quantum anomalies in noncommutative geometryth- · 2010-06-11 · quantum anomalies in...
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![Page 1: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/1.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Quantum Anomalies in Noncommutative Geometry
Micha l Eckstein
Jagiellonian University, Krakow
Zakopane, june 11, 2010
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 2: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/2.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Outline
1 “Standard” anomaliesWhat are anomalies?Approaches to anomalies
2 Noncommutative spacesThe axiomsExample
3 Anomalies on spectral triplesSome resultsGeometrical approachNoncommutative 2-torus
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 3: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/3.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Outline
1 “Standard” anomaliesWhat are anomalies?Approaches to anomalies
2 Noncommutative spacesThe axiomsExample
3 Anomalies on spectral triplesSome resultsGeometrical approachNoncommutative 2-torus
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 4: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/4.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Outline
1 “Standard” anomaliesWhat are anomalies?Approaches to anomalies
2 Noncommutative spacesThe axiomsExample
3 Anomalies on spectral triplesSome resultsGeometrical approachNoncommutative 2-torus
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 5: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/5.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Outline
1 “Standard” anomaliesWhat are anomalies?Approaches to anomalies
2 Noncommutative spacesThe axiomsExample
3 Anomalies on spectral triplesSome resultsGeometrical approachNoncommutative 2-torus
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 6: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/6.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Outline
1 “Standard” anomaliesWhat are anomalies?Approaches to anomalies
2 Noncommutative spacesThe axiomsExample
3 Anomalies on spectral triplesSome resultsGeometrical approachNoncommutative 2-torus
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 7: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/7.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Outline
1 “Standard” anomaliesWhat are anomalies?Approaches to anomalies
2 Noncommutative spacesThe axiomsExample
3 Anomalies on spectral triplesSome resultsGeometrical approachNoncommutative 2-torus
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 8: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/8.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Outline
1 “Standard” anomaliesWhat are anomalies?Approaches to anomalies
2 Noncommutative spacesThe axiomsExample
3 Anomalies on spectral triplesSome resultsGeometrical approachNoncommutative 2-torus
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 9: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/9.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Outline
1 “Standard” anomaliesWhat are anomalies?Approaches to anomalies
2 Noncommutative spacesThe axiomsExample
3 Anomalies on spectral triplesSome resultsGeometrical approachNoncommutative 2-torus
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 10: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/10.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Outline
1 “Standard” anomaliesWhat are anomalies?Approaches to anomalies
2 Noncommutative spacesThe axiomsExample
3 Anomalies on spectral triplesSome resultsGeometrical approachNoncommutative 2-torus
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 11: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/11.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Outline
1 “Standard” anomaliesWhat are anomalies?Approaches to anomalies
2 Noncommutative spacesThe axiomsExample
3 Anomalies on spectral triplesSome resultsGeometrical approachNoncommutative 2-torus
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 12: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/12.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Outline
1 “Standard” anomaliesWhat are anomalies?Approaches to anomalies
2 Noncommutative spacesThe axiomsExample
3 Anomalies on spectral triplesSome resultsGeometrical approachNoncommutative 2-torus
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 13: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/13.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Outline
1 “Standard” anomalies
2 Noncommutative spaces
3 Anomalies on spectral triples
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 14: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/14.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Heuristic introduction
Classical FT Quantum FT
imposed symmetries broken symmetries
L[Aµ] = L[Agµ] Z[Aµ] 6= Z[Agµ]y xconserved currents
−−−−→
quantum anomalies
Dµjµ = 0 Dµj
µ = A
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 15: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/15.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Heuristic introduction
Classical FT Quantum FT
imposed symmetries broken symmetries
L[Aµ] = L[Agµ] Z[Aµ] 6= Z[Agµ]y xconserved currents
−−−−→
quantum anomalies
Dµjµ = 0 Dµj
µ = A
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 16: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/16.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Heuristic introduction
Classical FT Quantum FT
imposed symmetries broken symmetries
L[Aµ] = L[Agµ] Z[Aµ] 6= Z[Agµ]y xconserved currents
−−−−→
quantum anomalies
Dµjµ = 0 Dµj
µ = A
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 17: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/17.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Heuristic introduction
Classical FT Quantum FT
imposed symmetries broken symmetries
L[Aµ] = L[Agµ] Z[Aµ] 6= Z[Agµ]y xconserved currents −−−−→ quantum anomalies
Dµjµ = 0 Dµj
µ = A
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 18: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/18.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Heuristic introduction
Classical FT Quantum FT
imposed symmetries broken symmetries
L[Aµ] = L[Agµ] Z[Aµ] 6= Z[Agµ]y xconserved currents −−−−→ quantum anomalies
Dµjµ = 0 Dµj
µ = A
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 19: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/19.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
QFT methods
Feynman diagrams
Figure: Triangle diagram appearing in the famous ABJ anomaly computation.
Path-integrals - the Fujikawa method
The non-invariance of the path-integral measure!
J [β,Aµ] = e−∫MβA
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 20: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/20.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
QFT methods
Feynman diagrams
Figure: Triangle diagram appearing in the famous ABJ anomaly computation.
Path-integrals - the Fujikawa method
The non-invariance of the path-integral measure!
J [β,Aµ] = e−∫MβA
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 21: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/21.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
QFT methods
Feynman diagrams
Figure: Triangle diagram appearing in the famous ABJ anomaly computation.
Path-integrals - the Fujikawa method
The non-invariance of the path-integral measure!
J [β,Aµ] = e−∫MβA
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 22: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/22.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Geometry of anomalies
BRST cohomology
BRST (non)invariance sL = 0, sW [Aµ] = ANilpotency s2 = 0 ⇒ s
∫MνaAaconsistent = 0
Wess-Zumino consistency conditions
Chain of descent equations =⇒ A2nconsistent ←→ A2n+2
singlet
Index theory
Singlet anomaly ←→ AS index theoremNon-abelian anomaly
AS index theorem for families of elliptic operators
AS index theorem in 2n+ 2 dimensions (AGG)The winding number of the fermionic determinant
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 23: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/23.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Geometry of anomalies
BRST cohomology
BRST (non)invariance sL = 0, sW [Aµ] = ANilpotency s2 = 0 ⇒ s
∫MνaAaconsistent = 0
Wess-Zumino consistency conditions
Chain of descent equations =⇒ A2nconsistent ←→ A2n+2
singlet
Index theory
Singlet anomaly ←→ AS index theoremNon-abelian anomaly
AS index theorem for families of elliptic operators
AS index theorem in 2n+ 2 dimensions (AGG)The winding number of the fermionic determinant
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 24: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/24.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Geometry of anomalies
BRST cohomology
BRST (non)invariance sL = 0, sW [Aµ] = ANilpotency s2 = 0 ⇒ s
∫MνaAaconsistent = 0
Wess-Zumino consistency conditions
Chain of descent equations =⇒ A2nconsistent ←→ A2n+2
singlet
Index theory
Singlet anomaly ←→ AS index theoremNon-abelian anomaly
AS index theorem for families of elliptic operators
AS index theorem in 2n+ 2 dimensions (AGG)The winding number of the fermionic determinant
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 25: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/25.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Geometry of anomalies
BRST cohomology
BRST (non)invariance sL = 0, sW [Aµ] = ANilpotency s2 = 0 ⇒ s
∫MνaAaconsistent = 0
Wess-Zumino consistency conditions
Chain of descent equations =⇒ A2nconsistent ←→ A2n+2
singlet
Index theory
Singlet anomaly ←→ AS index theoremNon-abelian anomaly
AS index theorem for families of elliptic operators
AS index theorem in 2n+ 2 dimensions (AGG)The winding number of the fermionic determinant
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 26: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/26.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Geometry of anomalies
BRST cohomology
BRST (non)invariance sL = 0, sW [Aµ] = ANilpotency s2 = 0 ⇒ s
∫MνaAaconsistent = 0
Wess-Zumino consistency conditions
Chain of descent equations =⇒ A2nconsistent ←→ A2n+2
singlet
Index theory
Singlet anomaly ←→ AS index theoremNon-abelian anomaly
AS index theorem for families of elliptic operators
AS index theorem in 2n+ 2 dimensions (AGG)The winding number of the fermionic determinant
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 27: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/27.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Geometry of anomalies
BRST cohomology
BRST (non)invariance sL = 0, sW [Aµ] = ANilpotency s2 = 0 ⇒ s
∫MνaAaconsistent = 0
Wess-Zumino consistency conditions
Chain of descent equations =⇒ A2nconsistent ←→ A2n+2
singlet
Index theory
Singlet anomaly ←→ AS index theoremNon-abelian anomaly
AS index theorem for families of elliptic operators
AS index theorem in 2n+ 2 dimensions (AGG)The winding number of the fermionic determinant
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 28: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/28.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Geometry of anomalies
BRST cohomology
BRST (non)invariance sL = 0, sW [Aµ] = ANilpotency s2 = 0 ⇒ s
∫MνaAaconsistent = 0
Wess-Zumino consistency conditions
Chain of descent equations =⇒ A2nconsistent ←→ A2n+2
singlet
Index theory
Singlet anomaly ←→ AS index theoremNon-abelian anomaly
AS index theorem for families of elliptic operators
AS index theorem in 2n+ 2 dimensions (AGG)The winding number of the fermionic determinant
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 29: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/29.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Geometry of anomalies
BRST cohomology
BRST (non)invariance sL = 0, sW [Aµ] = ANilpotency s2 = 0 ⇒ s
∫MνaAaconsistent = 0
Wess-Zumino consistency conditions
Chain of descent equations =⇒ A2nconsistent ←→ A2n+2
singlet
Index theory
Singlet anomaly ←→ AS index theoremNon-abelian anomaly
AS index theorem for families of elliptic operators
AS index theorem in 2n+ 2 dimensions (AGG)The winding number of the fermionic determinant
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 30: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/30.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Geometry of anomalies
BRST cohomology
BRST (non)invariance sL = 0, sW [Aµ] = ANilpotency s2 = 0 ⇒ s
∫MνaAaconsistent = 0
Wess-Zumino consistency conditions
Chain of descent equations =⇒ A2nconsistent ←→ A2n+2
singlet
Index theory
Singlet anomaly ←→ AS index theoremNon-abelian anomaly
AS index theorem for families of elliptic operators
AS index theorem in 2n+ 2 dimensions (AGG)The winding number of the fermionic determinant
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 31: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/31.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
What are anomalies?Approaches to anomalies
Geometry of anomalies
BRST cohomology
BRST (non)invariance sL = 0, sW [Aµ] = ANilpotency s2 = 0 ⇒ s
∫MνaAaconsistent = 0
Wess-Zumino consistency conditions
Chain of descent equations =⇒ A2nconsistent ←→ A2n+2
singlet
Index theory
Singlet anomaly ←→ AS index theoremNon-abelian anomaly
AS index theorem for families of elliptic operators
AS index theorem in 2n+ 2 dimensions (AGG)The winding number of the fermionic determinant
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 32: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/32.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
Outline
1 “Standard” anomalies
2 Noncommutative spaces
3 Anomalies on spectral triples
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 33: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/33.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
The axioms of noncommutative geometry
(A,H,D) - spectral triple
A - dense ∗-subalgebra of a C∗-algebra, unital
H - Hilbert space; by GNS construction∃ a faithful representation ρ(A) ' B(H)
D - the Dirac operator - selfadjoint, unbounded
D−1 - compact[D, ρ(a)] ∈ B(H) for all a ∈ A
γ ∈ B(H) - chirality - selfadjoint, unitary
γ2 = id, ∀a∈A γa = aγ, Dγ = −γDThe Z2-grading of the Hilbert space H = H+ ⊕H−
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 34: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/34.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
The axioms of noncommutative geometry
(A,H,D) - spectral triple
A - dense ∗-subalgebra of a C∗-algebra, unital
H - Hilbert space; by GNS construction∃ a faithful representation ρ(A) ' B(H)
D - the Dirac operator - selfadjoint, unbounded
D−1 - compact[D, ρ(a)] ∈ B(H) for all a ∈ A
γ ∈ B(H) - chirality - selfadjoint, unitary
γ2 = id, ∀a∈A γa = aγ, Dγ = −γDThe Z2-grading of the Hilbert space H = H+ ⊕H−
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 35: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/35.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
The axioms of noncommutative geometry
(A,H,D) - spectral triple
A - dense ∗-subalgebra of a C∗-algebra, unital
H - Hilbert space; by GNS construction∃ a faithful representation ρ(A) ' B(H)
D - the Dirac operator - selfadjoint, unbounded
D−1 - compact[D, ρ(a)] ∈ B(H) for all a ∈ A
γ ∈ B(H) - chirality - selfadjoint, unitary
γ2 = id, ∀a∈A γa = aγ, Dγ = −γDThe Z2-grading of the Hilbert space H = H+ ⊕H−
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
The axioms of noncommutative geometry
(A,H,D) - spectral triple
A - dense ∗-subalgebra of a C∗-algebra, unital
H - Hilbert space; by GNS construction∃ a faithful representation ρ(A) ' B(H)
D - the Dirac operator - selfadjoint, unbounded
D−1 - compact[D, ρ(a)] ∈ B(H) for all a ∈ A
γ ∈ B(H) - chirality - selfadjoint, unitary
γ2 = id, ∀a∈A γa = aγ, Dγ = −γDThe Z2-grading of the Hilbert space H = H+ ⊕H−
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 37: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/37.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
The axioms of noncommutative geometry
(A,H,D) - spectral triple
A - dense ∗-subalgebra of a C∗-algebra, unital
H - Hilbert space; by GNS construction∃ a faithful representation ρ(A) ' B(H)
D - the Dirac operator - selfadjoint, unbounded
D−1 - compact[D, ρ(a)] ∈ B(H) for all a ∈ A
γ ∈ B(H) - chirality - selfadjoint, unitary
γ2 = id, ∀a∈A γa = aγ, Dγ = −γDThe Z2-grading of the Hilbert space H = H+ ⊕H−
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 38: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/38.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
The axioms of noncommutative geometry
(A,H,D) - spectral triple
A - dense ∗-subalgebra of a C∗-algebra, unital
H - Hilbert space; by GNS construction∃ a faithful representation ρ(A) ' B(H)
D - the Dirac operator - selfadjoint, unbounded
D−1 - compact[D, ρ(a)] ∈ B(H) for all a ∈ A
γ ∈ B(H) - chirality - selfadjoint, unitary
γ2 = id, ∀a∈A γa = aγ, Dγ = −γDThe Z2-grading of the Hilbert space H = H+ ⊕H−
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 39: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/39.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
The axioms of noncommutative geometry
(A,H,D) - spectral triple
A - dense ∗-subalgebra of a C∗-algebra, unital
H - Hilbert space; by GNS construction∃ a faithful representation ρ(A) ' B(H)
D - the Dirac operator - selfadjoint, unbounded
D−1 - compact[D, ρ(a)] ∈ B(H) for all a ∈ A
γ ∈ B(H) - chirality - selfadjoint, unitary
γ2 = id, ∀a∈A γa = aγ, Dγ = −γDThe Z2-grading of the Hilbert space H = H+ ⊕H−
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 40: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/40.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
The axioms of noncommutative geometry
(A,H,D) - spectral triple
A - dense ∗-subalgebra of a C∗-algebra, unital
H - Hilbert space; by GNS construction∃ a faithful representation ρ(A) ' B(H)
D - the Dirac operator - selfadjoint, unbounded
D−1 - compact[D, ρ(a)] ∈ B(H) for all a ∈ A
γ ∈ B(H) - chirality - selfadjoint, unitary
γ2 = id, ∀a∈A γa = aγ, Dγ = −γDThe Z2-grading of the Hilbert space H = H+ ⊕H−
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 41: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/41.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
The axioms of noncommutative geometry
(A,H,D) - spectral triple
A - dense ∗-subalgebra of a C∗-algebra, unital
H - Hilbert space; by GNS construction∃ a faithful representation ρ(A) ' B(H)
D - the Dirac operator - selfadjoint, unbounded
D−1 - compact[D, ρ(a)] ∈ B(H) for all a ∈ A
γ ∈ B(H) - chirality - selfadjoint, unitary
γ2 = id, ∀a∈A γa = aγ, Dγ = −γDThe Z2-grading of the Hilbert space H = H+ ⊕H−
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 42: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/42.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
A commutative spectral triple
Example 1
Let M be compact Riemannian spin manifold, then
A = C∞(M) - dense in C0(M)
unital if M is compact
H = L2(S(M)
)representation: ρ(f)ξ = f(x) · ξ(x)
D = γµ∂µ - the Dirac operator
γ = γ5 - chirality operator
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 43: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/43.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
A commutative spectral triple
Example 1
Let M be compact Riemannian spin manifold, then
A = C∞(M) - dense in C0(M)
unital if M is compact
H = L2(S(M)
)representation: ρ(f)ξ = f(x) · ξ(x)
D = γµ∂µ - the Dirac operator
γ = γ5 - chirality operator
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 44: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/44.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
A commutative spectral triple
Example 1
Let M be compact Riemannian spin manifold, then
A = C∞(M) - dense in C0(M)
unital if M is compact
H = L2(S(M)
)representation: ρ(f)ξ = f(x) · ξ(x)
D = γµ∂µ - the Dirac operator
γ = γ5 - chirality operator
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 45: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/45.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
A commutative spectral triple
Example 1
Let M be compact Riemannian spin manifold, then
A = C∞(M) - dense in C0(M)
unital if M is compact
H = L2(S(M)
)representation: ρ(f)ξ = f(x) · ξ(x)
D = γµ∂µ - the Dirac operator
γ = γ5 - chirality operator
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 46: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/46.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
A commutative spectral triple
Example 1
Let M be compact Riemannian spin manifold, then
A = C∞(M) - dense in C0(M)
unital if M is compact
H = L2(S(M)
)representation: ρ(f)ξ = f(x) · ξ(x)
D = γµ∂µ - the Dirac operator
γ = γ5 - chirality operator
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 47: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/47.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
A commutative spectral triple
Example 1
Let M be compact Riemannian spin manifold, then
A = C∞(M) - dense in C0(M)
unital if M is compact
H = L2(S(M)
)representation: ρ(f)ξ = f(x) · ξ(x)
D = γµ∂µ - the Dirac operator
γ = γ5 - chirality operator
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 48: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/48.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
The axiomsExample
A commutative spectral triple
Example 1
Let M be compact Riemannian spin manifold, then
A = C∞(M) - dense in C0(M)
unital if M is compact
H = L2(S(M)
)representation: ρ(f)ξ = f(x) · ξ(x)
D = γµ∂µ - the Dirac operator
γ = γ5 - chirality operator
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 49: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/49.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Outline
1 “Standard” anomalies
2 Noncommutative spaces
3 Anomalies on spectral triples
4 Conclusions
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Some results
Moyal plane R4?
(J.M. Gracia-Bondıa, C.P. Martın, Phys. Lett. B 479 (2000) 321)
A = α TrTa, T bT c + β Tr[Ta, T b]T c
Geometric approach(D. Perrot, Contemp. Math. 434 (2007) 125.)
The role of projections in A.
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Some results
Moyal plane R4?
(J.M. Gracia-Bondıa, C.P. Martın, Phys. Lett. B 479 (2000) 321)
A = α TrTa, T bT c + β Tr[Ta, T b]T c
Geometric approach(D. Perrot, Contemp. Math. 434 (2007) 125.)
The role of projections in A.
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Perrot’s procedure
Assumption: p-summability i.e. Tr(D−p) <∞, r - biggest integer < p
The action of the “classical” theory
S(ψ, ψ) =⟨ψ, (Q+A)ψ
⟩, Q = D + γm, A ∈ B(H)
Path-integral quantization
Z(A) =
∫dψdψ e−S(ψ,ψ,A) = det(1 +Q−1A)
W (A) = Tr log(1 +Q−1A) =
∞∑n=1
(−1)n+1
nTr(Q−1A)n
Regularization - trace extension τ(T ) = Resz=0
1z Tr(T |Q|−2z)
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Perrot’s procedure
Assumption: p-summability i.e. Tr(D−p) <∞, r - biggest integer < p
The action of the “classical” theory
S(ψ, ψ) =⟨ψ, (Q+A)ψ
⟩, Q = D + γm, A ∈ B(H)
Path-integral quantization
Z(A) =
∫dψdψ e−S(ψ,ψ,A) = det(1 +Q−1A)
W (A) = Tr log(1 +Q−1A) =
∞∑n=1
(−1)n+1
nTr(Q−1A)n
Regularization - trace extension τ(T ) = Resz=0
1z Tr(T |Q|−2z)
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 54: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/54.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Perrot’s procedure
Assumption: p-summability i.e. Tr(D−p) <∞, r - biggest integer < p
The action of the “classical” theory
S(ψ, ψ) =⟨ψ, (Q+A)ψ
⟩, Q = D + γm, A ∈ B(H)
Path-integral quantization
Z(A) =
∫dψdψ e−S(ψ,ψ,A) = det(1 +Q−1A)
W (A) = Tr log(1 +Q−1A) =
∞∑n=1
(−1)n+1
nTr(Q−1A)n
Regularization - trace extension τ(T ) = Resz=0
1z Tr(T |Q|−2z)
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 55: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/55.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Perrot’s procedure
Assumption: p-summability i.e. Tr(D−p) <∞, r - biggest integer < p
The action of the “classical” theory
S(ψ, ψ) =⟨ψ, (Q+A)ψ
⟩, Q = D + γm, A ∈ B(H)
Path-integral quantization
Z(A) =
∫dψdψ e−S(ψ,ψ,A) = det(1 +Q−1A)
W (A) = Tr log(1 +Q−1A) =
∞∑n=1
(−1)n+1
nTr(Q−1A)n
Regularization - trace extension τ(T ) = Resz=0
1z Tr(T |Q|−2z)
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 56: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/56.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Perrot’s procedure
BRST operator - de Rham differential on S1
s : C∞(S1)⊗A → Ω1(S1)⊗A.
Idempotent loopsg = 1 + (β − 1)p, C∞(S1) 3 β = e2π i t, A 3 p = p2
Note: (A,H,D)←→(MN (A),H⊗ CN ,D ⊗ 1
)Mauer-Cartan form (ghost field) ω = g−1sg = 2π i p dt
The potential is pure gauge A = g−1Qg −Q
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 57: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/57.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Perrot’s procedure
BRST operator - de Rham differential on S1
s : C∞(S1)⊗A → Ω1(S1)⊗A.
Idempotent loopsg = 1 + (β − 1)p, C∞(S1) 3 β = e2π i t, A 3 p = p2
Note: (A,H,D)←→(MN (A),H⊗ CN ,D ⊗ 1
)Mauer-Cartan form (ghost field) ω = g−1sg = 2π i p dt
The potential is pure gauge A = g−1Qg −Q
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 58: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/58.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Perrot’s procedure
BRST operator - de Rham differential on S1
s : C∞(S1)⊗A → Ω1(S1)⊗A.
Idempotent loopsg = 1 + (β − 1)p, C∞(S1) 3 β = e2π i t, A 3 p = p2
Note: (A,H,D)←→(MN (A),H⊗ CN ,D ⊗ 1
)Mauer-Cartan form (ghost field) ω = g−1sg = 2π i p dt
The potential is pure gauge A = g−1Qg −Q
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 59: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/59.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Perrot’s procedure
BRST operator - de Rham differential on S1
s : C∞(S1)⊗A → Ω1(S1)⊗A.
Idempotent loopsg = 1 + (β − 1)p, C∞(S1) 3 β = e2π i t, A 3 p = p2
Note: (A,H,D)←→(MN (A),H⊗ CN ,D ⊗ 1
)Mauer-Cartan form (ghost field) ω = g−1sg = 2π i p dt
The potential is pure gauge A = g−1Qg −Q
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 60: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/60.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Perrot’s procedure
BRST operator - de Rham differential on S1
s : C∞(S1)⊗A → Ω1(S1)⊗A.
Idempotent loopsg = 1 + (β − 1)p, C∞(S1) 3 β = e2π i t, A 3 p = p2
Note: (A,H,D)←→(MN (A),H⊗ CN ,D ⊗ 1
)Mauer-Cartan form (ghost field) ω = g−1sg = 2π i p dt
The potential is pure gauge A = g−1Qg −Q
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 61: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/61.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Perrot’s procedure
The anomaly A(ω,A) := sWτ (A) =
r∑n=1
(−1)n+1
ns τ(Q−1A)n + (−1)r Tr
((ω+ −Q−1ω−Q)(Q−1A)r
),
Theorem (Perrot)
1
〈[D], [p]〉 =1
2π i
∮S1
A(ω,A) ∈ Z,
2 A(ω,A) is cohomologous, as a one-form, to the sum of residues
A(ω,A) := Resz=0
1
zTr(γωQ−2z
)+∑n≥1
∑k∈Nn
(−1)n+kc(k)×
× Resz=0
Γ(z + n+ k)
Γ(z + 1)Tr(
1+γ2
[ω,Q]A(k1)QA(k2)Q . . . A(kn)Q−2(z+n+k)).
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Perrot’s procedure
The anomaly A(ω,A) := sWτ (A) =
r∑n=1
(−1)n+1
ns τ(Q−1A)n + (−1)r Tr
((ω+ −Q−1ω−Q)(Q−1A)r
),
Theorem (Perrot)
1
〈[D], [p]〉 =1
2π i
∮S1
A(ω,A) ∈ Z,
2 A(ω,A) is cohomologous, as a one-form, to the sum of residues
A(ω,A) := Resz=0
1
zTr(γωQ−2z
)+∑n≥1
∑k∈Nn
(−1)n+kc(k)×
× Resz=0
Γ(z + n+ k)
Γ(z + 1)Tr(
1+γ2
[ω,Q]A(k1)QA(k2)Q . . . A(kn)Q−2(z+n+k)).
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Perrot’s procedure
The anomaly A(ω,A) := sWτ (A) =
r∑n=1
(−1)n+1
ns τ(Q−1A)n + (−1)r Tr
((ω+ −Q−1ω−Q)(Q−1A)r
),
Theorem (Perrot)
1
〈[D], [p]〉 =1
2π i
∮S1
A(ω,A) ∈ Z,
2 A(ω,A) is cohomologous, as a one-form, to the sum of residues
A(ω,A) := Resz=0
1
zTr(γωQ−2z
)+∑n≥1
∑k∈Nn
(−1)n+kc(k)×
× Resz=0
Γ(z + n+ k)
Γ(z + 1)Tr(
1+γ2
[ω,Q]A(k1)QA(k2)Q . . . A(kn)Q−2(z+n+k)).
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Results
The anomaly result A(ω,A) = A(ω,A) =
4π i dtResz=0
tr((β − 1)[D−, p][D+, p] + (2− β − β−1)[D−, p]p[D+, p]
)Q−2z−2
The index result
1
2π i
∮S1
A(ω,A) = 2Resz=0
tr(− [D−, p][D+, p] + 2[D−, p]p[D+, p]
)Q−2z−2
Agreement with Connes-Moscovici
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Results
The anomaly result A(ω,A) = A(ω,A) =
4π i dtResz=0
tr((β − 1)[D−, p][D+, p] + (2− β − β−1)[D−, p]p[D+, p]
)Q−2z−2
The index result
1
2π i
∮S1
A(ω,A) = 2Resz=0
tr(− [D−, p][D+, p] + 2[D−, p]p[D+, p]
)Q−2z−2
Agreement with Connes-Moscovici
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Results
The anomaly result A(ω,A) = A(ω,A) =
4π i dtResz=0
tr((β − 1)[D−, p][D+, p] + (2− β − β−1)[D−, p]p[D+, p]
)Q−2z−2
The index result
1
2π i
∮S1
A(ω,A) = 2Resz=0
tr(− [D−, p][D+, p] + 2[D−, p]p[D+, p]
)Q−2z−2
Agreement with Connes-Moscovici
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Projections
V U = e2π i θ UV, f(V ) =∑n∈Z
fn Vn
Power-Rieffel projection p = f(V )U + U−1f(V ∗) + g(V )
σ(p) = θ
〈[D], [p]〉 = −1
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Some resultsGeometrical approachNoncommutative 2-torus
Projections
V U = e2π i θ UV, f(V ) =∑n∈Z
fn Vn
Second order projectionp(2) = p2(V )U−2 + p1(V )U−1 + p0(V ) + Up1(V ∗) + U2p2(V ∗)
⟨[D], [p(2)]
⟩d2 = 0 d2 = 1
d3 = 0 2 4d3 = 1 −4 −2
σ(p(2)) d2 = 0 d2 = 1d3 = 0 θ 2θd3 = 1 1− 2θ 1− θ
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Conclusions
Anomaly cancellation condition
Classically: TrT a, T bT c
Moyal plane: TrT aT bT c
Generally: ?
The geometry of anomalies
The classical correspondence not clear
The role of projections
Thank you for your attention!
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Conclusions
Anomaly cancellation condition
Classically: TrT a, T bT c
Moyal plane: TrT aT bT c
Generally: ?
The geometry of anomalies
The classical correspondence not clear
The role of projections
Thank you for your attention!
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
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“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Conclusions
Anomaly cancellation condition
Classically: TrT a, T bT c
Moyal plane: TrT aT bT c
Generally: ?
The geometry of anomalies
The classical correspondence not clear
The role of projections
Thank you for your attention!
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 72: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/72.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Conclusions
Anomaly cancellation condition
Classically: TrT a, T bT c
Moyal plane: TrT aT bT c
Generally: ?
The geometry of anomalies
The classical correspondence not clear
The role of projections
Thank you for your attention!
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 73: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/73.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Conclusions
Anomaly cancellation condition
Classically: TrT a, T bT c
Moyal plane: TrT aT bT c
Generally: ?
The geometry of anomalies
The classical correspondence not clear
The role of projections
Thank you for your attention!
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 74: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/74.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Conclusions
Anomaly cancellation condition
Classically: TrT a, T bT c
Moyal plane: TrT aT bT c
Generally: ?
The geometry of anomalies
The classical correspondence not clear
The role of projections
Thank you for your attention!
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 75: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/75.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Conclusions
Anomaly cancellation condition
Classically: TrT a, T bT c
Moyal plane: TrT aT bT c
Generally: ?
The geometry of anomalies
The classical correspondence not clear
The role of projections
Thank you for your attention!
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry
![Page 76: Quantum Anomalies in Noncommutative Geometryth- · 2010-06-11 · Quantum Anomalies in Noncommutative Geometry Micha l Eckstein Jagiellonian University, Krak ow Zakopane, june 11,](https://reader034.vdocuments.site/reader034/viewer/2022050600/5fa80e7bbc0ed7137747ecc8/html5/thumbnails/76.jpg)
“Standard” anomaliesNoncommutative spaces
Anomalies on spectral triplesConclusions
Conclusions
Anomaly cancellation condition
Classically: TrT a, T bT c
Moyal plane: TrT aT bT c
Generally: ?
The geometry of anomalies
The classical correspondence not clear
The role of projections
Thank you for your attention!
Micha l Eckstein Quantum Anomalies in Noncommutative Geometry