Download Report

Research Trimester on
Multiple Zeta Values, Multiple Polylogarithms
and Quantum Field Theory1
ICMAT, September 15 - December 19, 2014
Workshop on Multiple Zeta Values,
Modular Forms and Elliptic Motives II
ICMAT, December 1-5, 2014
Speakers:
1. C. Bogner (HU Berlin, Germany)
2. D. Broadhurst (The Open University, UK)
3. F. Brunault (ENS Lyon, France)
4. B. Enriquez (CNRS, Strasbourg, France)
5. G. Felder (ETH Z¨
urich, Switzerland)
6. R. Friedrich (MPIM Bonn, Germany)
7. Y. Iijima (RIMS, Kyoto, Japan)
8. M. Kaneko (Kyushu Univ., Japan)
9. M. Kim (Univ. of Oxford, UK)
10. U. K¨
uhn (Univ. of Hamburg, Germany)
11. M. Levine (Univ. of Duisburg-Essen, Germany)
12. Y. Manin (MPIM Bonn, Germany)
13. H. Nakamura (Osaka Univ., Japan)
14. A. Salerno (Bates College, USA)
15. L. Schneps (CNRS, Paris, France)
16. S. Stieberger (MPI Physik, M¨
unchen, Germany)
17. T. Terasoma (Univ. of Tokyo, Japan)
´ Bures-Sur-Yvette, France)
18. P. Vanhove (IHES,
19. K. Vogtmann (Cornell Univ., USA and Univ. of Warwick, UK)
20. S. Wewers (Univ. of Ulm, Germany)
1 www.icmat.es/RT/MZV2014
1
Schedule:
Monday Tuesday Wednesday Thursday Friday Registration 10:30 K. Vogtmann 9:30 – 10:30 T. Terasoma 9:30 – 10:30 P. Vanhove 9:30 – 10:30 G. Felder 9:30 – 10:30 Coffee break 11:00 – 11:30 D. Broadhurst 11:30 – 12:30 Coffee break 10:30 – 11:00 F. Brunault 11:00 – 12:00 Coffee break 10:30 – 11:00 M. Levine 11:00 – 12:00 Coffee break 10:30 – 11:00 Y. Manin 11:00 – 12:00 Coffee break 10:30 – 11:00 S. Stieberger 11:00 – 12:00 C. Bogner 12:30 – 13:30 B. Enriquez 12:00 – 13:00 H. Nakamura 12:00 – 13:00 M. Kaneko 12:00 – 13:00 U. Kühn 12:00 – 13:00 L u n c h 13:30 – 15:30 R. Friedrich 15:30 – 16:30 L u n c h 13:00 – 15:30 A. Salerno 15:30 – 16:30 L u n c h 13:00 L u n c h 13:00 – 15:30 Y. Iijima 15:30 – 16:30 L u n c h 13:00 M. Kim 16:30 – 17:30 L. Schneps 16:30 – 17:30 S. Wewers 16:30 – 17:30 Titles + Abstracts:
C. Bogner
Sunrise integrals beyond multiple polylogarithms
The vast majority of Feynman integrals considered in particle physics
today can be expressed in terms of multiple polylogarithms, but for
some relevant cases, this class of functions seems not to be sufficient.
The massive two-loop sunrise integral is an important showcase for
this problem. We discuss recent progress in the computation of this
Feynman integral for the general case of arbitrary values of the particle
masses. By examination of a Picard-Fuchs differential equation and an
elliptic curve related to the Feynman graph, we arrive at two versions
of our result: one in terms of integrals over complete elliptic integrals
and the other in terms of a function resembling an elliptic dilogarithm.
D. Broadhurst
Multiple Deligne Values in Quantum Field Theory
Multiple Deligne Values (MDVs) are iterated integrals on the interval x ∈ [0, 1] of the differential forms d log(x), −d log(1 − x) and
−d log(1 − λx), where λ is a primitive sixth root of unity. MDVs of
weight 11 enter the renormalization of the standard model of particle physics at 7 loops, via a counterterm for the self coupling of the
2
Higgs boson. I shall review a recent and intensive investigation of all
of the118,097 MDVs with weights up to 11 and comment on the status of 6 conjectures on MDVs, one of which engages modular forms
at even weights w = 12 and w > 14.
F. Brunault
Beilinson’s conjecture for Rankin–Selberg products of modular forms
In this talk, we will investigate Beilinson’s conjecture for the Rankin–
Selberg convolution of two modular forms. We construct explicit
elements in the motivic cohomology of the product of two Kuga–
Sato varieties, and show that their regulator is proportional to noncritical L-values of Rankin products of modular forms, as predicted by
Beilinson’s conjecture. Further, we show that these elements extend
to the boundary of the Kuga–Sato variety. This is joint work with
Masataka Chida.
B. Enriquez
Flat connection on configuration spaces of surfaces
We construct an explicit bundle with flat connection on the configuration space of n points of a complex curve. This enables one to
recover the ‘formality’ isomorphism between the Lie algebra of the
prounipotent completion of the pure braid group of n points on a surface and an explicitly presented Lie algebra (Bezrukavnikov). When
the genus is one, one recovers a part of the elliptic analogue of the
Knizhnik–Zamolodchikov–Bernard connection, which is at the origin
of the theory of elliptic associators and MZVs.
G. Felder
Derived representation schemes and combinatorial identities
Representation schemes parametrize representations of associative algebras on a given vector space. I will review a derived version of this
theory, due to Berest, Khachatryan and Ramadoss, and present simple
examples, such as the algebra of polynomials in two variables, featuring phenomena that are visible in computer experiments, and only
partly understood mathematically. I will present some conjectures
3
that lead to new combinatorial identities partly proven and partly still
conjectural. (Based on joint work with Y. Berest and A. Ramadoss
and with Y. Berest, A. Patotski, A. Ramadoss and T. Willwacher.)
R. Friedrich
TBA
TBA
Y. Iijima
A pro-l version of the congruence subgroup problem
for mapping class groups of genus one
Let l be a prime number. Then the natural outer action of the mapping class group on the surface group induces an outer action of the
relative pro-l completion of the mapping class group on the pro-l completion of the surface group. In this talk, we discuss the faithfulness
of this pro-l outer action in the case where genus is 1. Our main
result is that the pro-2 case is faithful, but the pro-l case for l ≥ 11
is not faithful. In order to give a negative answer to the problem in
the case where l ≥ 11, we also consider the issue of whether or not
the image of the natural outer action of the absolute Galois group of
a certain number field on the geometric pro-l fundamental group of
a modular curve is a pro-l group. This is a joint work with Yuichiro
Hoshi.
M. Kaneko
Finite and symmetric multiple zeta values
Starting from the naive truncation and mod p of the usual multiple
zeta values, we define ”finite multiple zeta values” as elements in an
algebra over the rationals. We give some results and conjectures on
these values as well as their real analogues which we call symmetric or
”finite real” multiple zeta values. Our main conjecture predicts that
these two objects are beautifully connected with each other. This is
a joint work with Don Zagier.
4
M. Kim
Non-abelian reciprocity laws and iterated integrals
We formulate iterative non-abelian reciprocity laws with coefficients in
a hyperbolic curve and then give examples showing iterated integrals
emerging in the process of making them explicit.
U. K¨
uhn
On the generators of a certain algebra of q-multiple zeta values
The q-analogon of multiple zeta values given by the generating series
of bi-multiple divisor sums, also refered to as bi-brackets, naturally
contains the algebra of generating series of multiple divisor sums as
well as the ring of quasi-modular forms. The bi-brackets satisfy a
variation of the double shuffle relations. This allows us to study the
number of generators via the linearised version of these relations. We
obtain this way upper bounds for the dimension of this bi-filtered
algebra for small length similar as for multiple zeta values.
M. Levine
TBA
TBA
Y. Manin
Iterated integrals, Dedekind symbols, and Zeta polynomials
In the first part of the talk, I will explain how the notion of a generalized Dedekind symbol can be extended to non commutative groups
of values, and give examples of such symbols using iterated integrals
of modular forms. In the second part, I will describe the somewhat
mysterious construction of ”zeta polynomials” that can be thought
of as ”local L-functions in characteristic one”.
5
H. Nakamura
Monodromy of elliptic curves and Mordell transformations
in Grothendieck-Teichmueller theory
Classical Mordell transformation switches quartic models of elliptic
curves to cubic forms. We discuss related Grothendieck–Teichm¨
uller
theory and application.
A. Salerno
On the double shuffle Lie algebra structure: Ecalle’s approach
The real multiple zeta values are known to form a Q-algebra; they
satisfy a pair of well-known families of algebraic relations called the
double shuffle relations. To study the algebraic properties of multiple
zeta values, one can study the algebra of formal symbols subject only
to the double shuffle relations. Quotienting this algebra by products,
one obtains a vector space that Racinet proved carries the structure of
a Lie coalgebra. The dual of this space is thus a Lie algebra, known as
the double shuffle Lie algebra. Ecalle has developed a deep theory to
explore combinatorial and algebraic properties of the formal multizeta
values. In this talk, we will explain how Racinet’s theorem follows in
a simple and natural way from Ecalle’s theory. This is joint work with
Leila Schneps.
L. Schneps
TBA
TBA
S. Stieberger
Periods and superstring amplitudes
We present (some) connections and implications of superstring amplitudes from and to number theory. These relations include motivic multiple zeta values, single-valued multiple zeta values, Drinfeld,
Deligne associators and Lie algebra structures related to Grothendiecks
Galois theory. More concretely, we will show that tree-level superstring amplitudes provide a beautiful link between generalized hypergeometric functions and the decomposition of motivic multiple zeta
6
values. Furthermore, we establish relations between complex integrals
on P 1 \{0, 1, ∞} as single-valued projection of iterated real integrals
on RP 1 \{0, 1, ∞}.
T. Terasoma
Depth filtration and mixed elliptic motives
On the Q-vector space generated by (motivic) multiple zeta values,
there are two filtrations by weight and depth. The depth filtraion is defined by the numbers of dx/(1−x)’s in the iterated integral expression
of the multiple zeta values. The conjecture by Broadhurst–Kreimer,
it is strongly expected that this filtration is related to the space of
elliptic modular forms. In this talk, we will try to explain the relation
between the depth filtration and the representation of Tannaka’s fundamental group of mixed elliptic motives on the fundamental group
of the Tate curve.
P. Vanhove
Feynman integrals, Elliptic poylogarithms and beyond
We will discuss a class of Feynman integrals in two dimensions leading to a surprizingly rich mathematical structure. These Feynman
integrals arise as period of mixed of Hodge structures, that we will
describe in detail. At the lowest loop order the integrals are given by
elliptic polylogarithms, but a new class of function is needed at higher
loop order.
K. Vogtmann
Hairy graphs, modular forms and the cohomology of Out(Fn )
The rational cohomology of Out(Fn ) vanishes in high and low dimensions, but Euler characteristic calculations show that there must be
many classes in the middle range. In this talk I will summarize what
we know and then show how tools from topology, representation theory and modular forms can be used to construct a large number of
cocycles. Those within range of computer calculations have been
shown to give nontrivial cohomology classes, and it is conjectured
that essentially all of them should. This is joint work with J. Conant,
A. Hatcher and M. Kassabov.
7
S. Wewers
TBA
TBA
8