Noncommutative Gauge Theory - ncg2017.cpt.univ-mrs.frncg2017.cpt.univ-mrs.fr/DOCUMENTS/PDF/Vitale-Lecture-1.pdf · Noncommutative Gauge Theory Patrizia Vitale Dipartimento di Fisica

Noncommutative Gauge Theory

Patrizia Vitale

Dipartimento di Fisica Universita di Napoli Federico II and INFN

Noncommutative Geometry and Applications to Quantum PhysicsQuy-Nhon, Vietnam 13.7-22.7 2017

Patrizia Vitale Noncommutative Gauge Theory

Outline

Lecture I

What is a gauge transformation

Commutative, Abelian and non-Abelian Gauge theories

Local and global problems of commutative gauge theories

Motivations for NC geometry

Prototype of NC algebra: Moyal algebra

Derivation based differential calculus

Lecture IINoncommutative Moyal gauge theoriesLecture IIINoncommutative Lie algebra type gauge theories


Commutative Gauge Transformations [ Nair, Nash]

General setting for first part of this lecture:

M smooth manifold (space-time, rigorously, should becompact)G finite dimensional Lie groupπ : P → M principal G-bundle

Gauge theories are theories where dynamical variables areconnections of G-bundles.

They describe fundamental interactionsG qualifies the interaction (U(1) for Electromagneticinteraction, SU(2)× U(1) electroweak interactions, SU(3)strong interactions)The number of generators of G is equal to the number ofparticles which mediate the interaction (so called vectorbosons)For Electrodynamics this is 1 (the foton)Gauge transformations are special automorphsims of P whichare symmetries of the dynamics


Commutative Gauge Transformations

Definition Aut(P)

An automorphism of P is a diffeomorphism ϕ : P → P which isG-equivariant, that is ϕ(p · g) = ϕ(p) · g for all p ∈ P and g ∈ G .

Every ϕ ∈ Aut(P) induces a diffeomorphism ϕ on the basismanifold.Locally φ : p ' (x , g(x)) −→ φ(p) ' (φ(x), g ′(x))

The map H which associates ϕ ∈ Diff (M) to ϕ ∈ Aut(P) is agroup homomorphism:

H : φ ∈ Aut(P)→ φ ∈ Diff (M)

H(φ · ψ) = H(φ) H(ψ)


Definition G(P)

The gauge group of P is G(P) :=ker(H). Its elements are calledgauge transformations or also vertical automorphisms.

φ ∈ G(P) ⊂ Aut(P) if π(φ(p)) = π(p)

that is φ = id ∈ Diff (M)

−→Physics:G ' Map(M→ G) = smooth maps g : x ∈ M ∈ G- Physical request when M = Rn: g(x)→ 1 as |x | → ∞- Physical states of matter fields are invariant only if this isassumed- Rn → Sn (origin of the Gribov problem in non-Abelian gaugetheories)


A= space of gauge connections of (P,M,G ), locally mapsA : M → Ω1(M)⊗ g

Ag = gAg−1 + dgg−1

Matter fields coupled to interactions are sections of associatedvector (spinor) bundles (E (P),V ,M,G )

A gauge connection on P induces a connection and acovariant derivative for sections ψ : M → E

∇ : Der(Γ(E ))× Γ(E )→ Γ(E )

which obeys∇X (f ψ) = X (f )ψ + f∇X (ψ)∇fX (ψ) = f∇X (ψ) f ∈ F(M)∇X+Y (ψ) = ∇X (ψ) +∇Y (ψ)∇X (ψ + φ) = ∇X (ψ) +∇X (φ)

Locally ∇µfψ = ∂µfψ + A · fψPatrizia Vitale Noncommutative Gauge Theory

Classical Euclidean action for gauge fields- A pure theory of fundamental interactions (no matter fields) is atheory where the dynamical fields are the gauge connections- B = A/G space of physical configurations

S [A] = Tr

∫M

F ∧ ?HF =1

4Tr

∫FµνFµνdΩ

with F = dA + A ∧ A ∈ Ω2(M)⊗ g- Lie algebra g = u(N), su(N) or products (StandardModel), Lorentz group (general relativity) ...-

S [A] =1

2

∫AaµMµν

ab AbνdΩ

Mµνab = δab(−δµν + ∂µ∂ν)

not invertible: ∂νgg−1 is eigenvector of Mµν with zero eigenvalue- For Electrodynamics g = u(1) g(x) = exp α(x)Ag = gAg−1 + dgg−1 → Ag

µ = Aµ + ∂µα Mµν∂µα = 0


The gauge fixing

Quantum theory:Classical action S [A]→ Quantum action Γ[A]

Γ[A] = Leg(ln Z [J])[A]

Z [J] =

∫[dµ(A)]exp (−S [A] + J · A)

But we can’t perform the Gaussian integralunless we fix the gauge

δ(f (A)− h)

and restrict the integration to equivalence classes

[dµ(A)]→ [dµ(A/G)]


The gauge fixing

This amounts to choose a surface Σf ⊂ A which possibly intersectsthe gauge orbits only once: a section for the principal bundle

A(P) ← G↓B(P)

The choice of Σ is the gauge fixing; for example d ?H A = 0(∂µAµ = 0).The classical action is invariant −→ insensitive to the gauge fixing.


The gauge fixing: Faddeev-Popov determinant

How to restrict the integration to equivalence classes, namely howto obtain [dµ(B)]

Locally (ignore global issues for the moment) A ∼ B × G =⇒

[dµ(A)] = [dµ(B)] [dµ(G)]

for gauge transformations close to the identity [dµ(G)] ' [dα]

To perform the change of variable [dα]→ [df (A)]:insert the Jacobian

Detab∆FP(x, y) = Detδfa(x)

δαb(y)=⇒

[dµ(A)]Det∆ = [[dµ(B)][dα]Det∆ = [dµ(B)] [df]

and integrate over [df] with the delta function:

[dµ(A)] Det∆ δ(f(A)− h(x)) = [dµ(B)]

.Patrizia Vitale Noncommutative Gauge Theory

Topological obstructions

Back to global approach• If we assume A globally trivial bundle → Πj(A) = Πj(G) + Πj(B)• A is an affine space

Aτ = (1− τ)A1 + τA2 0 ≤ τ ≤ 1

withAgτ = gAτg−1 + dgg−1

=⇒ A homotopically trivial

• G = Map(S4,G) (g(x)→ 1, |x | → ∞)=⇒ Π1(G) = g : S5 → G

Π1(G) = Π5(G )

forG = U(N) Π5 = Z , N ≥ 3; Π5 = Z2, N = 2; Π5 = 0, N = 1


Topological obstructions

=⇒ G only homotopically trivial for QED G = U(1).

=⇒ A 6= G × B for G = U(N),N ≥ 2.

Summary: Non-Abelian gauge theories do not admit global sectionsPhysics: gauge fixing doesn’t single out one representative for eachequivalence class for non-Abelian gauge theories (Gribov ambiguityGribov, Singer, Narasimhan and Ramadas ’78-’79)

Problems of QFT:- UV divergences → short distance cut-off needed, renormalization;- IR divergences → Gribov-Zwanziger-Dell’Antonio modification ofpropagator

What is the situation for NCQFT?


Motivations

Motivations for NCG:

Gravity motivations:

Gedanken experiments putting together general relativity andquantum mechanics are not compatible withpseudo-Riemannian space-time [DFR94];Loop Quantum Gravity has discrete spectrum for geometricoperators

Regularization of Quantum Field Theory

Low energy regimes of strings with background B field.Double field theory (Manifest duality invariance).


Noncommutative QED on R2nθ

The Moyal algebra

The simplest noncommutative space is modeled on the phasespace of quantum mechanics:

First, go to dual description in terms of algebra of functions onclassical phase spaceQuantize (make it ”noncommutative phase space”)No smooth manifold anymoreNoncommutativity can be described in terms of a star product:quantum mechanics in the Moyal approach → blackboard

Do the same for space-time → [xi , xj ] = iθij- θ constant- replace operators with an algebra of functions on

space-time (assume it even-dim.), with noncommutative starproduct



The Moyal algebra

For coordinate functions

xi ? xj − xj ? xi = iθij

The Moyal star-product (asymptotic form):

(f ? g)(x) = f (x) exp

i

2θρσ←∂ρ→∂σ

g(x)

Notice:x i ? f − f ? x i = i θij∂j f∫

Rn

f ? g =

∫Rn

g ? f

(=

∫Rn

f · g)

the integral is a trace



The Moyal algebra

(F(R2n), ?θ) =: R2nθ is the Moyal algebra

It is the associative algebraL ∩ R = T ∈ S ′ : T ? f ∈ S, f ? T ∈ S, ∀f ∈ SThe star product is defined for Schwartz functions on R2n

f ? g(x) =1

(πθ)2n

∫f (x + y)g(x + z)e−2iyµΘ−1

µν zν

extended to S ′ × S 〈T ? f , h〉 = 〈T , f ? h〉 (sim. S × S ′ )

and to S ′ × S ′ 〈T ? T ′, h〉 = 〈T ,T ′ ? h〉 Θ is block diagonal,antisymmetric with θ real.

Θ = θ

0 −11 0

..


The Moyal algebra R2nθ

S(R2n) Schwartz functions

S ′(R2n) dual algebra of tempered distributions

L,R ⊂ S ′ left and right multiplier algebras

R2nθ is unital and involutive. It contains S, polynomials,

constants [Varilly, Gracia-Bondia, Soloviev arxiv-1012.0669 ]

f ?θ g(x) = exp( i

2Θµν ∂

∂yµ∂

∂wµ

)f (y)g(w)|y=w=x

[xµ, xν ]?θ = iθµν

which describes space-time noncommutativity and implies thepresence of a minimal area ' θ



The differential calculus

Minimal derivation based differential calculus over the Moyalalgebra. [DV88, DVM94, M08, W09, CMW11, MVZ]

For an associative algebra a differential calculus can always bedefined algebraically [IS67, LM90], once a Lie algebra of derivations,L, is given:- 1-forms α are linear maps from L to A.- The exterior derivative d is defined as

dα(X ,Y ) = ρ(X )(α(Y ))− ρ(Y )(α(X ))− α([X ,Y ])

- If ρ : L → Der(A) is a Lie algebra homomorphism, thend2 = d d is zero.- Higher forms are defined as skew-symmetric multilinear mapsfrom L to the associative algebra A.

We do the same for noncommutative algebrasPatrizia Vitale Noncommutative Gauge Theory


The differential calculus

∂µ = −iθ−1µν [xν , ·]? generate the minimal Lie algebra of derivations

of R2nθ

Other derivations can be chosen (for example the wholeinhomogeneous simplectic group ISp(4,R) or subalgebras)- inner- not a left module because

f ? ∂µ(g ? h) 6= f ? ∂µg ? h + g ? f ? ∂µh- d , i∂µ defined algebraically,

df ∧? dg(X ,Y ) = f (X ) ? g(Y )− f (Y ) ? g(X );


- forms are constructed by duality:Ω0 ≡ R2n

θ

Ω1 g ? df (X ) = g ? X (f ), iXα = α(X )

Ω2 : f ? dα(X1,X2) := f ?(

X1(α(X2))−X2(α(X1))− α([X1,X2]))

...analogously Ωn.They are left R2n

θ modules- derivations have to be “sufficient”:

df (Xµ) = 0 ∀µ→ f is central


Summary

We have reviewed standard gauge transformations in alanguage which can be generalized to NC setting

We have introduced NC space time of Moyal type

Derivation based differential calculus

In Lecture II we shall define NC gauge connection and NC gaugetransformations and we shall apply to NC gauge theory of Moyaltype


Bibliography

[Nair] V. Parameswaran Nair. Quantum Field Theory. A Modern Perspective SpringerEd. (2004)[Nash] C. Nash Differential topology and Quantum Field Theory” Elsevier 1992[Gribov] V. N. Gribov, Quantization of Nonabelian Gauge Theories, Nucl. Phys. B139 (1978) 1.[Singer] I. M. Singer Some Remarks on the Gribov Ambiguity Commun. Math. Phys.60, 7 1978)[NR79] M. S. Narasimhan and T. R. Ramadas, Geometry of SU(2) Gauge Fields,Commun. Math. Phys. 67 (1979) 121.[DFR94] S. Doplicher, K. Fredenhagen and J. E. Roberts, ”Space-time quantizationinduced by classical gravity ”Phys. Lett. B331 (1994) 39.[Varilly-Gracia] J. M. Gracia-Bondia and J. C. Varilly, “Algebras of distributionssuitable for phase space quantum mechanics. 1.,” J. Math. Phys. 29, 869 (1988).[Segal] I. E. Segal, Quantized differential forms, Topology, 8 (1967) 147; Quantizationof the de Rham complex, Proc. Sympos. Pure Math., 16 (1970) 205.

[DV88] M. Dubois-Violette, Derivations et calcul differentiel non commutatif, C.R.

Acad. Sci. Paris, Serie I, 307 (1988) 403.


[DVM94] M. Dubois-Violette, P.W. Michor, Derivations et calcul differentiel noncommutatif II, C.R. Acad. Sci. Paris, Serie I, 319 (1994) 927.[LM90] G. Landi and G. Marmo Algebraic differential calculus for gauge theoriesNucl.Phys.Proc.Suppl. 18A, (1990) 171.[MVZ] G. Marmo, P. Vitale and A. Zampini, Noncommutative differential calculus forMoyal sub-algebras”, J. Geom. Phys. 56 (2006) 611[M08] T. Masson, “Examples of derivation-based differential calculi related tononcommutative gauge theories,” Int. J. Geom. Meth. Mod. Phys. 5, 1315 (2008)[arXiv:0810.4815 [math-ph]].[W09] J.-C. Wallet, ”Derivations of the Moyal algebra and noncommutative gaugetheories”, SIGMA 5 (2009) 013.

[CMW] E. Cagnache, T. Masson and J-C. Wallet, Noncommutative Yang-Mills-Higgs

actions from derivation based differential calculus, J. Noncomm. Geom. 5 (2011) 39,


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Noncommutative Gauge Theory - ncg2017.cpt.univ-mrs.frncg2017.cpt.univ-mrs.fr/DOCUMENTS/PDF/Vitale-Lecture-1.pdf · Noncommutative Gauge Theory Patrizia Vitale Dipartimento di Fisica