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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- rically the explicit relation between genus filtrated s-
dictions from superstring the- loop means of the Gaussian matrix model and terms of
ory on its low energy effective the genus expansion of the Kontsevich-Penner matrix
theory using methods develo- model, which is the generating function for volumes of
ped in quantum information discretized (open) moduli spaces. It was proved that
theory. Ooguri proved that any this generating function is also the generating function
consistent gravitational theory for ancestor invariants of a cohomological field theory
whose spacetime is asympto- thus enjoying the Givental decomposition, which pro-
tic to the anti-de Sitter space vides a proof of (quasi)-polynomiality of the discrete
must satisfy a certain set of positivity
volumes.
conditions on its local matter energy
density. This result was published in
Physical Review Letters and selected
for the Editors Suggestion.
The Young diagram reformulation of topological recursion
Fig. 4. The graphical representation of recurrence relation (5.6): it clearly indicates the coproduct
(the first term in the right-hand side) and product (the second and third terms in the right-hand side)
operations, the arrowed line is the recursion kernel K(p 1 , q), and nonarrowed lines are the Bergmann
Isomonodromic deformations of logarithmic
connections and stability
Indranil Biswas published 37
per reviewed papers in 2015.
He computed amoung others
kernels B(p k , q) and B(p k , q).
(g)
4. We introduce the correlation functions W s (p 1 , . . . , p s ) as symmetric s-differentials defined rec
sively. We choose one of the variables (p r ) as a root. Then
(0)
W 3 (p 1 , p 2 , p 3 ) =
isomonodromic deformations
of an irreducible connection
on a Riemann surface con-
tains stable bundles, proved results on formality and
obstructions for Sasakian manifolds, classified homo-
geneous holomorphic hermitian principal bundles
on a hermitian symmetric space, investigated Nori
fundamental gerbes on a tame stack and produced
(1)
W 1 (p 1 ) =
les on a compact Riemann
surface. He showed that the
K(p 1 , q)B(p 2 , q) + B(p 2 , q)B(p 3 , q) + B(p 3 , q),
(5
Res dx=0
the automorphism group of
moduli of Higgs line bund-
(5
K(p 1 , q)B(q, q),
Res dx=0
W (g) (p 1 , p 2 , . . . , p s ) =
=
Res dx=0
s
(g)
B(p k , q) + B(p k , q) W s1 (q, p 2 , . . . , p k , . . . , p s ) +
K(p 1 , q)
(g1)
k=2
+ W s+1 (q, q, p 2 , . . . , p s ) +
g 1 +g 2 =g
IJ=p 2 ,...,p s
(g 1 )
(g 2 )
W I+1
(q, p i iI )W J+1
(q, p j jJ ) ,
(5
where the right-hand side is manifestly symmetric with respect to all p 2 , . . . , p s but not with respect
an optimal bound on the essential dimension of vector p 1 and Nitin
means
that research
we take only
stable 2015
terms focused
(those with on 2g Moduli
2 + s > 0) explicitly segregating
Nitsures
during
bundles on curves.
only nonstable
contribution
with B(p k of
, q) ideas
+ B(p inspired
,
q)).
The hat over a symbol indicates
theory for
bundles (the
and term
application
k
omission
the list of to arguments,
and in the
term, we sum matter
over all partitions of the set of argume
by from
geometry
experimental
soft last
condensed
p 2 , . . physics.
. , p s into Jointly
two nonintersecting
subsets Gurjar
I and J. (QGM
We show
the recurrence relation schematically
with Sudarshan
postdoc
Fig. 4. 2014-2015) he completed the paper: Schematic Har-
(g)
(0)
(1)
Using
(5.6), we construct
all higher W s for from
W 3 (p of
1 , p principal
2 , p 3 ) and W 1 (p 1 ).
der-Narasimhan
stratification
families
The
lemma in
in higher
4 states dimensions.
that although recurrence relations (5.6) are not explicitly symmetric w
bundles
respect to permutations of all p 1 , . . . , p s , the whole sum in the right-hand side is in fact symmetric.
5.1. The topological recursion for the Gaussian means. In any local theory satisfying the top
(g)
logical recursion, all stable W s (x 1 , . . . , x s ) have singularities only at the branch points. In the Gauss
case, we therefore conclude that the only singularities in the right-hand side of (5.6) in addition to hi
(g)
order poles at the branch points (for W s , the highest possible order of a pole is 6g+2s3) are simple p
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