Proc. Estonian Acad. Sci. Phys. Math., 2006, 55, 1, 3–15
Generalization of superconnection in noncommutative geometry Viktor Abramov Institute of Pure Mathematics, University of Tartu, J. Liivi 2, 51004 Tartu, Estonia;
[email protected] Received 21 November 2005, in revised form 22 December 2005 Abstract. We propose the notion of a ZN -connection, where N ≥ 2, which can be viewed as a generalization of the notion of a Z2 -connection or superconnection. We use the algebraic approach to the theory of connections to give the definition of a ZN -connection and to explore its structure. It is well known that one of the basic structures of the algebraic approach to the theory of connections is a graded differential algebra with differential d satisfying d2 = 0. In order to construct a ZN -generalization of a superconnection for any N > 2, we make use of a ZN -graded q -differential algebra, where q is a primitive N th root of unity, with N -differential d satisfying dN = 0. The concept of a graded q -differential algebra arises naturally within the framework of noncommutative geometry and the use of this algebra in our construction involves the appearance of q -deformed structures such as graded q -commutator, graded q -Leibniz rule, and q -binomial coefficients. Particularly, if N = 2, q = −1, then the notion of a ZN -connection coincides with the notion of a superconnection. We define the curvature of a ZN -connection and prove that it satisfies the Bianchi identity. Key words: superconnection, covariant derivative, graded differential algebra, graded q -differential algebra.
1. INTRODUCTION The concept of a superconnection was proposed by Mathai and Quillen [1 ] (see also [2 ]) in the 1980s to represent the Thom class of a vector bundle by a differential form having a Gaussian shape. Later, Atiyah and Jeffrey [3 ] proposed the geometric approach to a topological quantum field theory on a four-dimensional manifold [4 ] based on the superconnection formalism. Assuming that a vector bundle π : E → M has a Z2 -graded structure, i.e. it is a superbundle, the total grading of an E-valued differential form can be defined as the sum of two 3
gradings, one of which comes from the Z2 -graded structure of the algebra of differential forms on a base manifold M and the other from a Z2 -graded structure of a superbundle E. A superconnection is a linear mapping of odd degree with respect to this total grading, behaving like a graded differentiation with respect to the multiplication by differential forms. Consequently, if we wish to generalize the notion of a superconnection to any integer N > 2, we must have a ZN -graded analogue of an algebra of differential forms, and assuming that a vector bundle has also a ZN -graded structure, we can elaborate a generalization of a superconnection following the scheme proposed by Mathai and Quillen. In the present paper we introduce the notion of a ZN -connection, where N is any integer satisfying N ≥ 2, within the framework of an algebraic approach to the theory of connections. The first component of our construction is a ZN -graded q-differential algebra [5−8 ], where q is a primitive N th root of unity, denoted by B. This algebra plays the role of an analogue of an algebra of differential forms. It should be mentioned that a differential d of B satisfies dN = 0. The second component is a ZN -graded left module E over the subalgebra A ⊂ B of the elements of grading zero of B. From a geometric point of view, a module E can be considered as an analogue of the space of sections of a ZN -graded vector bundle. Taking the tensor product EB = B ⊗A E, which can be viewed as an analogue of a space of ZN -graded vector bundle valued differential forms, and defining the ZN -graded structure on this product, we give the definition of a ZN -connection D in the spirit of Mathai and Quillen. We show that the N th power of a ZN -connection is the grading zero endomorphism of the left B-module EB , and we define the curvature FD of a ZN connection by FD = DN . It is proved that the curvature of a ZN -connection satisfies the Bianchi identity.
2. GRADED q-DIFFERENTIAL ALGEBRAS In this section we describe a generalization of a graded differential algebra, which naturally arises in the framework of q-deformed structures. This generalization is called a graded q-differential algebra, where q is a primitive N th root of unity. We show that given a graded unital associative algebra over C with element v satisfying v N = e, where e is the identity element of this algebra, one can construct the graded q-differential algebra by means of a q-commutator. Let B = ⊕k∈Z B k be an associative unital Z-graded algebra over C. We shall denote the identity element of B by e and the grading of a homogeneous element ω ∈ B by |ω|, i.e. if ω ∈ Bk , then |ω| = k. An algebra B is said to be a graded q-differential algebra ([5,6 ]), where q is a primitive N th root of unity (N ≥ 2), if it is endowed with a linear mapping d : B k → B k+1 of degree 1 satisfying the graded q-Leibniz rule d(ω ω 0 ) = d(ω) ω 0 + q |ω| ω d(ω 0 ), where ω, ω 0 ∈ B, and dN (ω) = 0 for any ω ∈ B. A mapping d is called an N -differential of a graded q-differential algebra. It is easy to see that a graded q-differential algebra is a generalization of 4
the notion of a graded differential algebra, since a graded differential algebra is a particular case of a graded q-differential algebra if N = 2 and q = −1. From the graded structure of an algebra B it follows that the subspace B 0 ⊂ B of elements of grading zero is the subalgebra of an algebra B. The pair (B, d) is said to be an N -differential calculus on a unital associative algebra A if B is a graded q-differential algebra with N -differential d and A = B 0 . For any k ∈ Z the subspace B k of elements of grading k has the structure of a bimodule over the subalgebra B 0 and a graded q-differential algebra can be viewed as an N -differential complex ([6 ]) d k−1 d k d k+1 d ... → B →B →B →. . . ,
with differential d satisfying the graded q-Leibniz rule. If B is a Z-graded q-differential algebra, then we can define the ZN -graded structure on an algebra B by putting B p¯ = ⊕i∈Z B N i+p , where p = 0, 1, 2, . . . , N − 1, and p¯ is the residue class of an integer p modulo N . Then B = ⊕p∈ZN B p . In what follows, if a graded structure of an algebra B is concerned, we shall always mean the above-described ZN -graded structure of B. Since all graded structures considered in this paper are ZN -graded structures, we always assume that the values of each index related to a graded structure are elements of ZN . If there is no confusion, we shall denote the values of indices by 0, 1, 2, . . . , N − 1 meaning the residue classes modulo N . Let us now show that if a graded unital associative algebra contains an element v satisfying v N = e, where e is the identity element of this algebra, then one equips this algebra with the N -differential satisfying the graded q-Leibniz rule, turning this algebra into a graded q-differential algebra. Let A be an associative unital ZN -graded algebra over the complex numbers C and Ak ⊂ A be the subspace of homogeneous elements of a grading k. Given a complex number q 6= 1, one defines a q-commutator of two homogeneous elements w, w0 ∈ A by the formula 0
[w, w0 ]q = ww0 − q |w||w | w0 w. Using the associativity of an algebra A and the property |ww0 | = |w| + |w0 | of its graded structure, it is easy to show that for any homogeneous elements w, w0 , w00 ∈ A it holds that 0
[w, w0 w00 ]q = [w, w0 ]q w00 + q |w||w | w0 [w, w00 ]q .
(1)
Given an element v of grading 1, i.e. v ∈ A1 , one can define the mapping dv : Ak → Ak+1 by the formula dv w = [v, w]q , w ∈ Ak . It follows from the property of q-commutator (1) that dv is the linear mapping of degree 1 satisfying the graded q-Leibniz rule dv (ww0 ) = dv (w)w0 + q |w| wdv (w0 ), where w, w0 are homogeneous elements of A. Let [k]q = 1 + q + q 2 + . . . + q k−1 and [k]q ! = [1]q [2]q . . . [k]q . 5
Lemma 1. For any integer k ≥ 2 the kth power of the mapping dv can be written as follows: k X (k) dkv w = pi v k−i wv i , i=0
where w is a homogeneous element of A and (k) pi
i |w|i
= (−1) q
[k]q ! = (−1)i q |w|i [i]q ![k − i]q !
|w|i = i|w| +
k i
,
q
i(i − 1) . 2
The proof of this lemma is based on the following identities: (k)
(k+1)
p0 = p0 (k+1)
pi
= 1,
(k)
= pi
(k+1)
(k)
pk+1 = −q |w|+k pk , (k)
− q |w|+k pi−1 , 1 ≤ i ≤ k.
Theorem 1. If N is an integer such that N ≥ 2, q is a primitive N th root of unity, A is a ZN -graded algebra containing an element v satisfying v N = e, where e is the identity element of an algebra A, then A equipped with the linear mapping dv = [v, ]q is a graded q-differential algebra with N -differential dv , i.e. dv satisfies the graded q-Leibniz rule and dN v w = 0 for any w ∈ A. Proof. It follows from Lemma 1 that if q is a primitive N th root of unity, then (N ) for any integer l = 1, 2, . . . , N − 1 the coefficient pl contains the factor [N ]q which vanishes in the case of q being a primitive N th root of unity. This implies (N ) N N |w|N wv N . Taking into account that pl = 0. Thus dN v (w) = v w + (−1) q N |w|N )w = λ w. The coefficient v N = e, we obtain dN v (w) = (1 + (−1) q N |w| λ = 1 + (−1) q N vanishes if q is a primitive N th root of unity. Indeed, if N is an odd number, then 1 − (q N )(N −1)/2 = 0. In the case of an even integer N we have 1 + (q N/2 )N −1 = 1 + (−1)N −1 = 0, and this ends the proof of the theorem. For applications in differential geometry it is important to have a realization of a graded q-differential algebra as an algebra of analogues of differential forms on a geometric space. The proved theorem allows us to construct a graded q-differential algebra taking as a starting point a generalized Clifford algebra. The structure of a generalized Clifford algebra suggests that we shall get an analogue of an algebra of differential forms with an N -differential on a noncommutative space. Indeed, let us remind that a generalized Clifford algebra Cp,N is a unital associative algebra over C generated by γ1 , γ2 , . . . , γp which are subjected to the relations γi γj = q sg(j−i) γj γi , 6
γiN = 1,
i, j = 1, 2, . . . , p,
(2)
where q is a primitive N th root of unity and sg(x) is the usual sign function. The structure of a graded q-differential algebra in the case of the generalized Clifford algebra with two generators is studied in [9 ]. In this case the corresponding generalized Clifford algebra C2,N can be interpreted as an algebra of polynomial functions on a reduced quantum plane. Let us denote by x, y the generators of the algebra in this case. The relations (2) take on the form xy = q yx, xN = y N = 1. The algebra C2,N becomes a ZN -graded algebra if we assign the grading zero to the generator x, the grading 1 to the generator y and define the grading of any monomial made up of generators x, y as the sum of gradings of its factors. The differential d is defined by dw = [y, w]q , w ∈ C2,N . Since y N = 1, it follows from Theorem 1 that the algebra C2,N is a graded q-differential algebra and d is its N -differential. We give this graded q-differential algebra and its N -differential d the following geometric interpretation: the subalgebra of polynomials of grading zero is the algebra of functions on a one-dimensional space with “coordinate” x, and the elements of higher gradings expressed in terms of “coordinate” x and its “differential” dx are the analogues of differential forms with exterior differential d. We have dx = y∆q x = y(x − qx). Since dk 6= 0 for k < N , a differential k-form w may be expressed either by means of (dx)k or by means of dk x, where dk x =
[k]q (dx)k k(k−1)/2 q
x1−k .
If w = (dx)k f (x), where f (x) is a polynomial of grading zero, and dw = (k) (dx)k+1 δx (f ), then δx(k) (f ) = (∆q x)−1 (q −k f − q k A(f )), where A is the homomorphism of the algebra of polynomials of grading zero (k) determined by A(x) = qx. The higher-order derivatives δx have the property δx(k) (f g) = δx(k) (f ) g + q k A(f ) δx(0) (g), (0)
∂g ∂x (k) δx
k = 0, 1, 2, . . . , N − 1,
where δx (g) =
= (∆q x)−1 (g − A(g)) is the A-twisted derivative. A higher-
order derivative
can be expressed in terms of the derivative δx(k) = q k
∂ ∂x
as follows:
q −k − q k −1 ∂ + x . ∂x 1−q
The realization of a graded q-differential algebra as an algebra of analogues of differential forms on an ordinary (commutative) space is constructed in [10 ]. Let x1 , x2 , . . . , xn be the coordinates of an n-dimensional space Rn , C ∞ (Rn ) be the algebra of smooth C-valued functions, and dx1 , dx2 , . . . , dxn be the differentials of the coordinates. Let N = {1, 2, . . . , n} be the set of integers, I be a subset of N , and |I| be the number of elements in I. Given any subset I of N , i.e. 7
I = {i1 , i2 , . . . , ik } ⊂ N , 1 ≤ i1 < i2 < . . . ik ≤ n, we associate to I the formal monomial dxI , where dxI = dxi1 dxi2 . . . dxin and dx∅ = 1. Let Ω(Rn ) be the free left C ∞ (Rn )-module generated by all formal monomials dxI . It is evident that Ω(Rn ) has a natural Z-graded structure Ω(Rn ) = ⊕k Ωk (Rn ), where Ωk (Rn ) is the left C ∞ (Rn )-module freely generated by all dxI , where I contains k elements. An element of the module Ωk (Rn ) has the form ω=
X
fI dxI =
I,|I|=k
X
fi1 i2 ...ik dxi1 dxi2 . . . dxik ,
(3)
1≤i1
