

3/05

1.Introduction

It is widely believed that in weakly coupled string theory in?at spacetime,a generic high energy state with particular values of mass,angular momentum and other charges looks from afar like a black hole with those charges.As the energy of the state increases, the curvature at the horizon decreases,and the black hole description becomes more and more reliable.

Conversely,as the mass of a black hole decreases,string(α′)corrections in the vicinity of the horizon grow and the black hole picture becomes less and less appropriate.When α′e?ects at the horizon are large,e.g.if the string frame curvature at the horizon is much larger than the string scale,a more useful description of the state is in terms of weakly coupled strings and D-branes.The transition between the black hole picture valid for small horizon curvature,and the perturbative string picture valid for large curvature, which occurs when the curvature at the horizon is of order the string scale,is smooth.In particular,the black hole and perturbative string entropies match,up to numerical coe?-cients independent of the string coupling and charges.1This is known as the string/black hole correspondence principle[1,2].

The above discussion relies crucially on the fact that the curvature at the horizon of a black hole depends on its mass,and goes to zero for large mass.In this paper we will study situations where this is not the case.Two prototypical examples which we will discuss are the BTZ black hole in AdS3,and the SL(2,I R)/U(1)black hole in I R1,1with asymptotically linear(spacelike)dilaton.In the former case,the curvature is the same everywhere,and is proportional to the cosmological constant of the underlying anti-de-Sitter spacetime.In the latter,the mass of the black hole determines the value of the dilaton at the horizon,while the curvature there as well as the gradient of the dilaton depend solely on the cosmological constant.Thus,for these black holes varying the mass does not change the size of theα′corrections.

It is natural to ask whether there is an analog of the string/black hole transition in these cases.We will see below that the answer is a?rmative.To study the transition, we will take a slightly di?erent approach than that of?at spacetime.Instead of varying the energy of the string or black hole,we will study the dependence of the high energy spectrum on the cosmological constant,which controls theα′corrections.

For AdS3,this means varying the radius of curvature of the anti-de-Sitter space,which is related to the cosmological constantΛvia

1

Λ=?

,(1.2)

2l p

where l p is the three dimensional Planck length.The expression for the central charge,

√

(1.2),is valid when R AdS is much larger than l p,l s(l s=

c e?L0ˉc e?ˉL0

;J=L0?ˉL0.(1.4)

12

Modular invariance of the spacetime CFT,which is generally expected to be a property of quantum gravity in AdS3,implies that the e?ective central charge c e?is given by[5,6]

c e?=c?24?min,(1.5) where?min is the lowest scaling dimension in the spacetime theory.Unitarity of the theory implies that?min≥0,so that c e?≤c;c e?=c i?the SL(2,

R AdS

k=

We will see that the structure of the theory depends crucially on the magnitude of k.For k>1,the SL(2,

C)invariant vacuum,as well as BTZ black holes are non-normalizable;hence,c e?

The transition between the black hole and string phases occurs at the point where the radius of curvature,R AdS,is equal to l s(1.6),in agreement with general expectations from the correspondence principle.At the transition point,the black hole and perturbative string entropies match exactly.

A similar structure is found for the asymptotically linear dilaton case.The high energy behavior of the entropy is in general Hagedorn,but the nature of the high energy states depends on the linear dilaton slope,Q.Below a certain critical value of Q,the generic high energy states correspond to SL(2,I R)/U(1)black holes,or their charged cousins.Above this value,the black holes become non-normalizable,and the high energy states correspond to highly excited perturbative strings living in the linear dilaton region.Again,at the transition between the non-perturbative(black hole)and perturbative(string)phases,the entropies match,for both charged and uncharged states.

The fact that the AdS3and linear dilaton backgrounds exhibit very similar string/black hole transitions is not accidental.To see that,consider a background of the form I R1,1×I Rφ×N,where I Rφis the real line along which the dilaton varies linearly and

N is a compact unitary CFT.String theory in this background has in general a Hagedorn density of states and exhibits the transition described above as a function of Q,the gradient of the dilaton.One can add to the background Q1fundamental strings stretched in I R1,1, and study the low energy dynamics of the combined system.This theory is related by the AdS/CFT correspondence to string theory in AdS3×N,with string coupling g2s～1/Q1, and curvature(1.6)k=2/Q2.The string/black hole transition in AdS3×N occurs at

k=1for all Q1.Thus,it is natural to expect that the same transition should also occur √

at k=1(or Q=

The AdS3and linear dilaton theories are also related via matrix theory[9,10].The low energy theory on the fundamental strings living in the linear dilaton throat can be thought of as the discrete lightcone description of the throat,with the discretization parameter equal to Q1.This relation implies that the transitions seen in the two cases should occur at the same value of k.

2.AdS3

In this section we will study string theory on AdS3times a compact space.A large class of such backgrounds can be constructed as follows2.Consider type II string theory on

I R1,1×I Rφ×S1×M.(2.1) Here,I Rφis the real line labeled byφ,with a dilaton linear inφ

Q

Φ=?

2We will restrict here to a certain class of supersymmetric backgrounds,but most of what we say can be generalized to any stable linear dilaton or AdS3background.

The radius of curvature of AdS3,R AdS,is related to the level of the SL(2,I R)current algebra of AdS3,k,via the relation(1.6).The worldsheet central charge of the CFT on AdS3is given by3

6

c AdS=3+

2

.(2.9)

Q1

Thus,weakly coupled string theory in AdS3requires a large number of fundamental strings, Q1?1.

A familiar example of(2.5)is the system of k NS5-branes wrapped around a four-manifold M4(M4=T4or K3),and fundamental strings stretched in the uncompacti?ed direction of the?vebranes.In this case one has M=SU(2)k

,such that the background(2.5)becomes

U(1)

AdS3×S3×M4.(2.10) the levels of SL(2,I R),(2.6),and SU(2)are both equal to the number of?vebranes,k, which is an integer≥2.Thus,in the class of backgrounds(2.10),the question what happens for k<1does not arise.

Nevertheless,there are cases where k<1in(2.5).For example,we can take M to be the N=2minimal model with c M=3?6

n+1

<1.(2.11) Increasing the central charge of M beyond c M=3takes one to k>1[12].

The spacetime dynamics in an AdS3background of the form(2.5)corresponds to an N=2superconformal?eld theory with spacetime central charge[14-17,12]

c=6kQ1.(2.12) If the string theory in AdS3is weakly coupled,the spacetime CFT has a large central charge,proportional to1/g2s(2.9).In this semiclassical regime,the radius of curvature of AdS3in Planck units is large,R AdS?l p,(1.2).

The?rst question we would like to address concerns the normalizability of the ground state of the spacetime CFT.From perturbative studies of string theory in AdS3it is known [15]that the spacetime Virasoro algebra takes the form

T(x)T(y)?3kI

(x?y)2

+

?y T

k2 d2zJ(x;z)ˉJ(ˉx;ˉz)Φ1(x,ˉx;z,ˉz),(2.14) in the notations of[15].I is an operator which commutes with the spacetime Virasoro generators.In particular,it has spacetime scaling dimension zero.Acting with it on the vacuum of the worldsheet theory creates a state proportional to the SL(2,

The state(2.15)is a descendant of a principal discrete series state with j=0,|m|=1. Such states are normalizable i?the unitarity condition[18,19]

?12(2.16) is satis?ed.For k>1,j=0is in the range(2.16),while for k<1,it is not.This means that the SL(2,

R2

AdS ;8l p J=

2r+r?

4For recent reviews see e.g.[22,23].

the ground state I|0 is normalizable,while for k<1it is not,due to(2.16).Thus,in the latter case the BTZ black hole is not in the spectrum of the theory.

To recapitulate,we?nd that string theory in AdS3has two phases.In one,corre-sponding to R AdS>l s(k>1,(1.6)),the SL(2,

C)invariant vacuum is non-normalizable,c e?

It remains to calculate c e?for k<1,and to identify the states in string theory on AdS3that give rise to the Cardy entropy(1.3)in this case.This is the problem we turn to next.

As mentioned in the introduction,one might expect that when the curvature exceeds the string scale and BTZ black holes cease to exist as normalizable states,the high energy spectrum should be dominated by weakly coupled highly excited perturbative string states. To see whether this is indeed the case,we need to recall a few facts about the perturbative string spectrum in AdS3.

Low lying states in the background(2.5)belong to principal discrete series representa-tions.5The worldsheet mass-shell condition for these states relates the quadratic Casimir of SL(2,I R)to the excitation level N[14]:

?j(j+1)2.(2.19) The scaling dimension of the corresponding Virasoro primary in the spacetime CFT is

h=j+1.(2.20)

The unitarity condition(2.16)implies that there is an upper bound on the excitation level N for which(2.19)can be solved.For higher excitation levels and spacetime scaling dimensions,one needs to consider long string states,which wind around the circle near the boundary of AdS3[19].In a sector with given winding w,the worldsheet mass-shell condition becomes

?j(j+1)4w2+N=1

5Principal continuous series states correspond to“bad”tachyons,which are absent in the supersymmetric theories discussed here.

and the spacetime scaling dimension is

k

h=|m+

+ip;p∈I R is the momentum of the long

2

string in the radial direction of AdS3.There are also physical states of the form(2.21) with j real in the range(2.16),which correspond to strings that do not make it all the way to the boundary[19].

As a?rst step towards examining the dynamics of long strings,consider a single string with winding number w=1.It can be thought of as extended in the I R1,1in(2.1).The remaining,transverse,directions of space,I Rφ×S1×M,parametrize the target space of the low energy CFT living on the string.The central charge of that CFT is c l=6k[14,15]. The scalar?eldφparametrizing the location of the string in the radial direction of AdS3 is described near the boundary by an asymptotically linear dilaton CFT,withΦ=?Q l

2

+3Q2l +3

2

;substituting in(2.24)we?nd

k

2

Q2l=

highly excited perturbative strings in AdS3interact strongly and are unstable near the boundary(like widely separated quarks and gluons in QCD).

In particular,consider a state with winding number w consisting of w highly excited long strings near the boundary.One expects large interactions among the strings to lead to the formation of a bound state,whose properties can be very di?erent from those of free strings.A natural candidate for such a state is a large BTZ black hole,with the same energy and angular momentum as the original con?guration of strings.In this regime (k>1),the perturbative description in terms of highly excited strings satisfying(2.21)is not useful.A better description of the highly excited state is as a BTZ black hole,which is a non-perturbative state from the point of view of string theory in AdS3.

For k<1,the situation is reversed.The strongly coupled bound state,the BTZ black hole,is no longer normalizable.Correspondingly,the interactions among highly excited perturbative strings are now small near the boundary,since the gradient of the dilaton on the long strings(2.23)has the opposite sign.Therefore,the generic high energy state with winding number w consists of w weakly interacting long strings.The higher the energy, the smaller the interactions among the strings.Note that it is crucial for the consistency of the picture that the linear dilaton on long strings(2.23)change sign precisely at the same value of k at which the BTZ black hole ceases to be normalizable.The fact that this is indeed the case supports the overall picture.

Since for k<1the high energy states consist of weakly interacting strings,it should be possible to compute the high energy entropy,and in particular c e?(1.3),perturbatively.

A quick way to get the answer is to note that if on a single long string we have the CFT I Rφ×S1×M,then on w weakly interacting strings we expect to?nd the symmetric

product CFT

I Rφ×S1×M w/S w.(2.26) The fact that the interactions among the strings go to zero as one approaches the boundary of AdS3should translate in the spacetime CFT to the statement that(2.26)describes accurately the high energy(large L0,ˉL0)behavior of the theory of w strings.It may be modi?ed signi?cantly at low energies.

To calculate the high energy density of states of(2.26)one simply adds the e?ective central charges of the component CFT’s in(2.26):

c e?=w 32+c M =6w(2?1

The winding number w is bounded from above by Q1,the total number of strings used to make the background(2.5).Thus,the high energy density of states is

1

c e?=6Q1(2?

constructions(2.5),one is restricted to k≥2

2

,at which c e?vanishes.

Another way to arrive at(2.27)is the following.We saw before that in the sector with winding w,the worldsheet mass-shell condition takes the form(2.21).The spacetime scaling dimension(2.22)can thus be written as6

h=kw

w ?j(j+1)2

.(2.29)

The bulk of the high energy density of states comes from the entropy associated with the excitation level,N.To estimate it,we can assume that the term proportional to N dominates the r.h.s.of(2.29),so that

h?

N

k

).Using the fact that the spacetime scaling dimension h is related to N via(2.30),we see that the perturbative string entropy,2π

w +

k

w .(2.31)

This is precisely the type of expression one?nds in the w-twisted sector of a symmetric orbifold,whose building block has c=6k[26,27,25].

We see that a number of seemingly independent things happen at k=1:

(1)The BTZ black holes,which are normalizable for k>1,cease to be so for k<1.The

same is true for the SL(2,

)(2.28).

k

We argued above that these phenomena are in fact related.They are all manifestations of a sharp transition,which occurs at k=1,between a strongly coupled phase where the high energy states are large black holes,and a weakly coupled phase in which they are weakly interacting fundamental strings.The matching of the entropies suggests that the dynamics of large BTZ black holes changes smoothly into that of highly excited fundamental strings as we vary k.

We?nish this section with a few comments.So far,we discussed type II string theory in AdS3.One could ask whether the matching that we found of black hole and string entropies extends to other cases,such as the heterotic string.Heterotic strings in AdS3 were studied in[28],where it was shown that in this case the spacetime CFT inherits the chiral nature of the worldsheet one.While the left moving8central charge is still given by (2.12),c=6Q1k,the right moving one is equal toˉc=6Q1(k+2).In the black hole phase this means that the Cardy formula is(1.3),with c e?=c andˉc e?=ˉc.

In the perturbative string phase,the left-movers still satisfy(2.21),(2.22),(2.29), while for the right-movers one has the mass-shell condition

?j(j+1)4w2+ˉN=1.(2.32) For large excitation levelˉN one has,as in(2.30),ˉh?ˉN/w.To calculate the e?ective worldsheet central charge of the right-movers recall that they are described by the world-sheet CFT

AdS3×S1×M×Γ,(2.33)

whereΓis a CFT with central charge13,which completes the total central charge of(2.33) to26:

3+

6

k

.Multiplying by Q1we conclude that the analog of(2.28)for the right-movers in the heterotic string is

ˉc e?=6Q1(4?

1

k

)(2.28).According to(1.5),this means that

?min=Q1

k

=Q1

Q2l

Thus,it exhibits non-trivial vacuum structure.In particular,the torus partition sum is given by a sum of the partition sums of the individual N=2minimal models,Z= j Z n j.

In our case,it is possible that the spacetime CFT for k<1splits into sectors labeled by winding numbers(w1,w2,···),with j w j=Q1.In a given sector,the spacetime CFT is a direct sum of the individual theories corresponding to the di?erent w j.Each of those has central charge c(w j)(=6w j k),and c(w j)

(=6w j(2?1

e?

24 c(w)?c(w)e? =w k=w Q2l

Note that this behavior is consistent with the general picture described above:for k>1,perturbative strings in AdS3are strongly coupled at high energies,and the strength of their interactions can be modi?ed by changing the moduli of the spacetime CFT.For k<1,perturbative strings are weakly coupled at high energies and thus no marginal or relevant perturbations can make their interactions strong there.

3.Linear dilaton

In the previous section we have seen that the low energy physics in linear dilaton backgrounds such as(2.1),in the presence of Q1fundamental strings in the throat,is qualitatively di?erent for k<1and k>1.This was demonstrated for Q1?1,but we

expect it to be true for all Q1.It is natural to ask whether something similar happens for Q1=0,i.e.in the absence of strings in the throat.In this case,the theory under consideration is the Little String Theory(LST)of the throat[30,12],and one would like to know whether the nature of the states which dominate its high energy behavior changes

as k passes through the value k=1,or as the gradient of the dilaton Q passes through √

Q=

?ge?2Φ(R+4gμν?μΦ?νΦ?4Λ),(3.1) where,as before,Λis related to the level of SL(2,I R),k,by(1.1),(1.6).The equations of motion of(3.1)have the solution9[33,34]

ds2=f?1dφ2?fdt2,f=1?2M

9Here and below we often setα′=2.

gradient of the dilaton Q is related to the level k via(2.8).Note that,as mentioned in the introduction,the mass determines the strength of quantum corrections near the horizon, but theα′e?ects are insensitive to it.

The Bekenstein-Hawking entropy of these black holes is obtained in the usual way by Wick rotating(3.2)to Euclidean space and calculating the circumference of the Euclidean time direction near the boundary atφ=∞.After Wick rotation,the metric(3.2)can be written as[35,33]

ds2=kl2s dρ2+tanh2ρdτ2

(3.3)

Φ=Φ0?log coshρ,

whereτis periodic,with period2π.The corresponding entropy is

√

S bh=2πl s M

2M

k

,and one may expect a transition to a string phase at

k

k～1.In fact,it turns out that the curvature is not the most useful guide for estimating the size of string corrections.One way to see that is to consider the generalization of(3.5) to charged black holes.We will see below that in that case,as one varies the mass to charge ratio,the curvature at the outer horizon changes continuously and even vanishes for a particular ratio.This does not necessarily mean that string corrections are small there!

Indeed,the metric is not the only?eld that is excited in the black hole geometry.In the uncharged case(3.2)we also have a non-trivial dilaton,and for charged black holes a gauge?eld as well.We are looking for a measure of the size of theα′corrections which takes all these?elds into account.

It is natural to propose that the relevant quantity is the cosmological constantΛin (3.1).This parameter determines both the gradient of the dilaton,and the curvature of the metric,and it plays the same role here as in the AdS3discussion of section2.Furthermore, while the curvature(3.5)depends on position,the cosmological constant does not,so we do not have to face the issue where in the black hole spacetime to evaluate it.

A nice feature of the above proposal is that the black hole(3.2)is a coset of SL(2,I R), while the BTZ black hole discussed in section2is an orbifold of the same space.Therefore, it is natural to expect that theα′corrections should be determined by the underlying SL(2,I R)group manifold,and should therefore only depend on k,(1.1),(1.6),(2.8).It also makes it easy to generalize to other examples related to SL(2,I R),such as the charged black holes that will be discussed below.

Another question that one might ask at this point is the following.Whileα′corrections are expected to grow when Q increases(or k decreases),it is known[36]that the metric and dilaton(3.2),(3.3),which are obtained by studying the lowest order inα′action(3.1), give rise to a solution of the classical string equations of motion10,to all orders in Q.In what sense areα′corrections growing11as one increases Q?

The answer to this question is that for large Q,or small k,the sigma model description breaks down due to non-perturbative e?ects in Q or1/k.For example,to describe the Euclidean black hole(3.3)in the bosonic string,one needs to turn on,in addition to the metric and dilaton,certain winding modes of the tachyon.This was?rst shown in unpublished work by V.Fateev,A.Zamolodchikov and Al.Zamolodchikov,and is reviewed in[38];see also[39]for a recent discussion.In the fermionic string,which is the case relevant here,one has to turn on winding modes of the fermionic string tachyon[18].The leading one is the worldsheet superpotential

d2?e?1

k(τL?τR)+···.(3.7)

The worldsheet superpotential(3.6)is non-perturbative in Q and becomes more and more important as Q grows.Its e?ects are visible in the structure of correlation functions in the Euclidean black hole geometry,as discussed in[29].

An interesting question is what is the analog of the superpotential(3.6)for the Minkowski SL(2,I R)

black hole with

U(1)

the region behind the singularity[40,37].One can try to think of the analytic continuation of the winding mode(3.6)as a?nite energy excitation living“behind the singularity at r=0”in the extended black hole spacetime.An observer living far outside the horizon of such a black hole would not see this?eld turned on directly,but would presumably feel its e?ects on the physics.

Now that we have established that as k decreases,α′e?ects grow,and pointed out the close analogy to the SL(2,I R)case of section2,we can turn to a discussion of the string/black hole transition.The most important e?ect of the superpotential(3.6)is that it implies that for k<1,the Euclidean SL(2,I R)

>Q

1

2?

)Q1.The same comments as there

apply here.In particular,S bh≥S pert;the two are equal at the string/black hole transition point k=1.As in AdS3,the transition from black holes to strings at k=1occurs not because the string entropy becomes larger than the black hole one,but rather because the black holes cease to be normalizable.

英语中的比较级与最高级 详解

比较级与最高级 1.as...as 与(not) as(so)...as as...as...句型中，as的词性 第一个as是副词，用在形容词和副词的原级前，常译为“同样地”。第二个as是连词，连接与前面句子结构相同的一个句子(相同部分常省略)，可译为“同..... He is as tall as his brother is (tall) . (后面的as 为连词) 只有在否定句中，第一个as才可换为so 改错: He is so tall as his brother.（X） 2.在比较状语从句中，主句和从句的句式结构一般是相同的 与as...as 句式中第二个as一样，than 也是连词。as和than这两个连词后面的从句的结构与前面的句子大部分情况下结构是相同的，相同部分可以省略。 He picked more apples than she did. 完整的表达为: He picked more apples than she picked apples. 后而的picked apples和前面相同，用did 替代。 He walked as slowly as she did.完整表达为: He walked as slowly as she walked slowly. she后面walked slowly与前面相同，用did替代。

3.谓语的替代 在as和than 引导的比较状语从句中，由于句式同前面 主句相同，为避免重复，常把主句中出现而从句中又出现的动词用do的适当形式来代替。 John speaks German as fluently as Mary does. 4.前后的比较对象应一致 不管后面连词是than 还是as,前后的比较对象应一致。The weather of Beijing is colder than Guangzhou. x than前面比较对象是“天气”，than 后面比较对象是“广州”，不能相比较。应改为: The weather of Bejing is colder than that of Guangzhou. 再如: His handwriting is as good as me. 应改为: His handwriting is as good as mine. 5.可以修饰比较级的词 常用来修饰比较级的词或短语有: Much,even,far,a little,a lot,a bit,by far,rather,any,still,a great deal等。 by far的用法: 用于强调，意为“...得多”“最最...”“显然”等，可修饰形容词或副词的比较级和最高级，通常置于其后，但是若比较级或最高级前有冠词，则可置于其前或其后。

The way常见用法

The way 的用法 Ⅰ常见用法： 1)the way+ that 2)the way + in which（最为正式的用法） 3)the way + 省略（最为自然的用法） 举例：I like the way in which he talks. I like the way that he talks. I like the way he talks. Ⅱ习惯用法： 在当代美国英语中，the way用作为副词的对格，“the way+ 从句”实际上相当于一个状语从句来修饰整个句子。 1)The way =as I am talking to you just the way I’d talk to my own child. He did not do it the way his friends did. Most fruits are naturally sweet and we can eat them just the way they are—all we have to do is to clean and peel them. 2）The way= according to the way/ judging from the way The way you answer the question, you are an excellent student. The way most people look at you, you’d think trash man is a monster. 3）The way =how/ how much No one can imagine the way he missed her. 4）The way =because

人教版(新目标)初中英语形容词与副词的比较级与最高级

人教版（新目标）初中英语形容词与副词的比较级与最高级 （一）规则变化： 1．绝大多数的单音节和少数双音节词，加词尾-er ，-est tall—taller—tallest 2．以不发音的e结尾的单音节词和少数以-le结尾的双音节词只加-r，-st nice—nicer—nicest , able—abler—ablest 3．以一个辅音字母结尾的重读闭音节词或少数双音节词，双写结尾的辅音字母，再加-er，-est big—bigger—biggest 4．以辅音字母加y结尾的双音节词，改y为i再加-er，-est easy—easier—easiest 5．少数以-er，-ow结尾的双音节词末尾加-er，-est clever—cleverer—cleverest, narrow—narrower—narrowest 6．其他双音节词和多音节词，在前面加more，most来构成比较级和最高级 easily—more easily—most easily （二）不规则变化 常见的有： good / well—better—best ; bad (ly)/ ill—worse—worst ; old—older/elder—oldest/eldest many / much—more—most ; little—less—least ; far—farther/further—farthest/furthest

用法： 1．原级比较：as + adj./adv. +as（否定为not so/as + adj./adv. +as）当as… as中间有名字时，采用as + adj. + a + n.或as + many / much + n. This is as good an example as the other is . I can carry as much paper as you can. 表示倍数的词或其他程度副词做修饰语时放在as的前面 This room is twice as big as that one. 倍数+as+adj.+as = 倍数+the +n.+of Your room is twice as larger as mine. = Your room is twice the size of mine. 2．比较级+ than 比较级前可加程度状语much, still, even, far, a lot, a little, three years. five times,20%等 He is three years older than I (am). 表示“（两个中）较……的那个”时，比较级前常加the（后面有名字时前面才能加冠词） He is the taller of the two brothers. / He is taller than his two brothers. Which is larger, Canada or Australia? / Which is the larger country, Canada or Australia? 可用比较级形式表示最高级概念，关键是要用或或否定词等把一事物（或人）与其他同类事物（或人）相分离 He is taller than any other boy / anybody else.

英语中的比较级和最高级

大多数形容词有三种形式,原级,比较级和最高级, 以表示形容词说明的性质在程度上的不同。 形容词的原级: 形容词的原级形式就是词典中出现的形容词的原形。例如: poor tall great glad bad 形容词的比较级和最高级: 形容词的比较级和最高级形式是在形容词的原级形式的基础上变化的。分为规则变化和不规则变化。 规则变化如下: 1) 单音节形容词的比较级和最高级形式是在词尾加 -er 和 -est 构成。 great (原级) (比较级) (最高级) 2) 以 -e 结尾的单音节形容词的比较级和最高级是在词尾加 -r 和 -st 构成。wide (原级) (比较级) (最高级) 3)少数以-y, -er, -ow, -ble结尾的双音节形容词的比较级和最高级是在词尾加 -er 和 -est 构成。 clever(原级) (比较级) (最高级) 4) 以 -y 结尾,但 -y 前是辅音字母的形容词的比较级和最高级是把 -y 去掉,加上 -ier 和-est 构成. happy (原形) (比较级) (最高级) 5) 以一个辅音字母结尾其前面的元音字母发短元音的形容词的比较级和最高级是双写该辅音字母然后再加 -er和-est。 big (原级) (比较级) (最高级) 6) 双音节和多音节形容词的比较级和最高级需用more 和 most 加在形容词前面来构成。 beautiful (原级) (比较级) (比较级) difficult (原级) (最高级) (最高级) 常用的不规则变化的形容词的比较级和最高级: 原级------比较级------最高级 good------better------best many------more------most much------more------most bad------worse------worst far------farther, further------farthest, furthest 形容词前如加 less 和 least 则表示"较不"和"最不 形容词比较级的用法: 形容词的比较级用于两个人或事物的比较,其结构形式如下: 主语+谓语(系动词)+ 形容词比较级+than+ 对比成分。也就是, 含有形容词比较级的主句+than+从句。注意从句常常省去意义上和主句相同的部分, 而只剩下对比的成分。

The way的用法及其含义(二)

The way的用法及其含义（二） 二、the way在句中的语法作用 the way在句中可以作主语、宾语或表语： 1.作主语 The way you are doing it is completely crazy.你这个干法简直发疯。 The way she puts on that accent really irritates me. 她故意操那种口音的样子实在令我恼火。The way she behaved towards him was utterly ruthless. 她对待他真是无情至极。 Words are important, but the way a person stands, folds his or her arms or moves his or her hands can also give us information about his or her feelings. 言语固然重要,但人的站姿,抱臂的方式和手势也回告诉我们他(她)的情感。 2.作宾语 I hate the way she stared at me.我讨厌她盯我看的样子。 We like the way that her hair hangs down.我们喜欢她的头发笔直地垂下来。 You could tell she was foreign by the way she was dressed. 从她的穿著就可以看出她是外国人。 She could not hide her amusement at the way he was dancing. 她见他跳舞的姿势，忍俊不禁。 3.作表语 This is the way the accident happened.这就是事故如何发生的。 Believe it or not, that's the way it is. 信不信由你, 反正事情就是这样。 That's the way I look at it, too. 我也是这么想。 That was the way minority nationalities were treated in old China. 那就是少数民族在旧中

英语比较级和最高级的用法归纳

英语比较级和最高级的用法归纳 在学习英语过程中，会遇到很多的语法问题，比如比较级和最高级的用法，对于 这些语法你能够掌握吗？下面是小编整理的英语比较级和最高级的用法，欢迎阅读！ 英语比较级和最高级的用法 一、形容词、副词的比较级和最高级的构成规则 1.一般单音节词和少数以-er，-ow结尾的双音节词，比较级在后面加-er，最高级 在后面加-est; (1)单音节词 如：small→smaller→smallest short→shorter→shortest tall→taller→tallest great→greater→greatest (2)双音节词 如：clever→cleverer→cleverest narrow→narrower→narrowest 2.以不发音e结尾的单音节词，比较在原级后加-r，最高级在原级后加-st; 如：large→larger→largest nice→nicer→nicest able→abler→ablest 3.在重读闭音节(即：辅音+元音+辅音)中，先双写末尾的辅音字母，比较级加-er，最高级加-est; 如：big→bigger→biggest hot→hotter→hottest fat→fatter→fattest 4.以“辅音字母+y”结尾的双音节词，把y改为i，比较级加-er，最高级加-est; 如：easy→easier→easiest heavy→heavier→heaviest busy→busier→busiest happy→happier→happiest 5.其他双音节词和多音节词，比较级在前面加more，最高级在前面加most; 如：bea utiful→more beautiful→most beautiful different→more different→most different easily→more easily→most easily 注意：(1)形容词最高级前通常必须用定冠词 the，副词最高级前可不用。 例句： The Sahara is the biggest desert in the world. (2) 形容词most前面没有the，不表示最高级的含义，只表示"非常"。 It is a most important problem. =It is a very important problem.

(完整版)the的用法

定冠词the的用法： 定冠词the与指示代词this ,that同源,有“那(这)个”的意思,但较弱,可以和一个名词连用,来表示某个或某些特定的人或东西. (1)特指双方都明白的人或物 Take the medicine.把药吃了. (2)上文提到过的人或事 He bought a house.他买了幢房子. I've been to the house.我去过那幢房子. (3)指世界上独一无二的事物 the sun ,the sky ,the moon, the earth (4)单数名词连用表示一类事物 the dollar 美元 the fox 狐狸 或与形容词或分词连用,表示一类人 the rich 富人 the living 生者 (5)用在序数词和形容词最高级,及形容词等前面 Where do you live?你住在哪? I live on the second floor.我住在二楼. That's the very thing I've been looking for.那正是我要找的东西. (6)与复数名词连用,指整个群体 They are the teachers of this school.(指全体教师) They are teachers of this school.(指部分教师) (7)表示所有,相当于物主代词,用在表示身体部位的名词前 She caught me by the arm.她抓住了我的手臂. (8)用在某些有普通名词构成的国家名称,机关团体,阶级等专有名词前 the People's Republic of China 中华人民共和国 the United States 美国 (9)用在表示乐器的名词前 She plays the piano.她会弹钢琴. (10)用在姓氏的复数名词之前,表示一家人 the Greens 格林一家人(或格林夫妇) (11)用在惯用语中 in the day, in the morning... the day before yesterday, the next morning... in the sky... in the dark... in the end... on the whole, by the way...

英语比较级和最高级的用法

More than的用法 A. “More than+名词”表示“不仅仅是” 1)Modern science is more than a large amount of information. 2)Jason is more than a lecturer; he is a writer, too. 3) We need more than material wealth to build our country.建设我们国家，不仅仅需要物质财富. B. “More than+数词”含“以上”或“不止”之意，如： 4)I have known David for more than 20 years. 5)Let's carry out the test with more than the sample copy. 6) More than one person has made this suggestion. 不止一人提过这个建议. C. “More than+形容词”等于“很”或“非常”的意思，如： 7)In doing scientific experiments, one must be more than careful with the instruments. 8)I assure you I am more than glad to help you. D. more than + (that)从句，其基本意义是“超过（=over）”，但可译成“简直不”“远非”.难以，完全不能（其后通常连用情态动词can） 9) That is more than I can understand . 那非我所能懂的. 10) That is more than I can tell. 那事我实在不明白。 11) The heat there was more than he could stand. 那儿的炎热程度是他所不能忍受的 此外，“more than”也在一些惯用语中出现，如： more...than 的用法 1. 比……多，比……更 He has more books than me. 他的书比我多。 He is more careful than the others. 他比其他人更仔细。 2. 与其……不如 He is more lucky than clever. 与其说他聪明，不如说他幸运。 He is more （a）scholar than （a）teacher. 与其说他是位教师，不如说他是位学者。 注：该句型主要用于同一个人或物在两个不同性质或特征等方面的比较，其中的比较级必须用加more 的形式，不能用加词尾-er 的形式。 No more than/not more than 1. no more than 的意思是“仅仅”“只有”“最多不超过”，强调少。如： --This test takes no more than thirty minutes. 这个测验只要30分钟。 --The pub was no more than half full. 该酒吧的上座率最多不超过五成。-For thirty years，he had done no more than he （had）needed to. 30年来，他只干了他需要干的工作。 2. not more than 为more than （多于）的否定式，其意为“不多于”“不超过”。如：Not more than 10 guests came to her birthday party. 来参加她的生日宴会的客人不超过十人。 比较： She has no more than three hats. 她只有3顶帽子。（太少了） She has not more than three hats. 她至多有3顶帽子。（也许不到3顶帽子） I have no more than five yuan in my pocket. 我口袋里的钱最多不过5元。（言其少） I have not more than five yuan in my pocket. 我口袋里的钱不多于5元。（也许不到5元） more than, less than 的用法 1. （指数量）不到，不足 It’s less than half an hour’s drive from here. 开车到那里不到半个钟头。 In less than an hour he finished the work. 没要上一个小时，他就完成了工作。 2. 比……（小）少 She eats less than she should. 她吃得比她应该吃的少。 Half the group felt they spent less than average. 半数人觉得他们的花费低于平均水平。 more…than,/no more than/not more than (1)Mr．Li is ________ a professor; he is also a famous scientist． (2)As I had ________ five dollars with me, I couldn’t afford the new jacket then． (3)He had to work at the age of ________ twelve． (4)There were ________ ten chairs in the room．However, the number of the children is twelve． (5)If you tel l your father what you’ve done, he’ll be ________ angry． (6)－What did you think of this novel? －I was disappointed to find it ________ interesting ________ that one． 倍数表达法 1. “倍数+形容词（或副词）的比较级+than+从句”表示“A比B大（长、高、宽等）多少倍” This rope is twice longer than that one.这根绳是那根绳的三倍（比那根绳长两倍）。The car runs twice faster than that truck.这辆小车的速度比那辆卡车快两倍（是那辆卡车的三倍）。 2. “倍数+as+形容词或副词的原级+as+从句”表示“A正好是B的多少倍”。

“the way+从句”结构的意义及用法

“tｈｅwａｙ+从句”结构的意义及用法 首先让我们来看下面这个句子: Ｒead the foｌloｗｉnｇpassagｅａnd talkａbout ｉt wi ｔh your classmaｔes．Try tｏｔeｌl whaｔyｏu thiｎk ｏf Tom aｎd ｏｆｔhe ｗay the chiｌｄrenｔrｅated ｈim. 在这个句子中，ｔhe waｙ是先行词,后面是省略了关系副词that或in whｉch的定语从句。 下面我们将叙述“the ｗaｙ+从句”结构的用法。 1.tｈe wａy之后，引导定语从句的关系词是tｈat而不是hoｗ,因此，<<现代英语惯用法词典＞＞中所给出的下面两个句子是错误的：Ｔhｉs is ｔｈeｗayｈowｉtｈａppened. This is the way how he ａlwａｙs treats me. 2．在正式语体中,thaｔ可被in which所代替;在非正式语体中，ｔhat则往往省略。由此我们得到theｗay后接定语从句时的三种模式：1) tｈe ｗaｙ＋ｔhａt-从句２）the way +ｉn whiｃh－从句3) the ｗay +从句 例如：Ｔｈe way(in whiｃh ，thａt) ｔhｅｓｅｃomｒade ｓloｏｋａｔｐrobｌems is wrong．这些同志看问题的方法

不对。 Ｔｈｅｗａy(that ,ｉn ｗhiｃh)yoｕ’ｒe doinｇit is comple ｔeｌy crａzy.你这么个干法,简直发疯。 Wｅａdmiｒｅd him for thｅｗay ｉｎｗhｉch he fａｃeｓdiｆficultieｓ． Waｌlａｃe ａｎd Ｄarwiｎｇreed ｏn the way ｉｎwhi ｃh ｄｉfｆｅrｅnt forms ｏf life had ｂegｕn.华莱士和达尔文对不同类型的生物是如何起源的持相同的观点。 Thｉs is ｔｈe ｗaｙ（tｈａｔ) hｅdｉd it. I lｉkeｄｔhe waｙ(ｔhat) ｓｈeｏｒganized ｔhe ｍeetｉng． 3.theｗay(tｈat)有时可以与hｏw(作“如何”解）通用。例如： That’s tｈe ｗaｙ(tｈaｔ) shｅspoke. = Tｈat’s how shｅspoke.

初中英语比较级和最高级讲解与练习

初中英语比较级和最高级讲解与练习 形容词比较级和最高级 一．绝大多数形容词有三种形式,原级,比较级和最高级, 以表示形容词说明的性质在程度上的不同。 1. 形容词的原级: 形容词的原级形式就是词典中出现的形容词的原形。例如: poor tall great glad bad 2. 形容词的比较级和最高级: 形容词的比较级和最高级形式是在形容词的原级形式的基 础上变化的。分为规则变化和不规则变化。 二．形容词比较级和最高级规则变化如下: 1) 单音节形容词的比较级和最高级形式是在词尾加-er 和-est 构成。 great (原级) greater(比较级) greatest(最高级) 2) 以-e 结尾的单音节形容词的比较级和最高级是在词尾加-r 和-st 构成。 wide (原级) wider (比较级) widest (最高级) 3) 少数以-y, -er, -ow, -ble结尾的双音节形容词的比较级和最高级是在词尾加 -er 和-est构成。 clever(原级) cleverer(比较级) cleverest(最高级)， slow(原级) slower(比较级) slowest (最高级) 4) 以-y 结尾,但-y 前是辅音字母的形容词的比较级和最高级是把-y 去掉,加上-ier 和-est 构成. happy (原形) happier (比较级) happiest (最高级) 5) 以一个辅音字母结尾其前面的元音字母发短元音的形容词的比较级和最高级是双写该 辅音字母然后再加-er和-est。 原形比较级最高级原形比较级最高级 big bigger biggest hot hotter hottest red redder reddest thin thinner thinnest 6) 双音节和多音节形容词的比较级和最高级需用more 和most 加在形容词前面来构 成。 原形比较级最高级 careful careful more careful most careful difficult more difficult most difficult delicious more delicious most delicious 7)常用的不规则变化的形容词的比较级和最高级: 原级比较级最高级 good better best 好的 well better best 身体好的 bad worse worst 坏的 ill worse worst 病的 many more most 许多 much more most 许多 few less least 少数几个 little less least 少数一点儿 （little littler littlest 小的） far further furthest 远（指更进一步，深度。亦可指更远） far farther farthest 远（指更远，路程）

way 用法

表示“方式”、“方法”，注意以下用法： 1.表示用某种方法或按某种方式，通常用介词in(此介词有时可省略)。如： Do it (in) your own way. 按你自己的方法做吧。 Please do not talk (in) that way. 请不要那样说。 2.表示做某事的方式或方法，其后可接不定式或of doing sth。 如： It’s the best way of studying [to study] English. 这是学习英语的最好方法。 There are different ways to do [of doing] it. 做这事有不同的办法。 3.其后通常可直接跟一个定语从句(不用任何引导词)，也可跟由that 或in which 引导的定语从句，但是其后的从句不能由how 来引导。如： 我不喜欢他说话的态度。 正：I don’t like the way he spoke. 正：I don’t like the way that he spoke. 正：I don’t like the way in which he spoke. 误：I don’t like the way how he spoke. 4.注意以下各句the way 的用法： That’s the way (=how) he spoke. 那就是他说话的方式。 Nobody else loves you the way(=as) I do. 没有人像我这样爱你。 The way (=According as) you are studying now, you won’tmake much progress. 根据你现在学习情况来看，你不会有多大的进步。 2007年陕西省高考英语中有这样一道单项填空题： ——I think he is taking an active part insocial work. ——I agree with you_____. A、in a way B、on the way C、by the way D、in the way 此题答案选A。要想弄清为什么选A，而不选其他几项，则要弄清选项中含way的四个短语的不同意义和用法，下面我们就对此作一归纳和小结。 一、in a way的用法 表示：在一定程度上，从某方面说。如： In a way he was right.在某种程度上他是对的。注：in a way也可说成in one way。 二、on the way的用法 1、表示：即将来(去)，就要来(去)。如： Spring is on the way.春天快到了。 I'd better be on my way soon.我最好还是快点儿走。 Radio forecasts said a sixth-grade wind was on the way.无线电预报说将有六级大风。 2、表示：在路上，在行进中。如： He stopped for breakfast on the way.他中途停下吃早点。 We had some good laughs on the way.我们在路上好好笑了一阵子。 3、表示：(婴儿)尚未出生。如： She has two children with another one on the way.她有两个孩子，现在还怀着一个。 She's got five children，and another one is on the way.她已经有5个孩子了，另一个又快生了。 三、by the way的用法

英语比较级和最高级

形容词比较级和最高级的形式 一、形容词比较级和最高级的构成 形容词的比较级和最高级变化形式规则如下 构成法原级比较级最高级 ①一般单音节词末尾加 er 和 est strong stronger strongest ②单音节词如果以 e结尾，只加 r 和 st strange stranger strangest ③闭音节单音节词如末尾只有一个辅音字母， 须先双写这个辅音字母，再加 er和 est sad big hot sadder bigger hotter saddest biggest hottest ④少数以 y, er(或 ure), ow, ble结尾的双音节词， 末尾加 er和 est(以 y结尾的词，如 y前是辅音字母， 把y变成i，再加 er和 est，以 e结尾的词仍 只加 r和 st) angry Clever Narrow Noble angrier Cleverer narrower nobler angriest cleverest narrowest noblest ⑤其他双音节和多音节词都在前面加单词more和most different more different most different 1) The most high 〔A〕mountain in 〔B〕the world is Mount Everest，which is situated 〔C〕in Nepal and is twenty nine thousand one hundred and fourty one feet high 〔D〕 . 2) This house is spaciouser 〔A〕than that 〔B〕white 〔C〕one I bought in Rapid City，South Dakota 〔D〕last year. 3) Research in the social 〔A〕sciences often proves difficulter 〔B〕than similar 〔C〕work in the physical 〔D〕sciences. 二、形容词比较级或最高级的特殊形式：

高中英语的比较级和最高级用法总结

比较级和最高级 1.在形容词词尾加上―er‖ ―est‖ 构成比较级、最高级： bright（明亮的）—brighter—brightest broad（广阔的）—broader—broadest cheap（便宜的）—cheaper—cheapest clean（干净的）—cleaner—cleanest clever（聪明的）—cleverer—cleverest cold（寒冷的）—colder—coldest cool（凉的）—cooler—coolest dark（黑暗的）—darker—darkest dear（贵的）—dearer—dearest deep（深的）—deeper—deepest fast（迅速的）—faster—fastest few（少的）—fewer—fewest great（伟大的）—greater—greatest hard（困难的,硬的）—harder—hardest high（高的）—higher—highest kind（善良的）—kinder—kindest light（轻的）—lighter—lightest long（长的）—longer—longest loud（响亮的）—louder—loudest low（低的）—lower—lowest near（近的）—nearer—nearest new（新的）—newer—newest poor（穷的）—poorer—poorest quick（快的）—quicker—quickest quiet（安静的）—quieter—quietest rich（富裕的）—richer—richest short（短的）—shorter—shortest slow（慢的）—slower—slowest small（小的）—smaller—smallest smart（聪明的）—smarter—smartest soft（柔软的）—softer—softest strong（强壮的）—stronger—strongest sweet（甜的）—sweeter—sweetest tall（高的）-taller-tallest thick（厚的）—thicker—thickest warm（温暖的）—warmer—warmest weak（弱的）—weaker—weakest young（年轻的）—younger—youngest 2.双写最后一个字母,再加上―er‖ ―est‖构成比较级、最高级： big（大的）—bigger—biggest fat（胖的）—fatter—fattest hot（热的）—hotter—hottest red（红的）—redder—reddest sad（伤心的）—sadder—saddest thin（瘦的）—thinner—thinnest wet（湿的）—wetter—wettest mad（疯的）—madder—maddest 3.以不发音的字母e结尾的形容词,加上―r‖ ―st‖ 构成比较级、最高级：able（能干的）—abler—ablest brave（勇敢的）—braver—bravest close（接近的）—closer—closest fine（好的,完美的）—finer—finest large（巨大的）—larger—largest late（迟的）—later—latest nice（好的）—nicer—nicest ripe（成熟的）—riper—ripest

The way的用法及其含义(一)

The way的用法及其含义（一） 有这样一个句子：In 1770 the room was completed the way she wanted. 1770年，这间琥珀屋按照她的要求完成了。 the way在句中的语法作用是什么？其意义如何？在阅读时，学生经常会碰到一些含有the way 的句子，如：No one knows the way he invented the machine. He did not do the experiment the way his teacher told him.等等。他们对the way 的用法和含义比较模糊。在这几个句子中，the way之后的部分都是定语从句。第一句的意思是，“没人知道他是怎样发明这台机器的。”the way的意思相当于how；第二句的意思是，“他没有按照老师说的那样做实验。”the way 的意思相当于as。在In 1770 the room was completed the way she wanted.这句话中，the way也是as的含义。随着现代英语的发展，the way的用法已越来越普遍了。下面，我们从the way的语法作用和意义等方面做一考查和分析： 一、the way作先行词，后接定语从句 以下3种表达都是正确的。例如：“我喜欢她笑的样子。” 1. the way+ in which +从句 I like the way in which she smiles. 2. the way+ that +从句 I like the way that she smiles. 3. the way + 从句(省略了in which或that) I like the way she smiles. 又如：“火灾如何发生的，有好几种说法。” 1. There were several theories about the way in which the fire started. 2. There were several theories about the way that the fire started.

(完整版)初中英语比较级和最高级的用法

英语语法---比较级和最高级的用法 在英语中通常用下列方式表示的词:在形容词或副词前加more(如 more natural,more clearly )或加后缀 -er(newer,sooner )。典型的是指形容词或副词所表示的质、量或关系的增加。英语句子中，将比较两个主体的方法叫做“比较句型”。其中，像“A比B更……”的表达方式称为比较级；而“A最……”的表达方式则称为最高级。组成句子的方式是将形容词或副词变化成比较级或最高级的形态。 一、形容词、副词的比较级和最高级的构成规则 1．一般单音节词和少数以-er，-ow结尾的双音节词，比较级在后面加-er，最高级在后面加-est； （1）单音节词 如：small→smaller→smallest short→shorter→shortest tall→taller→tallest great→greater→greatest （2）双音节词 如：clever→cleverer→cleverest narrow→narrower→narrowest 2．以不发音e结尾的单音节词，比较在原级后加-r，最高级在原级后加-st； 如：large→larger→largest nice→nicer→nicest able→abler→ablest 3．在重读闭音节（即：辅音＋元音＋辅音）中，先双写末尾的辅音字母，比较级加-er，最高级加-est； 如：big→bigger→biggest hot→hotter→hottest fat→fatter→fattest 4．以“辅音字母＋y”结尾的双音节词，把y改为i，比较级加-er，最高级加-est； 如：easy→easier→easiest heavy→heavier→heaviest busy→busier→busiest happy→happier→happiest 5．其他双音节词和多音节词，比较级在前面加more，最高级在前面加most； 如：beautiful→more beautiful→most beautiful different→more different→most different easily→more easily→most easily