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22
Том 153, кн. 3
УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА
Физико-математические пауки
2011
UDK 512.81^514.823
PETROV CATEGORY OF NONCOMMUTATIVE EINSTEIN SPACES
S.S. Moskaliuk
Abstract
We give a definition of Pet.rov category Petrov-NC-Einst of noncommutat.ive Einstein spaces NC-Einst, which is used for construction of noncommutative topological quantum held theory (NC TQFT). We suggest extensions of these ideas, which may be useful for further development of NC TQFT. and apply them to higher dimensions.
Key words: noncommutative Einstein spaces, Pet.rov category, noncommutative topological quantum held theory.
Introduction
The subjects of the rioricorimmtative Einstein spaces, double category and TQFT have been studied in fl 10]. In [3. 11] some noncommutative geometric aspects of twisted deformations were described, and it was shown that the universal enveloping algebra of vector fields can be deformed in two different ways:
• U E*
This is a Hopf algebra [11] defined by deforming the structure functions of Ue:
u-kv = Ja(u)Ja (v),
A*(«) = u (g) 1 + Ra (X) ~Ra(u), e*(u) = e(u) = 0, S+(u) = — Ra(u)Ra,
where R°'(u) is the usual Lie derivative of u along the vector field R°.
There is a natural action of E* on the algebra of functions A* given in terms of the usual undeformed Lie derivative.
K(h) :=7a(u)(Ja(h)),
UE*
The ^^^^^^a of vector fields E* is generating the Hopf algebra UE*.
• U ef
We have the following structure maps:
F
u • v = u • v, SF (u) = S(u), eF (u) = e(u), Af (u) = FA(u)F
However, Ua^d Uturn out to be isomorphic Hopf algebras. The star-connection V* is defined to satisfy the following axioms:
v;+v z = V*uz + V*v z,
Vh*uv = h* VUv, (1)
V*u(h* v) = C*u(h) * v + Ra(h) * V^(u)t>,
where u, v and z are vector fields. Next, we define connection coefficients by
V* X —FCT *X
using the basis {<9M}. The action of the covariant derivative on a one-form can be obtained employing the star-dual pairing of a vector field v with a one-form u,
V* < v,iu >*= = (V*^,«;)* + {Ra(v),V^a{u)iu)it,
which eqnivalently can be written as
V»* = £^([l)<i?a (*'),«■')* - {V^(u)(Ra(v)),w}*- (2)
For a given metric
g = SV * dxM dxv, the connection that leaves it invariant is called a Levi Civita connection:
V*g = 0.
For general twist J= f°' ® fa, torsion and cnrvatnre tensors are given by [3] T{u,v) = V*v - V^a(v)Ra(u) -
R(u, v, z) ee R(u, v)z = V*uV*vz - V±p(u) - Vf^.s .
It is enough to calculate the tensor on a basis xM, because of the tensorial property, i.e.
T(u,v) = uv *T(dv, xM) *vM.
In this frame, the star-connection is given by
V*u = £*(«") * d„ + Ra(in *Ra(zr * r£„ * da. (3)
We will need to compute the components of the cnrvatnre tensor in this base. They can be expressed in the following way:
Rijkl = (R(di, dj,dk), dxk)v.
Consequently, we have for the deformed Ricci tensor
Rij Rijk .
Classical Einstein spaces have a Ricci tensor proportional to the metric. In the noncommntative case, we are looking for spaces satisfying the same property:
Rij cgij,
where c is some constant.
1. Noncommutative Einstein spaces
1.1. Weyl — Moyal plane R^. The metric is the usual Minkowski or Euclidean one: the twist is Abelian [11]:
t = exP e^ <g, d,
where 0*v = - G R. The covariant derivative is given by
V*u = z* * d*(uv) * dv + z* * uv * r*v * 3a . (4)
In a first step, let us show that the choice r*v = 0 is a good choice and renders the affirie connection to be a Levi Civita connection. Thus, the expression for the covariant derivative (4) becomes
V*u = z* *d*(uv) *dv. Let us show that the axioms (1) are satisfied:
VU+vZ = (u + v)* * d*(zv) * dv = VUz + V*z, (5)
Vh*uv = (h*u*) *d*(vv) *dv = h* (u* *d*vv *dv) = h* V^v, (6)
VU(h*v) = u* *d*(h*vv) *dv = LU(h) *v + u* *h* (d*vv) *dv =
= £*t(/?.) * v + R"(h) ■k~Ra{utl) * (<V>") * =
= + (7)
In a next step, we show that the curvature and torsion vanish. The torsion is given by
T(d*, dv) = V^dv - VVd* - [d*, dv]* = 0,
since the Christoflel symbols are all zero and the derivatives commute. Similarly, we see that the curvature tensor also vanishes:
R{dv, d0, dM) = VtVp, - V^V^d, - Vf^uz = 0. At last, we consider the covariant derivative of the metric:
V*g = V*(ga^ dxa 0 dx^) =
= ) dxa 0* dx^ - gapr^ dxCT 0* dx^ - dxa 0* dxCT = 0
since we get from the star-dual pairing (2)
V*dxa = -r^ * dxff = 0.
Among these metrics those that are classically Einstein metrics are also shown to be noncommutative Einstein metrics.
1.2. M);
relations fill
The algebra is generated by the coordinates x1,... ,x5 satisfying the
x1 —5
—5 x1
-2 —4 _ —4 —2
-2 —5 —5 —2 —4 —5 -1—5 —4
The coordinate —3 is central. Conjugation is given by
—— - x— - ——
-3*
x3.
(8)
Hence, the twist (for symmetrical ordering) reads
F = exP ^(Xi ® X2 - X2 ® xi,
where x1 and X2 are the following commuting vector fields:
Xi = x2d2 — x4d4, X2 = x1^! — xz8z. Thus, we have for the inverse R matrix
(9)
R-
R ® Ra = f°'fß ® faf
ß
n\ fk
'h\n+k \ I\!
E, 1 \m+k — l I \ V7V \ '' / n — m+l m+k — l ~ m+k — l n — m+l l-ij ^ Xl \2 (S>Xl \2
n
1.3. Twisted sphere. The twisted sphere is defined by the relations (8) and the additional condition [12]
r2 = 2(x1 x5 + x2 x4) + (x3)2. Using stereographic coordinates y\ i =1,2,4,5, the metric is given by
4r2
g
where
(r
(Cij
■kCij dyi dyj,
„2)2 ij
1\
1
1
VI 7
In order to simplify the notation, we introduce the following definitions. For the vector fields, let us define
■ d ■
U ■= y' g- = y'di;
note that here no summation over the index i is implied. Hence, we write for the twist
ih
T = exp ( - — tpijti ™tj
with
<fij — -<fji — -<fij
f 12 _ 1, f
ii fii
0
and i' = 6 — i. Furthermore, let us introduce Pj and its square,
i h
exP I —<Pij
q¿¿
P
j •
Using these definitions, we can write for the metric
4r2
g* = ^2 gij dy" =
(r2
j2)2 j j
Cj Pij dy" 0 dyj.
(10)
The Levi Civita connection can be obtained by demanding vanishing torsion and vanishing covariant derivative of the metric. The former condition reads
k
The latter condition then leads to
r*jk = \g'k {Ujdjgu + digij - digji). As a result, the universal connection is the same as in the undeformed case:
V* = V.
(ID
(12)
The converse is also true: assuming (12), we obtain (11) for the connection coefficients. Similarity, we obtain for the Riemann curvature:
R* = P,
and in terms of components
R* = R*ikimdvi dyk dm,
^ijki = (9ii9jk - qik gijQik) •
Now let us consider a possible transformation between 5d theta-deformed plane (see Subsection 1.1.) and a 5d q-deformed one (see Subsection 1.2.). The theta-deformed space is chosen in the following way: ] = ¿ftj with the coordinate x3 commuting
with all other coordinates and
( 0 h —h 0
—h 0 0 h
h 0 0 —h
—h h 0
Then with the map yi = exp(xi) we obtain the correct commutation relations (8). But unfortunately this map does not respect the complex structure, and the induced metric seems not to be the proper metric for the q-deformed plane. But another possible map is from the q-deformed sphere to a plane, via stereographic projection. Starting with the q-deformed sphere: commutation relations (8) and the constraint 2(xxx5 + x2x4) + (x3)2, one defines a map to the plane in the usual way by (xir)/(r — x3), i = 1, 2,4, 5. The induced metric is then given by (10).
3
x , y
2. Petrov category
Definition 1. A category is a quadruple (Obj, Mor, id, o) consisting of: (CI) a class Obj of objects;
(C2) a set Mor(A, B) of morphisms for each ordered pair (A, B) of objects; (C3) a morphism idA G Mor(A, A) for each object A: the identity of A; (C4) a composition law associating to each pair of morphisms f G Mor(A, B) and g G Mor(B, C): a morphism g o f G Mor(A, C); which is such that:
(Ml) h o (g o f) = (h o g) o f for all f G Mor(A, B), g G Mor(B, C) and h G Mor(C, D);
(M2) idB of = f o idA = f for all f G Mor(A, B); Mor(A, B)
Definition 2. The category Petrov-NC-Einst. Objects of the category Petrov-NC-Einst are noncommutative Einstein spaces NC Einst defined in Subsection 1.1.1.3. by the induced metric (10). For morphisms s, t (NC Einst ^ NC Einst') we define a map to the plane in the usual way by y3 = x3, yl = (xlr)/(r — x3), i = 1, 2,4, 5. A product of such morphisms of the category Petrov-NC-Einst is again a morphism of the category Petrov-NC-Einst. So, the category Petrov-NC-Einst is well defined.
The author is especially grateful to the Russian Academy of Sciences which, in the framework of the collaboration with the National Academy of Sciences of Ukraine, co-financed this research by the Joint Grant for Basic Research No 2010.
Резюме
С. С. Москалюк. Категория А.З. Петрова пекоммутативпых пространств Эйнштейна. В статье построены некоммутативные пространства Эйнштейна, которые являются объектами в определении категории А.З. Петрова. Концепция категории А.З. Петрова позволяет получить некоммутативные топологические теории поля и их расширения в многомерных пространствах.
Ключевые слова: некоммутативные пространства Эйнштейна, категория А.З. Петрова, некоммутативная топологическая квантовая теория поля.
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Поступила в редакцию 21.10.10
Moskaliuk, Stepan Stepanovich Doctor of Physics and Mathematics, Senior Scientific Researcher, Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine.
Москалюк Степан Степанович доктор физико-математических паук, старший научный сотрудник Института теоретической физики им. М.М. Боголюбова Национальной академии паук Украины, г. Киев, Украина.
E-mail: mss Qbitp.kiev. ua
