Ma148a Fall 2022 and Ma148b Winter 2023 Topics in Mathematical Physics: Motives and Periods in Quantum Field Theory
Fall 2022: Caltech, Linde Hall [NOTE: ROOM CHANGE, now meeting in Room 255],
Tuesday-Thursday 9:00-10:30 am
WINTER 2023: Linde Hall Room 187, Tuesday-Thursday 9:00-10:30 am [NOTE: SCHEDULE CHANGE, the class is keeping the same schedule as in the Fall term]
Instructor:
Matilde Marcolli
Brief Course Description
This course presents an introduction to motives and
periods of algebraic varieties and their occurrence
and role in Quantum Field Theory.
The main topics covered will
include a general introduction to
motives: pure motives, motivic
Galois groups,
periods of algebraic varieties, the Grothendieck ring
of varieties, motivic zeta functions, an introduction
to mixed motives, mixed Tate motives; applications of
the theory of motives to quantum field theory:
Feynman integrals of scalar field theories,
graph hypersurfaces and periods; Feynman integrals
in configuration and momentum space; Feynman integrals
in configuration space, the algebraic geometry of
wonderful compactifications and periods; Feynman
integrals in momentum space and determinant hypersurfaces;
algebraic renormalization via Hopf algebras and
Rota-Baxter algebras
Notes of Classes
scanned notes of classes will be posted here.
FALL TERM
- pdf introduction to perturbative QFT
- pdf introduction to perturbative QFT; parametric Feynman integrals
- pdf algebro-geometric preliminaries
- pdf origins of the theory of motives
- pdf Grothendieck rings of varieties and motives
- pdf
L-functions, Grothendieck ring and stable birational equivalence
- pdf graph hypersurfaces
- pdf Grothendieck classes of graph hypersurface, deletion-contraction relations
- pdf Grothendieck classes of sums of graphs, dichotomy
- pdf Graph complete intersections
- pdf Universality of matroids and Grothendieck classes
- pdf Parametric Feynman integrals and determinant hypersurfaces part 1
- pdf Parametric Feynman integrals and determinant hypersurfaces part 2
WINTER TERM
- pdf review of deRham cohomology and Hodge decomposition
- pdf Pure motives: algebraic cycles and equivalence relations, Weil cohomologies
- pdf construction of the category ofpure motives (and additional material not covered in class)
- pdf construction of the category of pure motives, examples, Manin's identity principle, motives of blowups, Grothendieck standard conjectures
- pdf Grothendieck standard conjectures (additional material), affine group schemes and Tannakian formalism
- pdf the abelian category of numerical motives, sign constraint and Tannakian structure, motivic Galois group
- pdf Mixed Motives, the Voevodsky category
- pdf Mixed Motives continuation
- pdf Mixed Motives, mixed Tate motives
- pdf Mixed Motives, supplementary notes on Nori motives
- pdf Periods and Motives
- pdf Periods and multiple zeta values
- pdf Multiple Zeta Values, continued
- pdf Feynman integrals, periods and the Igusa Zeta function
- pdf Algebraic Renormalization
- pdf Connes-Kreimer
theory of renormalization
Summary of Lectures
FALL TERM
- Tuesday September 27: a quick introduction to perturbative quantum field theory, finite dimensional Gaussian integrals, source and interaction terms, Green functions, expansion in graphs
- Thursday September 29: a quick introduction to perturbative quantum field theory, Euclidean scalar field theory, formal Gaussian integrals, expansion in Green functions, Feynman rules, Feynman graphs, Schwinger and Feynman parameters, parametric Feynman integrals
- Tuesday October 4: Feynman parameters, parametric Feynman integrals, graph hypersurfaces
- Thursday October 6: preliminary algebro-geometric notions for the theory of motives: algebraic varieties, algebraic cycles, homology-cohomology, categories
- Tuesday October 11: background to the theory of motives: universal cohomology theory; pure and mixed motives, and Grothendieck classes as universal Euler characteristic; Grothendieck ring of varieties, resolution of singularities and weak factorization, Bittner relations
- Thursday October 13: Grothendieck ring of varieties and stable birational equivalence classes
- Tuesday October 18: stable birational equivalence and Grothendieck ring, zero-divisors in the Grothendieck ring, graph hypersurfaces of Feynman graphs
- Thursday October 20: example of "banana graphs", Cremona tranformation and computation of Grothendieck classes, affine and projective graph hypersurfaces, differential forms on hypersurface complements and the parametric Feynman integral
- Tuesday October 25: singular locus of the graph hypersurfaces, hypersurface complements and abstract Feynman rules
- Thursday October 27: deletion-contraction relations: Tutte polynomial, Tutte-Grothendieck invariants, Kirchhoff polynomial deletion-contraction, partial deletion-contraction for Grothendieck classes, recursive formula for edge doubling
- Tuesday November 1: Grothendieck classes of sums of graphs are in the Tate subring
- Thursday November 3: Hypersurfaces for extended Faynman integrals fibering over the graph hypersurfaces with Tate classes; outline of Belkale-Brosnan
- Tuesday November 8: Belkale-Brosnan theorem, graph hypersurfaces and incidence schemes
- Thursday November 10: Belkale-Brosnan theorem, matroid schemes and universality
- Tuesday November 15: dichotomy in the Grothendieck ring and stable birational equivalence classes of graph hypersurfaces
- Thursday November 17: Feynman integrals on determinant hypersurfaces, combinatorial conditions for injectivity and connectivity of graphs
- Tuesday November 22: connectivity conditions, Grothendieck classes of determinant hypersurfaces
- Thursday November 24: Thanksgiving holiday (no class)
- Tuesday November 29: Grothendieck classes of varieties of flags and the Tate property
- Thursday December 1: Student presentations
- Tuesday December 6: Student presentations
- Thursday December 8: Student presentations
WINTER TERM
- Thursday January 5: review of de Rham and Hodge
- Tuesday January 10: algebraic cycles and equivalence relations, Weyl cohomologies
- Thursday January 12: construction of the category of pure motives
- Tuesday January 17: Grothendieck standard conjectures
- Thursday January 19: Tannakian formalism, sign constraint for numerical motives, motivic Galois groups
- Tuesday January 24: Bloch-Ogus cohomolgoies,
triangulated categories, construction of the
Voevodsky triangulated category of mixed motives
- Thursday January 26: mixed Tate motives, hearts of t-structures
in triangulated categories, mixed Tate motives over number fields
- Tuesday January 31: periods and motives, Grothendieck and Konstevich conjectures on periods
- Thursday February 2: periods and multiple zeta values, MZVs as periods of mixed Tate motives
- Tuesday February 7: structures and relations for multiple zeta values
- Thursday February 9: Feynman integrals and periods and the Igusa zeta function
- Tuesday February 14: renormalization: Connes-Kreimer Hopf algebra
- Thursday February 16: renormalization: Birkhoff factorization of loops and the Riemann-Hilbert problem
- Tuesday February 21: renormalization: mass scale dependence in dimensional regularization, classification of physical Feynman rules
- Thursday February 23: renormalization: flat equisingular connections
- Tuesday February 28: renormalization: Tannakian category of flat equisingular vector bundles, universal renormalization group flow
- Thursday March 2: Feynman integrals in configuration spaces
- Tuesday March 7: Feynman integrals in configuration spaces: wonderful compactifications
- Thursday March 9: student presentations
Some book references
There is no official textbook for this class, but the following books
will be useful references (available in the Caltech library)
- M.Marcolli, "Feynman Motives", World Scientific, 2010.
- A. Connes and M. Marcolli, "Noncommutative geometry, quantum fields and motives", American Mathematical Society, 2008.
- Yves Andre, "Une introduction aux motifs", Societe Mathematique de France, 2004.
Note: Yves Andre's book can be downloaded as free ebook
here
Other Reading Material
Suggested reading material including both material covered in the
lectures and possible suggestions for student presentations (more will
be added as the class progresses), see below for papers that have already
been used for persentations in the Fall term.
-
pdf S.Bloch, H.Esnault, D.Kreimer, "On motives associated to
graph polynomials"
- pdf S.Bloch, "Motives associated to graphs"
- pdf
S.Bloch, "Motives associated to graph sums"
- pdf
P.Aluffi, M.Marcolli, "Feynman motives of banana graphs",
Communications in Number Theory and Physics, Vol.3 (2009) N.1, 1-57
- pdf
P.Aluffi, M.Marcolli, "Algebro-geometric Feynman rules"
-
pdf P.Belkale, P.Brosnan, "Matroids, motives and a conjecture
of Kontsevich"
- pdf
P.Aluffi, M.Marcolli, "Graph hypersurfaces and a dichotomy in
the Grothendieck ring"
-
pdf S.Bloch, "A note on Hodge structures associated to graphs"
- pdf
O.Schnetz, "Quantum field theory over Fq"
- pdf
P.Belkale, P.Brosnan, "Periods and Igusa zeta function"
- pdf
Emad Nasrollahpoursamami, "Periods of Feynman Diagrams and GKZ D-Modules", arXiv:1605.04970
- pdf
Matilde Marcolli,
"Motivic renormalization and singularities", in "Quanta of Maths", pp.409-458, Clay Mathematics Institute and American Mathematical Society, 2010
- pdf
P.Aluffi, M.Marcolli, "Parametric Feynman integrals and
determinant hypersurfaces"
- pdf
O.Ceyhan, M.Marcolli, "Feynman integrals and motives of configuration
spaces" Communications in Mathematical Physics, Vol.313, N.1 (2012), Page 35-70
- pdf
O.Ceyhan, M.Marcolli, "Feynman integrals and periods in configuration
spaces", Clay Mathematics Proceedings Vol.21, Clay Mathematical Institute and American Mathematical Society, 2020, pp.35-102
- pdf
O.Ceyhan, M.Marcolli, "Algebraic renormalization and Feynman integrals in configuration
spaces", Advances in Theoretical and Mathematical Physics, Vol.18 (2014) 469-511
- pdf
D.J.Broadhurst, D.Kreimer, "Association of multiple zeta values with
positive knots via Feynman diagrams up to 9 loops"
- pdf J.R.Stembridge,
"Counting points on varieties over finite fields related to a
conjecture of Kontsevich"
- pdf
I.Kausz, "A modular compactification of the general linear group"
- pdf
Matilde Marcolli, Xiang Ni, "Rota-Baxter algebras, singular hypersurfaces,
and renormalization on Kausz compactifications", J Singularities 15 (2016) 80-117
- pdf
P.Aluffi, "Chern classes of graph hypersurfaces and deletion-contraction"
- pdf
F.Brown, O.Schnetz, "Modular forms in quantum field theory"
- pdf
F.Brown, O.Schnetz, K.Yeats,
"Properties of the c2 invariants of Feynman graphs"
- pdf
D.Doryn, "The c2 invariant is invariant"
- pdf
D.Doryn "On one example and one counterexample in counting rational points on graph hypersurfaces",
arXiv:1006.3533
- pdf
D.Doryn "Cohomology of graph hypersurfaces associated to certain Feynman graphs", arXiv:0811.0402
- pdf
F.Brown, O.Schnetz, "A K3 in phi4"
- pdf
Francis Brown, Dzmitry Doryn, "Framings for graph hypersurfaces", arXiv:1301.3056
- pdf
F.Brown, K.Yeats, "Spanning forest polynomials and the transcendental
weight of Feynman graphs"
- pdf
S.Mueller-Stach, S.Weinzierl, R.Zayadeh, "Picard-Fuchs
equations for Feynman graphs"
- pdf
Francis Brown, "Feynman Amplitudes, Coaction principle, and Cosmic Galois group",
arXiv:1512.06409
- pdf
Christian Bogner, Francis Brown, "Feynman integrals and iterated integrals
on moduli spaces of curves of genus zero", arXiv:1408.1862
- pdf
Paolo Aluffi, Matilde Marcolli, Waleed Qaisar, "Motives of melonic graphs",
arXiv:2007.08565
- pdf
Matilde Marcolli, Goncalo Tabuada, "Feynman quadrics-motive of the massive sunset graph",
arXiv:1705.10307
- pdf
Spencer Bloch, Matt Kerr, Pierre Vanhove, "Local mirror symmetry and the
sunset Feynman integral", arXiv:1601.08181
- pdf
Spencer Bloch, Pierre Vanhove, "The elliptic dilogarithm for the sunset graph",
arXiv:1309.5865
- pdf
Spencer Bloch, Dirk Kreimer, "Mixed Hodge Structures and Renormalization in Physics",
arXiv:0804.4399
- pdf
Paolo Aluffi, Matilde Marcolli, "Intersection theory, characteristic classes, and
algebro-geometric Feynman rules", MathemAmplitudes 2019
- pdf
Maxim Kontsevich and Don Zagier, "Periods", in `Mathematics unlimited
- 2001 and beyond', 771-808, Springer, 2001.
- pdf
Alexander Goncharov and Yuri Manin, "Multiple zeta-motives and
moduli spaces M_{0,n}", Compos. Math. 140 (2004), no. 1, 1-14,
arXiv:math/0204102.
- pdf
Francis Brown, "Mixed Tate motives over Z",
Ann. of Math. (2) 175 (2012), no. 2, 949-976,
arXiv:1102.1312
- pdf
Franziska Bittner, "The universal Euler characteristic for
varieties of characteristic zero", Compos. Math. 140 (2004), no. 4, 1011-1032,
arXiv:math/0111062.
- pdf
Michael Larsen, Valery Lunts, "Motivic measures and stable
birational geometry", Mosc. Math. J. 3 (2003), no. 1, 85-95, 259,
arXiv:math/0110255.
- pdf
Scholl, A. J. "Motives for modular forms",
Invent. Math. 100 (1990), no. 2, 419-430.
- pdf
Kahn, Bruno; Sebastian, Ronnie, "Smash-nilpotent
cycles on abelian 3-folds", Math. Res. Lett. 16 (2009),
no. 6, 1007-1010.
- pdf
Kahn, Bruno, "Zeta functions and motives", Pure
Appl. Math. Q. 5 (2009), no. 1, 507-570.
- pdf
Manin, Yuri I. "Moduli, motives, mirrors", European
Congress of Mathematics, Vol. I (Barcelona, 2000), 53-73,
Birkhauser, 2001.
- pdf
Manin, Yuri "Lectures on zeta functions and motives (according to Deninger and Kurokawa)", Asterisque,
228 (1995) N.4, 121-163.
- Deninger, Christopher, "On the Gamma-factors attached to motives",
Invent. Math. (1991) N.2, 245-261.
- pdf
Ramachandran, Niranjan, "Values of zeta functions at s=1/2", Int. Math.
Res. Not. (2005) N.25, 1519-1541.
- pdf
Niranjan Ramachandran, "Zeta functions, Grothendieck groups, and the Witt ring",
arXiv:1407.1813
- pdf
Niranjan Ramachandran, Goncalo Tabuada, "Exponentiation of motivic measures",
arXiv:1412.1795
- pdf Minhyong Kim, "Arithmetic Chern-Simons Part I"
- pdf Minhyong Kim at al. "Arithmetic Chern-Simons Part II"
- pdf
Nicholas Katz "One the differential equations satisfied by period matrices"
- pdf N.Katz T.Oda, "On the differentiation of De Rham cohomology classes with respect to parameters"
- pdf Yuri Manin "Algebraic curves over fields with differentiation"
- pdf
S.Bloch, H.Esnault "Gauss-Manin determinant connections and periods for irregular connections"
- pdf D.Dore "Monodromy in de Rham cohomology: analytic and algebraic theory"
- pdf H.Esnault P.H.Hai "The Gauss Manin connection and Tannaka duality"
- pdf
J.Fresan P.Jossen "Exponential motives"
- pdf
F. Brown "Notes on motivic periods"
- pdf
G.Ancona, "Some arithmetic and geometric aspects of algebraic cycles and motives"
Final Schedule of Presentations: Fall Term
THURSDAY Dec 1
- 8:00-8:25 Thorgal Hinault, Local mirror symmetry and sunset graph
- 8:30-8:55 Shaowu Zhang, Motivic renormalization and singularities
- 9:00-9:25 Stephanie Chen, Universal Euler Characteristic
- 9:30-9:55 Isaiah Curtis, Hodge polynomials of graph hypersurfaces
- 10:00-10:25 Joonhwi Kim, Yang-Mills as quantized gravity
TUESDAY Dec 6
- 8:15-8:40 Ethan Davis, Thermodynamics and Endomotives (Part 1)
- 8:45-9:45 BREAK
- 9:45-10:10 Pedro Guicardi, Motivic Multiple Zeta Values and Superstring Amplitudes
- 10:15-10:40 Justin Toyota, Quantum field theory over Fq
- 10:45-11:10 Yousuf Soliman, Multiple zeta motives and M0n
- 11:15-11:45 Brian Yang, Modular forms in QFT
THURSDAY Dec 8
- 8:00-8:25 Lee Roger Chevres Fernandez, Feynman Amplitudes, Coaction principle, and Cosmic Galois group
- 8:30-8:55 Kai Svenson, Picard-Fuchs equations and Feynman integrals
- 9:00-9:25 Geoffrey Pomraning, Counterexamples to polynomial countability
- 9:30-9:55 Daniel Murphy, Thermodynamics and Endomotives (Part 2)
- 10:00-10:25 Thomas Cleveland, Witt rings and zeta functions
Final Schedule of Presentations: Winter Term
THURSDAY MARCH 9: 8:30-12:30 Room 187
- 8:30-8:55 Ethan Davis, combinatorially non-local field theories and Hopf algebra of 2-graphs
- 9:00-9:25 Thorgal Hinault, GKZ D-modules
- 9:30-9:55 Shaowu Zhang, Exponential motives
- 10:00-10:25 Lee Roger Chevres Fernandez, Quantum Black Holes, Wall Crossing, and Mock Modular Forms
- 10:30-10:55 Arnav Das, Poincare dualities and categorical symmetries of QFTs
- 11:00-11:25 Chi Zhang, periods in mirror symmetry
- 11:30-11:55 Stephanie Chen, Motives of modular forms
- 12:00-12:25 Geoffrey Pomraning, The c2 invariant (modular forms and QFT)
Links to Papers of Assigned Student Presentations: Fall Term
- Motivic Multiple Zeta Values and Superstring Amplitudes pdf
- Picard-Fuchs equations for Feynman integrals pdf
- Motivic Renormalization and Singularities
pdf
- Local mirror symmetry and the sunset Feynman integral
pdf
- Yang-Mills as Quantized Gravity: Color-Kinematics Duality from Non-commutative Geometry pdf andpdf
- Sections 4.1-4.4 of "Noncommutative geometry, quantum fields and motives"
- The Universal Euler Characteristic for Varieties of Characteristic Zero pdf
- Quantum Field Theory over Fq pdf
- Multiple zeta-motives and
moduli spaces M0n pdf
- Hodge polynomials and graph hypersurfaces pdf
- Witt rings and zeta functions pdf
- Modular forms in QFT pdf
- Coaction principle and cosmic Galois group pdf
- Picard-Fuchs equations for Feynman graphs pdf
- Counterexamples to polynomial countability pdf
Links to Papers of Assigned Student Presentations: Winter Term