Noncommutative Geometry of Yang–Mills fields
(joint work with Walter D. van Suijlekom)
Jord Boeijink
Radboud University Nijmegen
2 June 2011
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
1 / 18
Goal of the talk
Goal
Describe classical P SU (N )-gauge theories on compact smooth
Riemannian spin manifolds using spectral triples.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
2 / 18
Goal of the talk
Goal
Describe classical P SU (N )-gauge theories on compact smooth
Riemannian spin manifolds using spectral triples.
This will be a generalisation of the Chamseddine-Connes description
of Einstein–Hilbert–Yang–Mills theory (CC97, The spectral action
principle).
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
2 / 18
Classical gauge theory
Definition
Let G be a Lie-group. A gauge theory on a manifold M is given by a
principal G-bundle π : P → M together with a connection ω on P .
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
3 / 18
Classical gauge theory
Definition
Let G be a Lie-group. A gauge theory on a manifold M is given by a
principal G-bundle π : P → M together with a connection ω on P .
Relation with physics
Connections are identified with gauge potentials
Sections of associated bundles are identified with particle fields.
Lagrangians give field equations.
Lagrangian is invariant under action of the gauge group
{φ : P → P equivariant , π ◦ φ = π}.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
3 / 18
Overview
1
Part I: reminder
Basics of noncommutative geometry
The Chamseddine-Connes spectral triple
2
Part II
Algebra bundles
Construction of spectral triples from algebra bundles
Application: P SU (N )-gauge theory
3
Conclusion
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
4 / 18
Spectral triple
Definition
A spectral triple is a triple (A, H, D) consisting of a unital ∗-subalgebra
A, a Hilbert-space H on which A is faithfully represented, and a densely
defined self-adjoint operator D on H that satisfies the following properties
the operator D has compact resolvent: (D − λ)−1 is compact for all
λ∈
/ R,
the operator [D, a] extends to a bounded operator on H for all a ∈ A.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
5 / 18
Real structure
Definition
A spectral triple is called real if there exists an anti-unitary operator J on
H which satisfies
J 2 = ε,
JD = ε0 DJ,
Jγ = ε00 γJ
and
[a, Jb∗ J ∗ ] = 0,
Jord Boeijink (Radboud University Nijmegen)
[[D, a], Jb∗ J ∗ ] = 0,
NCG of YM fields
(a, b ∈ A).
2 June 2011
6 / 18
Real structure
Definition
A spectral triple is called real if there exists an anti-unitary operator J on
H which satisfies
J 2 = ε,
JD = ε0 DJ,
Jγ = ε00 γJ
and
[a, Jb∗ J ∗ ] = 0,
[[D, a], Jb∗ J ∗ ] = 0,
(a, b ∈ A).
The numbers ε, ε0 , ε00 are ±1 and the signs determine the
KO-dimension.
We restrict to the even case, then ε0 = 1.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
6 / 18
Other ingredients
Given a spectral triple (A, H, D)
Gauge potentials
I
I
I
Generated by Morita self-equivalences.
∗
(A, H, D, J)
P7→ (A, H, DA = D + A + JAJ , J), where
∗
A = A = j aj [D, bj ], aj , bj ∈ A.
A is interpreted as a gauge potential
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
7 / 18
Other ingredients
Given a spectral triple (A, H, D)
Gauge potentials
I
I
I
Generated by Morita self-equivalences.
∗
(A, H, D, J)
P7→ (A, H, DA = D + A + JAJ , J), where
∗
A = A = j aj [D, bj ], aj , bj ∈ A.
A is interpreted as a gauge potential
Action
I
I
Action is determined by the spectral action principle.
The action depends on the operator DA .
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
7 / 18
Chamseddine-Connes example
The Chamseddine-Connes (real) spectral triple is given by (M is 4
dimensional)
(C ∞ (M ) ⊗ MN (C), L2 (M, S) ⊗ MN (C), D
/ ⊗ 1, JM ⊗ (·)∗ ),
where S denotes the spinor bundle.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
8 / 18
Chamseddine-Connes example
The Chamseddine-Connes (real) spectral triple is given by (M is 4
dimensional)
(C ∞ (M ) ⊗ MN (C), L2 (M, S) ⊗ MN (C), D
/ ⊗ 1, JM ⊗ (·)∗ ),
where S denotes the spinor bundle.
Inner fluctuations: A = iγ µ Aµ , where Aµ is a
ad(su(N )) ∼
= su(N )-valued 1-form on M .
Spectral action contains a Yang–Mills action with repect to the gauge
field Aµ .
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
8 / 18
Chamseddine-Connes example
The Chamseddine-Connes (real) spectral triple is given by (M is 4
dimensional)
(C ∞ (M ) ⊗ MN (C), L2 (M, S) ⊗ MN (C), D
/ ⊗ 1, JM ⊗ (·)∗ ),
where S denotes the spinor bundle.
Inner fluctuations: A = iγ µ Aµ , where Aµ is a
ad(su(N )) ∼
= su(N )-valued 1-form on M .
Spectral action contains a Yang–Mills action with repect to the gauge
field Aµ .
Observation
The triple (C ∞ (M ) ⊗ MN (C), L2 (M, S) ⊗ MN (C), D
/ ⊗ 1) determines a
trivial gauge theory (P, A), where P is a trivial P SU (N )-bundle and A a
connection on P (=connection induced by A).
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
8 / 18
Algebra bundles
Definition
A ∗-algebra bundle B is a vector bundle whose fibres have the structure of
finite-dimensional ∗-algebras, and a vector bundle morphism map
µ : B ⊗ B → B that satisfies
µ(1 ⊗ µ) = µ(µ ⊗ 1),
as a map from B ⊗ B ⊗ B → B.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
9 / 18
Algebra bundles
Definition
A ∗-algebra bundle B is a vector bundle whose fibres have the structure of
finite-dimensional ∗-algebras, and a vector bundle morphism map
µ : B ⊗ B → B that satisfies
µ(1 ⊗ µ) = µ(µ ⊗ 1),
as a map from B ⊗ B ⊗ B → B.
Γ∞ (M, B) is a involutive finitely generated projective
C ∞ (M )-module algebra (i.e. f.g.p. as a C ∞ (M )-module).
B is locally trivial as a vector bundle but not necessarily as an algebra
bundle.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
9 / 18
Serre-Swan for algebra bundles
Theorem
The map B 7→ Γ∞ (M, B) determines a bijective correspondence between
isomorphism classes of algebra bundles over M and finitely generated
projective C ∞ (M )-module algebras. (Actually, one has an equivalence of
categories.)
Remark
Same results hold if one considers a (unital) ∗-algebra bundle. That is, B
has a ∗-structure on the fibres such that s∗ (x) = s(x)∗ defines a involution
on Γ∞ (M, B).
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
10 / 18
Spectral triples associated to Γ∞ (M, B)
Assumptions
I
I
I
π : B → M a locally trivial unital ∗-algebra bundle where all fibres are
isomorphic as ∗-algebras.
for each x ∈ M , there exists a faihtful tracial state τx on Bx
for all s ∈ Γ∞ (M, B), the function x → τx (s(x)) is smooth.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
11 / 18
Spectral triples associated to Γ∞ (M, B)
Assumptions
I
I
I
π : B → M a locally trivial unital ∗-algebra bundle where all fibres are
isomorphic as ∗-algebras.
for each x ∈ M , there exists a faihtful tracial state τx on Bx
for all s ∈ Γ∞ (M, B), the function x → τx (s(x)) is smooth.
Then
I
I
Hermitian structure Γ∞ (M, B) × Γ∞ (M, B) → C ∞ (M ) given by
(s, t)B (x) := τ (s∗ (x)t(x)).
L2 (M, B ⊗ S) := compl (Γ∞ (M, B ⊗ S)) with respect to norm
induced by the hermitian structure on B ⊗ S is a Hilbert-space. S
denotes the spinor bundle.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
11 / 18
First proposition
Proposition
In the situation above, the triple
(Γ∞ (M, B), L2 (M, B ⊗ S), DB = γ(∇B ⊗ 1 + 1 ⊗ ∇S )), where ∇B is a
hermitian connection on B, is a spectral triple.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
12 / 18
First proposition
Proposition
In the situation above, the triple
(Γ∞ (M, B), L2 (M, B ⊗ S), DB = γ(∇B ⊗ 1 + 1 ⊗ ∇S )), where ∇B is a
hermitian connection on B, is a spectral triple.
Proof.
Boundedness of [DB , a] follows as in the trivial case, and compact
resolvent is immediate from the ellipticity of DB .
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
12 / 18
First proposition
Proposition
In the situation above, the triple
(Γ∞ (M, B), L2 (M, B ⊗ S), DB = γ(∇B ⊗ 1 + 1 ⊗ ∇S )), where ∇B is a
hermitian connection on B, is a spectral triple.
Proof.
Boundedness of [DB , a] follows as in the trivial case, and compact
resolvent is immediate from the ellipticity of DB .
Remark
If we introduce grading 1 ⊗ γ on L2 (M, B ⊗ S), where γ is a grading on
L2 (M, S), then the above construction is an example of an unbounded
internal Kasparov product.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
12 / 18
Real structure on the spectral triple
To get gauge theory we need to introduce a real structure J. Define
J(s ⊗ ψ) = s∗ ⊗ JM ψ
on Γ∞ (M, B ⊗ S).
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
13 / 18
Real structure on the spectral triple
To get gauge theory we need to introduce a real structure J. Define
J(s ⊗ ψ) = s∗ ⊗ JM ψ
on Γ∞ (M, B ⊗ S).
In general, for a hermitian connection ∇B ,
[[DB , a], Jb∗ J ∗ ] 6= 0.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
13 / 18
Real structure on the spectral triple
To get gauge theory we need to introduce a real structure J. Define
J(s ⊗ ψ) = s∗ ⊗ JM ψ
on Γ∞ (M, B ⊗ S).
In general, for a hermitian connection ∇B ,
[[DB , a], Jb∗ J ∗ ] 6= 0.
Need more assumptions: we assume that for all s, t ∈ Γ∞ (M, B):
∇B (st) = (∇B s)t + s∇B t,
Jord Boeijink (Radboud University Nijmegen)
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(∇s)∗ = ∇s∗ .
2 June 2011
13 / 18
Real structure on the spectral triple
To get gauge theory we need to introduce a real structure J. Define
J(s ⊗ ψ) = s∗ ⊗ JM ψ
on Γ∞ (M, B ⊗ S).
In general, for a hermitian connection ∇B ,
[[DB , a], Jb∗ J ∗ ] 6= 0.
Need more assumptions: we assume that for all s, t ∈ Γ∞ (M, B):
∇B (st) = (∇B s)t + s∇B t,
(∇s)∗ = ∇s∗ .
Remark
On any locally trivial (hermitian) ∗-algebra bundle there exist
(hermitian) connections ∇B that are ∗-derivations.
The conditions on ∇B are necessary for DJ = JD, [[D, a], JbJ ∗ ] = 0
to be satisfied.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
13 / 18
Main theorem
Theorem
(Γ∞ (M, B), L2 (M, B ⊗ S), DB = c(∇B ⊗ 1 + 1 ⊗ ∇S ), 1 ⊗ γ, (·)∗ ⊗ JM )
as defined above is a real and even spectral triple provided ∇B satisfied
the aforementioned conditions.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
14 / 18
P SU (N )-principal fibre bundle
Consider now the spectral triple belonging to a locally trivial unital
∗-algebra bundle B with fibres MN (C) and the fibrewise trace as smoothly
varying tracial state:
(Γ∞ (M, B), L2 (M, B ⊗ S), DB , 1 ⊗ γ, (·)∗ ⊗ J).
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
15 / 18
P SU (N )-principal fibre bundle
Consider now the spectral triple belonging to a locally trivial unital
∗-algebra bundle B with fibres MN (C) and the fibrewise trace as smoothly
varying tracial state:
(Γ∞ (M, B), L2 (M, B ⊗ S), DB , 1 ⊗ γ, (·)∗ ⊗ J).
Transition functions of B are ∗-automorphisms of MN (C), i.e.
P SU (N )-valued. One can use these transition functions to construct
a principal P SU (N )-bundle P such that P ×P SU (N ) MN (C) ∼
= B.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
15 / 18
Connection
We can construct a connection on the bundle P as follows:
Locally ∇B = d + ω, where ω ∈ Γ∞ (T ∗ M ⊗ su(N )) acts in the
adjoint representation.
By change of local trivialisation these ω transform as
−1
−1
ωu = guv
dguv + guv
ωv guv ,
where guv : U ∩ V → P SU (N ) denote the transition functions.
The set {ωu } induces a connection 1-form on P . This will be our
connection.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
16 / 18
Connection
We can construct a connection on the bundle P as follows:
Locally ∇B = d + ω, where ω ∈ Γ∞ (T ∗ M ⊗ su(N )) acts in the
adjoint representation.
By change of local trivialisation these ω transform as
−1
−1
ωu = guv
dguv + guv
ωv guv ,
where guv : U ∩ V → P SU (N ) denote the transition functions.
The set {ωu } induces a connection 1-form on P . This will be our
connection.
In 4D, under inner fluctuations DB 7→ c((∇B + A) ⊗ 1 + 1 ⊗ ∇S ),
where A ∈ Γ∞ (T ∗ M ⊗ ad P ).
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
16 / 18
PSU(N)-gauge theory
From the above spectral triple we have constructed a principal
P SU (N )-bundle P together with the connection on P . This
describes a P SU (N )-gauge theory. Applying the spectral action
principle we will obtain an action that contains a Yang–Mills
Lagrangian where the field strength tensor is given by the curvature of
∇B + A.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
17 / 18
PSU(N)-gauge theory
From the above spectral triple we have constructed a principal
P SU (N )-bundle P together with the connection on P . This
describes a P SU (N )-gauge theory. Applying the spectral action
principle we will obtain an action that contains a Yang–Mills
Lagrangian where the field strength tensor is given by the curvature of
∇B + A.
Given (P, ω), the above construction applied to
(P ×P SU (N ) MN (C), ∇B ), where ∇B is the connection on B induces
by the connection ω on P , will give (P, ω) again (modulo
isomorphism).
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
17 / 18
PSU(N)-gauge theory
From the above spectral triple we have constructed a principal
P SU (N )-bundle P together with the connection on P . This
describes a P SU (N )-gauge theory. Applying the spectral action
principle we will obtain an action that contains a Yang–Mills
Lagrangian where the field strength tensor is given by the curvature of
∇B + A.
Given (P, ω), the above construction applied to
(P ×P SU (N ) MN (C), ∇B ), where ∇B is the connection on B induces
by the connection ω on P , will give (P, ω) again (modulo
isomorphism).
Remark
Dixmier-Douady: a ∗-algebra bundle with continuously varying trace is an
endomorphism bundle if and only if the Dixmier-Douady class δ(Γ(M, B))
vanishes in H 3 (M, Z). In this case the P SU (N )-bundle P can be lifted to
a U (N )-bundle.
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
17 / 18
Results
Constructed spectral triples that describes arbitrary P SU (N )-gauge
theories on smooth compact Riemannian spin manifolds M .
The gauge group is isomorphic to Γ∞ (M, Ad P ).
The gauge potentials are parametrised by Γ∞ (M, ad P ).
In 4D this generalises the Chamseddine-Connes spectral triple.
The spectral triple (Γ∞ (M, B), L2 (M, B ⊗ S), DB ) comes from a
product in unbounded KK-theory between the triples
(Γ(M, B), Γ(M, B), 0) and (C(M ), L2 (M, B), D
/ ).
Jord Boeijink (Radboud University Nijmegen)
NCG of YM fields
2 June 2011
18 / 18