Table Of ContentDi(cid:27)erential gorms, di(cid:27)erential worms
4
0
0
2 Denis Ko
han, Pavol ’evera
n
a February 1, 2008
J
5
1
Abstra
t
]
G Westudy(cid:16)higher-dimensional(cid:17) generalizationsofdi(cid:27)erentialforms. Justasdi(cid:27)Ce∞re(nMtia)l
forms
anbede(cid:28)nedastheuniversal
ommutativedi(cid:27)erentialalgebra
ontaining ,
D
we
an de(cid:28)ne di(cid:27)erential gorms as the universal
ommutative bidi(cid:27)erential algebra
on-
. C∞(M)
h taining . From a mRor0e|1
→onM
eptual point of viewD,idffi(cid:27)(eRr0e|n1t)ial forms are fun
tions
t
a on the superspa
e of maps and the a
tion of on forms is equivalent
m todeRRh0a|m2 →di(cid:27)MerentialandtodegreesofformDs.iffG(oRrm0|s2)arefun
tions onthesuperspa
eof
[ maps and we study the a
tion of on gnorms; it
ontains more than
just degrees and di(cid:27)erentials. By repla
ing 2 with arbitrary , we get di(cid:27)erential worms.
2
v DiffW(Re0a|1ls)o studyDiaffg(eRn0e|rna)lizationn≥of2homologi
al algebra that one obtains by repla
ing
with for , and the
losely related question of forms (and
3
0 gorms and worms) on somegeneralized spa
es (
ontravariantfun
tors and sta
ks) and of
3 approximations of su
h (cid:16)spa
es(cid:17) in terms of worms.
7 Clearly, this is not a gormless paper.
0
3
0
/
h
t
a
m
:
v
i
X
r
a
Figure 1: A worm on a manifold
Contents
1 Introdu
tion 2
2 Di(cid:27)erential gorms as a universal bidi(cid:27)erential algebra 3
3 Appetizer: di(cid:27)erential forms 4
ΠTM
3.1 as the spa
e of odd
urves .D.iff. (.R.0|.1). . . . . . . . . . . . . . . . . . . . 4
3.2 De Rham di(cid:27)erential and a
tion of . . . . . . . . . . . . . . . . . . . 5
3.3 Cartan formula and its generalizations . . . . . . . . . . . . . . . . . . . . . . . 5
1
ΠTM
3.4 Ve
tor (cid:28)elds on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.5 Integration and Stokes formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Di(cid:27)erential gorms as fun
tions onDthiffe(sRp0a|2
)e of odd surfa
es 7
4.1 More than di(cid:27)erentials: a
tion of . . . . . . . . . . . . . . . . . . . . 8
4.2 Cartan gormulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 Cohomology of gorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.4 Integration of gorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Diff(R0|2)
5 Di(cid:27)erential gorms as a representation of 9
Mat(n) Vect Diffop
5.1 Irredu
ible representations of and of the
ategories and . . 10
Ω (M)
[2]
5.2 De
omposition of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Mat(2)
5.2.1 De
omposition to irredu
ibles . . . . . . . . . . . . . . . . . . . 10
T∗M
5.2.2 Generi
part: the bundles ~l (
otangent tetris) . . . . . . . . . . . . . 11
T⊞∗M f
5.2.3 An example: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.2.4 De
ompositionfof the generi
part . . . . . . . . . . . . . . . . . . . . . 12
5.2.5 The non-generi
part and di(cid:27)erential forms . . . . . . . . . . . . . . . . 12
5.2.6 The entire de
omposition . . . . . . . . . . . . . . .C.∞.(R. 0.|2.). . . .D.iff.(R103|2)
5.3 Derivations on gorms as a module of the
rossed produ
t of with 13
6 Integration and Euler
hara
teristi
14
7 Beyond homologi
al algebra 14
n = 1
7.1 Example: (the
ase of homologi
al algebra) . . . . . . . . . . . . . . . . . 15
7.2 Representability of fun
tors and their approximations . . . . . . . . . . . . . . . 16
7.3 Examples of approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.4 Approximations and representability of sta
ks . . . . . . . . . . . . . . . . . . 18
7.4.1 Groupoids over a
ategory and Hilsum-Skandalis morphisms . . . . . . . 19
7.4.2 The 2-
ategory of Lie groupoids and the sta
ks they represent . . . . . 20
7.4.3 Generalized Lie groupoids and the sta
ks they represent . . . . . . . . . 20
7.5 Examples of sta
ks and of generalized Lie groupoids . . . . . . . . . . . . . . . 21
7.5.1 (cid:16)Categori(cid:28)ed de Rham
omplex(cid:17) . . . . . . . . . . . . . . . . . . . . . . 21
7.5.2 Weil and Cartan models of equivariant
ohomology . . . . . . . . . . . 22
7.5.3 Quasi-Poisson groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1 Introdu
tion
M
There is a well known idea of regarding di(cid:27)erential forms on a manifold as fun
tions on
ΠTM
a supermanifold, namely on the odd tangent bundle . The de Rham di(cid:27)erential then
ΠTM
be
omes a ve
tor (cid:28)eld on .
ΠTMThis paper is based on the followRin0g|1r→emMarkable fa
t: one
an des
ribe the supermaniMfold
as the superspa
e of all maps Diff((i.Re.0|a1s) the superspa
e of all odd
urves in ).
More importantly, the a
tion of the group then gives rise to de Rham di(cid:27)erential
and to degrees of di(cid:27)erential forms. This point of view
an be found (more or less expli
itly)
at many pla
es in physi
s literature, but we took it expli
itly from [Kon℄.
We use this idea for two purposRes0.|nF→irstMly, it is natural tno make a generalizatiDoniffa(nRd0|nto)
study the superspa
es of all maps for arbitrary , and the a
tion of
2
on these superspa
es. To explain the title, the fun
tions on these map spa
es will be
alled
n = 2 n
di(cid:27)erential gorms in the
ase of , or generally di(cid:27)erential worms for arbitrary . Se
-
ondly, the idea is straightforwardly applied to some generalizations of manifolds, namely to
ontravariant fun
tors from the
ategory of manifolds, or more generally to sta
ks. A
on-
F
travariant fun
tor from the
ategory of manifolds is usually understood as a (cid:16)generalized
F(M) M F
spa
e(cid:17), su
h that is thFe(Rse0t|1o)f maps from to that spa
e. Di(cid:27)erFen(tRia0l|2f)orms on
should thus be fun
tions on (and di(cid:27)erential gorms fun
tions on , et
.). As
an example, we get a very simple interpretation of equivariant de Rham theory. The problem
of wormDsiffof(Rge0|n1e)ralized spa
es isD
ilffos(eRly0|nr)elated with a gneneralization of homologi
al algebra,
where is repla
ed by for arbitrary .
Here is the plan of the paper: In Se
tion 2 we shall des
ribe di(cid:27)erential gorms without use
of supermanifolds, as the universal
ommutative bidi(cid:27)erential algebra
ontaining the algebra
C∞(M)
. This pDoiinfft(oRf0v|2i)ew is not
ompletely satisfa
tory, as it doesn't reveal the a
tion of
the supergroup (only the sub-supergroup of a(cid:30)ne transformations
an be seen).
Then, as an appetiser, in Se
tion 3 we identify di(cid:27)erential forms with fun
tions on the spa
e
of all odd
urves and derive the basi
properties of di(cid:27)erential forms from this fa
t. In Se
tion
4 we stRa0r|t2t→o inMvestigate di(cid:27)erential gorms (and worms) as fun
tions on the suDpieffrs(pRa0
|e2)of all
maps . In Se
tion 5 we de
ompose gorms as a representation of and
in Se
tion 6 we prove a theorem
onne
ting the Euler
hara
teristi
with the integral of any
losed integrable gorm. In the (cid:28)nal, and possibly the most interesting Se
tion 7, we look at
generalized spa
es -
ontravariant fun
tors and sta
ks.
Finally we should add that the ideas used here are very simple; mu
h of the length of the
paper is
aused by our attempt to write expli
it
oordinate expressions.
Remarks on notation
Generally we denote even
oordinates on supermanifolds by latin letters and odd
oordinates
by greek letters. To avoid
onfusion with di(cid:27)erential forms, we denote Berezin integral with
x ξ f(x,ξ)dxdξ
respe
t to (say) and as .
R
A
knowledgement
We would like to thank to Marián Fe
ko for many dis
ussions, to Peter Pre²najder for use-
ful
omments on an early version of this paper, and to James Stashe(cid:27) for suggesting many
improvements after the (cid:28)rst version was posted in the arXives.
2 Di(cid:27)erential gorms as a universal bidi(cid:27)erential algebra
Ω(M)
One
an des
ribe the algebra of di(cid:27)erential forms as the universal graded-
ommutative
C∞(M)
di(cid:27)erential graded algebra
ontaining the algebra . That is, there is an algebra homo-
C∞(M) → Ω0(M) A
morphism , and if is any graded-
ommutative di(cid:27)erential graded algebra
C∞(M) → A0
with a homomorphism , there is a unique homomorphism of di(cid:27)erential graded
Ω(M) → A
algebras making the triangle
ommutative.
Ω (M)
[2]
In the same spirit, we
an de(cid:28)ne the algebra of di(cid:27)erential gorms as the universal
C∞(M)
graded-
ommutative bidi(cid:27)erential algebra
ontaining (by a bidi(cid:27)erentialZa2lgebra we
mean a bi
omplex with a
ompatible stru
ture of algebra; in parti
ular, it is -graded).
3
C∞(M) → Ω0,0(M) A
That is, there is an algebra homomorphism [2] , and if is any graded-
C∞(M) → A0,0
ommutative bidi(cid:27)erential algebra with a homomorphism , there is a unique
Ω (M) → A
[2]
homomorphism ofbidi(cid:27)erential algebras making thetriangle
ommutative. Just
Ω(M) Ω0,0(M) ∼= C∞(M)
as in the
ase of it turns out that [2] .
xi M
To make this abstra
t de(cid:28)nition down-to-earth, let us
hoose lo
al
oordinates on .
Ω (M)
[2]
The algebra is freely generated by the algebra of fun
tions, and by the elements
d xi d xi d d xi (1,0) (0,1) (1,1) d a = 1,2
1 2 1 2 a
, and (of bidΩegr(eMes) ,f and respde
ftiv=ely)∂,fwdhxerie ( )
are the two di(cid:27)erentials in [2] . If is a fun
tion then a ∂xi a and therefore
d d f = ∂2f d xid xj+ ∂f d d xi x˜i
1 2 ∂xi∂xj 1 2 ∂xi 1 2 . If is another system of lo
al
oordinates, we get from
here ∂x˜i
d x˜i = d xj
a ∂xj a
∂2x˜i ∂x˜i
d d x˜i = d xjd xk + d d xj.
1 2 ∂xj∂xk 1 2 ∂xj 1 2
Di(cid:27)erential gorms are not tensor (cid:28)elds, sin
e their transformation law involves se
ond deriva-
tives; they belong to 2nd order geometry.
n Ω (M)
[n]
Finally, for arbitrary we
an de(cid:28)ne the algebra of di(cid:27)erential worms of level n, ,
n C∞(M)
astheuniversalgraded-
ommutative -di(cid:27)erentialalgebra
ontaining . One
aneasily
n
ompute transformation laws for worms of any level; they
ontain -th derivatives at most.
Ω (M) Mat(n) C∞(M)
[n]
On we have an obvious a
tion of the semigroup : it leaves inta
t,
n d
a
and linearly transforms the di(cid:27)erentials . It turns out that the di(cid:27)erentials and the a
tion
Mat(n) Ω (M)
[n]
of are just the tip of an i
eberg: on we have an a
tion of the supersemi-
R0|n → R0|n Mat(n) R0|n
group of all maps (
orresponds to linear transformations of and the
di(cid:27)erentials totranslations). To seethiswillrequire anewpoint ofview ondi(cid:27)erential worms.
3 Appetizer: di(cid:27)erential forms
ΠTM
3.1 as the spa
e of odd
urves
ΠTM MR0|1 R0|1 → M
We denote by the supermanifold of all maps . It is
hara
terized by
Y Y → ΠTM
Rth0e|1×foYllow→inMg property: for any supermanΠifTold , a map is the sYam7→e aRs0|a1×mYap
(inotherwordsΠ,TthMefun
tor istherightadθjointofthefun
tor R0|1 x)i.
It is easy to understaMnd iRn0l|o1
→al M
oordinates. If is tYhe
oordinateRo0n|1×Y a→ndM
are lo
al
oordinates on , a map parametrized by , i.e. a map ,
is given by fun
tions
xi(θ,η) = xi(η)+θξi(η),
η Y θ
where denotes lo
al
oordinates on (we just used Taylor expansion in ). Su
h a map
Y xi
is therefore the same as a map from to a supermanifold with even
oordinates and odd
ξi ΠTM x˜i
oordinates . To (cid:28)nd globally, suppose is another system of lo
al
oordinates on
M
; then ∂x˜i
x˜i(θ)= x˜i(x(θ)) = x˜i(x+θξ)= x˜i(x)+θ ξj,
∂xj
ΠTM
i.e. the transition fun
tions on are
∂x˜i
x˜i = x˜i(x), ξ˜i = ξj.
∂xj
4
ΠTM M
Now we
an see that we
an identify fun
tions on with di(cid:27)erential forms on , by
ξi dxi
identifying with (we also see that our notation is
orre
t, i.e. that the supermanifold is
M
indeed the odd tangent bundle of ).
Diff(R0|1)
3.2 De Rham di(cid:27)erential and a
tion of
ΠTM R0|1 → M
SDiinff
e(R0|1) is the supermanifold of allDmiffap(Rs 0|1) , we have an a
tion of the supergroup
oΠnTiRt.0|1Let us (cid:28)rst deRs
0r|i1be→ R0|1 . It is an open sub-supergroupR0o|f1t→heRsu0p|1er-
semigroup θ′ = aθof+aβll maps . A (possibly parametrized) Rm1a|p1 is
of the forma , hen
e the sβupeDrisffem(Rig0r|o1)up is di(cid:27)eomaor6=ph0i
to , with one even
oordinate and one odΠdT
oRo0r|d1inate ; Diffis(Rg0iv|1e)n by ΠTM. x′ + θξ′ =
The right a
tion of (hen
e also of ) on is given by
x+(aθ+β)ξ
, i.e.
x′ = x+βξ, ξ′ = aξ.
The ve
tor (cid:28)elds generating this a
tion are
∂ ∂
E = ξi , d = ξi .
∂ξi ∂xi
E ΠTM M
When a
ts on a fun
tion on , i.e. on a di(cid:27)erential form on , it multiplies it by
d
its degree; on tΩh(eMo)ther hand, a
ts as the de Rham di(cid:27)erenDtiiaffl.(RT0h|1e)
anoni
al stru
ture of
a
omplex on is therefore equivalent to the a
tion of . The fa
t that it is a
omplex follows from the
ommutation relations
[E,d] = d, [d,d] = 0
Diff(R0|1)
in the Lie algebra of .
3.3 Cartan formula and its generalizations
L = di + i d
v v v
Now we will try to understand Cartan's formula from this point of view.
YX X → Y
It will be
onvenient to treat a general map spa
e of all maps between two
YX
(Xsu=perR)m0|nanifolds; we shall however pretend that is (cid:28)nite-dimensional (sin
e it is so for
) to avoidDainffa(lyXti)
⋉al(pDroiffbl(eYm)s).XAs wYeXshall see, CartanX's=forRm0|u1la
omes from the
a
tion of the group on (in the
ase of ).
ev : X ×YX → Y YX → YX
Let X = R0|1 be Ythe=evMaluation maYpX(ad=joiΠnTt Mto theevid:en(θti,txy,ξm)a7→p x + θξ. );
in the
ase of and (so that ), We
YX ev∗TY
1
an naturally identify ve
tor (cid:28)elds on with se
tions of the ve
tor bundle , and
X×YX YX f
therefore we
an multiply them with arbitrary fun
tions on (not just on ). If is
X X×YX w YX
a fun
tion on (and therefore also on ) and a ve
tor (cid:28)eld on , we shall denote
f ·w
their produ
t as .
u X v Y
If is a ve
tor (cid:28)eld on and a ve
tor (cid:28)eld on , we shall denote their natural lifts to
YX u♭ v♯ Diff(X) Diff(Y)
as and ; these arethe ve
tor (cid:28)eldsgenerating the left a
tions of and
YX YX
on . The followDinigfff(oXrm)⋉ula(sDeiffxp(Yre)s)sXthe fa
t that on we have a (left) a
tion of the
semidire
t produ
t :
[u♭,u♭]= [u ,u ]♭
1 2 1 2
YX YX →YX
1
toseethis,realizethatave
tore(cid:28)vel:dXon×YXis→anYin(cid:28)nitesimaldefoermv∗aTtiYonoftheidentitymap ,
i.e. anin(cid:28)nitesimal deformation of , i.e a se
tion of
5
[u♭,f ·v♯]= (uf)·v♯
[f ·v♯,f ·v♯]= (−1)|f2||v1|f f ·[v ,v ]♯;
1 1 2 2 1 2 1 2
we also have
f ·u♭ = (fu)♭.
X = R0|1 Y = M (∂ )♭ = −d (θ∂ )♭ = −E
θ θ
In the
ase of and we have , (the minus signs
d E θ ·∂ = −∂ θ ·∂ = 0
xi ξi ξi
are here sin
e and generate the right a
tion), (and therefore ).
v M v♯ = L θ · v♯ = −i
v v
Moreover, for any ve
tor (cid:28)eld on we have and . The equation
[(∂ )♭,θ·v♯] =v♯ di +i d= L
θ v v v
is Cartan's .
ΠTM
3.4 Ve
tor (cid:28)elds on
YX C∞(X)
As we have seen, the spa
e of ve
tor (cid:28)elds on is a module over , and it is also a
Diff(X) φ ∈ Diff(X) f ∈ C∞(X) w YX
representation of . If , and is a ve
tor (cid:28)eld on then
learly
φ·(f ·w) = (f ◦φ−1)·(φ·w);
YX
in other words, the spa
e of ve
tor (cid:28)elds on is a module over the
rossed produ
t of
C∞(X) Diff(X) u X
with . In(cid:28)nitesimally, if is a ve
tor (cid:28)eld on ,
[u♭,f ·w] = (uf)·w+(−1)|f||u|f ·[u♭,w].
X = R0|1 Y = M YX = ΠTM YX
Now let us return to the
ase of , , . Ve
tor (cid:28)elds on
Ω(M) θ·(θ·w) = 0 θ2 = 0
are then derivations of the algebra . Noti
e that (sin
e ) and that
[(∂ )♭,[(∂ )♭,w]] = 0 [(∂ )♭,(∂ )♭] = 0 θ (∂ )♭ (∂ )♭
θ θ θ θ θ θ
(sin
e ), i.e. both and a
t as di(cid:27)erentials;
θ
in
reases degree by 1, while de
reases it by 1. Finally,
w = [(∂ )♭,θ·w]+θ·[(∂ )♭,w],
θ θ
w w = w +w w = [(∂ )♭,θ·w] w = θ·[(∂ )♭,w]
1 2 1 θ 2 θ
i.e.any
anbeuniquely de
omposedas (by , )
[(∂ )♭,w ]= 0 θ·w = 0
θ 1 2
so that and .
w ΠTM p θ ·w = 0
In
oordinates, a ve
tor (cid:28)eld on , of degree and su
h that , is of the
form
w = Ak ξi1ξi2...ξip+1∂ ,
i1i2...ip+1 ξk
TM ⊗ p+1T∗M
i.e. it is a se
tion of .
V Der(Ω(M))
Let us summarize what we have found. The graded Lie algebra de
omposes
to a dire
t sum of graded ve
tor spa
es
Der(Ω(M)) = ker(θ)⊕ker(∂ )
θ
(they turn out to be subalgebras). The two subspa
es are naturally isomorphi
; the two
θ ker(∂ ) → ker(θ) ∂
θ θ
mutually inverse isomorphisms are the a
tion of ( ) and the a
tion of
ker(θ) → ker(∂ )
θ
( ). Moreover, they are both isomorphi
with the spa
e of ve
tor (cid:28)elds with
ker(∂ ) Ω(M)
θ
values in di(cid:27)erential forms. The Lie bra
ket on (the derivations of
ommuting
with the di(cid:27)erential) is the Fröli
her-Nijenhuis bra
ket.
6
3.5 Integration and Stokes formula
ΠTM
Thefa
tthat isamapspa
edoesn'tseemtoshedmu
hlightonintegrationofdi(cid:27)erential
ΠTM
forms, i.e. of fun
tions on . For this reason we just repeat the simple standard fa
ts:
dxdξ ΠTM xi
the volume measure on is independent of the
hoi
e of
oordinate system (up
d
to
hoi
e of orientation), and it is also invariant with respe
t to the ve
tor (cid:28)eld . For that
dα = 0 α χ
Ω
reason for any form with
ompa
t support. To get Stokes theorem, let be the
R Ω 0 = d(χ α) = (dχ )α+ χ dα =
Ω Ω Ω
hara
teristi
fun
tion of a
ompa
t domain ; then
− α+ dα R R R
∂Ω Ω .
R R R0|1 → M
It is
ertainly an interesting thing that on the spa
e of all maps there is a
natural volume measure (e.g. fromRth0|enpoint nof≥vie2w of quantum (cid:28)eld theoDriyff).(RA0s|nw)e will see,
the situation gets even better for with ; the measure will be -invariant
d E
(here it was -invariant, but not -invariant).
4 Di(cid:27)erential gorms as fun
tions on the spa
e of odd surfa
es
Everything we('ΠllTb)e2Mdoing here will be fairly analogous to thRe0p|2re→vioMus se
tioθn1, sθo2we
an
be brief. Let R0|2 dexniote the supermanifold of aMll maps . Let ,R0|2b→e tMhe
oordinates on and be lo
al
oordinates on . A (parametrized) map
θ
expanded to Taylor series in 's looks as
xi(θ1,θ2) = xi+θ1ξi +θ2ξi +θ2θ1yi,
1 2
(ΠT)2M xi yi ξi ξi
1 2
hastherefore lo
al
oordinates , (even
oordinates) and , (odd
oordinates).
x˜i M
If is another system of lo
al
oordinates on then (expanding to Taylor series)
x˜i(x(θ1,θ2)) = x˜i(x+θ1ξ +θ2ξ +θ2θ1y)=
1 2
∂x˜i ∂x˜i ∂x˜i ∂2x˜i
= x˜i(x)+θ1 ξj +θ2 ξj +θ2θ1 yj + ξjξk ,
∂xj 1 ∂xj 2 (cid:18)∂xj ∂xj∂xk 1 2(cid:19)
(ΠT)2M
i.e. the transition fun
tions on are
∂x˜i ∂x˜i ∂2x˜i
x˜i = x˜i(x), ξ˜i = ξj, y˜i = yj + ξjξk.
a ∂xj a ∂xj ∂xj∂xk 1 2
M
From this we
an see that we
an identify di(cid:27)erential gorms on (as de(cid:28)ned in se
tion 2)
(ΠT)2M y d xi ξi d d xi
a a 1 2
with fun
tions on polynomial in 's, by identifying with and with
yi
. We arrived to this identi(cid:28)
ation by a
omputation, but there is a simpler reason using
(ΠT)2M y
di(cid:27)erentials (see below). General fun
tions on (noet−n(de1
de2sxs)2adrilxydpoxlynomial in 's)
1 2
will be
alled pseudodi(cid:27)erential gorms (these are things like ).
(ΠT)2M TM
Let us noti
e that the body of the supermanifold is naturally isomorphi
to ;
ξi
a
indeed, we get the body by setting all the odd
oordinates to zero, and then the transition
x y TM
fun
tions for 's and 's be(
ΠoTm)entMho=seMofR0|n . n
Finally, let us des
ribe for higher 's and dire
tly identify its fun
tions
n M xi M
withdi(cid:27)erentialwormsoflevel on . If 'sarelo
al
oordinateson thenthe
oordinates
(ΠT)nM xi d xi (1 ≤ a ≤ n) d d xi (1 ≤ a < b ≤ n) d d ...d xi
a a b 1 2 n
on are , , , ... , (i.e. apply
d (1 ≤ a ≤ n) xi
a
the di(cid:27)erentials toR0|n's→inMall possible ways). These
oordinates are identi(cid:28)ed
with Taylor
oe(cid:30)
ients of a map by
xi(θ1,θ2,...,θn)= eθadaxi.
7
−(∂θa)♭ da n = 2
From this expression it is
lear that is equal to . For example, in the
ase of
−(∂θ1)♭ = ξ1i∂xi +yi∂ξi −(∂θ2)♭ = ξ2i∂xi −yi∂ξi −(∂θa)♭ = ξai∂xi +ǫabyi∂ξi
we have 2, 1, i.e. b.
Diff(R0|2)
4.1 More than di(cid:27)erentials: a
tion of
−(∂θa)♭
We have alreadRy0f|o2und(tΠhTe)v2eM
tor (cid:28)elds : they generate the a
tion of the group of
translations of on u♭ , and they are equual toR0t|h2e two di(cid:27)erentials on di(cid:27)erential
gormus. We
an easily (cid:28)nd for any ve
torR(cid:28)0e|l2d on : either we do it dire
tly, regard-
ing as an in(cid:28)nitesimal transformation on and (cid:28)nding the
orresponding in(cid:28)nitesimal
(ΠT)2M
transformation of from the formula
x′+θaξ′ +θ2θ1y′ = x+θ′aξ +θ′2θ′1y,
a a
(1)
(∂θa)♭
or use the known expression for and the identities
θa·∂ = −∂ , θ2θ1·∂ = ∂ .
xi ξi xi yi
a
The result is
da = −(∂θa)♭ = ξai∂xi +ǫabyi∂ξi
b
♭
Eab = − θb∂θa = ξai∂ξi +δbayi∂yi
(cid:16) (cid:17) b
Ra = −(θ2θ1∂θa)♭ = ξai∂yi.
diff(R0|2) (ΠT)2M
These ve
tor (cid:28)elds give us a right a
tion of the Lie algebra on (that is
u♭
the reason for the minus signs: give the left a
tion), i.e. a left a
tion on the algebra of
E1 E2 Eb
1 2 a
di(cid:27)erential gorms. and are the two degrees on di(cid:27)erential gorms, generate the
gl(2) R
a
a
tion of on gorms, but 's are something new. Similar formulas
an be easily found
for worms of arbitrary level. R0|2
(ΠT)2R0|2 = R0|2
If we want to know the right a
tion of the supersemigroup on
(ΠT)2M (cid:0) (cid:1)
(not just the in(cid:28)nRite0|s2im→alRa0
|2tion we have just derived), we
an easily (cid:28)nd it from
(1). A (parametrized) map is of the form
θ′1 = β1+a1θ1+a1θ2+γ1θ2θ1
1 2
θ′2 = β2+a2θ1+a2θ2+γ2θ2θ1,
1 2
(ΠT)2R0|2 R4|4 aa βa γa
i.e. is di(cid:27)eomorphi
to (with even
oordinates b and odd
oordinates , );
Mat(2) 2×2 Mat(2)
we also see that its body is the semigroup of -matri
es. The a
tion of
β γ
(i.e. when we set 's and 's to zero) is given by
x′ = x, ξ′ = abξ , y′ = det(A)y,
a a b
A aa
where is the matrix b. If it ever be
omes useful, the full result of (1) is
x′ = x+βaξ +β2β1y, ξ′ = abξ +ǫ βbacy, y′ = (det(A)+ǫ βbγc)y+γaξ .
a a a b bc a bc a
8
4.2 Cartan gormulas
L = di +i d v♯ = [(∂ )♭,θ·v♯]
v v v θ
Analogues of the Cartan formula (i.e. ) for worms of arbitrary
θ2θ1·∂ =
xi
levelwerealreadyfoundinse
tion3.3. Herewementionjustonespe
ial
ase: sin
e
∂ θ2θ1·v♯ =vi∂ v = vi∂ M θ2θ1·v♯ i
yi yi xi v
, we have for any ve
tor (cid:28)eld on ; let us denote as .
[d ,[d ,i ]] = L
1 2 v v
Then .
4.3 Cohomology of gorms
d
a
The
ohomologyofdi(cid:27)erentialgormswithrespe
ttoany isnaturallyisomorphi
todeRham
M (ΠT)2M ΠT(ΠTM)
ohomology of . To see this we just write as , i.e. the
ohomology of
ΠTM ΠTM M
di(cid:27)erential gorms is de Rham
ohomology of , and
an be
ontra
ted to .
R (R )2 = 0
1 1
We
analso
ompute
ohomologywithrespe
ttosay ,sin
e . This
ohomology
Ω(M) Ω (M) → Ω(M) d 7→ d d 7→ 0
[2] 1 2
isisomorphi
to . In fa
t, the proje
tion given by , , isa
Ω(M) Ω (M) R
[2] 1
quasiisomorphism, when istakenwithzerodi(cid:27)erentialand withdi(cid:27)erential .
[R ,d ]= −E1 E1
The reason is simple: 1 2 1, i.e. 1 a
ts trivially on
ohomology, i.e. the semigroup
x7→ x d x7→ λd x d x7→ d x d d x7→ λd d x
1 1 2 2 1 2 1 2
a
tion , , , is trivial on
ohomology, and setting
λ = 0
gives us the result. Noti
e that we
ould use the same argument to show that the
d Ω(M) d
2
ohomology with respe
t to isΠjTusMtth⊂e
(oΠhTomMo)l2ogy of with respe
t to .RIn0|2ge→omRet0r|i1
terms, we have an embedding
oming from the proje
tion ,
(θ1,θ2) 7→ θ2 (θ1,θ2) 7→ (λθ1,θ2)
; this proje
tion
an be obtained by the semigroup a
tion ,
λ = 0 θ1∂ θ1∂ = [θ2θ1∂ ,∂ ]
θ1 θ1 θ1 θ2
setting . The a
tion is generated by and .
4.4 Integration of gorms
Like di(cid:27)erential forms, gorms (and worms)
an be integrated. However we need pseudodif-
d d x
1 2
ferentialgorms(dependingnon-polynomiallyon 's)fortheintegraltobe(cid:28)nite. In
oordi-
dx1dξ1dξ1dy1...dxmdξmdξmdym
nates,theintegralisjusttheBerezinintegralwithvolumemeasure 1 2 1 2 ,
m M ξ
where is the dimension of . In other words, to integrate a gorm, expand it in 's, take
ξ1ξ1ξ2ξ2...ξmξm y x
1 2 1 2 1 2
tMhe=
oRe(cid:30)
ient in front of , and integrate it over 's and 's. For example, if
,
e−x2−(d1d2x)2d xd x= π.
1 2
Z
dxdξ dξ dy Diff(M)
1 2
The integral is independentDiofff(tRh0e|2
)hoi
e of
oordinates, i.e. is α-
invariant, and in fa
tuit isRa0ls|2o (-uin♭αva)r=ian0t. It means that for any integrable gorm
and any ve
tor (cid:28)eld on we have (cid:21) a form of Stokes theorem.
R n ≥ 2
(ΠTA)nsMimilar
laiDmiffis(Rtr0u|ne)for wDoriffm(sMw)ith any ; the
oordinate Berezin integral on
is both and -invariant.
Diff (R0|2)
5 Di(cid:27)erential gorms as a representation of
Ω (M) Diff(R0|2)
[2]
Ouraiminthisse
tionistode
ompose toinde
omposablerepresentationsof .
Ω(M)
Itshouldbe
omparedwiththedeRham
ohomologyof ; thelatteris
onne
tedwiththe
dα = β Ω(M) Ω (M)
[2]
problem "solve the equation in ", while the de
omposition of des
ribes
Ω (M) d Eb R
[2] a a a
solutions of all linear equations in that use the operators , and (i.e. the
Diff(R0|2) Ω (M)
[2]
a
tion of ) on .
9
Diff(R0|n) n ≥ 3
The representation theory of for was shown to be wild by N. Shomron
[Sho℄ (i.e. it is impossible to
lassify all (cid:28)nite-dimensional inde
omposable representations of
Ω n ≥ 3
[n]
this supergroup). The de
omposition of for remains an open problem for us.
Mat(n) Vect
5.1 IDrirffeodpu
ible representations of and of the
ategories and
Mat(n)
Let us re
all that irredu
ible representations of the semigroup , or more invariantly,
End(V) V n
of the semigroup , where is an -dimensional ve
tor spa
e, are
lassi(cid:28)ed by highest
n l ≥ l ≥ ··· ≥ l ≥ 0 n
1 2 n
weights, that is by -tuples of integers . Su
h an -tuple
an be
(5,2,1)
represented by a Young table; for example, the triple is represented by
~l = (l ,l ,...,l )
1 2 n
The representation with the highest weight
an be found in the de
om-
V⊗N N = l +l +···+l
1 2 n
position of the representation to irredu
ibles, where . Namely, let
W S
~l be the irredu
ible representation of the symmetri
group N,
orresponding to the Young
~l S V⊗N End(V)
N
table . a
ts also on~l , by permutations. The irredu
ible representation of
with the highest weight is then
V = Hom (W ,V⊗N).
~l SN ~l
Vect
In fa
t, this formula gives us a representation of the
ategory of (cid:28)nite-dimensional ve
tor
Vect → Vect V 7→ V
spa
es, i.e. a fun
tor , ~l.
M T∗M
For any manifold we
an
onsider the spa
e of se
tions of the ve
tor bundle ~l ;
M → M
it is a right representation of the s~elmigroup of dalilmsMmooth maps . This bundle is
non-zero i(cid:27) the numb~ler of rows of is at most , anTd∗Mthe=repNresTe∗nMtation is kNnown to
be irredu
ible unless has only one
olumn; in that
ase, ~l , where is the
N M V
length of the
olumn, and the spa
e of di(cid:27)erential -forms on has the invariant subspa
e
N
of all
losed -forms. Generally,
omplete redu
ibility doesn't hold for the representations of
these semigroups; we shall meet many examples soon.
M 7→ Γ(T∗M)
Let is noti
e that the
onstru
tion ~l is a
ontravariant fun
tor from the
Diff Γ(T∗)
ategory ofsmoothmanifoldstothe
ategoryofve
torspa
es. Thefun
tor ~l
analso
X
beappliedtosupermanifolds. Letushowever noti
ethatif isasupermanifold, thea
tionof
S (T∗)⊗NX a⊗b 7→ (−1)|a||b|b⊗a
N
on isTm∗oRd0i|(cid:28)ned by the si~gln rule: the transpnosition a
ts by .
This implies that ~l is zero i(cid:27) has more than
olumns. For example, there is no
R0|1 (cid:3)(cid:3) k k
Riemann metri
on (the Young table is ), but there are non-zero -forms for any .
Ω Diff
[2]
is also a right representation of the
ategory (i.e. a
ontravariant fun
tor from
R0|2
Diff R0|2
),anditisalsoarepresentationofthesemigroup . Ouraimwillbetode
ompose
(cid:0) (cid:1)
it to inde
omposable parts.
Ω (M)
[2]
5.2 De
omposition of
Mat(2)
5.2.1 De
omposition to irredu
ibles
R0|2
Ω (M) Mat(2) ⊂ R0|2
[2]
In this preliminary se
tion we de
ompose as a representation of .
(l ,l ) l −l (cid:0)≤ dim(cid:1) M
1 2 1 2
It
ontains only the representations with highest weights su
h that ;
10
