Comment on Two Dimensional O(N) and Sp(N) Yang Mills Theories as String Theories

a r X i v :h e p -t h /9307085v 2 30 J u l 1993YCTP-P16-93

hep-th/9307085

Comment on Two Dimensional O (N )and Sp (N )Yang Mills Theories as String Theories Sanjaye Ramgoolam Department of Physics Yale University,New Haven CT 06511-8167We write down all orders large N expansions for the dimensions of irreducible repre-sentations of O (N )and Sp (N ).We interpret all the terms in these expansions as symmetry factors for singular worldsheet con?gurations,involving collapsed crosscaps and tubes.We use it to complete the interpretation of two dimensional Yang Mills Theories with these gauge groups,on arbitrary two dimensional manifolds,in terms of a String Theory of maps of the type considered by Gross and Taylor.We point out some intriguing similarities to the case of U (N )and discuss their implications.

July 1993

1.Introduction

Gross and Taylor[1],and Minahan[2]have established that the partition function,of two-dimensional U(N)or SU(N)Yang Mills theory can be expanded in1/N to give terms that all have a geometrical interpretation in terms of maps of a string worldsheet to the target space.The partition function for an orientable closed manifold of genus G is[3][4]

Z M= [DAμ]exp[?1gT rFμνFμν]

= R(dim R)2?2G e?λA

n!χ

R

(?n),(1.2)

where R is the representation associated to a Young tableau with n boxes,and

dim(R

n!?n!χR

S)is the irreducible representation of largest dimension in the tensor product of R with the complex conjugate of S.

We will discuss in detail in the sections2-4the case of O(N)and in section5we describe the small modi?cation for Sp(N).In section2we quote a character formula for

irreducible tensor representations of O(N)and convert it to a form appropriate for our use.We specialise it to give the expansion of the dimension.We will not consider the spinor representations in detail in this section.They have casimirs of order N2[5]and do not contribute to the large N large A asymptotics but may have nonperturbative e?ects. In section3,we discuss the geometrical interpretation of the expansion.Some of the terms have a familiar interpretation similar to the case of the chiral expansion for U(N).In addition there is a set of permutations which we identify as describing multiply branched sheets equipped with a collapsed crosscap or with an in?nitesimal tube connecting two cycles.To prove that the coe?cients in the expansion are really the symmetry factors for the singular maps described above,we derive,in section4,a simple formula for a certain sum of characters of S n,called Xσ

1

,which appears in the character formula for O(N).We comment on the formula for the dimensions of the spinor representations.In section5, we write the large N expansion for Sp(N)and show that the geometrical interpretation carries over with one modi?cation,the‘collapsed crosscaps on branch points’do not come with a minus sign as in the case of O(N).

2.Formula for characters and dimensions of O(N)representations

The following formula gives the large N expansion for the dimension of a tensor representation of O(N)associated with a Young tableau with n boxes.

dim R= σ∈S nχR(σ)n2!|Tσ1||Tσ2|

n!χ

R

(?n).(2.1)

For each n1we are summing over conjugacy classes Tσ

1of S n

1

,with|Tσ

1|being the order

of the conjugacy class andσ1a representative of the conjugacy class.The condition

Tσ

1Tσ

2

=Tσmeans that the cycles ofσcan be separated into two sets one of which

characterises a conjugacy class in S n

1and the other a conjugacy class in S n

2

(n2=n?n1).

Xσ

1

is de?ned by

Xσ

1

=(?1)n1/2 R∈Y?n1χR(σ1)

where Y?n is a subset of the Young tableaux with n(even)boxes,which is described in the appendix.The expression for?n is

?n= σ∈S nσ n1≥0 Tσ1Tσ

1Tσ

2

=Tσ

N?n1Xσ

1

N Kσ2?n2

n!

|Tσ|

=(1+O(1/N)).(2.3)

The leading term in(2.1)is obtained,when n1is zero,andσ2is the identity element and the next term when it is an element belonging to the conjugacy class characterised by1 cycle of length2and all other cycles of length1(the class T2in the notation of[1]).These are easily seen to agree with the terms as computed in[5]from a hook length type formula.

We now prove equation(2.1).Littlewood[6](page240)gives the following formula for the characters of O(N)

χ

R

(U)={R}+ n1≥2 R1∈Y?n1 R2∈Y n2n1+n2=n(?1)n1/2g(R1,R2;R){R2}.(2.4)

R is a Young tableau having n boxes.R1runs over a certain subset of the Young tableaux

with n1(even)boxes we will call Y?n

1

and describe in detail in Appendix A.g(R1,R2;R)are the Littlewood-Richardson coe?cients.R2runs over all tableaux with n2boxes.{R}and {R2}are symmetric functions in the N eigenvalues of the O(N)matrix U.The symmetric function{R}can be written in terms of characters of S n:

{R}= σ∈S nχR(σ)

the delta function is 1if R and R ′are the same and zero otherwise.From the orthogonality relation, R ∈Y n

χR (σ)χR (τ)=δT σ,T τn !

n !.(2.9)

Now if σis a permutation of n elements and it can be written as σ1σ2in cycle notation,where σ1∈S n 1σ2∈S n 2,n 2=(n ?n 1)i.e σ1σ2also lives in the subgroup S n 1×S n 2of S n ,then we can write [7][8],

χR (σ)= R 1∈Y n 1,R 2∈Y n 2

g (R 1,R 2;R )χR 1(σ1)χR 2(σ2)(2.10)

The existence of such an expansion follows from the fact that the left hand side is a class function for the subgroups S n 1and S n 2.The fact that the coe?cients are precisely the

Littlewood-Richardson coe?cients follows from applications of the Frobenius reciprocity theorem [7].

Now we can use equations (2.4),(2.7)and (2.10),to write the O (N )character formula as follows:

χR (U )={R }+ n 1≥2 n 2≥0n 1+n 2=n

σ1∈S n 1,σ2∈S n 2X σ1Y σ2(U )

(n 2)!

χR (σ1σ2)

= σ∈S n χR (σ) n 1≥0

T σ1T σ1T σ2=T σX σ1Y σ2(U )|T σ|=

σ∈S n χR (σ)?Y

σ(U ).(2.12)In the second line the sum is over all ways of separating the cycles of each σ∈S n into 2sets of cycles 1.The ?rst set of cycles corresponds to a conjugacy class in S n 1,a representative

of which isσ1and the second set to a conjugacy class in S n?n

1

,a representative of which is σ2.The last line is a de?nition of?Yσ(U).By putting U=1we get the large N expansion for the dimension of the O(N)representation(2.1).

3.Geometrical interpretation

Having a formula for the?point in the form above,the same steps of Gross-Taylor can be used to write the2D Yang-Mills partition function on an arbitrary manifoldΣas a sum over homomorphisms fromΠ1(Σ\{punctures})into the symmetric group of permutations on the sheets of the cover,where there are punctures for the branch points and the|2?2G| omega points.The same argument as in[1]can be used to show that inverse powers of the dimension are written in terms of the inverse powers of?.The argument relies on the fact that the element?n of the group algebra of S n commutes with S n,which is also true here because?n is a sum of elements of S n with coe?cients that only depend on the cycle structure(conjugacy class)of the element,as is clear from inspection of(2.1).Negative powers of the omega point can be dealt with in a1/N expansion as in[1].For arbitrary closed non-orientable surfaces,the reality of the reps of S n[5]guarantees that the full partition function can be expressed as a sum of homomorphisms to S n.

We recall from[1]how the?point can be inserted in the partition function of the theory.Take the case of sphere,

Z= R(dim R)2e?λA

(n!)2

χ(?n)2e?λA d R?n)

=

∞

n=0 R∈Y n i≥0 p

1...p i∈T2

N2n?i

2e nλA i!

=

∞

n=0 i≥0 p

1...p i∈T2

N2n?i e?nλA2N(?1)i

(λA)i

n!

δ(p1...p i?2n)].(3.1)

In combining the product of the characters into a single character,we use the fact that ?n commutes with S n.The extra factor e nλA/2N compared to U(N)is accounted by in?nitesimal crosscaps being mapped to a point[5].We notice also that for either O(N) or SO(N)(whose lie algebras are isomorphic)the casimir does not give rise to collapsed handles and tubes as for SU(N).Finally as observed in[5]the above expansion,which

is the analog of chiral expansion in the case of unitary groups,su?ces to give non trivial anwers on both orientable and non orientable surfaces,unlike the SU(N)case.This is related to the reality of the reps of O(N).

For a manifold with G handles and b boundaries the appropriate observables for a stringy interpretation are

Z(G,λA,N;{σi})=< i?Yσi(U i)>

= dU1dU2...dU b Z(G,λA;U1,...,U b) i?Yσi(U i).(3.2) The orthogonality property of the?Yσwill follow from that of O(N)and S n characters,

as shown for the analogous functions in the unitary case[1].This

orthogonality will

guarantee[1]that the partition function de?ned above will count maps to the target space with permutations in the conjugacy class ofσi on the sheets covering the boundaries.

We now have to?nd a geometrical interpretation for all the coe?cients appearing in ?n of equation(2.3).For terms for which n1=0,n2=n,the coe?cient ofσ2=σ∈S n is N Kσ?n.This has a simple interpretation in terms of multiple branch points,and the power of N correctly gives the Euler character of a surface branched in the way prescribed by σ.The product of Tσ

n!

times the number of distinct homomorphisms describing the same con?guration of unlabelled sheets)with the coe?cient of N Kσ?nσin the omega point gives1

(n)!

Tσ(which is again1

n2!

.The latter term again correctly describes the genus and symmetry factors for multiple branch points described byσ2.The coe?cient ofσ1has a power of N which is smaller than the one expected from the branch points responsible for the permutation.A similar situation is met in the U(N)case for the coupled?point where there are collapsed tubes connecting cycles of the same length.So this is a useful ingredient,and we call the n1sheets the singular sector which we will now study.

We will need one more singular object.Let us look at a con?guration that can arise from the O(N)omega point,for concreteness,say on a target space which has the topology of a disc,and consider terms where there is no power of the area so that there are no branch points coming from the casimir.Let us further specialise to a term where n=n2andσ2is made of one cycle of length n2.This term contributes to dim R a factor N Kσ2=N1=N2?2g?b,where g and b are the number of handles and boundaries, respectively,of the worldsheet.If we specialised instead to a term where n=n1andσ1is made of one cycle only,then the associated power of N is zero.This can be understood if the branch point comes with a tiny crosscap as well.We also observe that n1is even, so odd total number of sheets do not appear in the singular sector(which we recall are the sheets on whichσ1are acting).This suggests that these crosscaps can only live on even cycles.The power of N associated withσ1is always correctly accounted for,if all the cycles either have a crosscap or share a tube with another cycle.We will consider the implications of this simplest set of singularities needed to account for the powers of N,for example we do not invoke pants connecting more than two cycles or such higher contact terms.And we will show that the few singularities can correctly account for the .Now what kind of tubes are allowed?Do they only connect cycles of the same length Xσ

1

as in the coupled U(N)case?We will look at some simple examples of the coe?cient to understand the symmetry factors associated with each singularity and then use the Xσ

1

geometrical picture that emerges to guess a general formula for Xσ

.We will prove this

1

formula in the next section.

We now show that for permutationsσ1,made of one or two cycles,the quantity |Tσ1|Xσ1≡T Xσ1(|Tσ1|is the number of elements in the conjugacy class ofσ1)is the symmetry factor for some singular worldsheet con?gurations involving multiple branch points,crosscaps,and tiny tubes connecting two cycles.We can use the sum of characters given in(2.9)whenσ1is a permutation made of one cycle only,or two cycles,appendix A describes some details.For one cycle of even length2m,T Xσ

=?1

1

.For two cycles of

2r

equal even length r,T X=?1

2r2

.The?rst term in each case comes from a collapsed tube connecting the2cycles.Except for the factor of2,the symmetry factor and the minus sign are exactly the same as for a tube connecting two sheets of opposite orientation in the coupled?point of U(N).The extra symmetry is expected because the sheets being connected are not equipped with di?erent orientations as in the case of U(N).The additional term in the case of two even cycles comes from con?gurations where the two even cycles are each carrying a crosscap.Note that we do not have separate factors for in?nitesimal Klein bottles and tori because in the singular limit they are the same.

We now summarise the complete set of singularities in the O(N)omega point.In the singular sector(described byσ1)then,an even cycle has to carry a crosscap or be connected to another cycle of the same length by a collapsed tube.An odd cycle must be connected to another cycle of the same length.Each crosscap or collapsed tube comes with a minus sign.These singular con?gurations come with the natural symmetry factors.The sheets not in the singular sector(described byσ2)come with multiple branch points.The only new singularity compared to the coupled U(N)case is the‘crosscap on even cycles’. These singular con?gurations are shown in?gure1.

This predicts that the T Xσshould factorise into separate factors for each cycle length. And the contribution is zero from an odd number of odd cycles.For an even number2m of odd cycles of length r,it should be

T X[(r2m)]=(?1)m

1

i!(n?2i)!2i r n?i

=int(n/2)

i=0(?1)n?2i i!r i2i,(3.4)

where,in the last line,we have separated out the factors for crosscaps and tubes.

4.Derivation of Formulae for the Sum of Characters Xσ

Directly summing the characters to get(3.3)and(3.4)in the case of arbitrary per-mutationsσ1appears hard.The following identity between symmetric functions[6]will be very useful:

i

=1+ n≥2 R∈Y?n(?1)n/2 σ∈S nχR(σ)

2m1(m1)!=(2m1)!T X[(12m

1)]

?T X[(12m1)]=(?1)m1

and,for even r1,as:

1≤i

1

(1?αr1iαr1j)r1 1≤i≤n1(1?αi2r1)r1/2(1?αi r1).(4.5)

Comparing coe?cients of the monomial gives the answer in(3.3)and(3.4).

Now we consider the proof of factorisation.Consider for a start the case of n1cycles of length r1and n2cycles of length r2.Choose the variables

α1···αr1n1+r2n2

as follows.Let

αi+k

1n1

=?αi e2πik1

r2,

1≤i≤n2,

0≤k2≤(r2?1)(4.6b). With this choice there is only one term on the RHS containing the monomial

?α1r1?α2r1···?αn1r1?β1r2?β2r2···?βn2r2.(4.7) And it is the polynomial

(αr11+αr12+···αr1r

1n1+r2n2)n1(αr21+αr22+···αr2r

1n1+r2n2

)n2,

whose coe?cient is T X[(r n11,r n22)].Now suppose without loss of generality that r2>r1. Then powers of?βin the monomial we are looking for cannot come from the?rst n1factors, so must be chosen from the last n2factors only.So the powers of?αhave to come from the ?rst n1.The combinatoric factor on the RHS is thus the product of those that determined the T X[(r n11)]and T X[(r n22)].Now on the LHS of(4.1)we can separate out the product into terms containing?αonly and terms containing?βonly,and mixed terms.The mixed terms can be computed to be

1≤i≤n1,1≤j≤n2

(1??αi r1r2r)r,

where r is the greatest common divisor of r1and r2.This does not contribute to the monomial(4.7).This argument for factorisation clearly generalises to the the case of an arbitrary number of cycle lengths.We have proved then,that there are no in?nitesimal tubes connecting cycles of di?erent lengths.

For spinor representations[6],the dimension can be written as2N times an expression

of the form(2.1)with Xσ

1replaced by?Xσ

1

whose generating function is the expression

[6],

i(1?αi) i

The same steps as above shows that similar expressions for?Xσ

1

can be written as a sum over con?gurations.The only di?erence compared to the case of tensor representations is that the crosscaps live on odd cycles only.

5.About Sp(N)

Many similarities between O(N)and Sp(N)have been observed in the context of matrix models[9]and loop equations[10].One interesting relation between the dimensions of O(N)and Sp(N)tensor representations will be useful here in giving a quick answer for the Sp(N)case using the result for O(N).From[11][12][13]we have

dim[Sp(N)]R=(?1)n dim O(?N)?R,(5.1) where?R is the conjugate representation,related to R by exchanging rows with columns, and dim O(?N)is meant in the sense of analytic https://www.doczj.com/doc/335382640.html,ing this equation and(2.1) we can write the following equation for the dimension of a representation of Sp(N) dim R= σ∈S nχ?R(σ)n2!(?1)n|Tσ1||Tσ2|

n! n1≥0

Tσ

1

Tσ

1

Tσ

2

=Tσ

(?1)Kσ1N Kσ2Xσ

1

n!

|Tσ|.(5.2)

We used the fact[6]thatχ?

R (σ)=χ

R

(σ)(?1)p where p is0if the permutation is even

and1if it is odd,and we wrote(?1)p=(?1)n?Kσ=(?1)n?Kσ1?Kσ2.This means that the geometrical interpretation of the?point of Sp(N)is identical except for a minus sign (?1)Kσ1.From the geometrical interpretation of Xσ

1

,this is equivalent to saying that the

collapsed crosscaps associated with branch points do not come with a minus sign.The natural generalisation of

(5.2)

to

arbitrary

characters

is,

χR (U )= σ∈S n χR (σ)n 2!

|T σ1||T σ2|

2?1(1+αi r 1).(5.6)

We observe that the terms which contribute to the monomial αr 11αr 12···αr 1n 1are unchanged

except that the terms which correspond to crosscaps sitting on branch points have a plus sign instead of a minus sign.The proof of factorisation is identical.Note that crosscaps coming from the casimir are also of opposite sign for O (N )and Sp (N )[5].6.Conclusions and Speculations

In performing the large N expansion of the partition function for the case of U (N )gauge group,the naive expansion analogous to (3.1)did not correctly reproduce the large N asymptotics.One of its most obvious weaknesses [1]was that it gave a trivial answer for nonorientable target spaces whereas there is no reason to expect the large N asymptotics

of the formulae derived in[14]to be trivial.The coupled?point[1]is necessary in order to get the correct large N,large A,approximation.For O(N)and Sp(N)the naive expansion does not give trivial answers[5]on nonorientable target spaces.Moreover we have found close similarities between the structure of the O(N)and Sp(N)?points with the coupled U(N)?point which suggests that,the‘naive’expansion might give the correct large N, large A asymptotics for the O(N)and Sp(N)gauge groups.The methods of[15],as applied to O(N)or Sp(N)could perhaps be used to see if the large N large A asymptotics for these gauge groups is indeed correctly given by the expansion in(3.1).For small A the spinor representations with the factor of2N in their dimension formula will probably have to be taken into account.

Our results on the?point together with those of[5],suggest that the string action of QCD2must be of a form that is generalisable to describe O(N)and Sp(N)Yang Mills. Whereas many questions about2D Yang Mills are most easily answered by starting with the original Yang Mills action,it seems that a satisfactory answer to the question of why there are no higher order contact terms than crosscaps and tubes connecting cycles of equal length can most naturally come from a string picture for all these theories.

One interesting fact about the in?nitesimal tubes is that they are not of two distinct kinds,Klein bottle and torus type(although the theory does contain worldsheets with both topologies[5]).This is naturally understood if the Omega point is associated with singular objects on the worldsheet rather than just being a singularity of the map from worldsheet to target space.A degenerated torus is indistinguishable from a degenerated Klein bottle, whereas singular maps from a Klein bottle to a point and a torus to a point are distinct. This suggests that the ingredients that go into the omega point,in?nitesimal crosscaps and in?nitesimal tubes,are closely connected to the local worldsheet physics,and perhaps the local operator content of the worldsheet theory.Further tests of this picture should be made,perhaps in the framework set up by Kostov[16].

Identifying the appropriate observables for manifolds with boundary may be of help in developing a Das-Jevicki like picture for the O(N)and Sp(N)case,as done for U(N) in[17][18].It should also allow a computation of Wilson loops by the gluing method of [1],adapted to O(N)and Sp(N).

Acknowledgements

I would like to thank Professor Gregory Moore for many discussions and for many comments on the preliminary drafts.I would like to thank Jim Horne and Ronen Plesser for many discussions.This work was supported by the DOE grant DE-AC02-76ER03075.

Appendix A.Determination of X[σ

by Summing Characters

1]

At this point we need to describe in more detail the set Y?,which is done conveniently using the Frobenius notation.The Frobenius notation for Young tableaux describes them by an array of pairs of numbers.The number of pairs is equal to the number of boxes on the leading diagonal of the tableaux.The upper number of the pair is the number of boxes to the right of that box and the lower number is the number of boxes below that box.A tableau with row lengths[6,4,3],for example,is described by

520

210 .

The set Y?consists of tableaux which are of the form

a+1b+1c+1...

a b c... .

Clearly they can only have an even number of https://www.doczj.com/doc/335382640.html,ing a formula for characters of a single cycle[7],only one tableau in Y?,with a single element along the diagonal,will contribute to Xσwhenσ=(2m).We?nd that X[(2m)]=?1

.

(2m)

For conjugacy classes with two cycles(n1,n2),n1=n2the sum over characters can still be done fairly easily,using for example the Murnaghan-Nakayama recursion formula for characters of S n[7].Only tableaux with at most two boxes on the leading diagonal contribute.And we?nd

1

T X[(n1,n2)]=

References

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[2]J.Minahan,‘Summing Over Inequivalent Maps in the String Theory of Two Dimen-

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Figure 1a.

Ordinary branch point. Cyclic

symmetry factor of 5 and weight

1/5.

Figure 1b.gives weight of (-1)/(2.5).Figure 1c.

An even branch point with a crosscap. Cyclic

symmetry factor of 4 and weight (-1)/4

in O(N) omega point.

2 cycles of equal length connected by a collapsed tube which carries a minus sign.

Cyclic symmetry factor of 5, and

symmetry of exchanging the 2 cycles

gives weight of (-1)/(10).

Figure 2a

A configuration of 10 single sheets , with collapsed tubes ( drawn as dotted lines) joining cycles of equal length (here 1) , in the O(N) omega point. Symmetry factor is (2^5 5!) and each tube comes with a minus sign, giving a weight of (-1)^5/(5! 2^5).

+

-

+

-

+

-

+

-

+

-

Figure 2b.

A configuration of 10 single sheets, with collapsed tubes connecting opposite orientations, in the coupled U(N) omega point. Symmetry factor is 5! and each tube comes with a minus sign, so the weight is (-1)^5/(5! ). No factor of 2^5 because the two cycles being connected by a tube have opposite orientation.

外研社英语三年级下册《I like football》教案

外研社英语三年级下册《I like football》教案 知识目标 1、掌握有关运动的单词：like, football, basketball, tabletennis, morning exercises。 2、学会使用句型：I like和I dont like谈论自己喜欢和不喜欢的运动。 能力目标 培养学生正确运用英语进行表述的能力。 情感目标 培养学生热爱运动，激发学生学习的兴趣。 教学重点： 1、单词：like, football, basketball, table tennis 2、能够正确运用句型：I like和I dont like谈论喜欢和不喜欢的运动项目。 教学难点： 1、单词：morning exercises 2、能正确运用本课句型谈论喜欢和不喜欢的事运动项目。 教学用具： word cards, balls, PPT, CD-ROM, stickers, forms 教学过程： Step 1 Warming up. 1、Greeting. T: Hello, boys and girls.

S: Hello, Miss Wu. T: How are you? S: Im fine , thank you. And how are you? T: Im fine ,too. Thank you. T：Now,letslookatthepicturesandthendotheactions. 2、Do and say. 教师播放课件，出示学生玩过的运动项目。学生在老师的带领下，伴随着欢快的音乐做运动。 T: The bell is ringing. Its time for sports.（教师描述图片的意思）Theyre skipping. Theyre playing basketball. Theyreplaying jianzi. Theyre playing football. Theyre running. Shesthewinner.Theyreswimming.Theyreplayingtabletennis.Theyredoingmorningexerc ises.Hesrunning.Theyreplayingtable tennis. Hes playing football. do morning exercises. （当听到运动员进行曲后，师生一块说英语做动作。） T:Standup,please.Letsdoandsay.football/football；run/run/； jump/jump;Swim/swim；Skip/Skip；basketball /basketball；table tennis/ table tennis. T: Are you happy? S: Yes！ T: I m very happy. T: Today we will talk about some sports games. T：We can talk about the sports games we like or dislike. T:Ihavemanyballs.Whichgroupcananswermyquestions. You will get a ball.

I like football教案

Module 3 Playground Unit 1 I like football 辽宁省庄河市大营镇中心小学 闫萍萍 教案背景： 面向学生：小学三年级 学科：英语 课时：1课时 学生课前准备：预习单词和课文，搜集喜欢的体育运动和不喜欢的体育运动。 教学背景： 本课时教学内容涉及的是体育运动的几个项目，贴近学生的学校生活，活动的实践性强。既易于学生表达个人对体育的喜好，又易于从体育项目扩展到其他知识，因此，还能巩固旧知识。 教学准备：多媒体、卡片 教学目标： 1、教材分析： 本课是《新标准英语》三年级起点第二册第三模块第一单元。《I like football.》的第一课时。本课要解决的主要知识点是“I like …” 和“I don’t like …”两个句型。并能围绕操场上的

运动项目：football, basketball, table tennis, morning exercises展开教学，进行句型交流，使学生在情景中掌握教学内容，巩固、运用这两个句型。 2、重点：（1）、单词：football, basketball, table tennis, morning exercise （2）、句型：“I like …” 和“I don’t like …” 难点：（1）、单词exercises的发音 （2）、在适当情景中灵活运用本课句型“I like …” 和“I don’t like …”表达个人喜好。 3、教学方法： 情景呈现 T: (PPT)Now, let’s go to the pla yground and do some sports.(创设操场运动的情景) 游戏法 Let’s play a game ,I call it “唱反调”。If I say “I like …”,you should say “I don’t like …” 全身反应法教师做动作让学生猜老师喜欢的运动 任务型教学法奥运畅想 教学过程： Step 1:Warmer

三年级下册《Ilikefootball》优秀教案

三年级下册《I like football 》优秀教Teaching objectives: 1. Word and phrases: football, basketball table tennis, morning exercises 2. Sentences: I like …I don ' t like Teaching properties: Tape-recorder, objects Teaching procedures: A. Song: Old MacDonald has a zoo. B. Free talk: 描述一下你家里的宠物或你画的小动 物 C. New concepts:

一. 老师对同学们说今年9 月县里要举办九运会, 学校为了响应九运会的号召, 准备在学校举办体育课外小组, 今天派我来调查一下你们都想参与什么项目, 并统计一下人数那么让我们来看看都有什么项目吧. 老师出示足球, 问What' s this ?引导同学们说football, 找会的同学当小老师, 教其他同学说此单词. 然后老师将单词写在黑板上, 让同学们一起书写, 用同样的方法教basketball ,morning exercises, table tennis 游戏:老师说出任意三个单词, 同学们找出老师没读的那个, 也可以同桌的同学玩此游戏. 二. 老师再拿起足球, 边玩边高兴的说I like football. 然后将足球扔给接受较快的同学, 让他模仿说此句型, 再由他扔给其他任何一个同学,以此类推来操练此句型. 老师将句型写在黑板上并画上笑脸.

I like football教案

《I like football》教学设计 一、教学目标 （一）、知识目标 1、单词：football、basketball、table tennis 、morning exercises 、like的听、说、读。 2、句型：I like …和I don’t like …的运用。 （二）、能力目标 善于运用I like …和 I don’t like …举一反三 （三）、情感目标 教育学生做事要有恒心，有毅力。 二、教学重点 1.单词football、basketball、table tennis、morning exercises、like的听、说、读。 2.句型：I like …和I don’t like …的运用 三、教学难点 善于运用I like …和 I don’t like …举一反三 四、教学过程 1.Learn new words 1、导入新课。 分别做踢足球、打篮球、打乒乓球、做早操的动作，课件分别出示单词及图片，领读。

2、Teach the new words： football\basketball\table tennis\morning exercises. (1)老师出示足球的图片，为问what’s this ?引导同学们说football，老师问，学生说此单词。然后小组读、男生女生读、个人读等进行操练.最后老师将单词写在黑板上，让同学们认读，用同样的方法教basketball ，morning exercises，table tennis. (2)游戏:listen ，stand up and say. (3)游戏:Clap your hands. 出示上、下、左、右各一个单词，然后教师说到哪个，学生就往哪个方面拍掌并说出相应单词。 3、Learn the text. (1)听录音，猜猜熊猫在做什么？ (2)题学生听录音，回答问题。跟读录音，老师带读，全班读，分组读。 4、不只是熊猫喜欢运动，我们也喜欢运动。 (1)教学句型I like…和I don’t like… (1)teach the key sentence: I like ... A 老师再拿起篮球并出示画有一张笑脸的图片边玩边高兴的说:“I like basketball.”并告诉学生如果你也喜欢篮球，可以说：“I like basketball”. B学说新句型“ I like…”. 手举笑脸的图片，让学生们说说自己喜欢的运动。

英语教案-I like football,(第二册)

英语教案:I like football，(第二册) 在英语课教学活动中运用信息技术丰富教学手段 —英语课《Ilikefootball.》教学设计 青岛永安路小学姜硕 一、教材简介 本课是《新标准英语》一年级起点第二册第七模块第一单元。《Ilikefootball.》的第二课时。在第一课时中，学生已经对六种运动的单词进行了学习，本课要解决的主要知识点是“Ilike…”和“Idon’tlike…”两个句型。并通过一系列的课堂活动，创设语境巩固、运用这两个句型。 二、学习者分析 一年级的学生活泼好动，好奇心强，喜欢表现自己，通过一段时间的学习，初步掌握了一些基本的单词。能根据教师的简单指令做游戏、做动作，能唱简单的英文歌曲，说简单的英语歌谣，能在图片、音像的帮助下听懂和读懂简单的小故事。能交流简单的个人信息，表达简单的情感和感觉。喜欢可爱生动的事物，

尤其喜欢观看动画，动画能吸引他们的注意力，激发他们的俄学习兴趣。能模仿老师或录音的发音，对新语言表现出好奇心和兴趣，但是往往掌握的不牢固。需要通过各种教学方式反复的练习、不断的强化才能巩固所学知识。 三、教学目标设计及其对应的课程标准 基础教育阶段英语课程的总目标对学生五个方面的能力做出了要求，分别包括语言技能、语言知识、情感态度、学习策略和文化意识等五方面。 本课的教学目标主要是让学生理解运用“Ilike…”和“Idon’tlike…”两个句型。让学生使用这两句话来简单的表达自己的情感。并通过多种练习方式，来实现上述五方面能力的培养。 1、语言技能 针对一年级的学生来说语言技能包括听、说、读三个方面的技能以及这三种技能的综合运用能力。本课的学习中，学生使用教学软件听、跟读本课的单词、句子，以及在传递足球说句子、涂色、猜动作等游戏中操练知识语言，有效的实现了这一技能的训练。 2、语言知识

《I like football》教案

UNIT1 I like football 教学目的： 1、基础知识： 听、说、认读新单词football 、basketball 、table tennis、cycling、swimming ,掌握句型“I like…….和I don’t like…….”。 2、基本技能： 能够在实际情景中熟练掌握句型“I like…….和I don’t like…….”。 3、思想渗透： 激发学生的学习兴趣，树立自信心，体验成就感，乐于与他人合作，培养集体荣誉感。 教学重、难点： 新单词football 、basketball 、table tennis、cycling、swimming 的认读. 句型“I like…….和I don’t like…….”的运用。 教具准备： 多媒体课件、录音机、录音带、一个篮球，一个足球，乒乓球拍，单词卡片，小贴纸（作为给学生的奖品）。 教学过程： Step 1 Warming up: 1、Greeting the students: Good afteroon , boys and girls. 2、Sing a song . T : Do you like songs? Ss : Yes. T : Let’s sing a “rainbow”song,ok? Ss : Ok. Step 2 Introduce and new sentences presentation. 一边展示课件，一边说。 T : Children , firstly I want to show you a wonderful picture. Look at carefully please. Oh, many lovely clouds. What colour are they? Ss : Green./Red./Pink./Black./Blue./Yellow. T : Oh,many colours. I like red.( 我喜欢红色。) 板书：I like……. （逐词练习句子）What do you like? Ss : I like red/green……. T : I like red, I don’t like black. Introduce “I don’t like…….”(作板书，加强逐词练习) How about you ? Ss : I don’t like green/pink………. Step 3 New words presentation . <1> T : Boys and girls , do you remember the chant “Bounce the ball”? 教师一边做动作一边chant. 学生们与教师一同参与。 教师从讲台后拿出篮球，说：“What’s this?”Ss : 篮球./Basketball. Introduce “basketball”.

I Like Football英语作文

I Like Football英语作文 我喜欢运动更喜欢足球。很多人问我为什么喜欢这个运动。我总是微笑着说因为我喜欢大家一起努力地奔跑。大家一起在球场洒下汗水一起在球场迎接团结胜利的喜悦。大家一起开心地笑是一件美好的事。我们在赛场上结识友谊结伴胜利。我们大声地笑回荡在这片绿地。友谊在这个游戏中渐渐地建立起来。我不会放弃不轻言败。这是我在足球场学到的。大家一起为比赛而准备。胜利一起笑失败了继续努力的样子是团结的美好。最特别的赛场上总会有我们特别的身影和笑声。我们互相搭肩开心地奔跑就如青春的友谊纯真一样。我不会在青春的路上停下我会奔跑。就算受伤了依然要爬起来。我不怕困难。我只怕我只有一个人在奔跑。 I like sports like football。 Many people asked me why I love this sport。 I always smile to say because I love work together to run。 Everyone in the stadium sweat together to greet the unity victory in the course of joy。 Everyone laughing is a good thing。 We meet together victory in the arena of friendship。 We laugh loudly echoed in this piece of green。 Friendship in this game gradually built up。 I will never give up never fail。 This is I learned at the football field。 Everyone together for the game and ready to。 Victory laugh

三年级下册《I like football》优秀教案

三年级下册《I like football》优秀教 案 Teaching content: Module 3 Playground Unit 1 I like football. Teaching objectives: 1. Word and phrases: football, basketball table tennis, morning exercises 2. Sentences: I like … I don’t like … Teaching properties: Tape-recorder, objects Teaching procedures: A. Song: Old MacDonald has a zoo. B. Free talk: 描述一下你家里的宠物或你画的小动物 C. New concepts: 一. 老师对同学们说今年9月县里要举办九运会,学校为了响应九运会的号召,准备在学校举办体育课外小组,今天派我来调查一下你们都想参与什么项目,并统计一下人数.那么让我们来看看都有什么项目吧. 老师出示足球,问What’s this ?引导同学们说football,找会的同学当小老师,教其他同学说此单词.然后老师将单词写在黑板上,让同学们一起书写,用同样的方法教basketball ,morning exercises, table tennis

游戏:老师说出任意三个单词,同学们找出老师没读的那个,也可以同桌的同学玩此游戏. 二. 老师再拿起足球,边玩边高兴的说I like football.然后将足球扔给接受较快的同学,让他模仿说此句型,再由他扔给其他任何一个同学,以此类推来操练此句型.老师将句型写在黑板上并画上笑脸. 老师再在黑板上写上I don’t like …的句型.并画上哭脸,然后拿篮球说 I don’t like basketball .以同样的传球游戏操练此句型. 三．老师让同学们听录音，听听Panpan的喜好变化，听听他最终喜欢什么体育项目，并引导同学们发现Panpan 是一个有困难就退缩的人，鼓励同学们要不怕困难，做事要坚持到底。 再让同学们听音模仿跟读课文，并练习扮演课文。 四．现在我们已经学完了四个体育项目，那么你们帮我来完成学校布置给我的任务吧。布置四个组长拿好他们各自的球，分别到各组调查同学们喜欢的项目，假如该同学喜欢就说 I like …并站到该组长的后面，假如不喜欢就说I don’t like…并等待下一位组长的到来，最后统计一下人数。 Homework: 1. 扮演课文 2. 依照单词表拼读新单词 Module 3 Unit 1

《Ilikefootball》教学实录

《I like football》教学实录 教材分析 本模块主要学习有关运动项目的词汇以及谈论自己喜欢和不喜欢的事物，Unit1中，我们将和Panpan一起了解一些运动项目，并在运动过程中学习喜欢与不喜欢，在语言上，主要以I like …和I don’t like…为重点。 重点 能听说认读本单元重点词句。 掌握重点句型I like …/ I don’t like…。 难点 谈论自己喜欢和不喜欢的运动项目或事物， 教学目标 认知目标 能听说认读本单元重点词汇（重点词汇有：football，basketball，table tennis，morning exercises） 能完整流利的认读本单元的课文容。 掌握重点句型I like …/ I don’t like…。 能力目标 能够谈论自己喜欢和不喜欢的运动项目或事物 情感目标 提升学生对运动的热爱 教师准备 PPT课件，图片卡纸，笑脸、哭脸，游戏用小纸条

学生准备 积极的学习态度 教学设计思路 本单元课文容主要讲述了熊猫Panpan参加了很多运动项目，有football，basketball，table tennis，morning exercises，在活动中体会like和don’t like 的含义。本文容简单，通过设计各种活动，游戏让学生对重点句型反复操练，利用学生好玩、好动的心理，让学生真正参与到课堂中，边玩边把重点句型掌握好。教学设计流程 Step1：Warming up（热身运动） T：Good afternoon，boys and girls 。 Ss: Good afternoon，Ms Yao T：sit down，please。First，it’s TPR time。All of you，stand up and line up。（呈现课件中TPR 视频，教师跟学生一起跟随视频做律动英语） Step2：Lead in（新课导入） T：Great，now go back to your seat。In the zoo，there are so many animals ，look at these pictures。 （课件呈现动物图片） T：What’s this ？ Ss：It’s a lion T：And what are they ？ Ss：They are lions T：And what are they ？ Ss：They are cats

小学一年级英语 Ilikefootball.(第二册)

Ilikefootball.(第二册) 一年级英语教案 在英语课教学活动中运用信息技术丰富教学手段 —英语课《 I like football. 》教学设计 青岛永安路小学 姜硕 一、 教材简介 本课是《新标准英语》一年级起点第二册第七模块第一单元。《 I like football. 》的第二课时。在第一课时中，学生已经对六种运动的单词进行了学习，本课要解决的主要知识点是 “I like …” 和 “I don’t like …”

两个句型。并通过一系列的课堂活动，创设语境巩固、运用这两个句型。●二、 学习者分析 一年级的学生活泼好动，好奇心强，喜欢表现自己，通过一段时间的学习，初步掌握了一些基本的单词。 能根据教师的简单指令做游戏、做动作，能唱简单的英文歌曲，说简单的英语歌谣，能在图片、音像的帮助下听懂和读懂简单的小故事。能交流简单的个人信息，表达简单的情感和感觉。喜欢可爱生动的事物，尤其喜欢观看动画，动画能吸引他们的注意力，激发他们的俄学习兴趣。能模仿老师或录音的发音，对新语言表现出好奇心和兴趣，但是往往掌握的不牢固。需要通过各种教学方式反复的练习、不断的强化才能巩固所学知识。 ●三、 教学目标设计及其对应的课程标准 基础教育阶段英语课程的总目标对学生五个方面的能力做出了要求，分别包括语言技能、语言知识、情感态度、学习策略和文化意识等五方面。 本课的教学目标主要是让学生理解运用

“I like …” 和 “I don’t like …” 两个句型。让学生使用这两句话来简单的表达自己的情感。并通过多种练习方式，来实现上述五方面能力的培养。 1、 语言技能 针对一年级的学生来说语言技能包括听、说、读三个方面的技能以及这三种技能的综合运用能力。本课的学习中，学生使用教学软件听、跟读本课的单词、句子，以及在传递足球说句子、涂色、猜动作等游戏中操练知识语言，有效的实现了这一技能的训练。 2、 语言知识 “I like …” 和 “I don’t like …” 两个句型是本课的主要知识语言，在教学中，教师通过展示动画课件（一只小熊猫在高兴的踢足球时，它说：“ I like football.”

I like football教学设计

《新标准英语》（三年级起始）Book 2 Module 3 Unit 1 I like football . 一、教材分析 本单元的课文内容主要讲述了熊猫Panpan参加的许多运动项目，有football,basketball,table tennis ,morning exercises,在活动中要求学生能够谈论自己喜欢和不喜欢的事物。题材贴近学生生活实际，学生比较感兴趣。 通过学习本课，学生能够掌握运动项目词汇：football , basktball , morning exercises ,table tennis, 语言结构上能用I like ……/ I don’t like ……描述自己喜欢和不喜欢的运动项目以及围绕此句型进行拓展。 二、教学目标 1、知识目标 （1）能听懂会说单词football,basketball,table tennis, morning exercises. （2）能听懂会说并熟练在情景中运用“I like.....”和”I don`t like....”句型谈论自己的喜好。 2、能力目标 培养学生在游戏及情景中熟练运用单词，句子的能力。通过各种游戏激发学生的学习兴趣。

3、情感目标 教育学生热爱体育运动，勤于体育锻炼,在遇到困难的时候不要退缩，要努力去克服。同时在积极参与教学活动中体会到学习的快乐。 三、教学重难点 重点： 1、能听懂、会说、会写单词和短语： football,basketball,table tennis,morning exercises. 2、能熟练的运用I like…..\I don`t like…..句型谈论自 己的喜好。 难点： 1、 morning exercises的读音； 2、将本课的知识运用到真实的情境中。 四、教具准备 Tape recorder、Word cards、Football、basketball、Table tennis、Multimedia PPT. 五、教学过程 Step 1 : warming-up 1.Greetings T ： Class begins ! Good morning ,boys and girls . Ss : Good morning ,Miss Ren . 2. T : What’s your favourite song ? Ss： It’s the ABC song .

I like football.教学设计

Module 3 Unit 1 Book 2 I like football.教案设计 一、教材分析 本课是《新标准英语》三年级起点第二册第三模块第一单元，教学内容以运动项目词为主，要求学生能用“I like …”和“I don’t like …”两个句型谈论简单的运动项目，表达自己的喜好。 二、学情分析 小学英语教学主要激发和培养学生的英语学习兴趣，让学生树立英语学习信心，养成良好的学习习惯。三年级的学生活泼好动，好奇心强，喜欢表现自己，通过一个学期的学习，初步掌握了一些基本的单词，对英语有一定的感知能力。。能根据教师的简单指令做游戏、做动作，能说简单的英语歌谣，能在图片、音像的帮助下听懂和读懂简单的小故事。 三、教学目标 根据《新课标》中的阶段性目标和中年级儿童的年龄特点，我制定了以下的教学目标： 一、教学目标： 1、知识与能力目标： 1）会听说读写单词 like football basketball, table tennis 会认读单词：don’t ，morning exercises 2）会用句子I like···/ I don’t like···描述自己的喜好。 3）能听说理解课文内容。 2、情感目标： 加强体育锻炼，强身健体. 教育学生做事要有恒心，坚持到底才能胜利。 二、教学重难点： 重点：在日常生活中能灵活运用 I like…和I don’t like…来表达自己的喜好。 难点：exercises的发音。 三、突破措施： 1、直观教学法。利用图片、实物、多媒体直观教学，激发学生学英语的兴趣，让学生在情境中学习单词和句型，感受词句的意义，从而理解课文的意思。

外研版三年级下册I like football教案设计

外研版小学英语三年级下册 Module 3 unit1 I like football教案 一、教学目标 （一）知识目标：能听、说、认读单词：like，football , basketball ,table tennis, ,morning exercises。能听懂会说并熟练在情景中运用“I like.....”和“I don`t like....”讨论自己喜欢和不喜欢的运动。 (二）技能目标：培养学生说英语，正确运用英语进行表述的能力。 （三）情感目标：培养学生运用英语表达自己的情感，积极参加课堂交流活动，激发学生学习英语的兴趣，增加学生对自己喜爱的运动的了解。 二、教学重难点 （一)教学重点：能够运用句型“I like...”和“I don’t like...”谈论自己喜欢和不喜欢的运动项目。 (二）教学难点：morning exercises的读音；将本课的所学的单词运用到真实的情境中。 三、教具： （一）教学课件，单词卡片 四、教学程序 （一)热身导入 1、（师生问候） T: Hello, boys and girls. S: Hello, Miss zheng. T: How are you? S: I’m fine , thank you. And how are you? T: I’m fine ,too. Thank you. T：Today,we are going to study English in this beautiful classroom（教师用手指多媒体大教室）,Are you happy? S:Yes! T:Ok!Let’s sing a song and do the action?

Ilikefootball教案

《I like football 》教学设计 一、教学目标 （一）、知识目标 1、单词：football 、basketball、table tennis 、morning exercises 、like 的听、说、读。 2、句型：I like …和I don ' t like …的运用。 （二）、能力目标 善于运用I like …和I don ' t like …举一反三 （三）、情感目标 教育学生做事要有恒心，有毅力。 二、教学重点 1. 单词football 、basketball 、table tennis 、morning exercises 、like 的听、说、读。 2. 句型：I like …和I don ' t like …的运用 三、教学难点 善于运用I like …和I don ' t like …举一反三 四、教学过程 1 、热身（Warm up）。 （1 ）播放上一模块的歌曲Old Madanle has a zoo. 学生跟唱 （2）学习本单元第一部分的韵句 2、导入新课（lead in）。

由动物引出熊猫盼盼，分别做踢足球、打篮球、打乒乓球、做早操的 动作，课件分别出示单词及图片，领读 football\basketball\table tennis\morning exercises. 3、任务呈现( Task presentation )。 (1) 老师出示足球的图片，问what 's this ?引导同学们说football 老师问，学生说此单词。然后小组读、男生女生读、个人读等进行操练. 最后老师将单词写在黑板上，让同学们认读，用同样的方法教basketball ，morning exercises ，table tennis. (2) 游戏:listen ，stand up and say. ⑶游戏:Clap your han ds. 出示上、下、左、右各一个单词，然后 教师说到哪个，学生就往哪个方面拍掌并说出相应单词。 4、课文学习( Learn the text )。 (1) 听录音，猜猜熊猫在做什么？ (2) 学生听录音，回答问题。 I __ football. Ouch! I __________ football. I ___ basketball. Ouch! I _______ basketball. I ___ table tennis.Ouch! I _____ table tennis. I ___ morning exercises. Ouch! I ______ m orning exercises. (3)跟读录音，老师带读，全班读，分组读。 (4)按图片顺序逐个让学生角色扮演朗读，注意配上动作。

I_like_football说课稿

Module 3 Unit 1《I Like Football》 教学内容：小学新标准英语三年级起点，第二册Module 3 Unit 1 《I like football》. 本课结合教材的重难点以及学科的特点，利用多种教学方法，在愉快轻松的气氛中进行教学，从视、听、说等方面使学生得到了语言的训练，提高了学生学习英语的兴趣。 一、说教材 本课围绕I like football. 这个话题，进行了句型I like …….I don’t like …….的学习，使学生通过学习能掌握如何表达自己喜欢的事物和不喜欢的事物，同时学习了football/ basketball/ table tennis/ morning exercises等运动类的英语单词，从而使学生可以准确描述对这些运动的喜好。并通过一系列的课堂活动，创设语境，巩固、运用这两个句型。 二、说目标 《英语课程标准》指出：激发和培养学生学习英语的兴趣，使学生树立自信心，养成良好的学习习惯和形成有效的学习策略，发展学生自主学习的能力和培养学生学习

英语的兴趣是小学英语教学的基本任务。在认真分析 教材的基础上,我针对学生实际，将本课时的教学目标及重,难点确定如下: （一）知识目标 １、学习单词like、football、basketball、table tennis 能口头运用并读懂以上单词。 ２、学习语句I like…／I don’t like…并能够表达自己对运动的喜好。 （二）能力目标： 能够培养学生说英语，正确运用英语进行口语交际的能力。 （三）情感目标 教育学生热爱体育运动，激发学生学习英语的兴趣， 使学生树立学习英语的自信心。 三、说教学重难点 根据《英语新课程标准》的要求以及本课在教材中所处的地位和作用,我确立了这样的教学重难点： (1) 能够正确运用句型“I like…” and “I don’t like…..”来表达自己的想法。 (2) 谈论自己喜欢和不喜欢的运动项目。 (3) 能听说单词 like\ don’t\ football\ basketball\ morning exercises\table tennis 教学难点：能够正确、灵活地运用句型 I like …and I don't like …。 四、说教法 教学方法贯穿整个教学过程的始终，教师只有采取灵活多样的方法，才能调动学生的积极性，才能使知识由浅入深，化难为易。为了顺利完成教学目标，更好地突出重点，