a r X i v :0708.2568v 1 [h e p -t h ] 20 A u g 2007Imperial/TP/2007/OC/03
Spacetime singularity resolution by M-theory ?vebranes:calibrated geometry,Anti-de Sitter solutions and special holonomy metrics
Ois′?n A.P.Mac Conamhna Theoretical Physics Group,Blackett Laboratory,Imperial College London,London SW72AZ,U.K.The Institute for Mathematical Sciences,Imperial College London,London SW72PE,U.K.Abstract The supergravity description of various con?gurations of supersymmetric M-?vebranes wrapped on calibrated cycles of special holonomy manifolds is studied.The description is provided by solutions of eleven-dimensional supergravity which interpolate smoothly between a special holonomy manifold and an event horizon with Anti-de Sitter geom-etry.For known examples of Anti-de Sitter solutions,the associated special holonomy metric is derived.One explicit Anti-de Sitter solution of M-theory is so treated for ?vebranes wrapping each of the following cycles:K¨a hler cycles in Calabi-Yau two-,three-and four-folds;special lagrangian cycles in three-and four-folds;associative three-and co-associative four-cycles in G 2manifolds;complex lagrangian four-cycles
in Sp (2)manifolds;and Cayley four-cycles in Spin (7)manifolds.In each case,the as-sociated special holonomy metric is singular,and is a hyperbolic analogue of a known metric.The analogous known metrics are respectively:Eguchi-Hanson,the resolved conifold and the four-fold resolved conifold;the deformed conifold,and the Stenzel four-fold metric;the Bryant-Salamon-Gibbons-Page-Pope G 2metrics on an R 4bundle over S 3,and an R 3bundle over S 4or CP 2;the Calabi hyper-K¨a hler metric on T ?CP 2;and the Bryant-Salamon-Gibbons-Page-Pope Spin (7)metric on an R 4bundle over S 4.By the AdS/CFT correspondence,a conformal ?eld theory is associated to each of the new singular special holonomy metrics,and de?nes the quantum gravitational physics of the resolution of their singularities.
1Introduction
The AdS/CFT correspondence[1]provides a conceptual framework for consistently encoding the
geometry of Anti-de Sitter and special holonomy solutions of M-/string theory in a quantum theory.
Though the class of spacetimes to which it can be applied is restricted,and unfortunately does not
include FLRW cosmologies,it provides the only complete proposal extant for the de?nition of a
quantum theory of gravity.For the prototypical example of AdS5×S5/R10and N=4super Yang-Mills,the Maldacena conjecture is by now approaching the status of proof[2],[3].The literature
on the correspondence is enormous,from applications in pure mathematics to phenomenological
investigations.On the phenomenological front,much e?ort has been devoted to extending the
AdS/CFT correspondence from N=4super Yang-Mills to more realistic?eld theories[4]and
even QCD itself[5],[6].Also,recent developments have raised the hope that we may soon be
able to use AdS/CFT to test M-/string theory in the lab[7]-[10].On the mathematical front,the
motivation provided by the AdS/CFT correspondence has stimulated spectacular progress in dif-
ferential geometry;early work on the correspondence showed that there is a deep interplay between
Anti-de Sitter solutions of M-/string theory,singular special holonomy manifolds and conformal
?eld theories[11],[12].This relationship has since been the topic of intense investigation;a recent
highlight has been the beautiful work on Sasaki-Einstein geometry,toric Calabi-Yau three-folds and
the associated conformal?eld theories[13]-[19].What has become clear is that the geometry of a
supersymmetric AdS/CFT dual involves an Anti-de Sitter manifold,a singular special holonomy
manifold1and a supergravity solution which,in a sense that will be made more precise,interpo-
lates smoothly between them.This geometrical relationship,between Anti-de Sitter manifolds and
singular special holonomy manifolds,in the context of the AdS/CFT correspondence in M-theory,
is the subject of this paper.
The canonical example of this relationship,from IIB,is that between conically singular Calabi-
Yau three-folds and Sasaki-Einstein AdS5solutions of IIB supergravity.Each of these geometries,
individually,is a supersymmetric solution of IIB,preserving eight supercharges.Furthermore,the
manifolds may be superimposed2to obtain another supersymmetric solution of IIB,admitting four
supersymmetries.This interpolating solution-the supergravity description of D3branes at a
conical Calabi-Yau singularity-has metric
d s2= A+B r4 1/2 d r2+r2d s2(SE5) ,(1.1) for constants A,B and a Sasaki-Einstein?ve-metric d s2(SE5).Setting B=0gives th
e IIB solution R1,3×CY3,while setting A=0gives the solution AdS5×SE5.For positive A,B,the solution
is globally smooth,and contains two distinct asymptotic regions:a spacelike in?nity where the
metric asymptotes to that of the Calabi-Yau,and an internal spacelike in?nity,where the metric
asymptotes to that of the Anti-de Sitter,on an event horizon at in?nite proper distance.The causal
structure of these solutions is discussed in detail in[20].The Calabi-Yau singularity is excised in the
interpolating solution,and removed to in?nity;an important feature of the interpolating solution
is that it admits a globally-de?ned SU(3)structure.
The AdS/CFT correspondence tells us how to perform this geometrical interpolation in a quan-
tum framework.Open string theory on the singular Calabi-Yau reduces,at low energies,to a
conformally invariant quiver gauge theory,at weak’t Hooft coupling.This is the low-energy e?ec-
tive?eld theory on the world-volume of a stack of probe D3branes located at the singularity.The
gauge theory encodes the toric data of the Calabi-Yau.The same quiver gauge theory,at strong’t
Hooft coupling,is identical to IIB string theory on the AdS5×SE5;by the AdS/CFT dictionary, the CFT also encodes the Sasaki-Einstein data of the AdS solution.Clearly,it can only do this for
both the Calabi-Yau and the AdS5if their geometry is intimately related.In the classical regime,
this relationship is provided by the interpolating solution.In the quantum regime,the relationship
is provided by the CFT itself;the interpolation parameter is the’t Hooft coupling.In e?ect,the
CFT is telling us how to cut out the Calabi-Yau singularity quantum gravitationally,and replace
it with an event horizon with the geometry of Anti-de Sitter.
The correspondence is best understood for branes at conical singularities of special holonomy
manifolds.However,starting from the work of Maldacena and Nu?n ez[21],many supersymmetric
AdS solutions of M-/string theory have been discovered,[22]-[29],[13],which cannot be interpreted
as coming from a stack of branes at a conical singularity.Instead,they have been interpreted as the
near-horizon limits of the supergravity description of branes wrapped on calibrated cyles of special
holonomy manifolds.The CFT dual of the AdS/special holonomy manifolds is the low-energy
e?ective theory on the unwrapped worldvolume directions of the branes.A brane,heuristically
envisioned as a hypersurface in spacetime,can wrap a calibrated cycle in a special holonomy man-
ifold,while preserving supersymmetry.A heuristic physical argument as to why this is possible is
that a calibrated cycle is volume-minimising in its homology class;as a probe brane has a tension,
it will always try to contract,and so a wrapped probe brane is only stable if it wraps a minimal
cycle.The supergravity description of a stack of wrapped branes,by analogy with that of branes
at conical singularities,should be a supergravity solution which smoothly interpolates between a
special holonomy manifold with an appropriate calibrated cycle,and an event horizon with Anti-de
Sitter geometry.As the notion of an interpolating solution is central to this paper,a more careful
de?nition of what is meant by these words will now be given.
De?nition1Let M AdS be a d-dimensional manifold admiting a warped-product AdS metric g AdS,that,together with a matter content F AdS,gives a supersymmetric solution of a supergravity theory in d dimensions.Let M SH be a d-dimensional manifold admitting a special holonomy metric g SH,which gives a supersymmetric vacuum solution of the supergravity with holonomy G?Spin(d?1).Let M I be a d-dimensional manifold admitting a globally-de?ned G-structure, together with a metric g I and a matter content F I that give a supersymmetric solution of the supergravity.Then we say that(M I,g I,F I)is an interpolating solution if for all?,ζ>0,there exist open sets O AdS?M AdS,O I,O′I?M I,O SH?M SH,such that for all points p AdS∈O AdS, p I∈O I,p′I∈O′I,p SH∈O SH,
|g AdS(p AdS)?g I(p I)|
We also de?ne the following useful pieces of vocabulary:
De?nition2If for a given pair(M AdS,g AdS,F AdS),(M SH,g SH,F SH),there exists an inter-polating solution,then we say that M SH is a special holonomy interpolation of M AdS and that M AdS is an Anti-de Sitter interpolation of M SH.Collectively,we refer to(M AdS,g AdS,F AdS)and (M SH,g SH,F SH)as an interpolating pair.
The objective of this paper is to derive candidate special holonomy interpolations of some of the wrapped?vebrane near-horizon limit AdS solutions of[22]-[25].In[31],candidate special holonomy interpolations of the AdS5M-theory solutions of[21]were derived.These AdS solutions describe the near-horizon limit of?vebranes wrapped on K¨a hler two-cycles in Calabi-Yau two-folds and three-folds.As these results?t nicely into the more extensive picture presented here,they will be reviewed brie?y below.The new special holonomy metrics that will be derived here are candidate interpolations of:the AdS3solution of[24],describing the near-horizon limit of?vebranes wrapped on a K¨a hler four-cycle in a four-fold;the AdS4solution of[23],interpreted in[24]as the near-horizon limit of?vebranes on a special lagrangian(SLAG)three-cycle in a three-fold;the AdS3 solution of[24],for?vebranes on a SLAG four-cycle in a four-fold;the AdS4solution of[22],for ?vebranes on an associative three-cycle in a G2manifold;the AdS3solution of[24],for?vebranes on a co-associative four-cycle in a G2manifold;the AdS3solution of[25],for?vebranes on a complex lagrangian(CLAG)four-cycle in an Sp(2)manifold;and the AdS3solution of[24],for?vebranes on a Cayley four-cycle in a Spin(7)manifold.This paper therefore provides one candidate interpolating pair for every type of cycle on which M-theory?vebranes can wrap,in all manifolds of dimension less than ten with irreducible holonomy,with the exception of K¨a hler four-cycles in three-folds and quaternionic K¨a hler four-cycles in Sp(2)manifolds,for which no AdS solutions are known to the author.
No interpolating solutions of eleven-dimensional supergravity which describe wrapped branes are known.However,based on various symmetry and supersymmetry arguments,the di?erential equations they satisfy are known,for all types of calibrated cycles in all special holonomy manifolds that play a r?o le in M-theory.These equations will be called the wrapped brane equations;there is an extensive literature on their derivation[32]-[41];the most general results are those of[39]-[41].The key point that will be exploited here is that both members of an interpolating pair should individually be a solution of the wrapped brane equations,with a suitable ansatz for the interpolating solution.This is just like what happens for an interpolating solution associated to a conical special holonomy manifold.
One of the many important results of[13]was to show how any AdS5solution of M-theory, coming from?vebranes on a K¨a hler two-cycle in a three-fold,satis?es the appropriate wrapped brane equations.The canonical frame of the AdS5solutions,de?ned by their eight Killing spinors, admits an SU(2)structure.The AdS5solutions may also be re-written in such a way that the canonical AdS5frame is obscured,but a canonical R1,3frame is made manifest.This frame admits an SU(3)structure,and is de?ned by half the Killing spinors of the AdS5solution.And it is this Minkowski SU(3)structure which satis?es the wrapped brane equations.By de?nition,any interpolating solution describing?vebranes on a K¨a hler two-cycle in a three-fold admits a globally-de?ned SU(3)structure;this structure smoothly matches on to the SU(3)structure of the Calabi-Yau and also to the canonical SU(3)structure of the AdS5solution.This construction has since been systematically extended to all calibrated cycles in manifolds with irreducible holonomy of relevance to M-theory in[39],[40],[41],and,starting from the wrapped brane equations,has been used to classify(ie,derive the di?erential equation satis?ed by)all supersymmetric AdS solutions of M-theory which have a wrapped-brane origin.
The strategy used here to construct candidate special holonomy interpolations of the AdS solu-tions is therefore the following.We?rst construct the canonical Minkowski frames and structures of the AdS solutions,which satisfy the appropriate wrapped brane equations.We then use these as a guide to formulating a suitable ansatz for an interpolating solution.It is then a(reasonably) straightforward matter to determine the most general special holonomy solution of the AdS-inspired ansatz for the interpolating solution.In each case,the special holonomy metric thus obtained is the proposed interpolation of the AdS solution.No attempt has been made to determine the inter-polating solutions themselves.It is therefore a matter of conjecture whether the special holonomy metrics obtained are indeed interpolations of the AdS solutions.However the results are su?ciently striking that it is reasonable to believe that for the proposed interpolating pairs an interpolating solution does indeed exist.
As an illustration of this procedure,consider the results of[31]for the proposed interpolation of the N=2AdS5solution of[21],describing the near-horizon limit of?vebranes on a K¨a hler
two-cycle in a two-fold.When re-written in the canonical Minkowski frame,the AdS solution is of the form
d s2=L?1 d s2(R1,3)+F
4 d s2(H2)+ 1R4?1 ?1d R2.(1.5)
The range of R is R∈(0,1].At R=1,an S2degenerates smoothly,and a H2bolt stabilises.At R=0,the metric is singular,where the K¨a hler H2cycle degenerates.In the probe-brane picture, the?vebranes should be thought of as wrapping the H2at the singularity.Otherwise,they can always decrease their worldvolume by moving to smaller R.This incomplete special holonomy metric is to be compared with the Eguchi-Hanson metric[42],which is
d s2(EH)=R2
R4 (dψ?P)2
+ 1?1
20R2d s2(H4)+
36
R10/3?1
D Y a D Y a+ 1
3Here,and throughout,d s2(AdS n),d s2(H n),d s2(S n),denote the maximally symmetric Einstein metrics on n-dimensional AdS manifolds,n-hyperboloids or n-spheres with unit radius of curvature,respectively.The cartesian metric on?at space will be denoted by d s2(R n).The volume form on a unit n-hyperboloid or n-sphere will be denoted by Vol[H n],Vol[S n],respectively.
where the Y a are constrained coordinates on an S3and D will be de?ned later.The range of R is R∈(0,1];at R=1the S3degenerates smoothly and a H4bolt stabilises.At R=0the metric is singular where the H4Cayley four-cycle degenerates.This metric is to be compared with the Spin(7)metric on an R4bundle over S4,?rst found by Bryant and Salamon[43]and later independently by Gibbons,Page and Pope[44]:
d s2(BSGPP)=9
100
R2 1?1R10/3 ?1d R2,(1.9)
This metric is complete in the range R∈[1,∞);at R=1an S4degenerates smoothly and a
Cayley S4bolt stabilises.
This relationship with known complete special holonomy metrics is a universal feature of all
the proposed special holonomy interpolations of this paper.As this series of incomplete special
holonomy metrics has so many features in common,they will be given a collective name,the Nτ
series.Though they have been derived here from the AdS M-theory solutions ab initio,they may be obtained in a much simpler way a posteriori,by analytic continuation of known complete metrics4.
In every case,they may be obtained from a known complete metric with a radial coordinate of semi-
in?nite range,at the endpoint of which an S m degenerates and a calibrated S n(or,as appropriate,
CP2)cycle stabilises.The Nτseries is obtained by changing the sign of the scalar curvature of
the bolt and analytically continuing the dependence of the metric on the radial coordinate.This
generates a special holonomy metric with a“radial”coordinate of?nite range,with a smoothly
degenerating S m and a stabilised H n(or Bergman)bolt at one endpoint,and a singular degeneration
at the other.For the Calabi-Yau Nτwith K¨a hler cycles in three-folds and four-folds,the analogous
known metrics are the resolved conifold of[45],[46],and its four-fold analogue(see[47]for useful
additional background on the resolved conifold).For the Calabi-Yau Nτwith SLAG cycles,the
analogous known metrics are the Stenzel metrics[48](see[49],[50]for useful background on the
Stenzel metrics).The Stenzel two-fold metric coincides with Eguchi-Hanson,and the Stenzel
three-fold metric coincides with the deformed conifold metric of[45](see[51],[47]for additional
background on the deformed conifold).For the G2Nτmetrics with co-associative cycles,the analogous known metrics are the BSGPP metrics[43],[44]on R3bundles over S4or CP2.For the
G2Nτmetric with an associative cycle,the analogous known metric is the BSGPP metric[43], [44]on an R4bundle over S3.See[52],[53],[50]for more background on the complete G2metrics. For the Sp(2)Nτmetric with a CLAG cycle,the analogous known metric is the Calabi metric on T?CP2[54];the Calabi metric is the unique complete regular hyper-K¨a hler eight-manifold of
co-homogeneity one[55];for further background on the Calabi metric,see[56].Finally,for the Spin(7)Nτmetric with a Cayley four-cycle,we have seen that the analogous known metric is the BSGPP metric on an R4bundle over S4;see[52],[53],[50]for more details.
What is most striking about the conjectured special holonomy interpolations obtained here is that they are all singular.As occurs in the conical context,the expectation is that the singularity of the special holonomy manifold is excised in the interpolating solution,and that the conformal dual of the geometry gives a quantum gravitational de?nition of this process.If this is correct, then a singularity of the special holonomy manifold is an essential ingredient of the geometry of AdS/CFT.It would also explain a hitherto rather puzzling feature of the AdS solutions studied here,all of which were originally constructed in gauged supergravity.While for the Nτseries it is possible to obtain the known special holonomy manifolds by replacing the H n factors with S n factors,for their AdS interpolations this does not seem to be possible;the AdS solutions exist only for hyperbolic cycles.This makes sense if an AdS/CFT dual can exist only for a singular special holonomy manifold;otherwise,if AdS solutions like those studied here,but with S n cycles,existed, their special holonomy interpolations would be non-singular.Another way of saying this is that it seems that a conformal?eld theory can be associated to the singular Nτseries of special holonomy metrics,but not to their non-singular known analogues.If this idea is correct,it means that what the AdS/CFT correspondence is ultimately describing is the quantum gravity of singularity resolution for special holonomy manifolds.We formalise the geometry of this idea in the following two conjectures.
Conjecture1Every supersymmetric Anti-de Sitter solution of M-/string theory admits a special holonomy interpolation.
Conjecture2With the exception of?at space,the metric on every special holonomy manifold admitting an Anti-de Sitter interpolation is incomplete.
The organisation of the remainder of this paper is as follows.In section two,as useful introduc-tory material,we will review the relationship between the canonical AdS and Minkowski frames for AdS solutions,how to pass from one to the other by means of a frame rotation,and the relationship between the AdS and wrapped brane structures.In section three,we will derive the conjectured special holonomy interpolations of AdS solutions for?vebranes wrapped on cycles in Calabi-Yau manifolds.Section four is devoted to the proposed Sp(2)interpolating pair,section?ve to the G2 interpolating pairs and section six to the Spin(7)interpolating pair.In section seven we conclude and discuss interesting future directions.
2Canonical Minkowski frames for AdS manifolds
In this section we will review how the canonical AdS frame de?ned by all the Killing spinors of a supersymmetric AdS solution is related to its canonical Minkowski frame de?ned by half its Killing spinors;for more details,the reader is referred to[13],[39]-[41].The canonical Minkowski structure of an AdS solution is the one which can match on to the G-structure of an interpolating solution. This phenomenon-the matching of the structure de?ned by half the supersymmetries of the AdS manifold to that of an interpolating solution-is another,more precise way of stating the familiar feature of supersymmetry doubling in the near-horizon limit of a supergravity brane solution.
We will in fact distinguish two cases,which will be discussed seperately.The AdS solutions we study for?vebranes on cycles in manifolds of SU(2),SU(3)or G2holonomy have purely magnetic ?uxes.This means that no membranes are present in the geometry.However,the AdS solutions for ?vebranes on four-cycles in eight-manifolds(Spin(7),SU(4)or Sp(2)holonomies)have both elec-tric and magnetic?uxes.In probe-brane language,we can think of a stack of?vebranes wrapped a four-cycle in the eight-manifold.We also have a stack of membranes extended in the three overall transverse directions to the eight-manifold.The membrane stack intersects the?vebrane stack in a string;the low-energy e?ective?eld theory on the string worldvolume is then the two-dimensional dual of the AdS3solutions that come from these geometries.The presence of the membranes com-plicates the relationship of the AdS and Minkowski frames a little,so?rst we will discuss the case of?vebranes alone,and purely magnetic?uxes.
2.1AdS spacetimes from?vebranes on cycles in SU(2),SU(3)and G2
manifolds
The metric of an interpolating solution describing a stack of?vebranes wrapped on a calibrated cycle in a Calabi-Yau two-or three-fold,or a G2manifold,takes the form
d s2=L?1d s2(R1,p)+d s2(M q)+L2 d t2+t2d s2 S10?p?q ,(2.1) wher
e M q admits a globally-de?ned SU(2),SU(3)or G2structure respectively.The Minkowski isometries are isometries o
f the full solution,and the?ux has no components alon
g the Minkowski directions.The dimensionality of M q is q=4,6,7,respectively.The dimensionality of the un-wrapped?vebrane worldvolume is p+1,so p=3for a K¨a hler two-cycle,p=2for a SLAG or associative three-cycle,and p=1for a co-associative four-cyle.The intrinsic torsion of the G-structure on M q must satisfy certain conditions,implied by supersymmetry and the four-form Bianchi identity.These conditions are what are called the wrapped brane equations;they will be
given for each case below,and need not concern us now.For more details,the reader is referred to
[39].
Our interest here is how to obtain a warped product AdS metric from the wrapped-brane metric (2.1),and vice versa.The ?rst step is to recognise that every warped-product AdS p +2metric,written in Poincar′e coordinates,may be thought of as a special case of a warped R 1,p metric.If the AdS warp factor is denoted by λ,and is independent of the AdS coordinates,then
λ?1d s 2(AdS p +2)=λ?1[e ?2r d s 2(R 1,p )+d r 2].(2.2)
Therefore our ?rst step is to identify L =λe 2r in (2.1),with r the AdS radial coordinate.The next step is to pick out the AdS radial direction ?r =λ?1/2d r from the space transverse to the R 1,p factor in (2.1).In the cases of interest to us,the AdS radial direction is a linear combination of the radial direction ?v =L d t on the overall transverse space,and a radial direction in M q ,transverse to the wrapped cycle.We denote this radial basis one-form on M q by ?u .Thus we can obtain the AdS radial basis one-form by a local rotation of the frame of (2.1):
?r =sin θ?u +cos θ?v ,(2.3)
for some local angle θwhich we take to be independent of r .Denoting the orthogonal linear combination in the AdS frame by ?ρ,we have
?ρ=cos θ?u ?sin θ?v .
(2.4)
Now,imposing closure of d t and r -independence of θ,we get
?ρ=λ2
e ?2r ,?ρ=λ
1?λ3ρ2d ρ.(2.6)
Finally,we impose that the metric on the space tranverse to the AdS factor is independent of the AdS radial coordinate,and (in deriving the AdS supersymmetry conditions from the wrapped brane equations)that the ?ux has no components along the AdS radial direction.Thus we obtain the (for our purposes)general AdS p +2metric contained in (2.1):
d s 2=λ?1 d s 2(AdS p +2)+λ31?λ3ρ
2+ρ2d s 2 S 10?p ?q +d s 2(M q ?1),(2.7)
where d s2(M q?1)is de?ned by
d s2(M q)=d s2(M q?1)+?u??u.(2.8) In addition,w
e have
?u=λ λ3d r+ 1?λ3ρ2ρ
J6∧J6+Re?6∧?u,(2.15)
2
with
d s2(M7)=d s2(M6)+?u??u,(2.16) and th
e SU(3)structure o
f the AdS frame is de?ned on M6.
2.2AdS spacetimes from?vebranes on four-cycles in eight-manifolds
of Spin(7),SU(4)or Sp(2)holonomy
As discussed above,because of the presence of non-zero electric?ux for AdS3solutions from?ve-branes on four-cycles in eight-manifolds,the relationship between the canonical AdS and Minkowski frames of the AdS solutions is a little more complicated.These systems are the subject of[41],to which the reader is referred for more details5.The metric of an interpolating solution describing a stack of?vebranes wrapped on a four-cycle in an eight-manifold,with a stack of membranes extended in the transverse directions,takes the form
d s2=L?1d s2(R1,1)+d s2(M8)+C2d t2.(2.17)
Again,the Minkowski isometries are isometries of the full solution,the electric?ux contains a factor proportional to the Minkowski volume form,and the magnetic?ux has no components along the Minkowski directions.The Minkowski directions represent the unwrapped?vebrane worldvolume directions;the membranes extend in these directions and also along d t.Note that in this case the warp factor of the overall transverse space(the R coordinatised by t)is independent of the Minkowski warp factor.The global G-structure is de?ned on M8;the structure group is Spin(7), SU(4)or Sp(2),as appropriate.Again,supersymmetry,the four-form Bianchi identity,and now, the four-form?eld equation imply restrictions on the intrinsic torsion of the global G-structure. These equations,the wrapped brane equations for these systems,are given in[41].
To obtain an AdS3metric from(2.17),we again require that that L=λe2r,with r the AdS radial coordinate andλthe AdS warp factor,which we require to be independent of the AdS coordinates.As before,we must now pick out the AdS radial direction?r=λ?1/2d r from the space transverse to the Minkowski factor.In the generic case of interest to us,the AdS radial direction is a linear combination of the overall transverse direction e9=C d t and a radial direction in M8 transverse to the cycle that we denote by e8.Thus,as before,we write the frame rotation relating
the Minkowski and AdS frames as
?r=sinθe8+cosθe9,
?ρ=cosθe8?sinθe9,(2.18) for a local rotation angleθwhich we take to be independent of the AdS radial coordinate.Imposing AdS isometries on the electric and magnetic?ux,and requiring that the metric on the space transverse to the AdS factor is independent of the AdS coordinates,we?nd that we may introduce an AdS frame coordinateρsuch that
λ?3/2cosθ=f(ρ),
?ρ=
λ
1?λ3f2
dρ,(2.19)
for some arbitrary function f(ρ).See[41]for a fuller discussion of this point.Then the general AdS metric contained in(2.17)is
d s2=1
4(1?λ3f2)
dρ2 +d s2(M7),(2.20)
where d s2(M7)is de?ned by
d s2(M8)=d s2(M7)+e8?e8.(2.21) Th
e basis one-forms o
f the Minkowski frame are given in terms of the basis one-forms of the AdS frame by
e8=λ λ3d r+ 1?λ3f2f
2
λdρ.(2.22) For an explicit AdS3solution we knowλand f explicitly,and so we can integrate these expressions to get an explicit coordinatisation of the AdS solution in the Minkowski frame.Thus we can freely pass between the canonical AdS and Minkowski frames for known AdS solutions.
As in the previous subsection,because we are picking out a preferred direction on M8,the Minkowski-frame structure on M8is reduced,in the AdS frame,to a structure on M7.A Spin(7) structure on M8is reduced to a G2structure on M7;the decomposition of the Cayley four-form is
?φ=Υ+Φ∧e8.(2.23)
An SU(4)structure on M8is reduced to an SU(3)structure on M7.The decomposition of the SU(4)structure forms is
J8=J6+e7∧e8,
?8=?6∧(e7+ie8),(2.24) with
d s2(M8)=d s2(M7)+e8?e8=d s2(M6)+e7?e7+e8?e8,(2.25) with th
e SU(3)structure forms de?ned on d s2(M6).Finally,an Sp(2)structure on M8reduces to an SU(2)structure on d s2(M7).The decomposition o
f the triplet of Sp(2)almost complex structures(which obey the algebra J A J B=?δAB+?ABC J C,A=1,2,3)under SU(2)is
J1=K3+e5∧e6+e7∧e8,
J2=K2?e5∧e7+e6∧e8,
J3=K1+e6∧e7+e5∧e8,(2.26) with
d s2(M8)=d s2(M4)+e5?e5+e6?e6+e7?e7+e8?e8,(2.27) and th
e K A are a triplet o
f self-dual SU(2)-invariant two-forms on M4,which satisfy the algebra6 K A K B=?δAB??ABC K C.Havin
g concluded the introductory review,we now move on to the main results of the paper.
3Calabi-Yau interpolating pairs
In this section,we will give conjectured interpolating pairs for?vebranes wrapped on calibrated cycles in Calabi-Yau manifolds.First we will discuss K¨a hler cycles,then SLAG cycles.In order to present a complete picture,we will summarise the results of[31]for K¨a hler two-cycles in two-folds and three-folds.In the new cases,we will?rst present the pair,and then give the derivation of the special holonomy interpolation from the AdS solution.
3.1K¨a hler cycles
In this subsection,the AdS solutions for which we give a conjectured special holonomy interpo-lation are:the half-BPS AdS5solution of[21],describing the near-horizon limit of?vebranes on
a two-cycle in a two-fold;the quarter-BPS AdS5solution of[21],for a two-cycle in a three-fold; and the AdS3solution of[24],admitting four Killing spinors,for a four-cycle in a four-fold.The special holonomy interpolations of the?rst two cases are derived in[31];here we will just describe the conjectured pair.All the other pairs given in this paper are new,and their derivation will be given.
3.1.1Two-fold
The conjectured interpolating pair The metric of the half-BPS AdS5solution of[21]is given by
d s2=1
2
d s2(H2)+(1?λ3ρ2)(dψ?P)2+
λ3
1?λ3ρ2
+ρ2d s2(S2) ,
λ3=
8
4 d s2(H2)+
1R4?1 ?1d R2.(3.3)
The range of R is R∈(0,1].At R=1,an S2degenerates smoothly,provided thatψhas the same period as in the AdS solution.At R=0,the metric is singular,where the K¨a hler H2cycle degenerates.
3.1.2Three-fold
The conjectured interpolating pair The metric of the quarter-BPS AdS 5solution of [21]is
d s 2=13d s 2(H 2)+14(1?λ3ρ2)
d ρ2 ,λ=4
3,2/√3,
an S 3degenerates smoothly,provided that ψis periodically identi?ed with period 4π.
The conjectured special holonomy interpolation of this manifold is
d s 2=d s 2(R 1,4)+d s 2(N τ),
(3.5)
where,up to an overall scale,
d s 2(N τ)=1
2(1+sin ξ)d s 2(S 2)+
13sin 3ξ+sin ξ=
23).At R =0(corresponding to ξ=π/2)an S 3degenerates
smoothly,provided that ψhas the same periodicity as for the AdS coordinate.The metric is singular at R =2/
√λ d s 2(AdS 3)+34(1?λ3f 2) d s 2(S 2)+(d ψ+P +P ′)
2 +λ312+ρ2
,f =2ρ
Here KE?4is an arbitrary negative scalar curvature K¨a hler-Einstein manifold,normalised such that the Ricci form R4is given by R4=??J4,with?J4the K¨a hler form of KE?4.In addition,
d P=Vol[S2],
d P′=R4.(3.8)
The range ofρisρ∈[?2,2];at the end-points,an S3smoothly degenerates,provided thatψis periodically identi?ed with period4π.These manifolds admit an SU(3)structure,which was obtained in[41],and will be given below(in somewhat more transparent coordinates),together with the magnetic?ux(the electric?ux,which is irrelevant to the discussion,can be obtained from [24]or[41]).
The conjectured special holonomy interpolation of these manifolds is
d s2=d s2(R1,2)+d s2(Nτ),(3.9) where,up to an overall scale,
d s2(Nτ)=1
2(1+sinξ)
d s2(S2)+
1
3
sin3ξ+sinξ=
2
3
2 2
This structure is a solution of the torsion conditions of[41]for the near-horizon limit of?vebranes on a K¨a hler four-cycle in a four-fold,which are
?ρ∧d(λ?1J6∧J6)=0,(3.13) d(λ?3/2
1?λ3f2 .(3.15) In addition it is a solution of the Bianchi identity for the magnetic?ux,d F mag=0,which in this case is not implied by the torsion conditions.The magnetic?ux is given by
F mag=
λ3/2
1?λ3f2
(λ3/2f+?8)(d[λ?3/2
3!
J6∧J6∧J6∧e7∧?ρ.(3.17)
The AdS solutions in the Minkowski frame Now we use the discussion of section2to frame-rotate the AdS solutions to the canonical Minkowski frame.De?ning the coordinates
t=?
1
3
H1/3
M5H2/3
M2
d s2(R1,1)+
H2/3
M5
H1/3
M5
3 F d u2+u2
These three functions have been chosen so that the metric takes a form reminiscent of the harmonic function superposition rule for intersecting branes,in line with the probe brane picture.The ?vebrane worldvolume directions are the Minkowski and KE?4directions;the membranes extend along the Minkowski and t directions.Also e2r is given in terms of t and u by a positive signature metric inducing root of the quartic
t6e8r? 1?3
H1/3
M5H2/3
M2
d s2(R1,1)+
H2/3
M5
H1/3
M5 α2F21F22d s2(KE?4)
+H1/3
M2H2/3
M5 14(dψ+P+P′)2
+u2
In any event,to determine the special holonomy metric,observe that closure of?8,with the obvious frame inherited from the AdS solution,is automatic.Closure of J8results in the pair of equations
α2?u(F21F22)+
u
4F22 ?u
α2u2
cos2ξ,
F22=u2
cos2ξ
,
?13?α2u4
4a6
,(3.28) the metric takes the form given above.
3.2Special Lagrangian Cycles
In this subsection we will give conjectured interpolating pairs for?vebranes wrapped on SLAG cycles in three-and four-folds.The AdS solutions for which a Calabi-Yau interpolation is derived are respectively the AdS4solution of[23],admitting eight Killing spinors;and the AdS3solution of[24],admitting four Killing spinors.In each case we will?rst give the conjectured pair,then the derivation of the Calabi-Yau interpolation from the AdS solution.
3.2.1Three-fold
The interpolating pair The eleven-dimensional lift of the AdS4solution of[23]was later inter-preted[24]as the near-horizon limit of?vebranes wrapped on a SLAG three-cycle in a three-fold. The metric is given by
d s2=1
4(1?λ3ρ2) dρ2+ρ2d s2(S1)
,
人教版九年级全册英语重点语法知识点复习提纲 一. 介词by的用法（Unit-1重点语法） 1. 意为“在……旁”，“靠近”。 Some are singing and dancing under a big tree. Some are drawing by the lake. 有的在大树下唱歌跳舞。有的在湖边画画儿。 2. 意为“不迟于”，“到……时为止”。 Your son will be all right by supper time. 你的儿子在晚饭前会好的。 How many English songs had you learned by the end of last term? 到上个学期末你们已经学了多少首英语歌曲? 3. 表示方法、手段,可译作“靠”、“用”、“凭借”、“通过”、“乘坐”等。 The monkey was hanging from the tree by his tail and laughing. 猴子用尾巴吊在树上哈哈大笑。 The boy’s father was so thankful that he taught Edison how to send messages by railway telegraph. 孩子的父亲是那么的感激,于是他教爱迪生怎样通过铁路电报来传达信息。 4. 表示“逐个”，“逐批”的意思。 One by one they went past the table in the dark. 他们一个一个得在黑暗中经过这张桌子。 5. 表示“根据”，“按照”的意思。 What time is it by your watch? 你的表几点了? 6. 和take , hold等动词连用，说明接触身体的某一部分。 I took him by the hand. 我拉住了他的手。
页脚.
. . 教学过程 一、课堂导入 本堂知识是初中最常见的介词by的一个整理与总结，让学生对这个词的用法有一个系统的认识。页脚.
. . 二、复习预习 复习上一单元的知识点之后，以达到复习的效果。然后给学生一些相关的单选或其他类型题目，再老师没有讲解的情况下，让学生独立思考，给出答案与解释，促进学生发现问题，同时老师也能发现学生的盲点，并能有针对性地进行后面的讲课。 页脚.
. . 三、知识讲解 知识点1： by + v.-ing结构是一个重点，该结构意思是“通过……，以……的方式”，后面常接v.-ing形式，表示“通过某种方式得到某种结果”，即表示行为的方式或手段。 I practice speaking English by joining an English-language club. 我通过加入一个英语语言俱乐部来练习讲英语。 Mr Li makes a living by driving taxis.先生靠开出租车为生。 页脚.
. . 页脚. 介词by + v.-ing 结构常用来回答How do you...?或How can I...?之类的问题。 —How do you learn English? 你怎样学习英语呢？ —I learn English by reading aloud. 我通过大声朗读来学英语。 —How can I turn on the computer? 我怎样才能打开电脑呢？ —By pressing this button. 按这个按钮。 知识点2：by 是个常用介词，其他用法还有： 1【考查点】表示位置，意思是“在……旁边”，“靠近……”，有时可与beside互换。 The girls are playing by (beside) the lake. 女孩们正在湖边玩。 此时要注意它与介词near有所不同，即by 表示的距离更“近”。比较： He lives by the sea. 他住在海滨。 He lives near the sea. 他住在离海不远处。
英语介词用法大全 TTA standardization office【TTA 5AB- TTAK 08- TTA 2C】
介词（The Preposition）又叫做前置词，通常置于名词之前。它是一种虚词，不需要重读，在句中不单独作任何句子成分，只表示其后的名词或相当于名词的词语与其他句子成分的关系。中国学生在使用英语进行书面或口头表达时，往往会出现遗漏介词或误用介词的错误，因此各类考试语法的结构部分均有这方面的测试内容。 1. 介词的种类 英语中最常用的介词，按照不同的分类标准可分为以下几类： (1). 简单介词、复合介词和短语介词 ①.简单介词是指单一介词。如： at , in ,of ,by , about , for, from , except , since, near, with 等。②. 复合介词是指由两个简单介词组成的介词。如： Inside, outside , onto, into , throughout, without , as to as for , unpon, except for 等。 ③. 短语介词是指由短语构成的介词。如： In front of , by means o f, on behalf of, in spite of , by way of , in favor of , in regard to 等。 (2). 按词义分类 {1} 表地点（包括动向）的介词。如： About ,above, across, after, along , among, around , at, before, behind, below, beneath, beside, between , beyond ,by, down, from, in, into , near, off, on, over, through, throught, to, towards,, under, up, unpon, with, within , without 等。 {2} 表时间的介词。如： About, after, around , as , at, before , behind , between , by, during, for, from, in, into, of, on, over, past, since, through, throughout, till(until) , to, towards , within 等。 {3} 表除去的介词。如： beside , but, except等。 {4} 表比较的介词。如： As, like, above, over等。 {5} 表反对的介词。如： againt ,with 等。 {6} 表原因、目的的介词。如： for, with, from 等。 {7} 表结果的介词。如： to, with , without 等。 {8} 表手段、方式的介词。如： by, in ,with 等。 {9} 表所属的介词。如： of , with 等。 {10} 表条件的介词。如：
1) leave的用法 1.“leave+地点”表示“离开某地”。例如： When did you leave Shanghai? 你什么时候离开上海的？ 2.“leave for+地点”表示“动身去某地”。例如： Next Friday, Alice is leaving for London. 下周五，爱丽斯要去伦敦了。 3.“leave+地点+for+地点”表示“离开某地去某地”。例如：Why are you leaving Shanghai for Beijing? 你为什么要离开上海去北京？ 2) 情态动词should“应该”学会使用
should作为情态动词用，常常表示意外、惊奇、不能理解等，有“竟会”的意思，例如： How should I know? 我怎么知道？ Why should you be so late today? 你今天为什么来得这么晚？ should有时表示应当做或发生的事，例如： We should help each other.我们应当互相帮助。 我们在使用时要注意以下几点： 1. 用于表示“应该”或“不应该”的概念。此时常指长辈教导或责备晚辈。例如： You should be here with clean hands. 你应该把手洗干净了再来。 2. 用于提出意见劝导别人。例如： You should go to the doctor if you feel ill. 如果你感觉不舒服，你最好去看医生。
3. 用于表示可能性。should的这一用法是考试中常常出现的考点之一。例如： We should arrive by supper time. 我们在晚饭前就能到了。 She should be here any moment. 她随时都可能来。 3) What...? 与Which...? 1. what 与which 都是疑问代词，都可以指人或事物，但是what仅用来询问职业。如： What is your father? 你父亲是干什么的？ 该句相当于： What does your father do? What is your father's job? Which 指代的是特定范围内的某一个人。如：
介词by用法详解 1.表示场所，意为“在……旁边”“在……近旁”“在……手边”，此时要注意它与介词near有所不同，即by 表示的距离更“近”。比较： He lives by the sea. 他住在海滨。 He lives near the sea. 他住在离海不远处。 2.表示动词执行者，主要用于被动语态，此时要注意它与介词with的区别：by 表示动作的主体，with 表示动作者的手段工具。如： The house was destroyed by fire. 此屋被大火烧毁。（fire是动作的主体，此句的主动形式为Fire destroyed the house.） The house was destroyed with fire. 此屋是（被人）用火烧毁的。（fire只是工具，动作的主体另有其人，此句可认为省略了一个by短语，如by someone之类的，其主动形式可以是Someone destroyed the house with fire.） 3.表示手段或方式等，注意以下用法： (1) 表示“乘”“坐”时，其后接交通工具（如bus, bike, train, plane, car, taxi, ship 等）或与交通工具密切相关的名词（如air, water, land, road等），在句中主要用作方式状语，其中通常不用冠词或其他限定词。如： They came here by the first bus. 他们是坐第一班车来的。 但是，若表示交通工具的名词前插有定语修饰语，则也可以用冠词。如： We’re going by the 9:30 train. 我们坐9:30的火车去。 We went to Shanghai by a large ship. 我们乘一艘大船去上海。 注意，汉语说“步行”，英语习惯上用on foot, 而不用by foot。 (2) 注意以下表示方式的有用表达，其中不用冠词或其他限定词： by phone 用电话by telegram 用电报 by letter 用信件by express 用快件 by air mail 用航空邮件by ordinary mail 用平信 by post 用邮寄by radio 用无线电 by hand 用手工by machine 用机器 注意下面两例用by与用with的区别： The letter was written by hand. 这封信是用手写的（即不是打印的）。（by hand表示一种抽象的手段，是无形的，注意其中没用冠词或其他限定词）
一. 介词by的用法 1. 意为“在……旁”，“靠近”。 Some are singing and dancing under a big tree. Some are drawing by the lake. 有的在大树下唱歌跳舞。有的在湖边画画儿。 2. 意为“不迟于”，“到……时为止”。 Your son will be all right by supper time. 你的儿子在晚饭前会好的。 How many English songs had you learned by the end of last term? 到上个学期末你们已经学了多少首英语歌曲? 3. 表示方法、手段,可译作“靠”、“用”、“凭借”、“通过”、“乘坐”等。 The monkey was hanging from the tree by his tail and laughing. 猴子用尾巴吊在树上哈哈大笑。 The boy’s father was so thankful that he taught Edison how to send messages by railway telegraph. 孩子的父亲是那么的感激,于是他教爱迪生怎样通过铁路电报来传达信息。 4. 表示“逐个”，“逐批”的意思。 One by>他们一个一个得在黑暗中经过这张桌子。 5. 表示“根据”，“按照”的意思。 What time is it by your watch? 你的表几点了? 6. 和take , hold等动词连用，说明接触身体的某一部分。 I took him by the hand. 我拉住了他的手。 7. 用于被动句中，表示行为主体,常译作“被”、“由”等。
词性—介词 定义：介词是一种用来表示词与词, 词与句之间的关系的词。在句中不能单独作句字成分。介词后面一般有名词代词或相当于名词的其他词类，短语或从句作它的宾语。介词和它的宾语构成介词词组，在句中作状语，表语，补语或介词宾语。 一、表示时间的介词： 1)in , on，at 在……时 in表示较长时间，如世纪、朝代、时代、年、季节、月及一般（非特指）的早、中、晚等。 如in the 20th century, in the 1950s, in 1989, i n summer, in January, in the morning, in the night, in one’s life , in one’s thirties等。 on表示具体某一天及其早、中、晚。 如on May 1st, on Monday, on New Year’s Day, on a cold night in January, on a fine morning, on Sunday afternoon等。 at表示某一时刻或较短暂的时间，或泛指圣诞节，复活节等。 如at 3:20, at this time of year, at the beginning of, at the end of …, at the age of …, at Christmas，at night, at noon, at this moment等。 注意：在last, next, this, that, some, every 等词之前一律不用介词。如：We meet every day. “at时间点，有on必有天，in指月季年，也和色相连” 就是说，有具体的时间点的时候用at，具体那一天用on，说到月份，季节，年份，就用in ；而且说谁穿了什么颜色的衣服的时候，也是用in XX（color） at用于某一具体时刻或重大节日之前 ①在五点钟___ ___②在中午___ _____③在夜晚_____ ___ ④在圣诞节____ ____⑤在午夜______ ___ (2)on用在具体某一天或某天的上午、下午、晚上之前 ①在国庆节___ ______②在周二晚上___ ______③在星期天____ _____ (3)in用在周、日、季节或泛指的上午、下午、晚上前 ①在一周内___ ______②在五月____ _____③在夏季______ ___ ④在2009年____ _____⑤在下午___ ______ 归纳总结 在初中阶段常见的固定短语 in English用英语in a minute一会儿、立刻in a short while一会儿、不久 in a hurry匆匆忙忙in danger在危险中in full全部地、详细地 in a word一句话in all总共in every case不管怎样 in the end最后in spite of尽管in person亲自 in fact事实上in good health身体健康的in front of在……前面 in some ways在某些方面in common共同的in public当众 考题再现：---Who was the first man with A(h1n1) flu in mainland China know for sure? ---________May 11,2009. A In B On C For D Since
介词用法知多少 介词是英语中最活跃的词类之一。同一个汉语词汇在英语中可译成不同的英语介词。例如汉语中的“用”可译成：（1）用英语（in English）；（2）用小刀（with a knife）；（3）用手工（by hand）；（4）用墨水（in ink）等。所以，千万不要以为记住介词的一两种意思就掌握了这个介词的用法，其实介词的用法非常广泛，搭配能力很强，越是常用的介词，其含义越多。下面就简单介绍几组近义介词的用法及其搭配方法。 一. in, to, on和off在方位名词前的区别 1. in表示A地在B地范围之内。如： Taiwan is in the southeast of China. 2. to表示A地在B地范围之外，即二者之间有距离间隔。如： Japan lies to the east of China. 3. on表示A地与B地接壤、毗邻。如： North Korea is on the east of China. 4. off表示“离……一些距离或离……不远的海上”。如： They arrived at a house off the main road. New Zealand lies off the eastern coast of Australia. 二. at, in, on, by和through在表示时间上的区别 1. at指时间表示： （1）时间的一点、时刻等。如： They came home at sunrise （at noon, at midnight, at ten o’clock, at daybreak, at dawn）. （2）较短暂的一段时间。可指某个节日或被认为是一年中标志大事的日子。如：He went home at Christmas （at New Year, at the Spring Festival, at night）. 2. in指时间表示： （1）在某个较长的时间（如世纪、朝代、年、月、季节以及泛指的上午、下午或傍晚等）内。如： in 2004, in March, in spring, in the morning, in the evening, etc （2）在一段时间之后。一般情况下，用于将来时，谓语动词为瞬间动词，意为“在……以后”。如： He will arrive in two hours. 谓语动词为延续性动词时，in意为“在……以内”。如： These products will be produced in a month. 注意：after用于将来时间也指一段时间之后，但其后的时间是“一点”，而不是“一段”。如： He will arrive after two o’clock. 3. on指时间表示： （1）具体的时日和一个特定的时间，如某日、某节日、星期几等。如： On Christmas Day（On May 4th）, there will be a celebration. （2）在某个特定的早晨、下午或晚上。如： He arrived at 10 o’clock on the night of the 5th. （3）准时，按时。如： If the train should be on time, I should reach home before dark.
初中英语语法速记口诀大全 很多同学认为语法枯燥难学，其实只要用心并采用适当的学习方法，我们就可以愉快地学会英语，掌握语法规则。笔者根据有关书目和多年教学经验，搜集、组编了以下语法口诀，希望对即将参加中考的同学们有所帮助。 一、冠词基本用法 【速记口诀】 名词是秃子，常要戴帽子， 可数名词单，须用a或an， 辅音前用a，an在元音前， 若为特指时，则须用定冠， 复数不可数，泛指the不见， 碰到代词时，冠词均不现。 【妙语诠释】冠词是中考必考的语法知识之一，也是中考考查的主要对象。以上口诀包括的意思有：①名词在一般情况下不单用，常常要和冠词连用；②表示不确指的可数名词单数前要用不定冠词a或an，确指时要用定冠词the；③如复数名词表示泛指，名词前有this，these，my，some等时就不用冠词。 二、名词单数变复数规则 【速记口诀】 单数变复数，规则要记住， 一般加s，特殊有几处： 末尾字母o，大多加s， 两人有两菜，es不离口， 词尾f、fe，s前有v和e； 没有规则词，必须单独记。 【妙语诠释】①大部分单数可数名词变为复数要加s，但如果单词以/t蘩/、/蘩/、/s/发音结尾(也就是单词如果以ch，sh，s，x等结尾)，则一般加es；②以o结尾的单词除了两
人(negro，hero)两菜(tomato，potato) 加es外，其余一般加s；③以f或fe结尾的单词一般是把f，fe变为ve再加s；④英语中还有些单词没有规则，需要特殊记忆，如child—children，mouse—mice，deer—deer，sheep—sheep，Chinese—Chinese，ox—oxen，man—men，woman—women，foot—feet，tooth —teeth。 三、名词所有格用法 【速记口诀】 名词所有格，表物是“谁的”， 若为生命词，加“’s”即可行， 词尾有s，仅把逗号择； 并列名词后，各自和共有， 前者分别加，后者最后加； 若为无生命词，of所有格， 前后须倒置，此是硬规则。 【妙语诠释】①有生命的名词所有格一般加s，但如果名词以s结尾，则只加“’”；②并列名词所有格表示各自所有时，分别加“’s”，如果是共有，则只在最后名词加“’s”；③如果是无生命的名词则用of表示所有格，这里需要注意它们的顺序与汉语不同，A of B要翻译为B的A。 四、接不定式作宾语的动词 【速记口诀】 三个希望两答应，两个要求莫拒绝； 设法学会做决定，不要假装在选择。 【妙语诠释】三个希望两答应：hope，wish，want，agree，promise 两个要求莫拒绝：demand，ask，refuse 设法学会做决定：manage，learn，decide 不要假装在选择：petend，choose 五、接动名词作宾语的动词
初中英语介词解题技巧讲解及练习题(含答案) 一、初中英语介词 1．— Is this Mike's dictionary? — No, it's mine. The thick one on the desk is . A. his B. yours C. hers D. theirs 【答案】 A 【解析】【分析】句意：——这是迈克的字典吗？——不，是我的。桌子上最厚的那个是他的。A他的，B你的，C，她的，D他们的，根据Is this Mike's，可知此处指代第三人称单数表示男性，故用his，故选A。 【点评】考查名词性物主代词辨析，注意指代第三人称单数表示男性用his的用法。 2．We will attend the junior high graduation ceremony ________ June 21st, 2019. A. in B. at C. on 【答案】 C 【解析】【分析】句意：在2019年6月21日我们将参加高中毕业典礼。in用在年、月的名词前；介词at用在时间点前面，在具体的某个日期前用介词on，故选C。 【点评】此题考查介词用法。掌握介词的使用规则。 3． , I found the job boring, but soon I got used to it. A. To start with B. First of all C. Without doubt D. After all 【答案】 A 【解析】【分析】句意：起初我觉得工作很无聊，但是我很快就适应了。A.起初，B.首先，C.毫无疑问，D.毕竟。根据后半句“不久以后才适应，前后句进行对比，说明刚开始是不适应的，句子缺少时间状语，用to start with符合题意，故答案选A。 【点评】考查短语辨析。注意识记to start with的词义和用法。 4．We communicate _____ each other in many ways, such as by e-mail or by phone. A. on B. through C. in D. with 【答案】 D 【解析】【分析】句意：我们用很多方法相互联系，比如通过电子邮件或者电话。communicate with，与某人联系，与某人保持联系，固定搭配，故答案是D。 【点评】考查介词辨析，注意识记固定搭配communicate with的用法。 5．A recent study in Australia shows that parents are the top five world's hardest jobs. A. between B. among C. from D. above 【答案】 B 【解析】【分析】句意：澳大利亚最近研究显示父母位于世界上最难的前五名工作。A.在……之间；表两者之间 B.在……之间，表三者或三者以上的之间；C.来自；D.在……上面。此处表示父母在世界上前五名最难的工作之间，超过两者之间，用among，在……之间，故
【备战高考】英语介词用法总结(完整) 一、单项选择介词 1． passion, people won't have the motivation or the joy necessary for creative thinking. A．For . B．Without C．Beneath D．By 【答案】B 【解析】 【详解】 考查介词辨析。句意：没有激情，人们就不会有创新思维所必须的动机和快乐。A. For 对于；B. Without没有； C. Beneath在……下面 ; D. By通过。没有激情，人们就不会有创新思维所必须的动机和快乐。所以空处填介词without。故填without。 2．Modern zoos should shoulder more social responsibility _______ social progress and awareness of the public. A．in light of B．in favor of C．in honor of D．in praise of 【答案】A 【解析】 【分析】 【详解】 考查介词短语。句意：现代的动物园应该根据社会的进步和公众的意识来承担更多的社会责任。A. in light of根据，鉴于；B. in favor of有利于，支持；C. in honor of 为了纪念；D. in praise of歌颂，为赞扬。此处表示根据，故选A。 3．If we surround ourselves with people _____our major purpose, we can get their support and encouragement. A．in sympathy with B．in terms of C．in honour of D．in contrast with 【答案】A 【解析】 【详解】 考查介词短语辨析。句意：如果我们周围都是认同我们主要前进目标的人，我们就能得到他们的支持和鼓励。A. in sympathy with赞成；B. in terms of 依据；C. in honour of为纪念； D. in contrast with与…形成对比。由“we can get their support and encouragement”可知，in sym pathy with“赞成”符合句意。故选A项。 4．Elizabeth has already achieved success_____her wildest dreams. A．at B．beyond C．within D．upon
由ｂｙｄｏｉｎｇｓｔｈ．结构谈介词ｂｙ的用法 在新目标九年级教材Unit1中有这样的句子： —How do you study for a test? —I study by working with a group. 这句话中by是介词，用来表示方法、手段、方式，意为“凭借；靠；用；通过”，后接动词的-ing形式。例如： They learn English by watching TV. 他们通过看电视学英语。 He succeeded by working hard. 他由于工作努力而获得了成功。 以下是by的其它基本用法归纳： 1. 表示静态的位置，意为“靠近……；在……旁边”。例如： His house stands by the river. 他住在河边。 2. 表示动态的位置，意为“从……旁边经过”、“路过……”。例如： He passed by me without greeting me. 他从我身边走过，但 没和我打招呼。 3. 表示时间、时限，意为“不迟于；在……之前”、“到……时 为止”。例如： They will be back by six. 他们将于6点钟以前回来。 4. 和take, hold等动词连用，表示接触身体/物体的某一部 位。例如： Don’t take the baby by the arm. She is too young. 别拽那个 小孩的胳膊，她太小了。 5. 表示“逐个”、“逐批”之意，常见于以下短语中： step by step 一步一步地；day by day 日复一日地；little by little一点一点地。 6. 用于被动语态中，后接动作的执行者，表示“被……”、“由……”。例如： English is spoken by many people. 许多人讲英语。 7. 表示判断的标准，意为“依照，根据”。例如： By my watch it is eight o’clock. (按)我的表显示的时间是八点。 8. 表示交通、传递的方式。例如： I often go to school by bus. 我经常坐公共汽车上学。 9. 表示原因，意为“由于……的结果，凭着”。例如： By good fortune, I succeeded the first time. 我侥幸地第一次就成功了。 10. 由by还可以构成以下短语： (1) (all) by oneself 意为“独自地”、“单独地”。 (2) by the way “顺便说一声”、“顺便问一下”，常用作插入语。 (3) by chance / by accident 意为“偶然地”。 (4) learn by heart 意为“背会、记住”。
七年级英语介词精讲与练习 1. in 在…之内 1) 表示地点，表示大地点。 She’s in China with her mom and dad. 她和妈妈爸爸一起在中国。 My mother is an English teacher in a university in Beijing. 我妈妈是北京一名大学的英语老师。 I’m in Class One. 我在一班。 The camel lives in the desert and eats grass. 骆驼住在沙漠里，它以草为生。 2) 表示时间，表示在某一个时间段内。 The weather is cold in winter. 冬天天气很冷。 3) 表示颜色和语言也用介词i n。 Do you know the girl in red? 你认识穿着红色衣服的女孩么？ I can write this article in English. 我能用英语写这篇文章。 2. from 从…中来 1) come from I come from China and I’m Chinese. 我从中国来，我是中国人。 There are camels from Africa. 骆驼从非洲来。2) download from I download music from the Internet. 我从网络上下载音乐。 3) be different from Lily’s habits are different from Linda’s. 莉莉的习惯与琳达的不同。 3. with和，用，与1) with sb 表示和…人在一起 I’m in Class One with Daming and Lingling. 我与大明和玲玲都是一班的学生。 Would you like to go to the cinema with Betty and me? 你愿意与贝蒂和我一起去看电影么？ 2）with sth.表示用某种工具 I write with my pen, and I see with my eyes. 我用我的笔写字，用我的眼睛来看。 4. at 在… 1) 表示地点，主要指小地点。 My father is a teacher at Beijing International School. 我爸爸是北京国际学校的一名老师。 My mother is a doctor at the hospital. 我妈妈是医院医生。 2)表示时间，主要指时间的一个点。 I get up at half past seven in the morning. 我早上七点半起床。 I usually do my homework at home and at the weekend.我通常周末在家里写作业。
九年级英语全册各单元知识点总结 Unit 1 How can we become good learners? 一、短语： 1.have conversation with sb. 同某人谈话 2.connect …with… 把…和…连接/联系起来 3.the secret to… ……的秘诀 4.be afraid of doing sth./to do sth. 害怕做某事 5.look up 查阅 6.repeat out loud 大声跟读 7.make mistakes in 在……方面犯错误8.get bored 感到厌烦 9.be stressed out 焦虑不安的10.pay attention to 注意；关注 11.depend on 取决于；依靠12.the ability to do sth. 做某事的能力 二、知识点： 1. by + doing：通过……方式（by是介词，后面要跟动名词，也就是动词的ing形式）； 2. a lot：许多，常用于句末； 3. aloud, loud与loudly的用法，三个词都与“大声”或“响亮”有关。 ①aloud是副词，通常放在动词之后。 ①loud可作形容词或副词。用作副词时，常与speak, talk, laugh等动词连用，多用于比较级， 须放在动词之后。 ①loudly是副词，与loud同义，有时两者可替换使用，可位于动词之前或之后。 4. not …at all：一点也不，根本不，not经常可以和助动词结合在一起，at all 则放在句尾； 5. be / get excited about sth.：对…感到兴奋； 6. end up doing sth：终止/结束做某事；end up with sth.：以…结束； 7. first of all：首先（这个短语可用在作文中，使得文章有层次）； 8. make mistakes：犯错make a mistake 犯一个错误； 9. laugh at sb.：笑话；取笑（某人）（常见短语） 10. take notes：做笔记/记录； 11. native speaker 说本国语的人； 12. make up：组成、构成； 13. deal with：处理、应付； 14. perhaps = maybe：也许； 15. go by：（时间）过去； 16.each other：彼此； 17.regard… as … ：把…看作为…； 18.change… into…：将…变为…； 19. with the help of sb. = with one's help 在某人的帮助下（注意介词of和with，容易出题） 20. compare … to …：把…比作… compare with 拿…和…作比较； 21. instead：代替，用在句末，副词； instead of sth / doing sth：代替，而不是（这个地方考的较多的就是instead of doing sth，也就是说如果of后面跟动词时，要用动名词形式，也就是动词的ing形式） 22.Shall we/ I + do sth.? 我们/我…好吗？ 23. too…to：太…而不能，常用的句型是too+形容词/副词+ to do sth.
介词讲解及练习(含答案) 一、单项选择介词 1．On Wednesday, the World Health Organization announced that the number of people killed by Ebola has now risen to over five thousand, with more than fourteen thousand ______. A．having infected B．to infect C．infected D．infecting 【答案】C 【解析】 试题分析：考查with的复合结构。With+宾语+宾补，宾语补足语可以是不定式（表将来），过去分词（表被动或完成），现在分词（表主动或进行）。主句时态为一般过去时，动作已完成。句意：世界卫生组织宣布因埃博拉病毒致死的人数截止上周三达到了五千多人，并有超过一万四千人感染病毒。故选C。 考点：考查with的复合结构 2．Bless your heart, I know you didn’t break the vase ________. Don’t cry！ A．on purpose B．by accident C．on business D．by mistake 【答案】A 【解析】 【详解】 考查介词短语。句意：好了好了，我知道你不是故意打破花瓶的。别哭了！A.故意地；B.偶然；C.出差；D.错误地。根据Don’t cry！可知，打破花瓶不是故意的。故选A。 3．A great man shows his greatness _____ the way he treats little man. A．under B．with C．on D．by 【答案】D 【解析】 考查介词的用法。“借助某种方法和手段”常用“by ”; 而"with" 强调使用工具,常与 with this method 搭配。 4．Soon after dinner, Wayne drove off ______ the direction of Paris. A．to B．at C．for D．in 【答案】D 【解析】 考查介词。in the direction of朝……方向。句意：晚餐后不久，Wayne开车朝巴黎方向驶去。 5．Before you pay a visit to a place of interest, look in your local library a book about it.
初三英语语法知识点总结大全 导读：我根据大家的需要整理了一份关于《初三英语语法知识点总结大全》的内容，具体内容：1 agree with sb 赞成某人2 all kinds of 各种各样 a kind of 一样3 a piece of cake =easy 小菜一碟(... 1 agree with sb 赞成某人 2 all kinds of 各种各样 a kind of 一样 3 a piece of cake =easy 小菜一碟(容易) 4 (see 、hear 、notice 、find 、feel 、listen to 、 look at (感官动词)+do eg:I like watching monkeys jump 5(比较级 and 比较级) 表示越来越怎么样 6 all over the world = the whole world 整个世界 7 along with同......一道，伴随...... eg : I will go along with you我将和你一起去 the students planted trees along with their teachers 学生同老师们一起种树 8 As soon as 一怎么样就怎么样 9 as you can see 你是知道的 10 ask for ......求助向...要...(直接接想要的东西) eg : ask you for my book 11 ask sb for sth 向某人什么 12 ask sb to do sth 询问某人某事 ask sb not to do 叫某人不要做
介词及介词短语 【考点直击】 1.常用介词及其词组的主要用法及意义 2.介词表示时间、方位、方式别的基本用法 3.一些易混介词的辨析 【语法讲解】 ◆介词的功能 介词是一种虚词，用来表示名词或相当于名词的其它词语句中其它词的关系，不能单独使用。介词可与名词或相当于名词的其它词构成介词短语。介词短语可在句中作定语，状语，表语和宾语补足语。例如： The boy over there is John’s brother.(定语) The girl will be back in two hours. (状语) ◆介词和种类 (1) 简单介词，常用的有at, in, on, about, across, before, beside, for , to, without等。 (2) 复合介词，如by means of, along with, because of, in front of, instead of等。 ◆不同介词的用法 （1）表时间的介词 1）at, in on 表示时间点用at。 例如：at six o’clock, at noon, at midnight。 表示在某个世纪，某年，某月，某个季节以及早晨，上午，下午，晚上时，用in。 例如：in the nineteenth century, in 2002, in may, in winter, in the morning, in the aftern oon等。表示具体的某一天和某一天的上午，下午，晚上时，用on。 例如：on Monday, on July 1st, on Sunday morning等。 2）since, after 由since和after 引导的词组都可表示从过去某一点开始的时段，但since词组表示的时段一直延续到说话的时