From Cayley-Klein double categories to TQFT
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ELEMENTARY PARTICLES AND FIELDS Theory
From Cayley–Klein Double Categories to TQFT* S. Moskaliuk** Bogoliubov Institute for Theoretical Physics of NAS of Ukraine, Kiev Received April 17, 2009
Abstract—New applications of categorical methods are connected with new additional structures on categories. One of such structures, the double category, is considered in this article. The Cayley–Klein double category structure is defined as generalization of the Cayley–Klein bicategory structure. It is shown that Cayley–Klein double categories exist in the topological quantum field theories (TQFT). DOI: 10.1134/S1063778810030105
1. INTRODUCTION An Euclidean vector space RRn+1 is a vector space over the field R of real numbers equipped with an inner product function (, ) : RRn+1 × RRn+1 → R which is bilinear, symmetric, and positive definite. These spaces are the objects of a category Euclid, with morphisms those linear maps which preserve inner product [1]. There are two functors ∗ : (Euclid)
U : Euclid → VctR ,
op
The transitions from the (n + 1)-dimensional real Cayley–Klein space RV n+1 (j) to the real Cayley– Klein space RV n+1 (j ), and from the groups SO(n + 1, R; j), Sp (n, R; j) to the groups SO(n + 1, R; j ), Sp (n, R; j ) as well, can be obtained from ψ : RRn+1 → RRn+1 (j) ≡ RVn+1 (j), ψx∗0 = x0 ,
ψx∗k = xk
jk = 1, ιk , i,
ψ x0 =
x0 ,
ψ xk =
xk
k = 1, 2, . . . , n,
k
(2)
−1 jm jm ,
m=1
by the same substitution of parameters jk for jk jk−1 , where j = (j1 , . . . , jn ) and each of parameters jk ∗ **
jm ,
(3)
k = 1, 2, . . . , n,
m=1
where z0∗ , zk∗ ∈ CRn+1 , z0 , zk ∈ CV n+1 (j) are complex Cartesian coordinates. The totality of all possible values of the parameter j gives 3n different real RV n+1 (j) and, correspondingly, complex CV n+1 (j) Cayley–Klein spaces. ∗ |2 of the The quadratic form (z∗ , z∗ ) = nm=0 |zm n+1 turns into the quadratic form space CR (z, z) = |z0 |2 =
n
|zk |2
k
2 jm
(4)
m=1
k=1
jm ,
ψ : RV n+1 (j) → RV n+1 (j ),
k
ψz0∗ = z0 ,
CRn+1 (j)
and the transitions
ψzk∗ = zk
(1)
m=1
j = (j1 , . . . , jn );
ψ : CRn+1 → CV n+1 (j),
→ VctR
to the category of real vector spaces: The (covariant) forgetful functor U “forget the inner product” and the contravariant functor “take the dual space” [2].
k
assuming three values: jk = 1, ιk , i. Similarly the permissibility of these transitions can be justified for complex Cayley–Klein space CV n+1 (j) which emerge from the (n + 1)-dimensional complex Euclidean space CRn+1 by the mapping
under the mapping (3). Here, of the space 2 2 1/2 is the absolute value (modulus) |zk | = (xk + yk ) of the complex number zk = xk + iyk , and z is the complex vector: z = (z0 , z1 , . . . , zn ). Let us define (formally) the transition from the space CV n+1 (j) and generators to the space CV n+1 (j ) by transformations, which can be obtained from the transformations (3), substituting in the latter the parameters jk for jk jk−1 , i.e., ψ : CV n+1 (j) → CV n+1 (j ), ψ zk = zk
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