14
RESEARCH REPORTS
PUBLIKATIONSNAVN
By Leonid
Chekhov
Visiting prof.
at QGM
In collaboration with Jørgen Andersen, Bob Penner, and
Paul Norbury (Melbourne University), we have applied
the technique of topological recursion developed in
20042006 by Bertrand Eynard, Nicolas Orantin and
myself to calculating n-point correlation functions of
(g)
Gaussian means (n-chord diagrams) W n (Z 1 , . . . , Z n )
with the genus-g filtration taken into account. Although
being perhaps the simplest ever example of applica-
tion of this powerful technique, it reveals deep con-
nections with geometry providing the generating func-
tion of cohomological invariants integrals over moduli
spaces M g,n of complex curves of genus g with n
marked points of special forms on these moduli spaces;
these integrals are subject to axiomatic relations of a
cohomological field theory and satisfy the Givental
decomposition relations. This reveals deep algeraic-
geometrical structures encoded in the Gaussian means
and provides effective methods of their calculation.
These models also find applications in bioinformatics
providing an effective tool for studying secondary struc-
tures of RNA this work is being performed in collab-
oration with J. Andresen, Penner, and Piotr Sulkowski
(Warsaw University).
The topological recursion method proved to be a very
effective tool for organising genus expansion calcu-
lations in numerous problems of mathematics and
physics. It is a rapidly developing domain of knowl-
edge; in last couple of years the QGM hosted several
workshops and master classes devoted to studying and
developing this method; the recent school in Leiden
(June 2015) and the forthcoming Oberwolfach meet-
ing (February 2016) will be devoted to applications
of this method to various problems of mathematics
and physics. It works for virtually every system mani-
festing conformal invariance of some sort and allows
obtaining topological invariants, or n-point correlation
functions for any n, including n = 0 corresponding
to a free energy term, on a surface of any genus g
provided we have solved the system for genus zero
(the planar limit), which provides the most important
ingredient of the method, which is the spectral curve
Σ(x, y) = 0 endowed with two meromorphic differen-
tials, dx and dy. All the higher correlation functions in
AFSNITSNAVN
Proved in 2015 joint with J.E:Andersen combinatorically the explicit
relation between genus filtrated s-loop means of the Gaussian matrix
model and terms of the genus expansion of the Kontsevich-Penner
matrix model, which is the generating function for volumes of discreti-
zed (open) moduli spaces. It was proved that this generating function
is also the generating function for ancestor invariants of a cohomo-
logical field theory thus enjoying the Givental decomposition, which
provides a proof of (quasi)-polynomiality of the discrete volumes.
all genera are then obtained by the recursion schemat-
ically represented in the picture below.
Z 2
Z 3
g
Z 1
.
.
.
Z n
Z 2
Z
= Z 1
Z
gh
h
J /I
I
+ Z 1
Z 3
Z
g1
Z
.
.
.
Z n
Picture illustrating the basic topological recursion relation
expressing W n (g) through correlation functions of lower g or n.
Recently, in collaboration with J. Ambjørn (Niels Bohr
Institute, Copenhagen) I successfully applied the to-
pological recursion method for deriving generating
functions for enumerations of Grothendiecks dessins
denfant and hypergeometric Hurwitz numbers count-
ing possible homotopical types of coverings of the pro-
jective line by a genus-g Riemann surface ramified
over the fixed number r of points on the line. Together
with B. Eynard and researchers in Saclay I have ex-
tended this method to describing quantum spectral
curves related to quantum Liouville theories and their
conformal blocksthe basic objects of study in confor-
mal field theories of 2D quantum gravity.
I continue my collaboration with M. Mazzocco (Lough-
borough University, UK) and M. Shapiro (Michigan State
University) on studying geodesic algebras and related
Poisson and quantum structures appearing in hyper-
bolic geometry of Riemann surfaces with holes. Our
recent results extend these geodesic algebras to the
case of surfaces with bordered cusps: in this case we
were able to establish relation of special quantum
geodesic functions to quantum cluster algebras of
Berenstein and Zelevinsky.
AFSNITSNAVN PUBLIKATIONSNAVN QGM HIGHLIGHTS 2015 1
Photo by Christine Dilling Sandbjerg Estate, Nov 2015 CONTENT Message from the Director ........................................................................................................... 3 About QGM.............................................................................................
AFSNITSNAVN 3 PUBLIKATIONSNAVN Message from the Director Photo: AU Communication By Jørgen Ellegaard Andersen 2015 was a very productive year for QGM in many Very talented mathematicians from all over the world ways. QGM researchers and close international col- was present at Aarhus University durin
CENTRE FOR QUANTUM GEOMETRY OF MODULISPACES (QGM) 4 PUBLIKATIONSNAVN AFSNITSNAVN QGM was established in 2009 as a Center of Ex- Mission statement cellence funded by the Danish National Research - To contribute to defining quantum field theory as a Foundation. The research objective is to address fun
AFSNITSNAVN PUBLIKATIONSNAVN 5 HELLOS ... QGM continues to be an attractive workplace for both national and international researchers. Aside from the permanent profes- sors, the Centre hosts a population of 7-8 postdocs and around 15 PhD students at any one time. Qiongling Li Alessandro Malusà Postd
6 = i k πpj 2 e p cos k k p=0 πi SCIENTIFIC BREAKTHROUGH PUBLIKATIONSNAVN AFSNITSNAVN 2 ρ k,1 (S)(ẽ j ) = k k1 sin p=1 πpj ẽ p k 2 ρ k,0 (T )(e j ) = e πi/4 e 2k j e j , πi 2 ρ k,1 (T )(ẽ j ) = e πi/4 e 2k j ẽ j . 2 2 2πk(bb)y Isomorphisms between the combinatorial of Complex Chern-Simons t
AFSNITSNAVN 7 PUBLIKATIONSNAVN With Soibelman, Maxim Kontsevich established an algebraic wall-crossing struc- ture (applicable e.g. to cluster algebras). Kontsevich further found a resurgence structure in the path integral for the free particle on the sphere. Together with F. Haiden, L. Katzarkov an
8 PUBLIKATIONSNAVN AFSNITSNAVN Locality of Gravitational Systems from Entangle- ment of Conformal Field Theories Topological recursion for Gaussian means and cohomological field theories One of Hirosi Ooguris current J.E. Andersen & Leonid Chekhov proved combinato- projects is to derive robust pre-
PHD RESEARCH PROJECTS PUBLIKATIONSNAVN 9 Construction of Modular Functors from Modular Tensor Categories Willam Petersen adapted, in joint 1 6 k 1 6 1 6 1 6 1 6 πi 2 πpj 2 e p ρ k,0 (T )(e j ) = e πi/4 e 2k j e j , ρ k,0 (S)(e j ) = i cos construction of modular functors k k p=0 from modular ten
10 AWARDS 2015 PUBLIKATIONSNAVN AFSNITSNAVN Maxim Kontsevich, Docteur Honoris Causa de l'Universite de Vienne, Austria Maxim Kontsevich, Member of the Selection Committee for the 2015 Breakthrough Prize in Fundamental Physics and the New Horizons in Physics Prize Edward Frenkel, The Euler Book Prize
INTERDISCIPLINARY COLLABORATION WITH iNANO AFSNITSNAVN PUBLIKATIONSNAVN Center leader Jørgen Ellegaard Andersen joined the A molecular biological system comprises the building Modelling and the Medicine Group at iNANO, Aarhus blocks of all living systems. A description of the struc- University in Ma
12 CONFERENCE ON NEW DEVELOPMENTS IN TQFT PUBLIKATIONSNAVN AFSNITSNAVN The historical background for the conference New developments in TQFT is the seminal work of Vaughan Jones and Edward Witten. First Jones defined his knot polynomial based on his tower construction in von Neumann algebras in the
AFSNITSNAVN PUBLIKATIONSNAVN 13 In total 59 participants joined the conference New developments in TQFT SPEAKERS Dror Bar-Natan (University of Toronto) Christian Blanchet (Institut de Mathématiques de Jussieu, U. Paris Diderot) Gaëtan Borot (MPI, Bonn) Steven Bradlow (Univ. of Illinois at Urbana-Cha
14 RESEARCH REPORTS PUBLIKATIONSNAVN By Leonid Chekhov Visiting prof. at QGM In collaboration with Jørgen Andersen, Bob Penner, and Paul Norbury (Melbourne University), we have applied the technique of topological recursion developed in 20042006 by Bertrand Eynard, Nicolas Orantin and myself to calc
AFSNITSNAVN By Yang Huang Postdoc at QGM PUBLIKATIONSNAVN 15 In 2015 Yang Huang constructed, in a joint work in progress with Prof. K. Honda, a contact Morse function in the critical level, which may play a fundamental role in the intrinsic understanding of contact mani- folds in any dimension. Anot
16 PUBLIKATIONSNAVN AFSNITSNAVN PHD PROFILE By Simone Marzioni Simone Marzioni is presently working on his PhD thesis. He expects to finish 31 July 2016. I started my PhD program at the QGM in 2013 as a student of Jørgen Ellegaard Andersen, after a master degree at the University of Bologna. My prev
AFSNITSNAVN 17 PUBLIKATIONSNAVN QGM EVENTS 2015 Conferences, Masterclasses & workshops CENTRE Masterclass by Ivan Loseu (Northeastern University) Quantized quiver March 2015 varieties Title : Quantized quiver varieties Q G M for QUANTUM GEOMETRY of MODULI SPACES D E P A R T M E N T of M A T H E M A
18 PUBLIKATIONSNAVN AFSNITSNAVN PUBLICATIONS 2015 Journal articles Abad, C.A. & Schätz, F., Higher holonomies: comparing two constructions, Differential Geometry and its Applications, vol. 40, 1442 Alexeev, N.V., Andersen, J.E., Penner, R.C. & Zograf P., Enumeration of chord diagrams on many interva
AFSNITSNAVN PUBLIKATIONSNAVN 19 Mandel, T., Tropical theta functions and log CalabiYau surfaces, Paper accepted for publication in Selecta Mathematica Martens, J., Thaddeus, M., Compactifications of reductive groups as moduli stacks of bundles, Compositio Mathematica, vol. 152, no. 1, 62- 98 Mayer,
Center for Quantum Geometry of Moduli Spaces (QGM) Aarhus University Ny Munkegade 118, bldg. 1530 DK-8000 Aarhus C Denmark Phone: +45 8715 5141 Fax: +45 8613 1769 http://qgm.au.dk Center of Excellence funded by the Danish National Research Foundation