Ma 148b Topics in Mathematical Physics, Winter 2016: Noncommutative Geometry Models
for Particle Physics and Cosmology
Instructor:
Matilde Marcolli
Schedule: the class meets on Tuesday-Thursday 10:30-11:55 am in SLN 159
Brief Course Description
This class will cover recent approaches to geometric modeling for
particle physics and cosmology, based on Noncommutative Geometry
and the Spectral Action functional.
References
Books:
- Walter van Suijlekom, "Noncommutative Geometry and Particle Physics",
Springer 2014
- Wim Beenakker, Thijs van den Broek, Walter van Suijlekom, "Supersymmetry
and Noncommutative Geometry", Springer, 2016
(pdf)
- Alain Connes, Matilde Marcolli, "Noncommutative geometry, quantum fields
and motives", American Mathematical Society, 2008
- Matilde Marcolli, "Noncommutative Cosmology" (book in preparation): draft
notes will be handed out in class
Papers:
- pdf
A.H.Chamseddine, A.Connes, "The Spectral Action Principle",
Commun.Math.Phys.186:731-750,1997
- pdf
A.H.Chamseddine, A.Connes, M.Marcolli, "Gravity and the Standard
Model with Neutrino Mixing", Advances in Theoretical and Mathematical Physics,
11 (2007) 991-1090
- pdf A.H.Chamseddine, A.Connes,
"Resilience of the Spectral Standard Model", JHEP 1209 (2012) 104
- pdf
C.Estrada, M.Marcolli, "Asymptotic safety, hypergeometric functions and the
Higgs mass in spectral action models", International Journal of Geometric
Methods in Modern Physics, Vol.10 (2013) N.7, 1350036
- pdf A.H.Chamseddine, A.Connes,
W. van Suijlekom, "Beyond the Spectral Standard Model: Emergence of Pati-Salam
Unification", JHEP 1311 (2013) 132
- pdf A.H.Chamseddine, A.Connes,
W. van Suijlekom, "Grand Unification in the Spectral Pati-Salam Model", arXiv:1507.08161
-
pdf M.Marcolli, W.D.van Suijlekom, "Gauge Networks in Noncommutative
Geometry", Journal of Geometry and Physics, Vol.75 (2014) 71-91
- pdf
J.W.Barrett, L.Glaser, "Monte Carlo simulations of
random non-commutative geometries", arXiv:1510.01377
- pdf
M.Marcolli, E.Pierpaoli, "Early Universe models from Noncommutative Geometry"
Advances in Theoretical and Mathematical Physics, Vol.14 (2010) 1373-1432
- pdf
M.Marcolli, E.Pierpaoli, K.Teh, "The spectral action and cosmic topology",
Communications in Mathematical Physics, 304 (2011) 125-174
- pdf
M.Marcolli, E.Pierpaoli, K.Teh, "The coupling of topology and inflation
in noncommutative cosmology", Communications in Mathematical Physics,
Vol.309 (2012) N.2, 341-369
-
pdf B.Cacic, M.Marcolli, K.Teh, "Coupling of gravity to matter, spectral
action and cosmic topology", Journal of Noncommutative Geometry,
Vol.8 (2014) N.2, 473-504
- pdf
A.H.Chamseddine, A.Connes, "Spectral Action for Robertson-Walker metrics",
JHEP 1210 (2012) 101
- pdf
W.Fan, F.Fathizadeh, M.Marcolli, "Spectral Action for Bianchi Type-IX Cosmological Models",
arXiv:1506.06779
- pdf
W.Fan, F.Fathizadeh, M.Marcolli, "Modular forms in the spectral action
of Bianchi IX gravitational instantons", arXiv:1511.05321
- pdf
A.Ball, M.Marcolli, "Spectral Action models of gravity on Packed Swiss
Cheese Cosmology", arXiv:1506.01401
Suggested Readings for student presentations
- pdf B.Cacic,
"A reconstruction theorem for almost-commutative spectral triples",
Lett. Math. Phys. 100 (2012), no. 2, 181-202
- pdf
W.D.van Suijlekom, "Renormalizability conditions for almost-commutative geometries",
Ann. Henri Poincare' 15 (2014), no. 5, 985-1011
- pdf
S.Farnsworth, L.Boyle, "Non-Associative Geometry and the Spectral Action Principle",
JHEP 1507 (2015) 023
- pdf
A.H.Chamseddine, A.Connes, V.Mukhanov, "Geometry and the Quantum: Basics",
JHEP 12 (2014) 098
- pdf
A.H.Chamseddine, A.Connes, "Noncommutative Geometric Spaces with Boundary:
Spectral Action", J.Geom.Phys.61:317-332,2011
- pdf
B.Iochum, C.Levy, "Spectral triples and manifolds with boundary",
J. Funct. Anal. 260 (2011), no. 1, 117-134
- pdf
B.Iochum, C.Levy, "Tadpoles and commutative spectral triples",
J.Noncommut.Geom. 5:299-329,2011
- pdf
B.Iochum, C.Levy, D.Vassilevich, "Spectral action beyond the weak-field approximation",
Commun. Math. Phys. 316, 595-613 (2012)
- pdf
M.A.Kurkov, F.Lizzi, M.Sakellariadou, A.Watcharangkool,
"Spectral action with zeta function regularization", Phys.Rev. D91 (2015) 6, 065013
- pdf
G.Lambiase, M.Sakellariadou, A.Stabile, "Constraints on NonCommutative
Spectral Action from Gravity Probe B and Torsion Balance Experiments",
JCAP12(2013)020
- pdf
W.Nelson, J.Ochoa, M.Sakellariadou
"Gravitational Waves in the Spectral Action of Noncommutative Geometry"
Phys.Rev.D82:085021,2010
- pdf
M.Buck, M.Fairbairn, M.Sakellariadou,
"Inflation in models with Conformally Coupled Scalar fields: An application
to the Noncommutative Spectral Action", Phys.Rev.D82:043509,2010
- pdf
A.H.Chamseddine, A.Connes, "The Uncanny Precision of the Spectral Action",
Commun.Math.Phys.293:867-897,2010
- pdf
A.H.Chamseddine, A.Connes, "Inner fluctuations of the spectral action",
J.Geom.Phys.57:1-21,2006
- pdf
A.H.Chamseddine, A.Connes, "Scale Invariance in the Spectral Action",
J.Math.Phys.47:063504,2006
- pdf
W.D. van Suijlekom, "Perturbations and operator trace functions",
J.Funct.Anal. 260 (2011), no. 8, 2483-2496
- pdf
F.Fathizadeh, A.Ghorbanpour, M.Khalkhali, "Rationality of Spectral Action
for Robertson-Walker Metrics", JHEP 12 (2014) 064
- pdf
F.Fathizadeh, M.Khalkhali, "The Algebra of Formal Twisted Pseudodifferential
Symbols and a Noncommutative Residue", Lett.Math.Phys. 94 N.1 (2010) 41-61
- pdf
W.Kalau, M.Walze, "Gravity, Non-Commutative Geometry and the Wodzicki Residue",
J.Geom.Phys. 16 (1995) 327-344
- pdf
D.Denicola, M.Marcolli, A.Zainy al-Yasry, "Spin foams and
noncommutative geometry", Class.Quant.Grav.27:205025,2010
- pdf
M.Marcolli, A.Zainy al-Yasry, "Coverings, correspondences, and noncommutative geometry", J. Geom. Phys. 58 (2008), no. 12, 1639-1661
- pdf
J.W.Barrett, "Matrix geometries and fuzzy spaces as finite spectral triples",
arXiv:1502.05383
- pdf
K.van den Dungen, W.D.van Suijlekom, "Electrodynamics from Noncommutative Geometry",
J. Noncommut. Geom. 7, 433-456 (2013)
- pdf
J.Boeijink, W.D.van Suijlekom, "The noncommutative geometry of Yang-Mills fields",
J.Geom.Phys.61:1122-1134,2011
- pdf
G.V.Dunne, "Heat Kernels and Zeta Functions on Fractals",
J.Phys. A45 (2012) 374016
- pdf
M.Eckstein, B.Iochum, A.Sitarz, "Heat trace and spectral action
on the standard Podles sphere", Commun. Math. Phys. 332 (2014)
N.2, 627-668
Additional Reading Material
- pdf
S.Brain, B.Mesland, W.D.van Suijlekom, "Gauge Theory for Spectral Triples
and the Unbounded Kasparov Product", arXiv:1306.1951
- pdf
M.V.Gargiulo, M.Sakellariadou, G.Vitiello, "Noncommutative spectral geometry,
Bogoliubov transformations and neutrino oscillations", arXiv:1502.02919
- pdf
M.A.Kurkov, M.Sakellariadou, "Spectral Regularisation: Induced Gravity and the
Onset of Inflation", JCAP 01(2014)035
- pdf
S.Farnsworth, L.Boyle, "Rethinking Connes' approach to the standard model
of particle physics via non-commutative geometry", New J. Phys. 17, 023021 (2015)
- pdf
L.Boyle, S.Farnsworth, "Non-Commutative Geometry, Non-Associative Geometry
and the Standard Model of Particle Physics", New J. Phys. 16, 123027 (2014)
-
pdf C.Estrada, M.Marcolli, "Noncommutative Mixmaster Cosmologies",
International Journal of Geometric Methods in Modern Physics, Vol.10 (2013) N.1
- pdf
D.Kolodrubetz, M.Marcolli, "Boundary conditions of the RGE flow in the noncommutative geometry approach to particle physics and cosmology", Physics Letters B 693 (2010) 166-174
- pdf
M.Marcolli, "Building cosmological models via noncommutative geometry",
International Journal of Geometric Methods in Modern Physics, Vol.8, N.5
(2011) 1131-1168
- pdf
Mairi Sakellariadou, "Cosmology within Noncommutative Spectral Geometry",
arXiv:1101.2174, PoS CNCFG2010:028,2010
- pdf
Mairi Sakellariadou, "Cosmological consequences of the noncommutative spectral
geometry as an approach to unification", J.Phys.Conf.Ser.283:012031,2011
- pdf
W.Nelson, J.Ochoa, M.Sakellariadou, "Constraining the Noncommutative Spectral
Action via Astrophysical Observations", Phys.Rev.Lett.105:101602,2010
- pdf
W.Nelson, M.Sakellariadou, "Natural inflation mechanism in asymptotic
noncommutative geometry", Phys.Lett.B680:263-266,2009
- pdf
P.Olczykowski, A.Sitarz, "On spectral action over Bieberbach manifolds",
Acta Phys.Polon.B42:1189-1198,2011
- pdf
A.H.Chamseddine, A.Connes, "Quantum Gravity Boundary Terms from Spectral
Action of Noncommutative Space" Phys.Rev.Lett.99:071302,2007
- pdf M.A.Kurkov, F.Lizzi
"Higgs-Dilaton Lagrangian from Spectral Regularization",
Mod. Phys. Lett. A, Vol. 27, No. 35 (2012) 1250203
- pdf
A.A.Andrianov, M.A.Kurkov, F.Lizzi, "Spectral action, Weyl anomaly and the
Higgs-Dilaton potential", arXiv:1106.3263
- pdf
H.Grosse, F.Lizzi, H.Steinacker, "Noncommutative gauge theory and symmetry
breaking in matrix models", Phys.Rev.D81:085034,2010
- pdf
A.A. Andrianov, F.Lizzi, "Bosonic Spectral Action Induced from Anomaly Cancellation"
JHEP 1005:057,2010
- pdf
T. Krajewski, "Classification of Finite Spectral Triples", J.Geom.Phys. 28 (1998) 1-30
- pdf
B.Iochum, T.Schucker, C.Stephan, "On a Classification of Irreducible Almost
Commutative Geometries, J.Math.Phys. 45 (2004) 5003-5041
Notes of classes
- Tuesday January 5:
General introduction: review of manifolds,
vector bundles, tensors, and connections; the action principle in physics:
classical mechanics, general relativity, Yang-Mills, the pertubative QFT
approach; the Standard-Model of elementary particles, quick overview
- Thursday January 7:
Introduction to noncommutative geometry:
quantum commutation relations, geometry of bad quotients, different
occurrences in physics, vector bundles and finite projective modules,
connections, spin geometry and Dirac operator, spectral triples
- Tuesday January 12:
Finite Spectral
Triples: algebras and representations, unitary equivalence, structure
space (spectrum), modules and bimodules, Hilbert modules, Morita equivalence,
metric, 1-forms, unitary and Morita equivalence of spectral triples,
compatible connection, classification, diagrams, real structure, KO-dimension,
unitary and Morita equivalence, classification and Krajewski diagrams, real
algebras, moduli spaces of Dirac operators
- Thursday January 14:
Models based on finite spectral triples: Gauge networks: categories
of finite spectral triples, quiver representations in finite spectral
triples, braid Bratteli diagrams, gauge networks and correspondences,
spectral action for field theory on a lattice (graph), continuum limit,
twisted Dirac, quartic spectral action, Wilson loop action with Higgs field;
Random noncommutative geometries: quartic spectral action, Clifford
modules types, Monte Carlo simulations, density of states
- Tuesday January 19:
The
Spectral Geometry of the Standard Model: the left-right symmetric
finite real spectral triples, odd bimodules and fermions, matter-antimatter
involution (real structure), grading and KO-dimension, subalgebra and
order one condition, left-right symmetry breaking
- Thursday January 21: continuation of the standard model spectral
triple, structure of Dirac operator, moduli space of Dirac operators,
mixing matrices and Yukawa parameters, gauge group and hypercharges,
gauge symmetries and diffeomorphisms, product of even real
spectral triples
- Tuesday January 26:
The
Spectral Action and the Standard Model: Spectral action
functional, heat kernel and zeta function, residues and Seeley deWitt
coefficients, asymptotic expansion, fermionic part of the spectral action,
Grassmann variables, spectral triple of the standard model: expansion
of the spectral action, gravity, Higgs, and Yang-Mills terms
- Thursday January 28: continuation of The Spectral Action:
Yukawa parameters and momenta of the test function, Gilkey's theorem
and Seeley deWitt coefficients, normalization of kinetic terms,
coupling constants at unification, 1-loop Renromalization Group Equations,
approximations, Higgs self coupling and top quark Yukawa parameter,
geometric constraints at unification, see-saw mechanism, low energy limits,
excessive Higgs mass problem
- Tuesday February 2: student presentations
- Thursday February 4: student presentations
- Tuesday February 9:
The Higgs Mass in the NCG Standard Model and
Spectral Triples and Supersymmetry
- Thursday February 11:
Adinkras, Supersymmetry, and Spectral Triples
- Tuesday February 16:
Early Universe Models
- Thursday February 18:
Spectral Action and Cosmic Topology
- Tuesday February 23:
Spectral Action of Swiss Cheese Cosmology
- Thursday February 25:
Symbolic dynamics, modular curves, and Bianchi IX cosmologies
- Tuesday March 1:
Bianchi IX cosmologies, spectral action, and modular forms
- Thursday March 3: student presentations
- Tuesday March 8: Spectral triples and Pati-Salam GUT
- Thursday March 10: student presentations
Syllabus
The course is offered Pass/Fail. Requirements: one oral presentation
on reading materials chosen from the list provided above (first come
first serve) and attendance of (most) lectures.
Schedule of students presentations
We will have a seminar time for presentations on Thursday
afternoon, 5-6 pm, Room SLN 159
- Thursday January 14: Yujie Xu: Coverings and Correspondences
- Thursday January 21: no presentation today
- Thursday January 28: Konrad Pilch:
Matrix Geometries and Fuzzy Spaces as Finite Spectral Triples
(CANCELLED! will be rescheduled)
- Tuesday February 2 (class time): Ryan Yoo: Gravitational
Waves and the Spectral Action
- Thursday February 4 (class time): Gabriel Foster:
Non-Associative Geometry and the Spectral Action Principle
- Thursday February 11: Wentao Fan:
The noncommutative geometry of Yang-Mills fields
- Thursday February 18: Konrad Pilch:
Matrix Geometries and Fuzzy Spaces as Finite Spectral Triples
- Thursday February 25: Thomas Norton: Electrodynamics from
Noncommutative Geometry
- Thursday March 3 (class time): Mike McAneny:
Heat Kernels and Zeta Functions on Fractals
- Thursday March 3 (seminar time 5pm): Joshua Lieber: holography
and algebraic curves
- Thursday March 10 (class time): Oscar Mickelin:
The Uncanny Precision of the Spectral Action
- Thursday March 10 (class time): Karthik Siva:
Boundary conditions of the RGE flow