Morita Equivalence of Noncommutative Supertori

0.22 MB·English

by  Ee Chang-Young

KIAS-P09044
Morita Equivalence of Noncommutative Supertori
Ee Chang-Young1
Department of Physics, Sejong University, Seoul 143-747, Korea
Hoil Kim2
Department of Mathematics, Kyungpook National University
9
Taegu 702-701, Korea
0
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2 and

v
Hiroaki Nakajima3
o
N
Department of Physics and Institute of Basic Science

1 Sungkyunkwan University, Suwon 440-746, Korea

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h
t
-
p ABSTRACT
e
h
[
In this paper we study the extension of Morita equivalence of noncommutative tori to the

3
v supersymmetric case. The structure of the symmetry group yielding Morita equivalence
3
appears to be intact but its parameter field becomes supersymmetrized having both body
1
6 and soul parts. Our result is mainly in the two dimensional case in which noncommutative
5
. supertori have been constructed recently: The group SO(2,2,V0), where V0 denotes Grass-
9 Z Z
0 mann even number whose body part belongs to Z, yields Morita equivalent noncommutative
9
0 supertori in two dimensions.
:
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i
X PACS: 02.40.Gh 11.30.Pb
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[email protected]
[email protected]
[email protected]
1 Introduction
In the work of Connes, Douglas, and Schwarz [1] Morita equivalence of two dimensional
noncommutative tori resulting from toroidal compactifications of M(atrix) theory was men-
tioned in relation with T-duality in string theory [2, 3]. Then it was proved by Rieffel and
Schwarz [4] that the actions of the group SO(n,n,Z) on an antisymmetric n n matrix θ
×
which represents noncommutativity parameters for an n-dimensional noncommutative torus
yield Morita equivalent tori. Then Schwarz [5] showed that compactifications on Morita
equivalent tori are physically equivalent, corresponding to T-duality in string theory.
Noncommutative geometry [6] naturally appears in string theory: Low-energy effective
theory of D-branes in a background NSNS B-field becomes the noncommutative field the-
ory where the spacetime coordinates xµ are noncommutative, [xµ,xν] = 0 [1, 7, 8]. If
6
we turn on the background RR field, the low-energy effective theory of D-branes becomes
the field theory on non(anti)commutative superspace of which the fermionic coordinate θα
has nontrivial commutation relation θα,θβ = 0 [9, 10, 11, 12, 13]. Gauge theories on
{ } 6
non(anti)commutative superspace are studied extensively [14, 15, 16, 17]. Toroidal com-
pactification in string theory with the above mentioned background field then naturally
leads to noncommutative supertorus. Although the noncommutative torus has been a very
well known subject for quite a long time, its supersymmetric version, the noncommutative
supertorus, has been constructed only recently in the two dimensional case [18].
Recently, Berkovits and Maldacena [19] showed that tree level superstring theories on
certainsupersymmetric backgroundsarerelatedbytheshift symmetry ofacertainfermionic
coordinate,which theycalled“fermionic”T-duality. Thisisverysimilar tobosonicT-duality
in the sense that it is a symmetry under a shift of a coordinate. Bosonic T-duality is related
with torus compactification. This naturally leads us to an expectation that the fermionic
T-duality might be related to supertorus compactification. On the other hand, the fermionic
T-duality relates different RR field backgrounds [19]. Furthermore, the background RR field
leads to a non(anti)commutative superspace [12]. This makes us think that the fermionic
T-duality might be related with the compactification on noncommutative supertorus rather
than on commutative supertorus. In order to further investigate this idea, we have to
understand Morita equivalence of noncommutative supertori first, since in the bosonic case
it has been shown that T-duality corresponds to Morita equivalence of noncommutative tori
[5]. This is the motivation behind our present work.
Commutative supertorus was constructed by Rabin and Freund [20] based on the work
of Crane and Rabin [21] on super Riemann surfaces. The supertorus was obtained as a
quotient of superplane by a subgroup of superconformal group Osp(1 2) which acts properly
|
2
discontinuously on the plane together with the metrizable condition. These two conditions
boil down to proper latticing of the superplane, and can be expressed as appropriate trans-
lation properties along the cycles of the torus. Noncommutative tori can be constructed by
embedding a lattice [22, 23, 24, 25]into Heisenberg group [26, 27, 28]. The lattice embedding
determines howthegeneratorsofnoncommutative torus, which correspondtothetranslation
operators along the cycles of commutative torus, would act on the module of noncommuta-
tive torus. Heisenberg group can be regarded as a central extension of commutative space,
which is equivalent to a deformation of space by constant noncommutativity. Super Heisen-
berg groupcan be regarded as a central extension of ordinarysuperspace, which is equivalent
to a deformation of superspace by constant noncommutativity and nonanticommutativity.
Based on the construction of super Heisenberg group [29] as a central extension of or-
dinary superspace, the embedding maps for noncommutative supertori in two dimensions
were obtained in [18]. Guided by the classical translation properties of commutative su-
pertorus along its cycles, noncommutative supertorus was constructed in [18] by expressing
the translations along the cycles of supertorus in operator language, and by implementing
the spin structures of supertorus under even(bosonic) and odd(fermionic) translations with
appropriate representations of spin angular momentum operator. Based on the above con-
structionofnoncommutative supertori, herewestudyMoritaequivalence ofnoncommutative
supertori by investigating symmetry actions that give dual embedding maps, i.e., finding the
endomorphisms of the module that yield Morita equivalent tori.
This paper is organized as follows. In section 2, we review Morita equivalence in rela-
tion with the group SO(n,n,Z) in the bosonic noncommutative n-tori case. In section 3,
we recall the construction of noncommutative supertori. In section 4, we first consider the
Morita equivalence of noncommutative supertori in a general setting and identify symmetry
operators yielding Morita equivalent tori. When restricted to the two dimensional case, this
yields the symmetry group SO(2,2,V0) where V0 denotes Grassmann even number whose
Z Z
body part belongs to Z. We conclude in section 5.
2 Morita equivalence in the bosonic case
In this section we review Morita equivalence of noncommutative tori in the bosonic case in
line with the work of Rieffel and Schwarz [4]. In general, Morita equivalence relates two
algebras and ˆ in such a way that for every -module R there exists ˆ-module Rˆ where
A A A A
the correspondence R Rˆ is an equivalence of categories of -modules and ˆ-modules.
→ A A
More specifically, (strong) Morita equivalence of C∗-algebra can be defined as follows. If we
3
consider a finite projective right module over a C∗-algebra A, then the algebra End of
A
E E
endomorphisms of has a canonical structure as a C∗-algebra. We say that a C∗-algebra A′
E
is (strongly) Morita equivalent to A if it is isomorphic to End for some finite projective
A
E
module .
E
An n-dimensional noncommutative torus is an associative algebra with involution having
unitary generators U ,...,U obeying the relations
1 n
U U = e2πiθijU U , i,j = 1,...,n, (1)
i j j i
where (θ ) form a real n n anti-symmetric matrix Θ. One can consider noncommutative
ij
×
torus as a C∗-algebra An, the universal C∗-algebra generated by n unitary operators U
θ { j}
obeying the above relations. One can also consider a smooth version of An consisting of
θ
formal series
n = a Ui1 Uin a = (a ) (Zn) , (2)
Aθ { i1···in 1 ··· n | i1···in ∈ S }
where (Zn) is Schwartz spXace of sequences with rapid decay.
S
Every projective module over a smooth algebra n can be represented by a direct sum of
Aθ
modulesoftheform (Rp Zq F)[22],thelinearspaceofSchwartz functionsonRp Zq F,
S × × × ×
where 2p+q = n and F is a finite abelian group. Let D be a lattice in = M Mˆ, where
G ×
M = Rp Zq F and Mˆ is its dual. The embedding map Φ under which D is the image of
× ×
Zn determines a projective module E on which the algebra of noncommutative torus acts.
In the Heisenberg representation the operators U’s are defined by
U f(r) = e2πi<r,sˆ>f(r+m), m,r M, sˆ Mˆ, f E, (3)
(m,sˆ)
∈ ∈ ∈
where < r,sˆ > is a usual inner product between M and Mˆ. Here we would like to note
that the vector (m,sˆ) can be mapped into an element of Heisenberg group which we explain
below. In this representation, we can easily get the antisymmetric matrix Θ characterizing
noncommutative torus as follows.
U U f(r) = e2πi(<r,sˆ+tˆ>+<m,tˆ>)f(r+m+n),
(m,sˆ) (n,tˆ)
= e2πi<m,tˆ>U f(r), (4)
(m+n,sˆ+tˆ))
where m,n,r M, sˆ,tˆ Mˆ, f E, and thus
∈ ∈ ∈
U U = e2πi(<m,tˆ>−<n,sˆ>)U U . (5)
(m,sˆ) (n,tˆ) (n,tˆ) (m,sˆ)
Therefore, if we denote ~e := (m,sˆ), ~e := (n,tˆ), then we can express θ as follows.
i j ij
0 I
θ =~e J ~e , where J = . (6)
ij i 0 j 0
· I 0!
−
4
Notice that with the embedding map Φ, which is given by Φ = (~e ,~e , ,~e ), the above
1 2 n
···
relation can be expressed in terms of Θ as
0 I
Θ = ΦtJ Φ, where J = . (7)
0 0
I 0!
−
When M in (3) is given by M = Rp, Heisenberg group Heis(R2p,ψ) is defined as follows.
For t,t′ U(1), and (u,v),(u′,v′) R2p, we define the product for (t,u,v),(t′,u′,v′)
∈ ∈ ∈
Heis(R2p,ψ),
(t,u,v) (t′,u′,v′) = (t+t′ +ψ(u,v;u′,v′),u+u′,v +v′), (8)
·
where ψ : R2p R2p R, satisfies the cocycle condition
× −→
ψ(u,v;u′,v′)ψ(u+u′,v+v′;u′′,v′′) = ψ(u,v;u′+u′′,v′+v′′)ψ(u′,v′;u′′,v′′), (9)
which is a necessary and sufficient condition for the multiplication to be associative. There
is an exact sequence
0 R i Heis(R2p,ψ) j R2p 0, (10)
→ → → →
which is called a central extension, with the inclusion i(t) = (t,0,0) and the projection
j(t,x,y) = (x,y), where i(R) is the center in Heis(R2p,ψ). The previously appeared vector
(m,sˆ) in (3) corresponds to a vector (u,v) R2p in the above description of Heisenberg
∈
group.
In a canonical form, we can define the embedding map as
Θ 0
Φ = := (x ), (11)
i,j
0 I!
b
where Θ = diag(θ , ,θ ), i,j = 1, ,2p. The Heisenberg representation is given by
1 p
··· ···
p
b
(U f)(s , ,s ) := (U f)(~s) = exp(2πi s x )f(~s+~x ), for j = 1, ,2p, (12)
j 1 ··· p ~ej k k+p,j j ···
k=1
X
where ~e = (x ,...,x ), ~s = (s , ,s ), and ~x = (x , ,x ). Here, U ’s satisfy
j 1,j 2p,j 1 p j 1,j p,j j
··· ···
U U = e2πiθjU U , otherwise U U = U U . (13)
j j+p j+p j j k k j
Applying the relation (7), one can also easily check the above relation:
t
Θ 0 0 I Θ 0 0 Θ
Θ = = . (14)
0 I! I 0! 0 I! Θ 0!
− −
b b b
5 b
In the general embedding case, one can also use the Manin’s representation [24] in which
(12) becomes
p p
1
(U f)(~s) := exp(2πi s x + x x )f(~s+~x ), for j = 1, ,2p. (15)
~ej k k+p,j 2 k,j k+p,j j ···
k=1 k=1
X X
This corresponds to the second representation of Heisenberg group in Ref. [18].
In order to consider the endomorphisms of An, we first consider the group O(n,n,R)
θ
on the space of of real antisymmetric n n matrices. The group O(n,n,R) can be
n
T ×
considered as a group of linear transformations of the space R2n preserving the quadratic
form x x +x x + +x x . It is convenient to consider coordinates in R2n as two n-
1 n+1 2 n+2 n 2n
···
dimensional vectors (a1,...,an,b ,...,b ) so that the quadratic form on R2n can be written
1 n
as aib . Thus we write the elements of O(n,n,R) in 2 2 block form
i
×
A B
g = . (16)
C D!
The blocks A,B,C,D are n n matrices satisfying AtC +CtA = BtD +DtB = 0, AtD +
×
CtB = 1, where t denotes transpose. The action of O(n,n,R) on the space is defined by
n
T
the formula
Θ′ = gΘ := (AΘ+B)(CΘ+D)−1. (17)
Note that the above action is defined only on the subset of where CΘ+D is invertible.
n
T
Now, let
0 = Θ : gΘ is defined for all g SO(n,n,Z) . (18)
Tn { ∈ Tn ∈ }
Then the following theorem holds [4].
Theorem. For Θ 0 and g SO(n,n,Z) the noncommutative torus corresponding to
∈ Tn ∈
gΘ is Morita equivalent to the noncommutative torus corresponding to Θ.
This remains true for the smooth version of noncommutative tori.
Nowwebrieflyreviewtheproofoftheabovetheorem. Fortheproofoftheabovetheorem,
we first consider a suitable set of generators for SO(n,n,Z). In [4], it was shown as a lemma
that the three elements, ρ(R),ν(N), and σ , which we explain below generate the group
2
SO(n,n,Z).
1) For every matrix R GL(n,Z), ρ(R) SO(n,n,Z) defines the following transformations
∈ ∈
a′i = Riaj, b′ = (R−1)jb . (19)
j i i j
The action of ρ(R) on x = (a,b) with a,b Rn can be simply expressed as
∈
Rt 0 a
ρ(R)x = . (20)
0 R−1! b!
6
2) For an antisymmetric n n matrix N whose entries are n Z the transformation
ij
× ∈
ν(N) SO(n,n,Z) is defined as follows.
∈
a′i = ai +nijb , b′ = b . (21)
j i i
3) For every integer k an element σ O(n,n,Z) defines the following transformations
k
∈
a′i = b for 1 i k, a′i = ai for k < i n,
i
≤ ≤ ≤
b′ = ai for 1 i k, b′ = b for k < i n. (22)
i ≤ ≤ i i ≤
Note that σ O(n,n,Z) only when k is even, since no element of is invertible when
k n
∈ T
n is odd. We refer the proof of the above lemma to Ref. [4], and proceed to the proof of
the theorem. The proof of the theorem was done by showing that each action of the above
generators for SO(n,n,Z) yields Morita equivalent torus.
For g = ρ(R), the noncommutative torus determined by Θ′ = RΘRt is Morita equivalent
to the torus corresponding to Θ. This is because the two tori are isomorphic as we see
below. For a general embedding, one can express an embedding vector ~x in terms of basis
vectors ~e as ~x = nx~e where x Z, and ~e ’s satisfy the relation (6). Then θ for two
{ i} 1 i i i ∈ i xy
embedding vectors ~x,~y is given by
P
n n
θ = (x~e ) J (y ~e ) = x θ y . (23)
xy i i 0 j j i ij j
·
i,j=1 i,j=1
X X
Namely, we get
U U = exp(2πixtΘy)U U , (24)
x y y x
where x = (x ,...,x ) and y = (y ,...,y ) belong to Zn. Under the action of ρ(R), the
1 n 1 n
transformed x is given by x′ = ρ(R)x = Rtx, and thus
Ux′Uy′ = exp(2πi(Rtx)tΘ(Rty))Uy′Ux′ = exp(2πixtRΘRty)Uy′Ux′. (25)
Therefore, the two tori with Θ and Θ′ = RΘRt are isomorphic.
For ν(N) SO(n,n,Z), Θ′ = ν(N)Θ is given by replacing θ with θ′ = θ + n for
∈ ij ij ij ij
n Z. This does not change the commutation relations among U , therefore An and An
ij ∈ x θ′ θ
correspond to the same noncommutative torus.
To prove that An with Θ′ = σ Θ is Morita equivalent to An, we first prove it for any σ
θ′ 2 θ 2p
with a positive integer p obeying 2p n. For this we consider Θ in 2 2 block form
n
≤ ∈ T ×
whose top left part is a 2p 2p matrix denoted by θ ,
11
×
θ θ
11 12
Θ := . (26)
θ21 θ22!
7
Then
θ−1 θ−1θ
Θ′ = σ Θ = 11 − 11 12 , (27)
2p θ21θ1−11 θ22 −θ21θ1−11θ12!
and the noncommutative tori corresponding to Θ′ and Θ are Morita equivalent.
To prove this we first choose an invertible matrix T such that Tt J T = θ where
11 11 0 11 − 11
0 I
p
J = .
0
Ip 0!
−
Then set T = θt , and T be any q q matrix such that θ = Tt T where q := n 2p.
31 12 32 × 22 32− 32 −
Now, set a (n+q) n matrix T as
×
T 0
11
T := 0 I , (28)
 q 
T T
31 32
 
 
and a (n+q) (n+q) square matrix J as
×
J 0 0
0
J := 0 0 I , (29)
 q
0 I 0
q
 − 
 
then it can be shown that TtJT = Θ. Note that a linear transformation T carries Z2p Zq
− ×
into R2p Zq Rq. Thus T satisfies the conditions in [22] for being an embedding map. This
× ×
means that we can view T as a homomorphism from Zn into = R2p Zq Tq, then the
G × ×
ˆ
range of T is a lattice in , which we denoted by D. From the relation = M M, here
G G ×
M = Rp Zq. As we saw before, carries the Heisenberg cocycle β defined by
× G
β((m,sˆ),(n,tˆ)) = exp(2πi < m,tˆ>), (30)
ˆ
where <,> denotes the pairing between M and M. The corresponding skew cocycle ρ is
defined by
ρ((m,sˆ),(n,tˆ)) = exp(2πi(< m,tˆ> < n,sˆ>)). (31)
−
Notethat ρ(x,y) = exp(2πixtJy) for the J defined above. Since TtJT = Θ, the restriction
−
of ρ to D and so to Zn is exactly (x,y) exp( 2πixtΘy). Thus the algebra A = (Zn,β∗),
7→ − S
the space of Schwartz functions on Zn with convolution twisted by the cocycle β∗, is isomor-
phic to the noncommutative torus An and has a natural right action on the space (M).
θ S
Now we like to know the endomorphism algebra of this right A-module, which is Morita
equivalent to A via this module. In [22], this endomorphism algebra is given by (D⊥,β)
S
where
D⊥ = w : ρ(w,z) = 1 for all z D , (32)
{ ∈ G ∈ }
8
and β is restricted to D⊥.
Our next step is to determine D⊥ as the image of Zn under some embedding map, such
that from this embedding map we can get Θ′ which on Zn gives the cocycle corresponding
to the restriction of ρ to D⊥. Any given x will be in D⊥, exactly if xtJTz Z for all
∈ G ∈
z Zn, i.e., exactly if TtJx Zn. For a natural isomorphism from Zn to this D⊥, we define
∈ ∈
an invertible (n+q) (n+q) matrix
×
T 0 0
11
T¯ := 0 I 0 . (33)
 q 
T T I
31 32 q
 
It can be checked that TtJx Zn exactly if T¯tJx Zn+q. Since T¯,J are invertible, the
∈ ∈
following holds viewed in .
G
D⊥ = (T¯tJ)−1(Zn+q). (34)
Since the inverse of (T¯tJ) is given by
J (Tt )−1 0 J (Tt )−1Tt
− 0 11 0 11 31
(T¯tJ)−1 = 0 0 I , (35)
 q 
−
0 I Tt
 q − 32 
 
we get
(T¯tJ)−1(0 Zq 0) = 0 0 Zq, (36)
× × × ×
which is 0 in = R2p Zq Tq. Thus omitting the second column of (T¯tJ)−1 and up to an
G × ×
inessential sign, we get the following desired embedding map
J (Tt )−1 J (Tt )−1Tt
0 11 − 0 11 31
S = 0 I , (37)
 q 
0 Tt
 32 
 
which gives an isomorphism from Zn onto D⊥. Now one can easily show that
StJS = σ Θ. (38)
2p
So far we have shown that the noncommutative torus corresponding to σ Θ is Morita
2p
equivalent to that of Θ. However, in the above proof we have a freedom of decomposing
the matrix Θ. Namely we have a freedom to choose the size of the component θ in (26),
11
which is given by a 2p 2p matrix, for any p within the range of 2 2p n. This means
× ≤ ≤
that the noncommutative tori with M = Rp Zq of different p’s can have the same Θ in
×
this set-up. Thus the noncommutative tori with Θ′ = σ2pΘ and Θ′′ = σ2p′Θ are Morita
equivalent. Therefore, the noncommutative tori corresponding to σ Θ and σ Θ are Morita
2p 2
equivalent. This proves the last part of the theorem.
9
3 Noncommutative supertori
Inthissectionwereviewtheconstructionofnoncommutativesupertoriintwodimensions[18]
and comment on its possible extension to higher dimensions. In order to construct noncom-
mutative supertori, we first consider the ambient noncommutative superspace. In the case of
= (1,1) ambient superspace, we found that the only deformation which preserves a part
N
of supersymmetry is just giving the noncommutativity for bosonic coordinates whereas the
fermionic coordinates are anticommuting each other [18]. So we will not consider this case.
When we consider the case of = (2,2) superspace spanned by (X1,X2,θ±,θ¯±)1, there are
N
a few ways to give the non(anti)commutativity in this superspace. Here we consider the case
of Q-deformation, where the Moyal product is given by the supercharge Q
±
∗
i ←−∂−−−−∂→ C
= exp Θǫµν ←Q−−Q→+←Q−−Q→ . (39)
∗ 2 ∂Xµ∂Xν − 2 + − − +
" #
(cid:16) (cid:17)
Here the supercharges Q , Q¯ are defined by
± ±
∂ θ¯+ ∂ ∂ ∂ θ¯− ∂ ∂
Q = i Q = +i
+ ∂θ+ − 2 ∂X1 − ∂X2 − ∂θ− − 2 ∂X1 ∂X2
(cid:18) (cid:19) (cid:18) (cid:19)
∂ θ+ ∂ ∂ ∂ θ− ∂ ∂
Q¯ = i Q¯ = +i . (40)
+ ∂θ¯+ − 2 ∂X1 − ∂X2 − ∂θ¯− − 2 ∂X1 ∂X2
(cid:18) (cid:19) (cid:18) (cid:19)
We note that there is another non(anti)commutative deformation, called D-deformation.
But the noncommutative supertorus with D-deformation does not have the structure of
super Heisenberg group [18]. Now we move to the operator formalism. In Q-deformation,
the nontrivial commutation relations among supercoordinates are
i 1 1
[X1,X2] = iΘ Cθ¯+θ¯−, [X1,θ+] = Cθ¯−, [X1,θ−] = Cθ¯+,
− 2 2 2
i i
[X2,θ+] = Cθ¯−, [X2,θ−] = Cθ¯+, [θ+,θ−] = C, (41)
2 −2
We note that the fermionic coordinates θ¯+,θ¯− appear as the central elements. This algebra
can be represented on the module f(s,η,θ¯+,θ¯−) as
∂ 1 ∂ 1
X1f(s,η,θ¯+,θ¯−) = iΘ + √Cθ¯+ + √Cθ¯−η f(s,η,θ¯+,θ¯−),
∂s 2 ∂η 2
(cid:18) (cid:19)
i ∂ i
X2f(s,η,θ¯+,θ¯−) = s √Cθ¯+ + √Cθ¯−η f(s,η,θ¯+,θ¯−),
− 2 ∂η 2
(cid:18) (cid:19)
∂
θ+f(s,η,θ¯+,θ¯−) = √C f(s,η,θ¯+,θ¯−),
∂η
θ−f(s,η,θ¯+,θ¯−) = √Cηf(s,η,θ¯+,θ¯−). (42)
1the convention of complex conjugation is (θ+)∗ =θ−, (θ¯+)∗ =θ¯−.
10