学无止境英文
a r X i v :h e p -t h /0303148v 1 17 M a r 2003Energy-momentum conrvation laws in higher-dimensional Chern–Simons models
G.Sardanashvily
Department of Theoretical Physics,Moscow State University,117234Moscow,Russia E-mail:sard@grav.phys.msu.su
URL:webcenter.ru/∼sardan/
Abstract.Though a Chern–Simons (2k −1)-form is not gauge-invariant and it depends on a background connection,this form en as a Lagrangian of gauge theory on a (2k −1)-dimensional manifold leads to the energy-momentum conrvation law.
The local Chern–Simons (henceforth CS)form en as a Lagrangian of the gauge theory on a 3-dimensional manifold is well known to lead to the local conrvation law of the canonical energy-momentum tensor.Generalizing this result,we show that a global higher-dimensional CS gauge theory admits an energy-momentum conrvation law in spite of the fact that its Lagrangian depends on a background gauge potential and that one can not ignore its gauge non-invariance.The CS gravitation t
heory here is not considered (e.g.,[1,3]).We derive Lagrangian energy-momentum conrvation laws from the first variational formula (e.g.,[5,7,8,9,10]).Let us consider a first order field theory on a fibre bundle Y →X over an n -dimensional ba X .Its Lagrangian is defined as a density L =L d n x on the first order jet manifold J 1Y of ctions of Y →X .Given bundle coordinates (x λ,y i )on a fibre bundle Y →X ,its first and cond order jet manifolds J 1Y and J 2Y are equipped with the adapted coordinates (x λ,y i ,y i µ)and (x λ,y i ,y i µ,y i λµ),respectively.We will u the notation ω=d n x and ωλ=∂λ⌋ω.
baby是什么意思英文Given a Lagrangian L on J 1Y ,the corresponding Euler–Lagrange operator reads
δL =δi L θi ∧ω=(∂i L −d λ∂λi )L θi ∧ω,(1)
where θi =dy i −y i λdx λare contact forms and d λ=∂λ+y i λ
∂i +y i λµ∂µi are the total derivatives,which yield the total differential
d H ϕ=dx λ∧d λϕ(2)
acting on the pull-back of exterior forms on J 1Y onto J 2Y .The kernel Ker δL ⊂J 2Y of the Euler–Lagrange operator (1)defines the Euler–Lagrange equations δi L =0.Awebrvices
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英孚网络课程网址Lagrangian L is said to be variationally trivial ifδL=0.This property holds iffL=h0(ϕ), whereϕis a clod n-form on Y and h0is the horizontal projection
h0(dxλ)=dxλ,h0(dy i)=y iλdxλ,h0(dy iµ)=y iλµdxλ.(3) The relation d H◦h0=h0◦d holds.
Any projectable vectorfield
u=uλ(xµ)∂λ+u i(xµ,y j)∂i(4) on Y→X is the infinitesimal generator of a local one-parameter group of bundle auto-morphisms of Y→X,and vice versa.Its prolongation onto J1Y is
J1u=uλ∂λ+u i∂i+(dλu i−y iµ∂λuµ)∂λi.(5) Then,the Lie derivative of a Lagrangian L along J1u reads
L J1u L=J1u⌋dL+d(J1u⌋L)=[∂λ(uλL)+u i∂i L+(dλu i−y iµ∂λuµ)∂λi L]ω.(6) Thefirst variational formula provides its canonical decomposition
L J1u L=u V⌋δL+d H h0(u⌋H L)=(7)
(u i−y iµuµ)δi Lω−dλ[(uµy iµ−u i)∂λi L−uλL]ω,
where u V=(u⌋θi)∂i,H L is the Poincar´e–Cartan form,and
J u=−h0(u⌋H L)=Jλuωλ=[(uµy iµ−u i)∂λi L−uλL]ωλ(8) is the symmetry current along u.On KerδL,thefirst variational formula(7)leads to the weak equality
L J1u L≈−d H J u,(9)
∂λ(uλL)+u i∂i L+(dλu i−y iµ∂λuµ)∂λi L≈−dλ[(¸uµy iµ−u i)∂λi L−uλL].
Let the Lie derivative(6)reduces to the total differential
L J r u L=d Hσ,(10) e.g.,if L is a variationally trivial Lagrangian or L is invariant under a one-parameter group of bundle automorphisms of Y→X generated by u.Then,the weak equality(9) takes the form
0≈−d H(J u+σ),(11)
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regarded as a conrvation law of the modified symmetry current
of the tangent bundle T P of P with respect to the canonical action of G on P.Given a basis{ǫr}for the right Lie algebra g r of the group G,let{∂λ,e r}be the corresponding fibre bas for the vector bundle T G P.Then,a ctionξof T G P→X reads
ξ=ξλ∂λ+ξr e r.(16) The infinitesimal generator of a one-parameter group of vertical automorphisms is a G-invariant vertical vectorfield on P identified to a ctionξ=ξr e r of the quotient
V G P=V P/G⊂T G P
of the vertical tangent bundle V P of P→X by the canonical action of G on P.The Lie bracket of two ctionsξandηof the vector bundle T G P→X reads
[ξ,η]=(ξµ∂µηλ−ηµ∂µξλ)∂λ+(ξλ∂ληr−ηλ∂λξr+c r pqξpηq)e r,
where c r pq are the structure constants of the Lie algebra g r.Puttingξλ=0andηµ=0, we obtain the Lie bracket
[ξ,η]=c r pqξpηq e r(17) of ctions of the vector bundle V G P→X.A glance at the expression(17)shows that the typicalfibre of V G P→X is the Lie algebra g r.The structure group G acts on g r by the adjoint reprentation.
Note that the connection bundle C(14)is an affine bundle modelled over the vector bundle T∗X⊗V G P,and elements of C are reprented by local V G P-valued1-forms a rµdxµ⊗e r.Bundle automorphisms of a principal bundle P→X generated by the vector field(16)induce bundle automorphisms of the connection bundle C who generator is
ξC=ξλ∂λ+(∂µξr+c r pq a pµξq−a rν∂µξν)∂µr.(18) The connection bundle C→X admits the canonical V G P-valued2-form
1
F=(da rµ∧dxµ+
F rλµdxλ∧dxµ⊗e r,F rλµ=∂λA rµ−∂µA rλ+c r pq A pλA qµ,(20)
2
of F onto X is the strength form of a gauge potential A.
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Let I k(ǫ)=b r
< k ǫr1···ǫr k be a G-invariant polynomial of degree k>1on the Lie
algebra g r written with respect to its basis{ǫr},i.e.,
I k(ǫ)= j kǫr1···c r j pq e p···ǫr k= k c r1pqǫpǫr2···ǫr k=0.
Let us associate to I(ǫ)the2k-form
P2k(F)=b r
< k
F r1∧···∧F r k(21) on C.It is a clod form which is invariant under vertical automorphisms of C.Let A be a ction of C→X.Then,the pull-back
奥运新闻P2k(F A)=A∗P2k(F)(22) of P2k(F)is a clod characteristic form on X.Recall that the de Rham cohomology of C equals that of X since C→X is an affine bundle.It follows that P2k(F)and P2k(F A) posss the same cohomology classtypography
[P2k(F)]=[P2k(F A)](23) for any principal connection A.Thus,I k(ǫ)→[P2k(F A)]∈H∗(X)is the familiar Weil homomorphism.
Let B be afixed ction of the connection bundle C→X.Given the characteristic form P2k(F B)(22)on X,let the same symbol stand for its pull-back onto C.By virtue of the equality(23),the difference P2k(F)−P2k(F B)is an exact form on C.Moreover, similarly to the well-known transgression formula on a principal bundle P,one can obtain the following transgression formula on C:
P2k(F)−P2k(F B)=d S2k−1(B),(24)
S2k−1(B)=k
1eyquem
P2k(t,B)dt,(25)
P2k(t,B)=b rget loo
< k (a r1µ
1
−B r1µ
1
)dxµ1∧F r2(t,B)∧···∧F r k(t,B),
F r j(t,B)=[d(ta r jµ
j +(1−t)B r jµ
j
)∧dxµj+
1
on X.For instance,if P2k(F A)is the characteristic Chern2k-form,then S2k−1(A,B)is the familiar CS(2k−1)-form.Therefore,we agree to call S2k−1(B)(25)the CS form on the connection bundle C.In particular,one can choo the local ction B=0.Then, S2k−1=S2k−1(0)is the local CS form.Let S2k−1(A)denote its pull-back onto X by means of a ction A of C→X.Then,the CS form S2k−1(B)admits the decomposition
S2k−1(B)=S2k−1−S2k−1(B)+dK2k−1(B).(26)
Let J1C be thefirst order jet manifold of the connection bundle C→X equipped with the adapted coordinates(xλ,a rµ,a rλµ).Let us consider the pull-back of the CS form (25)onto J1C denoted by the same symbol S2k−1(B),and let
S2k−1(B)=h0S2k−1(B)(27)
be its horizontal projection.This is given by the formula
S2k−1(B)=k
1
P2k(t,B)dt,
P2k(t,B)=b r
< k (a r1µ
1
−B r1µ
1
)dxµ1∧F r2(t,B)∧···∧F r k(t,B),
F r j(t,B)=
2020高考英语卷子1
2c r j pq(ta pλ
j
+(1−t)B pλ
j
)(ta qµ
stockings worldj
+(1−t)B qµ
j
]dxλj∧dxµj⊗e r.
The decomposition(26)induces the decomposition
S2k−1(B)=S2k−1−S2k−1(B)+d H K2k−1(B),K2k−1(B)=h0K2k−1(B).(28) Now,let us consider the CS gauge model on a(2k−1)-dimensional ba manifold X who Lagrangian
L CS=S2k−1(B)(29) is the CS form(27)on J1C.Let
ξC=(∂µξr+c r pq a pµξq)∂µr(30) be a vertical vectorfield on C.We have shown that
L J1ξ
C
S2k−1(B)=d Hσ(31)
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