SL(2,R) Chern-Simons Theories with Rational Charges and Two-dimensional Conformal Field The

a r X i v :h e p -t h /9208016v 1 5 A u g 1992GEF-TH 5/1992

SL(2,R)CHERN-SIMONS THEORIES WITH RATIONAL CHARGES

AND 2-DIMENSIONAL CONFORMAL FIELD THEORIES

CAMILLO IMBIMBO

I.N.F.N.,Sezione di Genova Via Dodecaneso 33,I-16146Genova,Italy ABSTRACT We present a hamiltonian quantization of the SL (2,R )3-dimensional Chern-Simons theory with fractional coupling constant k =s/r on a space manifold with torus topology in the “constrain-?rst”framework.By generalizing the “Weyl-odd”projection to the fractional charge case,we obtain multi-components holomorphic wave functions whose components are the Kac-Wakimoto characters of the modu-lar invariant admissible representations of ?A 1current algebra with fractional level.The modular representations carried by the quantum Hilbert space satisfy both Verlinde’s and Vafa’s constraints coming from conformal ?eld theory.They are the “square-roots”of the representations associated to the conformal (r,s )mini-

mal models.Our results imply that Chern-Simons theory with SO (2,2)as gauge group,which describes 2+1-dimensional gravity with negative cosmological con-stant,has the modular properties of the Virasoro discrete series.On the way,we show that the 2-dimensional counterparts of Chern-Simons SU (2)theories with

half-integer charge k =p/2are the modular invariant D p +1series of ?A

1current algebra of level 2p ?2.

GEF-TH 5/1992

March 1992

1.Introduction

Three-dimensional Chern-Simons topological gauge theory[1]with SL(2,R)as gauge group has attracted considerable interest for various reasons[2]-[5]including its relationship to both2-dimensional[6]and3-dimensional quantum gravity[7].

Chern-Simons gauge theories with non-compact gauge groups are not ex-pected to present any special pathology as3-dimensional quantum?eld theories. Their hamiltonian being identically zero and their action being linear in time derivatives,one expects on general grounds that they de?ne perfectly unitary quan-tum theories[8].Therefore it appears that their2-dimensional counterparts cannot be the non-unitary Wess-Zumino-Witten models on non-compact group manifolds: if2-dimensional quantum?eld theories associated to Chern-Simons theories with non-compact gauge group do exist,they are likely to represent some,possibly yet unknown,generalization of current algebra constructions.Understanding such generalization is another motivation to study SL(2,R)Chern-Simons gauge the-ory.

Unfortunately,the extension of the Hamiltonian quantization techniques which allowed a non-perturbative solution of Chern-Simons theories with compact gauge groups to theories with non-compact gauge groups is revealed to be prob-lematic[8].Canonical quantization in the holomorphic“quantize-?rst”scheme[9]-[11]has been essential for establishing the correspondence between3-dimensional Chern-Simons theory with compact gauge groups and2-dimensional current alge-bras,but this approach is not viable for the real non-compact SL(2,R)due to the lack of a gauge invariant polarization.Analyses based on polarizations which are not gauge invariant[2]-[3]provided some intriguing information about SL(2,R) Chern-Simons theory,but were di?cult to carry out at explicit and less formal levels and were limited to the case of trivial space topology.Recent perturbative computations[5]have stressed the substantially novel features that non-compact gauge groups introduce into the quantization of topological Chern-Simons theories.

In this paper we will present a canonical quantization of Chern-Simons the-ory with SL(2,R)as gauge group in the“constrain-?rst”framework[9],[12].This approach,being gauge invariant ab initio,avoids the di?culties of non-gauge in-variant polarizations a?ecting the“quantize-?rst”method.We will limit ourselves

to the case when the“space”manifoldΣis a2-dimensional torus;such limitation has been su?cient to unravel the underlying2-dimensional current algebra struc-ture in the compact case.

The starting point in the“constrain-?rst”approach is the classical gauge invariant phase space M,the space of?at gauge connections on the space manifold Σ.Since M is?nite-dimensional,the canonical quantization problem actually has a?nite number of degrees of freedom.However,the fact that M is not in general a smooth manifold,makes its quantization rather non-standard.Even in the case of compact gauge groups,M has singularities of?nite order which are associated to important quantum-mechanical e?ects,such as the“shift”of the central charge in the Sugawara construction for2-dimensional algebras[12].When the gauge group is SL(2,R),the singularities of M are of a more general type,as we will shortly see:they play a central role in the quantization of the SL(2,R)Chern-Simons theory which we consider here.

WhenΣis a torus,the problem of quantizing M is reduced to the problem of quantizing the moduli space of?at-connections of an abelian gauge group[11]. This makes the computation for genus one drastically simpler than for higher genus,where non-abelian Chern-Simons theory appears to be vastly more complex than abelian.On the other hand,the factorization properties of2-dimensional conformal?eld theories suggest that the torus topology already contains most,if not all,of the complexities of higher genus.The solution of this apparent paradox is that M for a torus is almost the space of?at connections of an abelian group, but not quite:it is the space of abelian?at connections modulo the action of a discrete group whose?xed points give rise to orbifold singularities.It is only here that the quantization of non-abelian Chern-Simons theory(with compact gauge group)for genus one di?ers from the computationally trivial abelian case. Thus,in some sense,the singularities of M for genus one must encode much of the information about the theory on higher genus surfaces,at least for compact non-abelian gauge groups.That this remains true for non-compact gauge groups like SL(2,R)is plausible,though yet to be proven.

In comparison with M SU(2),the distinctive feature of the phase space M SL(2,R)is its non-compactness.Even if one restricts oneself(as we will essen-

tially do in this paper)to the sector of M SL(2,R)corresponding to?at connections

which lie in the“compact”Cartan subgroup of SL(2,R),one has to deal e?ectively with the non-compact smooth manifold obtained by deleting the singularities from the compact non-smooth phase space.One consequence of the non-compactness of the(e?ective)phase space is that the integrality condition on the Chern-Simons charge k[1],[9],[10]disappears.Another consequence is that in?nite-dimensional Hilbert spaces emerge,in general,upon quantization.However,if one takes k to be rational,the Hilbert space of quantum states becomes?nite-dimensional[13]. Therefore,it is reasonable to think that theories with rational k correspond to rational conformal2-dimensional?eld theories,or to some“deformation”of them, which,when k becomes an integer,reduce to the familiar non-abelian current alge-bras.Recently,abelian Chern-Simons theories with rational charge k=p/q have been investigated because of their(possible)relevance to the theory of quantum Hall e?ect and to a new mechanism for superconductivity[14].Wave functions are represented by q-dimensional multiplets of theta-functions of level pq,[15],[16] which can be thought of as describing holomorphic sections of a holomorphic“line bundle”with rational Chern class p/q on the non-compact phase space.When k is an integer,by projecting the abelian Hilbert space to the“Weyl”odd sector, one obtains the non-abelian wave functions,that is the Kac-Weyl characters of the integrable current algebra representations[11],[12].In this paper we will show that the appropriate generalization of such projection to the fractional charge case leads to the modular invariant Kac-Wakimoto[17]characters of the(non-integrable and non-unitary)representations of?A1current algebra with fractional admissible level. We will also discover that the modular representations acting on the Hilbert space of states of the SL(2,R)Chern-Simons theory are identical to one of the two factors into which the modular representations of the conformal minimal models factorize.This suggests that the2-dimensional counterpart of the SL(2,R)Chern-Simons theory might be non-conformal.It also implies that Chern-Simons theory with gauge group SL(2,R)×SL(2,R)≈SO(2,2)and rational charges(k,1

2.Geometric Quantization of M SL(2,R)

Flat SL(2,R)connections on a torus correspond to pairs(g1,g2)of commuting SL(2,R)elements,modulo overall conjugation in SL(2,R).g1and g2represent the holonomies of the?at connections around the two non trivial cycles of the torus. SL(2,R)has a non-trivial Z2center and SO(1,2)≈SL(2,R)/Z2.Therefore,M is a four-cover of the space M′of SO(1,2)?at connections,since to each SO(1,2)?at connection correspond four SL(2,R)?at connections whose holonomies di?er by elements of the center Z2.It is convenient to describe M in terms of the simpler M′.Let us think of SL(2,R)as the group of unimodular2×2real matrices.The

basic fact of M(or M′)is that it is the union of three“sectors”

M= i=1,2,3M i,(1)

where the M i’s,i=1,2,3,are the space of SL(2,R)?at connections whose respective holonomies have two imaginary(and conjugate)eigenvalues(i=1), two eigenvectors with real(and reciprocal)eigenvalues(i=2),and one single eigenvector with unit eigenvalue(i=3).

When i=1,both holonomies can be simultaneously brought by conjugation into the compact U(1)subgroup of SL(2,R).Therefore M1≈T(1),the two dimensional torus.Let us introduce the real coordinates(θ1,θ2)for M1.In our normalization,the periodic coordinatesθ1,2lie in the unit real interval when the gauge group is SO(1,2);for SL(2,R),these take values in the enlarged interval of length2.

For i=2,the holonomies can be conjugated into a diagonal form.However, one can still conjugate diagonal holonomies by an element of the gauge group which permutes the eigenvalues.Therefore,when the gauge group is SO(1,2), M2≈R(2)/Z2.If(x,y)are cartesian coordinates on the real plane R(2),the Z2 action is the re?ection around the origin,mapping(x,y)onto(?x,?y).If the gauge group is SL(2,R),M2consists of four copies of R(2)/Z2.

Finally,when i=3,holonomies can be conjugated into an upper triangular form with units on the diagonal.Conjugation allows one to rescale the(non-vanishing)elements in the upper right corner by an arbitrary positive number.

Thus,M3≈S1,the real circle.Being odd-dimensional,S1cannot be a genuine non-degenerate symplectic space.In fact,the symplectic form on the space of ?at connections coming from the Chern-Simons action,when pushed down to M3 vanishes identically.M3represents a“null”direction for the symplectic form of the SL(2,R)Chern-Simons theory,re?ecting the inde?niteness of the SL(2,R) Killing form.Since M3is a disconnected piece of the total phase space M(or M′), it is consistent to consider the problem of quantizing M1∪M2independently of M3.After all,modding out by the“null”directions(such as those originated by gauge symmetries),is the common recipe for dealing with degenerate symplectic forms.Hence,in what follows we concentrate on M1∪M2,though it is conceivable that the“light-like”sector M3merits further investigation.

To summarize,the space of gauge?at connections on the torus(disregarding M3)looks as follows:a torus(M1)with planes(M2)“attached”to it at the points z s in a discrete set N≡M1∩M2,representing?at connections with holonomies in the center of the gauge group.For the SO(1,2)case,N contains a single point, whose M1and M2coordinates are(θ1(s),θ2(s))=(0,0)and(x(s),y(s))=(0,0), respectively.When the gauge group is SL(2,R),N consists of four points,with (θ1(s),θ2(s))=(±1,±1).The M2planes are“folded”by the Z2re?ections around the points in N.

The distinctive feature of classical phase space M is that it ceases to be a smooth manifold around the points in N.The quantization of the classical phase space M involves considering smooth functions or smooth sections of appropriate line bundles on M,raising the question of the meaning of“smooth”sections on a non-smooth manifold such as M.Our strategy is to consider?rst the smooth, non-compact manifold M/N obtained by deleting the singular points in N.M/N consists of two disconnected smooth components,M1/N and M2/N.We will then consider quantizations of M1/N and M2/N which admit sections that can be“glued”at the points in N.The?nal Hilbert space will be the span of those “glued”sections.Our“intuitive”approach could conceivably be substantiated with more rigorous methods of algebraic geometry.

We will perform the quantization of M1/N and M2/N in the holomorphic scheme[9]-[11]since,as is familiar from the study of the compact Chern-Simons

theory[1],M admits a natural family T of K¨a hler polarizations[18].T is the Siegel upper complex plane,because the choice of a complex structure on the 2-dimensional space manifoldΣinduces a complex structure on the space of con-nections onΣand,by projection,on M.Forτ∈T,let us introduce holomorphic coordinates on both M1/N

z=θ1+τθ2,ˉz=θ1+ˉτθ2,(θ1,θ2)∈M1/N(2)

and M2/N

z=x+τy,ˉz=x+ˉτy,(x,y)∈M2/N.(3)

Then the symplectic form which descends from the Chern-Simons action with

charge k

S=k

3

A∧A (4)

can be written both on M1and M2in the coordinates systems(2)and(3)as follows:

ω=

ikπ

4τ2

(z?ˉz)2.(6)

In the context of K¨a hler quantization,the Hilbert space of quantum states is the span of square integrable holomorphic sections of a holomorphic line bundle with hermitian structure whose curvature two-form is the symplectic formωin (5).

The quantization of M2/N is rather straightforward.Since M2/N is not simply connected,the holomorphic wave functionsψ(z)can acquire an arbitrary phase e2πi?i when moving around the singular points z i=0of N.The B¨o hm-Aharonov phases e2πi?i should be regarded as free parameters of the quantization.

A further two-fold ambiguity of the M2/N quantization stems from the fact that the gauge invariant M2/N is the quotient of the complex plane(with the origin deleted)by the action of the re?ection around the origin.Thus,physical wave

functions should be invariant under the action of the unitary operator implement-ing the re?ection around the origin.Since there are two ways of implementing re?ections according to the“intrinsic”parity of the wave functions,one concludes that the wave functions on each of the four“sheets”of M2/N are

ψ(?,±)(z)=z?χ(±)(z),(7)

whereχ(±)(z)is holomorphic,even(odd)around the origin,and each choice of

.

(?,±)is associated to a di?erent quantum Hilbert space H(?,±)

M2/N

Let us now turn to M1/N.The crucial di?erence between quantum mechanics on the non-compact M1/N and on the compact torus M1originates from the fact that the homotopy groupπ1(M1/N)is non-abelian:

ab=baδ,[a,δ]=[b,δ]=0,(8)

where a and b are the non-trivial cycles of the compact torus andδ= iδi is the product of the cyclesδi surrounding the singularities z i in N.In this case,quan-tum states are represented by multi-components wave functionsΨ(z)=(ψα(z)),α=0,1,...,q?1,transforming in some irreducible unitary,q-dimensional rep-resentation of the homotopy groupπ1(M1/N).Let us consider a basis for such representation which diagonalizes theδi’s.For the representation to be?nite-dimensional and irreducible,theδi’s must be represented by rational phases.More-over,we take allδi’s to be the same,since we require that modular transformations (which mix the singular points in N)act on the Hilbert space of wave functions. In conclusion we takeδ=exp(?2πip/q)with p integer,coprime with q.

In the holomorphic quantization scheme,wave functionsΨ(z)should be holo-morphic and,in the trivialization corresponding to(6),should have the periodicity properties of theta functions with fractional“level”k:

Ψ(z+2m+2nτ)=exp(?2πikτn2?2πikzn)a m b nΨ(z),(9)

where a and b are q×q unitary matrices which provide a representation of ho-motopy relations(8).Note that on the compact torus M1,a and b would be

one-dimensional phases and the consistency(cocycle)condition for the transi-tion functions in(9)would require2k to be an integer[9]-[11].In our case,the consistency condition coming from(8)relates the Chern-Simons charge k to the monodromy of the wave functions around the singular points:

e2πi2k=e2πip/q.(10)

Therefore,we restrict ourselves henceforth to the case of k rational:

2k=2s/r=p/q,(11)

with s and r integers,relatively prime,and r chosen to be positive.It should be stressed that the restriction to k rational is motivated by the interest to investigate the connection between SL(2,R)Chern-Simons theory and2-dimensional rational conformal?eld theories.When k is irrational one expects an in?nite-dimensional Hilbert space of holomorphic wave functions:an interesting possibility,which we do not pursue here.

The holomorphic componentsψα(z)of the wave functionsΨ(z)can be thought of as representing holomorphic sections of the holomorphic line bundle L(k)o on the non-compact M1/N with fractional“Chern-class”p/q.Locally,a

sectionψof L(k)o would be given by the q-root of a theta function of level p.ψwould have non-trivial mondromyδ=e2πip/q around the singular points in N, but would be single-valued when holomorphically extended to a q-cover?M1of the torus M1.If M1is the complex torus with modular parameterτ,the q-cover ?M1is a torus with modular parameter qτ.The q componentsψα(z)of the wave functionΨ(z)should be identi?ed with the di?erent holomorphic extentions ofψto?M1:they should therefore be theta functions of level q×p/q=p on the torus with modular parameter qτ.We will verify that this is in fact the case.In the following however we will simply think ofΨ(z)as hoomorphic sections of a vector bundle on the compact torus M1with?bers of dimensions q.

It follows from(9)that inequivalent holomorphic quantizations of M1/N with the same k are in one-to-one correspondence with classes of inequivalent,unitary and irreducible representations of the‘t Hooft algebra

ab=bae2πip/q.(12)

Let us denote by R(?a,?b)

the following q-dimensional representation of(12):

p/q

(a)αβ=e2πi?a e?2πip/qαδα,β

(13)

(b)αβ=e2πi?bδα,β+1,α,β=0,1,...,q?1.

It is easy to check that the“characteristics”(?a,?b)modulo(m/q,n/q)(with m,n relative integers)label all the inequivalent unitary irreducible representations of (12).The space of classes of inequivalent(irreducible,unitary)representations of (12)is therefore isomorphic to a2-dimensional torus T p/q.

It is has been stated[9]-[10]that for the modular group to act on the Hilbert space of holomorphic wave functions(9)one needs both pq even and the character-istics?a,b≡0modulo1/q.Since this is not quite correct,let us pause to discuss the issue of modular invariance in some detail.(See also[13]-[15].)Let us denote by s,t,c the following external automorphisms of the‘t Hooft algebra(12):

s: a→b?1

b→a

(14)

t: a→a b→e?iπp/q ab

c: a→a?1

b→b?1.

One can to verify that s,t,c satisfy the modular group relations,s2=c and (st)3=1and that the“conjugation”operator c commutes with the modular group generators,sc=cs,tc=ct.The automorphisms s,t,c map representations of(12)onto generically inequivalent representations;therefore they induce a non-trivial action on the torus T p/q,the space of classes of inequivalent(irreducible, unitary)representations of the‘t Hooft algebra(12).This action,however,is not the“standard”action of the modular group on the2-dimensional torus,which is linear and homogenous in the coordinates(q?a,q?b).Denoting by s?,t?,c?the action of s,t,c induced on T p/q one can explicitly calculate from(13)that t?has an inhomogenous term:

c?:(q?a,q?b)→(?q?a,?q?b)

(15)

s?:(q?a,q?b)→(?q?b,q?b)

t?:(q?a,q?b)→(q?a,q?a+q?b+pq/2),

where q?a,b are real numbers modulo integers.The vector space of a representation ,belonging in an equivalence class which is invariant under s?,t?,c?,carries R(?a,?b)

p/q

a(unitary)representation of the modular group whose generators we will denote by S,T and C.Such a representation R(?a,?b)

de?nes through(9)a vector space of

p/q

holomorphic wave functionsΨ(z)which supports a(unitary)representation of the modular group.The generators of this modular representation will be indicated below by U(s),U(t)and U(c).From(15)it follows that if pq is even the only(up to equivalence)modular invariant quantization corresponds to the(equivalence class representation of the‘t Hooft algebra(12),a fact already recognized

of the)R(0,0)

p/q

in the earlier literature on Chern-Simons theory[9],[10].However eq.(15)also implies that modular invariance can be mantained for pq odd as well by choosing

.This a representation of the‘t Hooft algebra in the equivalence class of R(1/2,1/2)

p/q

was?rst realized in[13]in the context of the abelian Chern-Simons theory.We will show in the following that in the non-abelian theory the choice k=p/2with pq=p odd(disregarded in[9],[10]on modular invariance grounds)does actually lead to the characters forming the D p+1series of modular invariants for?A1current algebra[19].Since these conformal models are well-de?ned on Riemann surfaces of arbitrary topology it is likely that a modular invariant quantization of Chern-Simons theory with k integer and odd,extending to all genuses the quantization that we will exhibit here for the torus topology,does exist.

In geometric quantization,in order to implement canonical transformations which do not leave the polarization invariant(such as modular transformations), the wave functionsΨτ(z)are also regarded as dependent on the polarizationτ∈T. Theτdependence is determined by the requirement that quantum Hilbert spaces Hτrelative to di?erentτ’s be unitarily equivalent with respect to the hermitian

forms

Ψ(1)τ,Ψ(2)τ = M1dz dˉzτ?1/22e?kπ

even or odd.For pq even,an orthonormal p-dimensional basis for the q-components parallel wave functionsΨ(z)of the quantization of M1/N is given by:

(ΨN(τ;z))α≡ψαN(τ;z)=θqN+pα,pq/2(τ;z/q),N=0,1,...,p?1,(17) where theθn,m(τ;z)(n integer modulo2m)are level m SU(2)theta functions [20]:

θn,m(τ;z)≡ j∈Z e2πimτ(j+n2m).

For pq odd,we have seen that modular invariance requires the representation of

the‘t Hooft algebra(12)to be(equivalent to)R(1/2,1/2)

p/q

.With this choice,an orthogonal p-dimensional basis of parallel holomorphic wave functions is: (ΨN(τ;z))α=(?1)qN+pα j∈Z e iπpqτ(j+N/p+q/α+1/2)2+iπp(z?q)(j+N/p+q/α+1/2) =θq(2N+p)+2pα,2pq(τ;z/2q)?θq(2N?p)+2pα,2pq(τ;z/2q),

?p/2

(18)

Among classical canonical transformations,re?ections?c around the singular points in N

?c:z→?z.(19) are of special interest for our purposes.?c will be implemented on the Hilbert space of wave functionsΨ(z)by a unitary operator U(c):

U(c):Ψ(z)→CΨ(?z),(20) where C is a q×q unitary matrix acting on the“internal”indicesα,which im-plements the automorphism c de?ned in(14)on the vector space of c?-invariant representations of(12):

Ca m b n=a?m b?n C.(21)

For a,b in both the representation R(0,0)

p/q (when pq is even)and R(1/2,1/2)

p/q

(when

pq odd)the solution of(21)is:

(C)α,β=δα,?β.(22)

The Hilbert spaces H M

spanned by the sections(17)and(18)split under

1/N

.For pq even an the action of U(c)into“even”and“odd”subspaces H±

M1/N

orthogonal parallel basis of H±

is

M1/N

(τ;z)=θqN+pα,pq/2(τ;z/q)±θ?qN+pα,pq/2(τ;z/q),(23) (Ψ±N)α≡ψα,(±)

N

while for pq odd one has:

(Ψ±N)α=(θq(2N+p)+2pα,2pq(τ;z/2q)±θ?q(2N+p)+2pα,2pq(τ;z/2q))?

(24)

(θq(p?2N)+2pα,2pq(τ;z/2q)±θ?q(p?2N)+2pα,2pq(τ;z/2q)).

Were we simply trying to quantize M1/N,we would keep both the even and the odd sector since canonical transformations(19)in the M1sector do not correspond to gauge transformations of the original Chern-Simons SL(2,R)the-ory.However,we really want to quantize the union M1∪M2.There are no “rigorous”ways to quantize a phase space consisting of di?erent branches with a non-zero intersection.Phase spaces of this sort have appeared in the context of 2-dimensional gravity in[21].It seems reasonable to think of a wave function on the union M1∪M2as a pair(ψ1,ψ2)of wave functions,withψ1∈H M1/N and ψ2∈H M2/N,“agreeing”in some sense on the intersection N.Our proposal is that ψ1andψ2should have the same behaviour around the points in N.Sinceψ1and ψ2are represented by holomorphic functions,this implies that the pair(ψ1,ψ2) should be determined uniquely byψ1and that most of the statesψ2in the in?nite-

should be discarded.Moreover,the B¨o hm-Aharonov phase dimensional H M

2/N

e2πi?in the M2branch should coincide with the analogous quantity e2πip/q in

have the same behaviour under the M1sector.However,all statesψ2in H M

2/N

re?ections?c around singular points,since?c corresponds to a gauge transformation of the Chern-Simons theory in the M2branch.This should put a restriction on the statesψ1,which,in order to“agree”withψ2,should also have de?nite parity under?c.In conclusion the(only)r?o le of M2should be“transmitting”to M1 the de?nite?c-parity projection.The phase space M1∪M2admits therefore two inequivalent quantizations,with Hilbert spaces isomorphic to H±

.A similar

M1/N ambiguity is present in the SU(2)case[12],but it is the“odd”quantization which is related to2-dimensional conformal?eld theories for generic k.In fact,only the

“odd”projection gives positive integer fusion rules for k generic,suggesting that this is the quantization of the Chern-Simons theory on the torus which generalizes, in some appropriate sense,to higher genus space manifolds[11].In our case as well,“odd”quantization gives positive,integer fusion rules for generic k,as we will shortly see,though we do not yet know its2-dimensional interpretation.

are related to the Kac-Wakimoto characters[17] Wave functions in H?

M1/N

of irreducible,modular invariant representations of SL(2,R)current algebra with fractional central charge m≡t/u(t,u coprime integer relative numbers,u positive) satisfying the admissibility condition

2u+t?2≥0.(25) The Kac-Wakimoto characters are de?ned as follows:

e2πiτL o+2πizJ3o,(26)

χj(N′,α′);m(z,τ)=tr H

j,m

where H j,m is the highest weight irreducible representation of SL(2,R)current algebra with level m and spin j.j=j(N′,α′)ranges over the following set:

j=1/2(N′?α′(m+2)),N′=1,2,...,2u+t?1,α′=0,1,...u?1.(27) In order to exhibit the explicit relation between Kac-Wakimoto characters and Chern-Simons wave functionsΨ(z)one has to distinguish the cases when:

(i)p is even and q is odd,so that p=2s and q=r;

(ii)p is odd and q is even,so that p=s and r=2q is a multiple of4;

(iii)both p and q are odd,so that p=s and r=2q≡2mod4.

In case(i)the“odd”orthogonal wave functions in(23)can be written in terms of the Kac-Wakimoto charactersχj;m of level m given by:

m+2=k,(28) i.e.u=q=r and p/2=s=2u+t.The explicit relation is:

(τ;z)

ψα,(?)

N

whereΠ(τ;z)is the Kac-Wakimoto denominator:

Π(τ;z)=θ1,2(τ,z)?θ?1,2(τ,z).(30)

Π(τ;z)is holomorphic and non-vanishing on M1/N.Therefore,the wave func-tionsΨ(?)

N

(τ;z)and the wave functions

Ψ′N(τ;z)=Ψ(?)

N

(τ;z)

Π(τ;z)= χj(2N,α);m(τ;z)ifα∈{0,1,...,r/4?1}

χj(s?2N,α?r/4);m(τ;z)ifα∈{r/4,...,r/2?1}.(33)

Finally,if(iii)is true,the level m of the SL(2,R)current algebra is still given by eq.(32),but u=q=r/2and2u+t=2p=2s.The relation between wave functions and characters becomes:

(τ;z)

ψα,(?)

N

=χp+2N;2(p?1)(τ;z/2)+χp?2N;2(p?1)(τ;z/2),(35)Π(τ;z)

from which one sees that wave functions are precisely those linear combinations of ?A

charactersχn;2(p?1)of level2(p?1)which form the D p+1-series of[19].

1

3.Modular Transformations

Let us use?s and?t to denote the canonical transformations of the classical phase space M1/N which generate the modular group SL(2,Z)of the torus

?s:(τ,z)→(?1/τ,z/τ)

?t:(τ,z)→(τ+1,z),

(36)

and satisfy the relations:

?s2=?c,(?s?t)3=1.(37)?s and?t will be represented on the space of the multi-components wave functions (Ψ(z))α≡ψα(τ;z)by means of unitary operators U(s)and U(t):

U(s):Ψ(τ;z)→(S?1Ψ)(?1/τ;z/τ)

U(t):Ψ(τ;z)→(T?1Ψ)(τ+1;z),

(38)

where S≡(S)αβand T≡(T)αβare unitary q×q matrices acting on the“internal”indices and implenting the modular transformations(14)on the representation

space the‘t Hooft algebra(12).Choosing the R(0,0)

p/q

representation of(12)when

pq is even and R(1/2,1/2)

p/q

when pq is odd,one obtains the following expressions for matrices T and S:

(T(p;q))αβ=δα,β(?1)pqαe2πi p

√qαβ,

α,β=0,1,...,q?1.

(39)

The phaseθ(p;q)in(39)is determined from the SL(2,Z)relation(ST)3=1, which gives:

e2πiθ(p;q)=1

q

q?1

n=0(?1)pqn e2πi p

R(1;2K)is the representation of the modular group associated to the conformal blocks of a2-dimensional scalar?eld compacti?ed on a circle of radius R2=2m/n with m and n integers and K=mn.R(1;2K)is also the representation of the modular group that one obtains upon quantization of the abelian Chern-Simons theory on a torus[9]-[10].For p=1,the sum in(40)is a generalized Gauss sum which has not yet appeared in conformal?eld theory and which we calculate in the Appendix.Some properties ofθ(p;q)follow immediately from its de?nition (40):

θ(p+2q;q)≡θ(p;q)mod1

(41)

θ(p′;q)≡θ(p;q)mod1,if p′≡pn2mod2q

for n integer.The explicit formula forθ(p;q)derived in the Appendix implies that

e8πiθ(p;q)=?1,(42) i.e.,that the allowed values forθ(p;q)are±1/8and±3/8(mod1).

The representations of the modular group acting on the quantum Hilbert and H CS can now be derived from the modular properties of the spaces H M

1/N

theta functions in(17),(18),(23),(24)and from the representation(39)acting on

carries the p?dimensional representation R(q;p) the“internal”indices.H M

1/N

“dual”to the representation R(p;q)de?ned in(39):

=(?1)Npq e2πi q

T(q;p)

N,M

(43)

√p NM N,M=0,1,...,p?1.

This representation is equivalent to the representation of the modular group ob-tained in[13],[15]by quantizing abelian Chern-Simons theory with fractional cou-pling constant.For q≡n2mod2p its interpretation in terms of2-dimensional conformal?eld theories is still obscure.We concentrate,however,on the represen-tations carried by H CS≡H±M1/N.Note that(S(q;p))2=C,with(C)N,M=δN,?M being the“charge conjugation”matrix.Since C commutes with the matrices in

,even and odd under C: (43),R(q;p)decomposes into two representations R(q;p)

±

R(q;p)=R(q;p)

+⊕R(q;p)?,(44)

where R(q;p)

±

is p/2±1dimensional if p is even(i.e.,if r is odd)and(p±1)/2-

dimensional if p is odd(i.e.,if r is even).Since(S(q;p)

?

)2=?1,it is convenient

to de?ne a?R(q;p)

?by?S(q;p)

?

=?iS(q;p)

?

and?T(q;p)

?

=iT(q;p)

?

such that the charge

conjugation matrix is equal to the identity.When q=1the“odd”representation ?R(1;p)

?

is the one associated to modular invariants of?A1current algebra(to the

diagonal A p/2?1series of level p/2?2if p is even,to the D p+1series of level2p?2if p is odd.).The fusion rules associated to the“even”representation are not positive and integer-valued for generic p,suggesting that the“even”quantization does not extend to Chern-Simons theories de?ned on higher genus surfaces[11],[12].The

same remains true for generic q.This motivates the choice H CS=H?

M1/N

carrying

the modular representation R(s/r)

CS≡?R(q;p)?which has the following explicit matrix representation:

T CS

N,M

=i(?1)Npq e2πi q

√p NM N,M=1,...,[(p?1)/2],

(45)

where[x]is the largest integer≤x.Therefore,the central charge c and the conformal dimensions h N of a hypothetical2-dimensional conformal?eld theory underlying the3-dimensional SL(2,R)Chern-Simons theory should satisfy the equation

h N?c/24=N2/4k?θ(q;p)/3+1/4mod1.(46) If such theory were unitary and had a unique identity operator corresponding to the conformal block labelled byˉN∈{1,2,...,[(p?1)/2]},one would have:

c=2?12ˉN2q/p+8θ(q;p)mod8

h N=

q

When pq is odd,a similar equation holds with the l.h.s.given by the modular

representation acting on the Kac-Wakimoto characters which appear in(34)and which de?ne a generalization of the D-series to the fractional level case.Eq.(48) encodes the relationship between Chern-Simons theories and Wess-Zumino-Witten models when2k is fractional.For integer2k(i.e.,for q=1),the left factor on the r.h.s.of(48)is trivial,and one obtains the well-established correspondence between Chern-Simons states and current algebra blocks.For fractional2k,Eq.

(48)can be phrased by saying that the2-dimensional theory underlying SL(2,R) Chern-Simons theory is the“quotient”of SL(2,R)current algebra by some yet unknown generalization of the gaussian model whose modular properties are given by R(p;q).

It was discovered in[23]that the Kac-Wakimoto characters are related by means of a certain projection to the Rocha-Caridi characters of the c<1conformal discrete series.This suggests that the modular representation R(s/r)

CS

in(45)has

something to do with the representation R(r;s)

V ir

relative to the(r,s)minimal model of Belavin-Polyakov-Zamolodchikov.This in fact turns out to be the case and one can establish,when pq is even,the following equation:

R(r;s)

V ir =R(r/4s)

CS?R(s/r)CS,(49)

where r must be chosen odd.(This is always possible since r and s are coprime integers:however,the r.h.s.of the Eq.(49)is not symmetric under the interchange of s and r if one of them is even.The equation as written is not valid for r even.) In order to understand how(49)comes about,let us consider the abelian Chern-Simons theory with even integer charge K=pq whose algebra of observables O K is generated by the holonomies A and B around the non-trivial cycles of the torus [9],[10]:

AB=BAe2πi/K.(50)

The quantum Hilbert space is K-dimensional and spanned by SU(2)theta func-tionsθλ,K/2(withλ∈Z K)of level K/2.It carries the representation R(1;K)of the modular group.Now the crucial fact is that

O K≈O p/q×O q/p,(51)

With的用法全解

With的用法全解 with结构是许多英语复合结构中最常用的一种。学好它对学好复合宾语结构、不定式复合结构、动名词复合结构和独立主格结构均能起很重要的作用。本文就此的构成、特点及用法等作一较全面阐述，以帮助同学们掌握这一重要的语法知识。 一、 with结构的构成 它是由介词with或without+复合结构构成，复合结构作介词with或without的复合宾语，复合宾语中第一部分宾语由名词或代词充当，第二部分补足语由形容词、副词、介词短语、动词不定式或分词充当，分词可以是现在分词，也可以是过去分词。With结构构成方式如下： 1. with或without-名词/代词+形容词； 2. with或without-名词/代词+副词； 3. with或without-名词/代词+介词短语； 4. with或without-名词/代词 +动词不定式； 5. with或without-名词/代词 +分词。 下面分别举例： 1、 She came into the room，with her nose red because of cold.（with+名词+形容词，作伴随状语）

2、 With the meal over ， we all went home.（with+名词+副词，作时间状语） 3、The master was walking up and down with the ruler under his arm。（with+名词+介词短语，作伴随状语。） The teacher entered the classroom with a book in his hand. 4、He lay in the dark empty house，with not a man ，woman or child to say he was kind to me.（with+名词+不定式，作伴随状语）He could not finish it without me to help him.（without+代词 +不定式，作条件状语） 5、She fell asleep with the light burning.（with+名词+现在分词，作伴随状语） Without anything left in the with结构是许多英 语复合结构中最常用的一种。学好它对学好复合宾语结构、不定式复合结构、动名词复合结构和独立主格结构均能起很重要的作用。本文就此的构成、特点及用法等作一较全面阐述，以帮助同学们掌握这一重要的语法知识。 二、with结构的用法 with是介词，其意义颇多，一时难掌握。为帮助大家理清头绪，以教材中的句子为例，进行分类，并配以简单的解释。在句子中with结构多数充当状语，表示行为方式，伴随情况、时间、原因或条件（详见上述例句）。 1.带着，牵着…… (表动作特征)。如： Run with the kite like this.

with的复合结构和独立主格结构

1. with+宾语+形容词。比如：. The boy wore a shirt with the neck open, showing his bare chest. 那男孩儿穿着一件衬衫，颈部敞开，露出光光的胸膛。Don’t talk with your mouth full. 嘴里有食物时不要讲话。 2. with+宾语+副词。比如：She followed the guide with her head down. 她低着头，跟在导游之后。 What a lonely world it will be with you away. 你不在，多没劲儿呀！ 3. with+宾语+过去分词。比如：He was listening to the music with his eyes half closed. 他眼睛半闭着听音乐。She sat with her head bent. 她低着头坐着。 4. with+宾语+现在分词。比如：With winter coming, it’s time to buy warm clothes. 冬天到了，该买些保暖的衣服了。 He soon fell asleep with the light still burning. 他很快就睡着了，（可）灯还亮着。 5. with+宾语+介词短语。比如：He was asleep with his head on his arms. 他的头枕在臂膀上睡着了。 The young lady came in, with her two- year-old son in her arms. 那位年轻的女士进来了，怀里抱着两岁的孩子。 6. with+宾语+动词不定式。比如： With nothing to do in the afternoon, I went to see a film. 下午无事可做，我就去看了场电影。Sorry, I can’t go out with all these dishes to wash. 很抱歉，有这么多盘子要洗，我不能出去。 7. with+宾语+名词。比如： He died with his daughter yet a school-girl.他去逝时，女儿还是个小学生。 He lived a luxurious life, with his old father a beggar . 他过着奢侈的生活，而他的老父亲却沿街乞讨。(8)With so much work to do ,I can't go swimming with you. (9)She stood at the door,with her back towards us. (10)He entered the room,with his nose red with cold. with复合结构与分词做状语有啥区别 [ 标签：with, 复合结构, 分词状语] Ciro Ferrara 2009-10-18 16:17 主要是分词形式与主语的关系 满意答案好评率：100%

with复合结构专项练习96126

with复合结构专项练习（二） 一请选择最佳答案 1）With nothing_______to burn，the fire became weak and finally died out. A.leaving B.left C.leave D.to leave 2）The girl sat there quite silent and still with her eyes_______on the wall. A.fixing B.fixed C.to be fixing D.to be fixed 3）I live in the house with its door_________to the south.（这里with结构作定语） A.facing B.faces C.faced D.being faced 4）They pretended to be working hard all night with their lights____. A.burn B.burnt C.burning D.to burn 二：用with复合结构完成下列句子 1）_____________（有很多工作要做），I couldn't go to see the doctor. 2）She sat__________（低着头）。 3）The day was bright_____.（微风吹拂） 4）_________________________，（心存梦想）he went to Hollywood. 三把下列句子中的划线部分改写成with复合结构。 1）Because our lessons were over，we went to play football. _____________________________. 2）The children came running towards us and held some flowers in their hands. _____________________________. 3）My mother is ill，so I won't be able to go on holiday. _____________________________. 4）An exam will be held tomorrow，so I couldn't go to the cinema tonight. _____________________________.

with的用法大全

with的用法大全----四级专项训练with结构是许多英语复合结构中最常用的一种。学好它对学好复合宾语结构、不定式复合结构、动名词复合结构和独立主格结构均能起很重要的作用。本文就此的构成、特点及用法等作一较全面阐述，以帮助同学们掌握这一重要的语法知识。 一、 with结构的构成 它是由介词with或without+复合结构构成，复合结构作介词with或without的复合宾语，复合宾语中第一部分宾语由名词或代词充当，第二部分补足语由形容词、副词、介词短语、动词不定式或分词充当，分词可以是现在分词，也可以是过去分词。With结构构成方式如下： 1. with或without-名词/代词+形容词; 2. with或without-名词/代词+副词; 3. with或without-名词/代词+介词短语; 4. with或without-名词/代词+动词不定式; 5. with或without-名词/代词+分词。 下面分别举例：

1、 She came into the room，with her nose red because of cold.(with+名词+形容词，作伴随状语) 2、 With the meal over ， we all went home.(with+名词+副词，作时间状语) 3、The master was walking up and down with the ruler under his arm。(with+名词+介词短语，作伴随状语。) The teacher entered the classroom with a book in his hand. 4、He lay in the dark empty house，with not a man ，woman or child to say he was kind to me.(with+名词+不定式，作伴随状语) He could not finish it without me to help him.(without+代词 +不定式，作条件状语) 5、She fell asleep with the light burning.(with+名词+现在分词，作伴随状语) 6、Without anything left in the cupboard， she went out to get something to eat.(without+代词+过去分词，作为原因状语) 二、with结构的用法 在句子中with结构多数充当状语，表示行为方式，伴随情况、时间、原因或条件(详见上述例句)。

5种基本句型和独立主格结构讲解

英语中的五种基本句型结构 一、句型1：Subject (主语) ＋Verb (谓语) 这种句型中的动词大多是不及物动词，所谓不及物动词，就是这种动词后不可以直接接宾语。常见的动词如：work, sing, swim, fish, jump, arrive, come, die, disappear, cry, happen等。如： 1) Li Ming works very hard.李明学习很努力。 2) The accident happened yesterday afternoon.事故是昨天下午发生的。 3）Spring is coming. 4) We have lived in the city for ten years. 二、句型2：Subject (主语) ＋Link. V(系动词) ＋Predicate(表语) 这种句型主要用来表示主语的特点、身份等。其系动词一般可分为下列两类： (1)表示状态。这样的词有：be, look, seem, smell, taste, sound, keep等。如： 1) This kind of food tastes delicious.这种食物吃起来很可口。 2) He looked worried just now.刚才他看上去有些焦急。 (2)表示变化。这类系动词有：become, turn, get, grow, go等。如： 1) Spring comes. It is getting warmer and warmer.春天到了，天气变得越来越暖和。 2) The tree has grown much taller than before.这棵树比以前长得高多了。 三、句型3：Subject(主语) ＋V erb (谓语) ＋Object (宾语) 这种句型中的动词一般为及物动词, 所谓及物动词，就是这种动词后可以直接接宾语，其宾语通常由名词、代词、动词不定式、动名词或从句等来充当。例： 1) He took his bag and left.（名词）他拿着书包离开了。 2) Li Lei always helps me when I have difficulties. (代词)当我遇到困难时，李雷总能给我帮助。 3) She plans to travel in the coming May Day.（不定式）她打算在即将到来的“五一”外出旅游。 4) I don’t know what I should do next. (从句)我不知道下一步该干什么。 注意：英语中的许多动词既是及物动词，又是不及物动词。 四、句型4：Subject(主语)＋Verb(谓语)＋Indirect object(间接宾语)＋Direct object (直接宾语) 这种句型中，直接宾语为主要宾语，表示动作是对谁做的或为谁做的，在句中不可或缺，常常由表示“物”的名词来充当；间接宾语也被称之为第二宾语，去掉之后，对整个句子的影响不大，多由指“人”的名词或代词承担。引导这类双宾语的常见动词有：buy, pass, lend, give, tell, teach, show, bring, send等。如： 1) Her father bought her a dictionary as a birthday present.她爸爸给她买了一本词典作为生日礼物。 2）The old man always tells the children stories about the heroes in the Long March. 老人经常给孩子们讲述长征途中那些英雄的故事。上述句子还可以表达为： 1）Her father bought a dictionary for her as a birthday present. 2）The old man always tells stories about the heroes to the children in the Long March. 五、句型5：Subject(主语)＋Verb (动词)＋Object (宾语)＋Complement(补语) 这种句型中的“宾语＋补语”统称为“复合宾语”。宾语补足语的主要作用或者是补充、说明宾语的特点、身份等；或者表示让宾语去完成的动作等。担任补语的常常是名词、形容词、副词、介词短语、分词、动词不定式等。如： 1）You should keep the room clean and tidy. 你应该让屋子保持干净整洁。（形容词） 2) We made him our monitor.(名词)我们选他当班长。 3) His father told him not to play in the street.(不定式)他父亲告诉他不要在街上玩。

with的复合结构

基本用法 它是由介词with或without+复合结构构成，复合结构作介词with或without的复合宾语，复合宾语中第一部分宾语由名词或代词充当，第二部分补足语由形容词、副词、介词短语或非谓语动词充当 一、with或without+名词/代词+形容词 例句:1.I like to sleep with the windows open. 我喜欢把窗户开着睡觉。(伴随情况) 2.With the weather so close and stuffy, ten to one it'll rain presently. 大气这样闷，十之八九要下雨(原因状语) 二、with或without+名词/代词+副词 例句:1.She left the room with all the lights on. 她离开了房间，灯还亮着。(伴随情况) 2.The boy stood there with his head down. 这个男孩低头站在那儿。(伴随情况) 三、with或without+名词/代词+介词短语 例句:1.He walked into the dark street with a stick in his hand. 他走进黑暗的街道时手里拿着根棍子。(伴随情况) 2. With the children at school, we can't take our vacation when we want to. 由于孩子们在上学，所以当我们想度假时而不能去度假。(原因状语) 四、with或without+名词/代词+非谓语动词 1、with或without+名词/代词+动词不定式，此时，不定式表示将发生的动作。 例句: 1.With no one to talk to, John felt miserable. 由于没人可以说话的人，约翰感到很悲哀。(原因状语)

with用法归纳

with用法归纳 （1）“用……”表示使用工具，手段等。例如： ①We can walk with our legs and feet. 我们用腿脚行走。 ②He writes with a pencil. 他用铅笔写。 （2）“和……在一起”，表示伴随。例如： ①Can you go to a movie with me? 你能和我一起去看电影'>电影吗？ ②He often goes to the library with Jenny. 他常和詹妮一起去图书馆。 （3）“与……”。例如： I’d like to have a talk with you. 我很想和你说句话。 （4）“关于，对于”，表示一种关系或适应范围。例如： What’s wrong with your watch? 你的手表怎么了？ （5）“带有，具有”。例如： ①He’s a tall kid with short hair. 他是个长着一头短发的高个子小孩。 ②They have no money with them. 他们没带钱。 （6）“在……方面”。例如： Kate helps me with my English. 凯特帮我学英语。 （7）“随着，与……同时”。例如： With these words, he left the room. 说完这些话，他离开了房间。 [解题过程] with结构也称为with复合结构。是由with+复合宾语组成。常在句中做状语，表示谓语动作发生的伴随情况、时间、原因、方式等。其构成有下列几种情形： 1.with+名词(或代词)+现在分词 此时，现在分词和前面的名词或代词是逻辑上的主谓关系。 例如:1)With prices going up so fast, we can't afford luxuries. 由于物价上涨很快，我们买不起高档商品。（原因状语） 2)With the crowds cheering, they drove to the palace. 在人群的欢呼声中，他们驱车来到皇宫。（伴随情况） 2.with+名词(或代词)+过去分词 此时，过去分词和前面的名词或代词是逻辑上的动宾关系。

独立主格结构练习题及解析

独立主格结构练习题及解析 1. I have a lot of books, half of ___ novels. A. which B. that C. whom D. them 2. __ more and more forests destroyed, many animals are facing thedanger of dying out. A. because B. as C. With D. Since 3. The bus was crowded with passengers going home from market, most of __ carrying heavy bags and baskets full of fruit and vegetables they hadbought there. A. them B. who C. whom D. which 4. The largest collection ever found in England was one of about 200,000 silverpennies, all of ___ over 600 years old. A. which B. that C. them

D. it 5. The cave __ very dark, he lit some candles ___ light. A. was; given B. was; to give C. being; given D. being; to give 6. The soldier rushed into the cave, his right hand __ a gun and his face ____ with sweat.A held; covered B. holding; covering C. holding; covered D. held; covering 7. The girl in the snapshot was smiling sweetly, her long hair ___ . A. flowed in the breeze B. was flowing in the breeze C. were flowing in the breeze D. flowing in the breeze 8. The children went home from the grammar school, their lessons ____ for the day. A. finishing B. finished C. had finished D. were finished 9. On Sundays there were a lot of children playing in the park, ___ parents seated together joking.

With的复合结构

With的复合结构 介词with without +宾语+宾语的补足语可以构成独立主格结构，上面讨论过的独立主格结构的几种情况在此结构中都能体现。 1. with+名词代词+形容词 He doesn’t like to sleep with the windows open. = He doesn’t like to sleep when the windows are open. He stood in the rain, with his clothes wet. = He stood in the rain, and his clothes were wet. With his father well-known, the boy didn’t want to study. 2. with+名词代词+副词 Our school looks even more beautiful with all the lights on. = Our school looks even more beautiful if when all the lights are on. The boy was walking, with his father ahead. = The boy was walking and his father was ahead. 3. with+名词代词+介词短语 He stood at the door, with a computer in his hand. He stood at the door, computer in hand. = He stood at the door, and a computer was in his hand. Vincent sat at the desk, with a pen in his mouth. Vincent sat at the desk, pen in mouth. = Vincent sat at the desk, and he had a pen in his mouth. 4. with+名词代词+动词的-ed形式 With his homework done, Peter went out to play. = When his homework was done, Peter went out to play. With the signal given, the train started. = After the signal was given, the train started. I wouldn’t dare go home without the job finished. = I wouldn’t dare go home because the job was not finish ed. 5. with+名词代词+动词的-ing形式 The girl hid her box without anyone knowing where it was. = The girl hid her box and no one knew where it was. Without anyone noticing, he slipped through the window. = When no one was noticing, he slipped through the window. 6. with+名词代词+动词不定式 The little boy looks sad, with so much homework to do. = The little boy looks sad because he has so much homework to do. with the window closed with the light on with a book in her hand with a cat lying in her arms with the problem solved with the new term to begin

with用法小结

with用法小结 一、with表拥有某物 Mary married a man with a lot of money . 马莉嫁给了一个有着很多钱的男人。 I often dream of a big house with a nice garden . 我经常梦想有一个带花园的大房子。 The old man lived with a little dog on the lonely island . 这个老人和一条小狗住在荒岛上。 二、with表用某种工具或手段 I cut the apple with a sharp knife . 我用一把锋利的刀削平果。 Tom drew the picture with a pencil . 汤母用铅笔画画。 三、with表人与人之间的协同关系 make friends with sb talk with sb quarrel with sb struggle with sb fight with sb play with sb work with sb cooperate with sb I have been friends with Tom for ten years since we worked with each other, and I have never quarreled with him . 自从我们一起工作以来，我和汤姆已经是十年的朋友了，我们从没有吵过架。 四、with 表原因或理由 John was in bed with high fever . 约翰因发烧卧床。 He jumped up with joy . 他因高兴跳起来。 Father is often excited with wine . 父亲常因白酒变的兴奋。 五、with 表“带来”，或“带有,具有”，在…身上，在…身边之意

独立主格结构图表解析

独立主格结构 一、概念 “独立主格结构”就是由一个相当于主语的名词或代词加上非谓语动词、形容词（副）词或介词短语构成的一种独立成分。该结构不是句子，也不是从句，所以它内部的动词不能考虑其时态、人称和数的变化，它与主句之间不能通过并列连词连接，也不能由从句阴道词引导，通常用逗号与主句隔开。独立主格结构在很多情况下可以转化为相应的状语从句或者其他状语形式，但很多时候不能转化为分词形式，因为它内部动词的逻辑主语与主句主语不一致。 二、独立主格的特点

1.当独立主格结构中的being done表示“正在被做时”，being不可以被省略。 2.当独立主格结构的逻辑主语是it, there时，being不可以省略。 三、独立主格结构的用法。 一般放在句首，表示原因时还可放在句末；表伴随状况或补充说明时，相当于一个并列句，通常放于句末。

四、非谓语动词独立主格结构。 “名词或代词+非谓语动词”结构构成的独立主格结构称为非谓语动词的独立主格结构。名词或代词和非谓语动词具有逻辑上的主谓关系。 1.不定式构成的独立主格结构 不定式构成的独立主格结构往往表示还未发生的行为或状态，在句中常 作原因状语，有时做条件状语。 Lots of homework to do, I have to stay home all day. 由于很多作业要做，我只好待在家里。 So many children to look after, the mother has to quit her job. 如此多的孩子要照顾，这个妈妈不得不辞掉她的工作。 2.动词+ing形式的独立主格结构 动词-ing形式的句中作状语时，其逻辑主语必须是主句的主语，否则就 是不正确的。动词-ing形式的逻辑主语与主句的主语不一致时，就应在 动词的-ing形式前加上逻辑主语，构成动词-ing 形式的独立主格结构，逻辑主语与动词间为主谓关系，是分词的动作执行者，分词表示的动作 时逻辑主语发出的动作。 We redoubled our efforts, each man working like two. 我们加倍努力，每个人就像在干两个人的活。 The governor considering the matter, more strikers gathered across his path. 总督思考这个问题时，更多的罢工工人聚集到他要通过的路上。 The guide leading the way, we had no trouble getting out of the forest. 在向导的带领下，我们轻松地走出了森林。 3.过去分词形式的独立主格 过去分词形式的独立主格结构是由“逻辑主语+过去分词”构成。逻辑主 语与动词之间为动宾关系，它是分词的动作承受者，这一结构在句中作 时间状语，原因状语、伴随状语、条件状语等。 This done, we went home.做完这个，我们就回家了。 All our savings gone, we started looking for jobs. 积蓄用完后，我们都开始找工作。 More time and money given, we can finish the work in advance. 如果给予更多的时间和金钱，我们能提前完成这个工作。 五、其他形式的独立主格结构

with复合宾语的用法(20201118215048)

with+复合宾语的用法 一、with的复合结构的构成 二、所谓"with的复合结构”即是"with+复合宾语”也即"with +宾语+宾语补足语” 的结构。其中的宾语一般由名词充当（有时也可由代词充当）；而宾语补足语则是根据 具体的需要由形容词，副词、介词短语，分词短语（包括现在分词和过去分词）及不定式短语充当。下面结合例句就这一结构加以具体的说明。 三、1、with +宾语+形容词作宾补 四、①He slept well with all the windows open.（82 年高考题） 上面句子中形容词open作with的宾词all the windows的补足语， ②It' s impolite to talk with your mouth full of food. 形容词短语full of food 作宾补。Don't sleep with the window ope n in win ter 2、with+宾语+副词作宾补 with Joh n away, we have got more room. He was lying in bed with all his clothes on. ③Her baby is used to sleeping with the light on.句中的on 是副词，作宾语the light 的补足语。 ④The boy can t play with his father in.句中的副词in 作宾补。 3、with+宾语+介词短语。 we sat on the grass with our backs to the wall. his wife came dow n the stairs,with her baby in her arms. They stood with their arms round each other. With tears of joy in her eyes ,she saw her daughter married. ⑤She saw a brook with red flowers and green grass on both sides. 句中介词短语on both sides 作宾语red flowersandgreen grass 的宾补， ⑥There were rows of white houses with trees in front of them.，介词短语in front of them 作宾补。 4、with+宾词+分词（短语 这一结构中作宾补用的分词有两种，一是现在分词，二是过去分词，一般来说，当分词所表 示的动作跟其前面的宾语之间存在主动关系则用现在分词，若是被动关系，则用过去分词。 ⑦In parts of Asia you must not sit with your feet pointing at another person.（高一第十课），句中用现在分词pointing at…作宾语your feet的补足语，是因它们之间存在主动关系，或者说point 这一动作是your feet发出的。 All the after noon he worked with the door locked. She sat with her head bent. She did not an swer, with her eyes still fixed on the wall. The day was bright,with a fresh breeze（微风）blowing. I won't be able to go on holiday with my mother being ill. With win ter coming on ,it is time to buy warm clothes. He soon fell asleep with the light still bur ning. ⑧From space the earth looks like ahuge water covered globe,with a few patches of land stuk ing out above the water而在下面句子中因with的宾语跟其宾补之间存在被动关系，故用过去分词作宾补：

(完整版)with的复合结构用法及练习

with复合结构 一. with复合结构的常见形式 1.“with+名词/代词+介词短语”。 The man was walking on the street, with a book under his arm. 那人在街上走着，腋下夹着一本书。 2. “with+名词/代词+形容词”。 With the weather so close and stuffy, ten to one it’ll rain presently. 天气这么闷热，十之八九要下雨。 3. “with+名词/代词+副词”。 The square looks more beautiful than even with all the light on. 所有的灯亮起来，广场看起来更美。 4. “with+名词/代词+名词”。 He left home, with his wife a hopeless soul. 他走了，妻子十分伤心。 5. “with+名词/代词+done”。此结构过去分词和宾语是被动关系，表示动作已经完成。 With this problem solved, neomycin 1 is now in regular production. 随着这个问题的解决，新霉素一号现在已经正式产生。 6. “with+名词/代词+-ing分词”。此结构强调名词是-ing分词的动作的发出者或某动作、状态正在进行。 He felt more uneasy with the whole class staring at him. 全班同学看着他，他感到更不自然了。 7. “with+宾语+to do”。此结构中，不定式和宾语是被动关系，表示尚未发生的动作。 So in the afternoon, with nothing to do, I went on a round of the bookshops. 由于下午无事可做，我就去书店转了转。 二. with复合结构的句法功能 1. with 复合结构，在句中表状态或说明背景情况，常做伴随、方式、原因、条件等状语。With machinery to do all the work, they will soon have got in the crops. 由于所有的工作都是由机器进行，他们将很快收完庄稼。（原因状语） The boy always sleeps with his head on the arm. 这个孩子总是头枕着胳膊睡觉。（伴随状语）The soldier had him stand with his back to his father. 士兵要他背对着他父亲站着。（方式状语）With spring coming on, trees turn green. 春天到了，树变绿了。（时间状语） 2. with 复合结构可以作定语 Anyone with its eyes in his head can see it’s exactly like a rope. 任何一个头上长着眼睛的人都能看出它完全像一条绳子。 【高考链接】 1. ___two exams to worry about, I have to work really hard this weekend.（04北京） A. With B. Besides C. As for D. Because of 【解析】A。“with+宾语+不定式”作状语，表示原因。 2. It was a pity that the great writer died, ______his works unfinished. (04福建) A. for B. with C. from D.of 【解析】B。“with+宾语+过去分词”在句中作状语，表示状态。 3._____production up by 60%, the company has had another excellent year. (NMET) A. As B.For C. With D.Through 【解析】C。“with+宾语+副词”在句中作状语，表示程度。

(完整版)独立主格结构完整讲解

一、独立主格结构的概念 独立主格结构（Absolute Structure ）是由名词或代词加上分词等构成的一种独立 结构，用于修饰整个句子。独立主格结构中的名词或代词与其后的分词等构成逻辑上的主谓关系。这种结构与主句不发生句法上的联系，它的位置相当灵活，可置于主句前、主句末或主句中，常由逗号将其与主句分开。在句中作状语，相当于一个状语从句。 需特别注意的是，独 二、独立主格结构基本构成形式 名词（代词）+现在分词；过去分词；形容词；副词；不定式；名词；介词短语） 1. 名词（代词）+现在分词 例句：The storm drawing near ，the navvy decided to call it a day The storm drawi ng n ear 在句中作：原因状语 =Si nee the storm was draw ing n ear , the n avvy decided to call it a day 由于暴风雨即将来临，那个挖土小工决定收工。（call it a day 今天到此为止） 例句：Win ter comin g, it gets colder and colder. Win ter comi ng 在句中作：伴随状语 =The win ter comes, and it gets colder and colder. 冬天来了，天气越来越冷了。 造句：时间允许的话，我就和你一起走。 Time permitti ng, I will go with you. 独立主格作：条件状语 =改写：f time permits, I will go with you. ___________ 造句：那个姑娘望着他，他不知道说什么好。 The girl staring at him, he didn ' t know what to say. 独立主格作：时间状语 =改写：As the girl stared at him, he didn ' t know what to say. 2.名词（代词）+过去分词 例句：He was listening attentively in class, his eyes fixed on the blackboard. 他上课专心听讲，眼睛紧盯着黑板。 例句：The meeti ng gone over, every one tired to go home earlier. 会议结束后，每个人都想早点回家。 造句：工作完成后，他回家了。 The work done, he went back home.