Ma 191: Motives and Quantum Field Theory
Spring 2014, Caltech Math Department, Tuesday-Thursday 1:00-2:25 pm,
Room SLN 257.
Brief Course Description
The class will focus on the occurrence of motives in
quantum field theory: the main topics covered will
include 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; Feynman amplitudes and motives
in super Yang-Mills theory.
If one is willing to take certain facts about
motives as a black box, the class can be followed
independently of the previous term 191b class on
"Motives and Periods".
Additional references to papers will be added as the class progresses
- Matilde Marcolli, "Feynman Motives", World Scientific, 2010.
pdf S.Bloch, H.Esnault, D.Kreimer, "On motives associated to
- pdf S.Bloch, "Motives associated to graphs"
S.Bloch, "Motives associated to graph sums"
P.Aluffi, M.Marcolli, "Algebro-geometric Feynman rules"
pdf P.Belkale, P.Brosnan, "Matroids, motives and a conjecture
P.Aluffi, M.Marcolli, "Graph hypersurfaces and a dichotomy in
the Grothendieck ring"
pdf S.Bloch, "A note on Hodge structures associated to graphs"
O.Schnetz, "Quantum field theory over Fq"
P.Belkale, P.Brosnan, "Periods and Igusa zeta function"
P.Aluffi, M.Marcolli, "Parametric Feynman integrals and
O.Ceyhan, M.Marcolli, "Feynman integrals and motives of configuration
Additional suggested readings
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"
I.Kausz, "A modular compactification of the general linear group"
P.Aluffi, "Chern classes of graph hypersurfaces and deletion-contraction"
F.Brown, O.Schnetz, "Modular forms in quantum field theory"
F.Brown, O.Schnetz, K.Yeats,
"Properties of the c2 invariants of Feynman graphs"
D.Doryn, "The c2 invariant is invariant"
F.Brown, O.Schnetz, "A K3 in phi4"
F.Brown, K.Yeats, "Spanning forest polynomials and the transcendental
weight of Feynman graphs"
S.Mueller-Stach, S.Weinzierl, R.Zayadeh, "Picard-Fuchs
equations for Feynman graphs"
- Tuesday April 1: Feynman integral and perturbative expansion:
finite dimensional model, Feynman graphs.
- Thursday April 3: Scalar quantum field theory, partition function,
Green functions, Euclidean and Lorentzian, asymptotic expansion, feynman
graphs, Feynman rules, Schwinger and Feynman parameters,
parametric Feynman integral.
- Tuesday April 8: Projective and affine graph hypersurfaces, Landau
varieties, Feynman integral in affine and projective coordinates, differential
forms on hypersurface complements (affine and projective), singularities
of graph hypersurfaces, irreducibility, hypersurface complements and
disjoint unions of graphs, hypersurface complements and 1PI components,
planar graphs, dual graphs, Cremona transformation and graph
- Thursday April 10: Deletion contraction formulae: example of the
Tutte polynomials, deletion-contraction for the graph polynomial,
deletion-contraction type formula for the classes in the Grothendieck
ring of the graph hypersurface complements, the role of the intersection
between the hypersurfaces of deletion and contraction
- Tuesday April 15: presentation by Emad Nasrollahpoursamani
- Thursday April 17: presentation by Emad Nasrollahpoursamani (continued)
- Tuesday April 22: Operations on graphs that inductively define mixed
Tate classes, splitting edges and multiplying edges, generating functions
of the classes of the graph hypersurface complements, chains of polygons give
mixed Tate classes.
- Thursday April 24: Belkale-Brosnan result: graph hypersurface classes
additively generate a localization of the Grothendieck ring, incidence
schemes, matroid representations schemes, universality of matroids
- Tuesday April 29: Bloch's motives of sums of graphs; stable
birational equivalence classes of graph hypersurfaces; graph complete
intersections and the parametric Feynman integral revisited for trivial
- Thursday May 1: Doryn and Schnetz's explicit counterexamples
to the Kontsevich conjecture; Schnetz's quantum field theory over
finite fields; dimensional regularization of parametric integrals
- Tuesday May 6: local Igusa zeta functions and periods
- Thursday May 8: parametric Feynman integrals and determinant
hypersurfaces, combinatorics of graphs and embeddings, reformulation
of the Feynman integral, motives of determinant hypersurfaces
- Tuesday May 13: varieties of frames, flag varieties, intersections
of unions of Schubert cells, motives of varieties of frames, explicit
computations of Grothendieck classes for two and three spaces.
- Thursday May 15: Feynman integrals in configuration spaces, potential
theory (Riesz and Bessel potentials), Euclidean massless and massive
propagators, Fourier transforms and relations between momentum space
and configuration space Feynman amplitudes
- Tuesday May 20: Green functions of Laplacian and expansion in
Gegenbauer orthogonal polynomials, properties of Gegenbauer polynomials
and ultraspherical harmonics, expansion of the Feynman amplitude in
Gegenbauer polynomials, angular and radial integrals, Feynman integral
on polygons and polylogarithm function, general case and reduction to
trivalent vertices, Gaunt coefficients, Racah factorization formula
- Thursday May 22: Angular integrals and summation domains,
reduction to Mordell-Tornheim and Apostol-Vu summations via Euler-MacLaurin
formula, case of dimension 4, matching half-edges and multiple polylog
type summations for the leading term (ell=0)
- Tuesday May 27: Complexified problem, wonderful compactifications
of configuration spaces, forms with logarithmic poles, pole subtraction
procedure via principal value currents, via deformation to the normal
cone and via Rota-Baxter algebras of meromorphic forms
- Thursday May 29: student presentation
- Tuesday June 3: student presentation
- Thursday June 5: student presentation
A grade for the class will be assigned on the basis
of participation in class and of an oral presentation
based on assigned reading material.
Schedule of student presentations
- May 29, Ingar Saberi: A K3 in phi4
- June 3, Siqi He: Picard-Fuchs equations for Feynman integrals
- June 5, Xiang Ni: A modular compactification of the general linear group
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