Petrov Noncommutative Topological Quantum Field Theory
This paper gives a definition of category NC-Einst of noncommutative Einstein spaces, and a Petrov noncommutative topological quantum field theory (NC TQFT) is constructed. We suggest extensions of these ideas which may be useful to further NC TQFT and apply it in higher dimensions.
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irk-123456789-561912014-02-14T03:10:35Z Petrov Noncommutative Topological Quantum Field Theory Moskaliuk, S.S. Wohlgenannt, M. Поля та елементарні частинки This paper gives a definition of category NC-Einst of noncommutative Einstein spaces, and a Petrov noncommutative topological quantum field theory (NC TQFT) is constructed. We suggest extensions of these ideas which may be useful to further NC TQFT and apply it in higher dimensions. У статтi дано означення категорiї некомутативних просторiв Ейнштейна NCEinst та побудовано некомутативну топологiчну квантову теорiю поля (НКТП) типу Петрова. Автори вважають корисним ознайомлення з iдеями даної роботи з метою дальшого розвитку НКТП та її застосування у просторах вищої розмiрностi. 2010 Article Petrov Noncommutative Topological Quantum Field Theory / S.S. Moskaliuk, M. Wohlgenannt // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 505-514. — Бібліогр.: 23 назв. — англ. 2071-0194 PACS 11.10.Nx, 02.40.Gh, 04.20.Gz, 04.60.Rt, 04.62.+v http://dspace.nbuv.gov.ua/handle/123456789/56191 en Український фізичний журнал Відділення фізики і астрономії НАН України |
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Поля та елементарні частинки Поля та елементарні частинки Moskaliuk, S.S. Wohlgenannt, M. Petrov Noncommutative Topological Quantum Field Theory Український фізичний журнал |
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This paper gives a definition of category NC-Einst of noncommutative Einstein spaces, and a Petrov noncommutative topological quantum field theory (NC TQFT) is constructed. We suggest extensions of these ideas which may be useful to further NC TQFT and apply it in higher dimensions. |
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Moskaliuk, S.S. Wohlgenannt, M. |
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Moskaliuk, S.S. Wohlgenannt, M. |
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Moskaliuk, S.S. |
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Petrov Noncommutative Topological Quantum Field Theory |
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Petrov Noncommutative Topological Quantum Field Theory |
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Petrov Noncommutative Topological Quantum Field Theory |
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Petrov Noncommutative Topological Quantum Field Theory |
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Petrov Noncommutative Topological Quantum Field Theory |
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petrov noncommutative topological quantum field theory |
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2010 |
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Поля та елементарні частинки |
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Petrov Noncommutative Topological Quantum Field Theory / S.S. Moskaliuk, M. Wohlgenannt // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 505-514. — Бібліогр.: 23 назв. — англ. |
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Український фізичний журнал |
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AT moskaliukss petrovnoncommutativetopologicalquantumfieldtheory AT wohlgenanntm petrovnoncommutativetopologicalquantumfieldtheory |
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2025-07-05T07:25:25Z |
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1836790921381806080 |
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PETROV NONCOMMUTATIVE TOPOLOGICAL QUANTUM FIELD THEORY
PETROV NONCOMMUTATIVE TOPOLOGICAL QUANTUM
FIELD THEORY
S.S. MOSKALIUK,1 M. WOHLGENANNT2
1Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
(14b, Metrolohichna Str., Kyiv 03143, Ukraine)
2Vienna University of Technology, Institute for Theoretical Physics
(Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria)
PACS 11.10.Nx, 02.40.Gh,
04.20.Gz, 04.60.Rt, 04.62.+v
c©2010
This paper gives a definition of category NC-Einst of noncommu-
tative Einstein spaces, and a Petrov noncommutative topological
quantum field theory (NC TQFT) is constructed. We suggest ex-
tensions of these ideas which may be useful to further NC TQFT
and apply it in higher dimensions.
1. Introduction
The subjects of the double category, TQFT, and non-
commutative Einstein spaces have been studied in [1–
13]. Let us describe some noncommutative geometric
aspects of twisted deformations. Consider a Lie algebra
g over C, and its associated universal enveloping algebra
Ug. A general twist F is an element F ∈ Ug⊗Ug in the
tensor product of a Hopf algebra (Ug, ·,Δ, S, ε) given by
F = fα ⊗ fα, F−1 = f̄α ⊗ f̄α , (1)
and satisfying the conditions
F12(Δ⊗ id)F = F23(id⊗Δ)F , (2)
(ε⊗ id)F = 1 = (id⊗ ε)F , (3)
where the elements fα, fα, f̄
α
, f̄ α belong to Ug, Δ de-
notes the coproduct and ε the co-unit of the respective
Hopf algebra [14–16].
Then, the universal R matrix is defined by
R = F21F−1 = Rα ⊗Rα, R−1 = R̄α ⊗ R̄α . (4)
Using the R matrix, we obtain, for functions h and g,
h ? g = R̄α(g) ? R̄α(h) . (5)
Our strategy is to deform a product ◦ of some objects A
and B by replacing it with a twisted product ◦?:
A ◦? B := f̄α(A) ◦ f̄α(B) . (6)
The universal enveloping algebra of vector fields can be
deformed in two different ways:
– UΞ?
This is a Hopf algebra [16] defined by deforming the
structure functions of UΞ:
u ? v = f̄α(u)f̄α(v) , (7)
Δ?(u) = u⊗ 1 + R̄α ⊗ R̄α(u) , (8)
ε?(u) = ε(u) = 0 , (9)
S?(u) = −R̄α(u)R̄α , (10)
where R̄α(u) is the usual Lie derivative of u along the
vector field R̄α.
There is a natural action of Ξ? on the algebra of functions
A? given in terms of the usual undeformed Lie derivative,
L?u(h) := f̄α(u)(f̄α(h)) , (11)
which can be extended to UΞ?.
The ?-Lie algebra of vector fields Ξ? generates the Hopf
algebra UΞ?.
– UΞF
We have the following structure maps:
u ·F v = u · v , (12)
SF (u) = S(u) , (13)
εF (u) = ε(u) , (14)
ΔF (u) = FΔ(u)F−1 . (15)
However, UΞ? and UΞF turn out to be isomorphic Hopf
algebras.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 505
S.S. MOSKALIUK, M. WOHLGENANNT
The star-connection ∇? is defined to satisfy the fol-
lowing axioms:
∇∗u+vz = ∇∗uz +∇∗vz ,
∇h?uv = h ?∇∗uv ,
∇∗u(h ? v) = L∗u(h) ? v + R̄α(h) ?∇∗R̄α(u)v , (16)
where u, v and z are vector fields. Next, we define the
connection coefficients by
∇?µ∂̂ν := Γσµν ? ∂̂σ , (17)
using the basis {∂̂µ}. The action of the covariant deriva-
tive on a one-form can be obtained employing the star-
dual pairing of a vector field v with a one-form ω,
∇∗u〈v, w〉? = L∗u〈v, w〉? =
= 〈∇∗uv, w〉? + 〈R̄α(v),∇∗R̄α(u)w〉?, (18)
which can be written equivalently as
〈v,∇∗uw〉? = LR̄α(u)〈R̄α(v), w〉?−
−〈∇∗R̄α(u)(R̄α(v)), w〉? . (19)
For a given metric
g = gµν ? dx̂
µ ⊗? dx̂ν , (20)
the connection that leaves it invariant is called a Levi-
Civita connection:
∇?µg = 0. (21)
For a general twist F−1 = f̄α ⊗ f̄α, the torsion and
curvature tensors are given by [13]
T (u, v) = ∇∗uv −∇∗R̄α(v)R̄α(u)− [u, v]∗, (22)
R(u, v, z) ≡ R(u, v)z =
= ∇∗u∇∗vz −∇∗R̄α(v)∇
∗
R̄α(u)z −∇
∗
[u,v]∗
z . (23)
It is enough to calculate the tensor on a basis ∂̂µ because
of the tensorial property, i.e.,
T (u, v) = uν ? T (∂̂ν , ∂̂µ) ? vµ. (24)
In this frame, the star-connection is given by
∇∗zu = L∗z(uν) ∗ ∂̂ν + R̄α(uν) ∗ R̄α(z)µ ∗ Γσµν ∗ ∂̂σ . (25)
We will need to compute the components of the curva-
ture tensor in this base. They can be expressed in the
following way:
Rijk
l = 〈R(∂̂i, ∂̂j , ∂̂k), dx̂k〉∗ . (26)
Consequently, we have, for the deformed Ricci tensor,
Rij = Rijk
k . (27)
Classical Einstein spaces have a Ricci tensor propor-
tional to the metric. In the noncommutative case, we
are looking for spaces satisfying the same property:
Rij = cgij ,
where c is some constant.
2. Noncommutative Einstein Spaces
2.1. Weyl–Moyal plane R4
θ
The metric is the usual Minkowski or Euclidean one; the
twist is Abelian [16]:
F = e−
i
2 θ
µν∂µ⊗∂ν , (28)
where θµν = −θνµ ∈ R. The covariant derivative is given
by
∇∗zu = zµ ? ∂µ(uν) ? ∂ν + zµ ? uν ? Γσµν ? ∂σ . (29)
In a first step, let us show that the choice Γσµν = 0 is
a good choice and renders the affine connection to be
a Levi-Civita connection. Thus, the expression for the
covariant derivative (29) becomes
∇∗zu = zµ ? ∂µ(uν) ? ∂ν . (30)
Let us show that axioms (16) are satisfied:
• ∇∗u+vz = (u+ v)µ ? ∂µ(zν) ? ∂ν = ∇∗uz +∇∗vz , (31)
• ∇h?uv = (h ? uµ) ? ∂µ(vν) ? ∂ν
= h ? (uµ ? ∂µvν ? ∂ν) = h ?∇∗uv , (32)
• ∇∗u(h ? v) = uµ ? ∂µ(h ? vν) ? ∂ν =
506 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
PETROV NONCOMMUTATIVE TOPOLOGICAL QUANTUM FIELD THEORY
= L∗u(h) ? v + uµ ? h ? (∂µvν) ? ∂ν =
= L∗u(h) ? v + R̄α(h) ? R̄α(uµ) ? (∂µvν) ? ∂ν =
= L∗u(h) ? v + R̄α(h) ?∇∗R̄α(u)v . (33)
In a next step, we show that the curvature and the tor-
sion vanish. The torsion is given by
T (∂µ, ∂ν) = ∇∗µ∂ν −∇∗ν∂µ − [∂µ, ∂ν ]∗ = 0 , (34)
since the Christoffel symbols are all zero, and the deriva-
tives commute. Similarly, we see that the curvature ten-
sor also vanishes:
R(∂ν , ∂β , ∂µ) =
= ∇∗ν∇∗β∂µ −∇∗R̄α(∂β)∇
∗
R̄α(∂ν)
∂µ −∇∗[∂ν ,∂β ]∗
z = 0 . (35)
At last, we consider the covariant derivative of the
metric:
∇∗µg = ∇∗µ(gαβ dxα ⊗∗ dxβ) =
= ∂µ(gαβ)dxα ⊗∗ dxβ − gαβΓαµσdxσ ⊗∗ dxβ−
−gαβdxα ⊗∗ Γβµσdx
σ = 0 , (36)
since the star-dual pairing (19) yields
∇∗µdxα = −Γαµσ ? dx
σ = 0 .
Among these metrics, those that are classically Ein-
stein metrics are also shown to be noncommutative Ein-
stein metrics.
2.2. R5
q
The algebra is generated by the coordinates x̂1, . . . , x̂5
satisfying the relations [16]
x̂1x̂2 = q x̂2x̂1, x̂1x̂4 = q−1x̂4x̂1,
x̂1x̂5 = x̂5x̂1, x̂2x̂4 = x̂4x̂2,
x̂2x̂5 = q x̂5x̂2, x̂4x̂5 = q−1x̂5x̂4. (37)
The coordinate x̂3 is central. The conjugation is given
by
x̂1∗ = x̂5, x̂2∗ = x̂4, x̂3∗ = x̂3 .
Hence, the twist (for the symmetric ordering) reads
F = exp
(
ih
2
(χ1 ⊗ χ2 − χ2 ⊗ χ1)
)
, (38)
where χ1 and χ2 are the following commuting vector
fields:
χ1 = x2∂2 − x4∂4, χ2 = x1∂1 − x5∂5 .
Thus, we have, for the inverse R matrix,
R−1 = R̄α ⊗ R̄α = fαf̄β ⊗ fαf̄β =
=
∑
(−1)m+k−l(
h
2
)n+k
(
n
m
)(
k
l
)
n!k!
χn−m+l
1 χm+k−l
2 ⊗
⊗χm+k−l
1 χn−m+l
2 . (39)
2.2.1. Note on Hermitian generators
Let us introduce Hermitian generators for the algebra
R5
q:
x̂1 = ẑ1 + iẑ2, x̂5 = ẑ1 − iẑ2
x̂2 = ŷ1 + iŷ2, x̂4 = ŷ1 − iŷ2 , (40)
with ŷ∗i = ŷi and ẑ∗i = ẑi, i = 1, 2. Inserting these
identifications into the commutation relations (37) yields
the identical relations
ẑ1ŷ1 = q ŷ1ẑ1, ẑ1ŷ2 = q−1ŷ2ẑ1,
ẑ1ẑ2 = ẑ2ẑ1, ŷ1ŷ2 = ŷ2ŷ1,
ŷ1ẑ2 = q ẑ2ŷ1, ŷ2ẑ2 = q−1ẑ2ŷ2 , (41)
in the case where q is a square root of unity.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 507
S.S. MOSKALIUK, M. WOHLGENANNT
2.2.2. Geometry
Again, we propose
Γµαβ = 0 (42)
and show that this definition leads to a sensible covariant
derivative and geometric tensors. The covariant deriva-
tive (25) is given by
∇∗zu = L∗z(uν) ? ∂̂ν . (43)
This satisfies the axioms for a affine connection, since
• ∇∗u+vz = L∗u+v(z
ν) ? ∂̂ν = L∗u(zν) ? ∂̂ν
+L∗v(zν) ? ∂̂ν = ∇∗uz +∇∗vz (44)
• ∇∗h?uv = L∗h?u(vν) ? ∂̂ν
= h ? L∗u(vν) ? ∂̂ν = h ?∇∗u(v) (45)
• ∇∗u(h ? v) = L∗u(h ? vν) ? ∂̂ν = L∗u(h) ? v
+R̄α(h) ? L∗R̄α(u)(v
ν) ? ∂̂ν
= L∗(h) ? v + R̂α(h) ?∇∗R̄α(u)(v) (46)
The torsion T is given by
T (u, v) = ∇∗uv −∇∗R̄α(v)R̄α(u)− [u, v]?
= L∗u(vν) ? ∂̂ν − L∗R̄α(v)(R̄α(u)ν) ? ∂̂ν − [u, v]? (47)
Computing the torsion for frame elements, we see explic-
itly that
T (∂̂µ, ∂̂ν) = 0. (48)
This is due to the tensorial property and
[∂̂µ , ∂̂ν ]∗ = [f̄α(∂̂µ), f̄α(∂̂ν)] = 0 , (49)
since the Lie derivative of ∂̂µ along f̄ , and consequently
also R̄, is again a vector field with constant coefficients:
cνµ∂̂ν , cνµ ∈ R.
Next, we compute the curvature tensor:
R(u, v, z) = ∇∗u∇∗vz −∇∗R̄α(v)∇
∗
R̄α(u)z −∇
∗
[u,v]?
z =
= L∗u(L∗v(zν)) ? ∂ν − L∗R̄α(v)(L
∗
R̄α(u)(z
ν)) ? ∂ν−
−L∗[u,v]?(z
ν) ? ∂ν=L∗u?v(zν) ? ∂ν−L∗R̄α(v)?R̄α(u)(z
ν) ? ∂ν−
−L∗[u,v]?(z
ν) ? ∂ν=L∗u?v−R̄α(v)?R̄α(u)−[u,v]?
(zν) ? ∂ν=0.
(50)
The Riemann curvature tensor vanishes identically. In
a next step, we show that this connection is a metric
one. We have to evaluate the covariant derivative of the
metric:
∇∗µg = ∇∗µ(gαβdx̂α ⊗? dx̂β) ,
where
(gαβ) =
1
1
1
1
1
.
In the present case, we again obtain, from the star-
dual pairing (19), that
∇∗µdx̂σ = 0.
Therefore, we get
∇∗µ g = gσβ×
×(∇∗µdx̂σ ⊗∗ dx̂β + R̄α(dx̂σ)⊗∗ ∇∗R̄α(∂̂µ)
dx̂β) = 0 . (51)
2.3. Glq(N)
The quantum space for Glq(N) [12] is defined by
x̂ix̂j = qx̂j x̂i, i < j . (52)
Therefore, we have, for the twist,
F−1 = exp
− ih
2
∑
i<j
(x̂j ∂̂j ⊗ x̂i∂̂i − x̂i∂̂i ⊗ x̂j ∂̂j)
.
(53)
In the same way as before, we can show that the trivial
connection satisfies all requirements and defines a Levi-
Civita connection with vanishing curvature tensor.
508 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
PETROV NONCOMMUTATIVE TOPOLOGICAL QUANTUM FIELD THEORY
2.4. Twisted sphere
The twisted sphere is defined by relations (37) and the
additional condition [15]
r2 = 2(x̂1x̂5 + x̂2x̂4) + (x̂3)2 . (54)
With the use of the stereographic coordinates yi, i =
1, 2, 4, 5, the metric is given by
g∗ =
4r2
(r2 + κ2)2
? Cijdy
i ⊗∗ dyj , (55)
where
(Cij) =
1
1
1
1
.
In order to simplify the notation, we introduce the fol-
lowing definitions: For the vector fields, let us define
ti := yi
∂
∂yi
= yi∂i (56)
(we note that no summation over the index i is implied).
Hence, we write, for the twist,
F = exp
(
− ih
2
ϕijti ⊗ tj
)
(57)
with
ϕij = −ϕji = −ϕij′ , (58)
ϕ12 = 1, ϕii = ϕii′ = 0 , (59)
and i′ = 6− i. Furthermore, let us introduce Pij and its
square,
Pij = e
ih
2 ϕij , qij = P 2
ij . (60)
Using these definitions, we can write, for the metric,
g∗ =
∑
ij
gijdy
i⊗∗dyj =
4r2
(r2 + κ2)2
∑
i,j
CijPij dy
i⊗dyj .
(61)
The Levi-Civita connection can be obtained by demand-
ing the vanishing torsion and the vanishing covariant
derivative of the metric. The former condition reads
Γ∗ij
k = qijΓ∗ji
k . (62)
The latter condition then leads to
Γ∗ij
k =
1
2
glk (qij∂jgil + ∂iglj − ∂lgji) . (63)
As a result, the universal connection is the same as that
in the undeformed case:
∇∗ = ∇ . (64)
The converse is also true: Assuming (64), we obtain (63)
for the connection coefficients.
Similarily, we obtain, for the Riemann curvature,
R∗ = R (65)
and, in terms of components,
R∗ = R∗ikl
mdyi ⊗∗ dyj ⊗∗ dyk ⊗∗ ∂m , (66)
R∗ijkl =
1
r2
(gligjk − qik gljgik) . (67)
Now let us consider a possible transformation between
a 5d theta-deformed plane (see Section 2.1) and a 5d
q-deformed one (see Section 2.2). The theta-deformed
space is chosen in the following way: [xi, xj ] = iθij with
the coordinate x3 commuting with all other coordinates
and
θij =
0 h −h 0
−h 0 0 h
h 0 0 −h
0 −h h 0
.
Then, with the map yi = exp(xi), we obtain the cor-
rect commutation relations (37). But, unfortunately,
this map does not respect the complex structure, and
the induced metric seems not to be the proper met-
ric for the q-deformed plane. But another possible
map is from the q-deformed sphere to a plane, via a
stereographic projection. Starting with the q-deformed
sphere, commutation relations (37), and the constraint
r2 = 2(x1x5+x2x4)+(x3)2, we define a map to the plane
in the usual way by y3 = x3, yi = (xir)/(r − x3), i =
1, 2, 4, 5. The induced metric is then given by (55).
3. Double (Bi-)Category
Definition 1. A category is a quadruple (Obj, Mor,
id, ◦) consisting of:
(C1) a class Obj of objects;
(C2) a set Mor(A,B) of morphisms for each ordered
pair (A,B) of objects;
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 509
S.S. MOSKALIUK, M. WOHLGENANNT
(C3) a morphism idA ∈ Mor(A,A) for each object A:
the identity of A;
(C4) a composition law associating, to each pair of mor-
phisms f ∈Mor(A,B) and g ∈Mor(B,C), a morphism
g ◦ f ∈Mor(A,C);
which is such that:
(M1) h ◦ (g ◦ f) = (h ◦ g) ◦ f for all f ∈ Mor(A,B),
g ∈Mor(B,C) and h ∈Mor(C,D);
(M2) idB ◦f = f ◦ idA = f for all f ∈Mor(A,B);
(M3) the sets Mor(A,B) are pairwise disjoint.
Example 1. The category NCEinst. Objects of the
category NCEinst are noncommutative Einstein spaces
NC Einst defined in Sections 2.1–2.4 by the induced met-
ric (55). For a morphisms s, t: NC Einst → NC Einst′,
we define a map to the plane in the usual way by
y3 = x3, yi = (xir)/(r − x3), i = 1, 2, 4, 5.
Definition 2. Let X and Y be two categories. A func-
tor from X to Y is a family of functions F which as-
sociates, to each object A in X, an object FA in Y
and, to each morphism f ∈ MorX(A,B), a morphism
Ff ∈MorY(FA,FB) which is such that
(F1) F(g ◦ f) = Fg ◦ Ff for all f ∈ MorX(A,B) and
g ∈MorY(B,C);
(F2) F idA = idFA for all A ∈ Obj(X).
Definition 3. A double category D consists of:
(1) A category D0 of objects Obj(D0) and morphisms
Mor(D0) of 0-level.
(2) A category D1 of objects Obj(D1) of 1-level and mor-
phisms Mor(D1) of 2-level.
(3) Two functors d, r : D1
−→→D0.
(4) A composition functor
∗ : D1 ×D0 D1 → D1,
where the bundle product is defined by the commutative
diagram
D1 ×D0 D1
π2→ D1
π1 ↓ ↓ d
D1
r→ D0
.
(5) A unit functor ID : D0 → D1 which is a section of
d, r.
The above data is subject to Associativity Axiom
and Unit Axiom. If both of them are fulfilled only up
to the equivalence, then the double category is called a
weak double category, and if they are fulfilled strictly,
then it is a strong double category.
Here, we see that, for two objects A,B ∈ Obj(D0),
there are 0-level morphisms D0(A,B) which are noted
by ordinary arrows f : A → B, and 1-level morphisms
D(1)(A,B) which are noted by the arrows ξ : A ⇒ B,
for A = d(ξ) and B = r(ξ). So, with a 2-level morphism
α : ξ → ξ′, where ξ : A ⇒ B and ξ′ : A′ ⇒ B′, we can
associate the diagram
A
ξ⇒ B ξ
d(α) ↓ ↓ r(α) 7−→ ↓ α
A′
ξ′⇒ B′ ξ′
and the arrow α : d(α)⇒ r(α)
The composition on 2-level is associated with the dia-
gram
A
ξ⇒ B ξ
d(α) ↓ ↓ r(α) ↓ α
A′
ξ′⇒ B′ 7−→ ξ′
d(α′) ↓ ↓ r(α′) ↓ α′
A′′
ξ′′⇒ B′′ ξ′′
Now we can define, for double categories, double
(category) functors and their morphisms, double
subcategories, the categoryDCat of double categories,
equivalence of double categories, dual double cate-
gories (changed direction of 1-level morphisms, i.e. d, r
are transposed), and so on [1, 22].
Definition 4. [4] The theory of bicategories is the cate-
gory (with finite limits) Th(Bicat) given by the follow-
ing data:
• Objects Ob, Mor, 2Mor
• Morphisms s, t : Ob→Mor and s, t : Mor→ 2Mor
• composition maps ◦ : MPairs→Mor and · :
BPairs→2Mor, satisfying the interchange law
(the requirement that this be a functor means that
the interchange law holds):
(α ◦ β) · (α′ ◦ β′) = (α · α′) ◦ (β · β′) , (68)
where MPairs = Mor×Ob Mor and BPairs =
2Mor×Mor 2Mor are the equalizers of diagrams of
the form:
Mor
t
""EEEEEEEE
MPairs
i // Mor2
π1
;;wwwwwwww
π2
##GGGGGGGG Ob
Mor
s
<<zzzzzzzz
(69)
and similarly for BPairs.
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PETROV NONCOMMUTATIVE TOPOLOGICAL QUANTUM FIELD THEORY
• the associator map a : Triples→2Mor, where
Triples = ×Ob Mor×Ob Mor is the equalizer of a
similar diagram for involving Mor3 such that a sat-
isfies s(a(f, g, h)) = (f ◦ g) ◦ h and t(a(f, g, h)) =
f ◦ (g ◦ h)
• unitors l, r : Ob→Mor with s ◦ l = t ◦ l = idOb
and s ◦ r = t ◦ r = idOb
These data are subject to the conditions that the asso-
ciator is subject to the pentagon identity [23], and the
unitors obey certain unitor laws
(g ◦ 1y) ◦ f
ag,1y,f
//
rg×1f
��
g ◦ (1 ◦ f)
1g×lf
wwppppppppppp
g ◦ f
. (70)
Definition 5. [4] A double bicategory consists of:
• bicategories Obj of objects, Mor of morphisms,
2Mor of 2-morphisms
• source and target maps s, t : Mor→Obj and
s, t : 2Mor→Mor
• partially defined composition functors ◦ :
Mor2→Mor and · : 2Mor2→2Mor, satisfying
the interchange law (68)
• partially defined associator a : Mor3→2Mor
with s(a(f, g, h)) = (f ◦ g) ◦ h and t(a(f, g, h)) =
f ◦ (g ◦ h)
• partially defined unitors l, r : Obj→Mor with
s(l(x)) = t(l(x)) = x and s(r(x)) = t(r(x)) = x .
All the partially defined functors are defined for com-
posable pairs or triples, for which the source and target
maps coincide in the obvious way. The associator should
satisfy the pentagon identity [23], and the unitors should
satisfy the unitor laws (70).
4. Action of a Double Category
Double categories are the categorical variants of usual
monoids (and groups), and thus we have the correspond-
ing variant for their actions. Below, the definition of
action of a double category d, r : D1 → D0 on cat-
egories over D0 is given. Thus, we get an analog of
group-theoretic methods in categorical frames.
Definition 6. (Left) action of a double category d, r :
D1 → D0 on a category p : M → D0 over D0 is a functor
ϕ such that
(1)The diagram is commutative
D1 ×D0 M
ϕ→ M
r ◦ π1 ↘ ↓ p
D0
,
where the bundle product D1 ×D0 M is defined by the
diagram
D1 ×D0 M
π2→ M
π1 ↓ ↓ p
D1
r→ D0
.
(2)The diagram is commutative to within an isomor-
phism
(D1 ×D0 D1)×D0 M
∼=→ D1 ×D0 (D1 ×D0 M)
idD1×D0ϕ−→ D1 ×D0 M
⊗×D0 idM ↓ ↓ ϕ
D1 ×D0 M
ϕ−→ M
,
and there exists a functor isomorphism ϕ such that
∀ ξ, ξ′ ∈ Obj(D1), m ∈ Obj(M1)
ϕξ,ξ′,m : (ξ ∗ ξ′) ∗m→ ξ ∗ (ξ′ ∗m).
(3) For the unit functor, we have a functor isomorphism
χ : ϕ ◦ (ID × idM )−̃→idM or for objects
∀ A ∈ Obj(D0), m ∈ Obj(M1) χA,m : IDA ∗m−̃→m.
So we have the map of a pair of objects ξ ∈ Obj(D1),
m ∈ Obj(M) (A
ξ⇒ p(m),m) 7→ ϕ(ξ,m) such that
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S.S. MOSKALIUK, M. WOHLGENANNT
p(ϕ(ξ,m)) = A, and of morphisms α ∈ D1(ξ, ξ′), u ∈
M(m,m′)
ξ A
ξ⇒ p(m) ϕ(ξ,m)
α ↓ f = d(α) ↓ ↓ r(α) = p(u) 7−→ ↓ ϕ(α, u)
ξ′ A′
ξ′⇒ p(m′) ϕ(ξ′,m′)
,
where p(ϕ(α, u)) = f.
The definition of a right action is evident.
5. Cobordism and Double Categories
Let Md be the category of oriented compact d-
dimensional smooth manifolds (with boundary) and
piecewise smooth maps (we do not define the sense of
the condition more exactly here; this may be such con-
tinuous maps f : M → Y that are smooth on a dense
open subset Uf ⊂ M ), let CMd be its subcategory
of closed (with empty boundary) manifolds and smooth
maps, CMd ⊂Md.
There are the following functors:
(1) Disjoint union
∪ : Md ×Md →Md : (X,Y ) 7→ X ∪ Y.
(2) Changing of the orientation of manifolds on the op-
posite one
(−) : Md →Md : X 7→ −X.
(3) Boundary operator
∂ : Md+1 → CMd : X 7→ ∂X.
(4) Multiplication on the unit segment I = [0, 1]
I × . : CMd →Md+1 : X 7→ I ×X.
Now we define a double category C(d) with
(1) C(d)0 = CMd.
(2) 1-level morphisms C(d)(1)(X,X
′) are a set of
pairs (Y, f), where Z is an oriented compact (d + 1)-
dimensional smooth manifold with the boundary ∂Y, and
f is a diffeomorphism
f : (−X) ∪X ′ → ∂Y,
where ∪ stands for the disjoint union of −X and X ′.
Thus, we write (Y, f) : X ⇒ X ′.
(3) The composition of (Y, f) : X ⇒ X ′ and (Y ′, f ′) :
X ′ ⇒ X ′′ is the morphism
(Y ∪X′ Y ′, (f |X) ∪ (f ′|X′)) : X ⇒ X ′′,
where (Y ∪X′ Y ′) denotes the union (Y ∪ Y ′) after the
identification of each point f(y) ∈ f(Y ) with the point
f ′(y) ∈ f ′(Y ) for all y ∈ Y and smoothing this topolog-
ical manifold.
(4) The 1-level identical morphism IDX is (X ×
[0; 1], id(−X)∪X), because ∂(X × [0; 1]) = (−X) ∪X.
(5) 2-level morphisms of C(d)1(ξ, ξ′) from ξ = (Y, f :
X ′ ∪ (−X) → ∂Y ) : X ⇒ X ′ to ξ′ = (Y ′, f ′ :
X ′′ ∪ (−X ′) → ∂Y ′) : X ′ ⇒ X ′′ are such triples of
smooth maps (f1, f2, f3) that the following diagram is
commutative:
(−X) ∪X ′ f−→ ∂Y ⊂ Y
↓ f1 ∪ f2 ↓ f3
(−X ′) ∪X ′′ f ′−→ ∂Y ′ ⊂ Y ′
.
It is easy to see that the functors ∪ and (−) may be
expanded to double category functors
∪ : C(d)→ C(d),
(−) : C(d)→ C(d)◦,
and (−) is an equivalence of the double categories.
Remark. Two following formulas for 1-level mor-
phisms in algebras and cobordisms [18–20] are of inter-
est:
f : A⊗k B◦ → Endk(N) f : (−X) ∪ Y → ∂Z,
where we have correspondence between the functors
(_)◦ ←→ −(_),
⊗k ←→ ∪,
Endk ←→ ∂.
6. Petrov Noncommutative Topological
Quantum Field Theory
The Petrov Noncommutative Topological Quantum
Field Theory (NC TQFT) is a 2-functor Z from a
certain bicategory of double cobordisms [7] CM(d)
of d-dimensional manifolds into the double bicategory
NCEinst of noncommutative Einstein spaces, and some
axioms are satisfied [1, 17, 22, 23].
Thus, Petrov NC TQFT in dimension d is a 2-functor,
Z : C(d)→Mor(NCEinst),
between double bicategories such that
(1) the disjoint union in C(d) goes to the tensor product
∪ 7→ ⊗,
512 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
PETROV NONCOMMUTATIVE TOPOLOGICAL QUANTUM FIELD THEORY
where (_)∗ : NC Einst → NC Einst◦ is a dualization
of noncommutative Einstein spaces,
(2) changing the orientation in C(d)0 goes to the dual-
ization
(−) 7→ (.)∗.
Thus, as a consequence of double bicategorical func-
torial properties, we get
(1) for each compact closed oriented smooth d-
dimensional manifold X ∈ Obj(C(d)0), the value
of the functor Z(X) is a noncommutative Einstein
space over the field C of the complex numbers,
(2) for each (Y, f) : X ⇒ X ′ from Obj(C(d)1),
the value of the functor Z(Y, f) is a homomor-
phism Z(X)→ Z(X ′) of noncommutative Einstein
spaces,
and the following axioms of Petrov NC TQFT are satis-
fied:
A(1) (involutivity) Z(−X) = Z(X)∗, where −X de-
notes the manifold with the opposite orientation,
and ∗ denotes the dual noncommutative Einstein
space.
A(2) (multiplicativity) Z(X ∪ X ′) = Z(X) ⊗ Z(X ′),
where ∪ denotes a disconnected union of manifolds.
A(3) (associativity) For the composition
(Y ′′, f ′′) = (Y, f) ∗ (Y ′, f ′) of cobordisms, the fol-
lowing relation holds:
Z(Y ′′, f ′′) =
= Z(Y ′, f ′) ◦ Z(Y, f) ∈MorC(Z(X),Z(X′′)).
(Usually, the identifications
Z(X ′−X) ∼= Z(X)∗⊗Z(X ′) ∼= MorC(Z(X),Z(X′))
allow one to identify Z(Y, f) with the element
Z(Y, f) ∈ Z(∂Y ).
A(4) For the initial object, ∅ ∈ Obj(C(d)0) Z(∅) = C.
A(5) (trivial homotopy condition)
Z(X × [0, 1]) = idZ(X).
7. Conclusions
We have studied the noncommutative counterparts of
the so-called Einstein spaces (such as twisted 4-spheres)
in the framework of twisted gravity. Their Ricci tensor
is proportional to the metric. We have computed the
deformed Riemannian tensor and the scalar curvature
in the formalism of twisted gravity. We could already
see, for some examples, the remarkable property that
being an Einstein space seems to be stable under defor-
mation, using a Killing vector field in the twist. The
deformed Levi-Civita connection and the deformed Rie-
mann tensor are just the undeformed ones. Deformed
spherical symmetric spaces are very important with re-
spect to, e.g., the Black-Hole solutions and are related to
cosmological problems. As a generalization, one should
study star geometries, where the vector fields are not
Killing vectors. On the other hand, the main result of
this paper can be summarized as that the construction
of Petrov NC TQFT is a 2-functor from a certain bi-
category of double cobordisms CM(d) of d-dimensional
manifolds into the double bicategory NCEinst of non-
commutative Einstein spaces.
The authors are especially grateful to the Austrian
Academy of Sciences and the Russian Foundation for
Fundamental Research which in the framework of the
collaboration with the National Academy of Sciences
of Ukraine co-financed this research. The authors
also could have not succeeded in pursuing this pro-
gram without the collaborations for many years with
Prof. W. Kummer and Prof. J. Wess.
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Received 10.02.09
НЕКОМУТАТИВНА ТОПОЛОГIЧНА КВАНТОВА ТЕОРIЯ
ПОЛЯ ТИПУ ПЕТРОВА
С.С. Москалюк, М. Волґенант
Р е з ю м е
У статтi дано означення категорiї некомутативних просторiв
Ейнштейна NCEinst та побудовано некомутативну топологi-
чну квантову теорiю поля (НКТП) типу Петрова. Автори вва-
жають корисним ознайомлення з iдеями даної роботи з метою
дальшого розвитку НКТП та її застосування у просторах ви-
щої розмiрностi.
514 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
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