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
theory
COMPLEX
QUANTUM
CHERNSIMONS
5
η t,0
(S)(f )(x) Construction
= i 2k e 2πk(bb)x
f (y) cos(4πkyx)e
dy,
R
and geometric construction of the funda-
2
2
2πk(bb)x
2πk(bb)y
sin(4πikyx)e
i
mental TQFT's
η t,1
f
(y)
(S)(f )(x) = 2k
e
dy,
η t,0
(T )(f )(x) =
and Kenji Ueno from Kyoto
University have provided an
explicit isomorphism from the
modular functor underlying
the skein-theoretic model
for the WittenReshetik-
hinTuraev TQFT due to C.
Blanchet, N. Haebegger, G.
Masbaum and P. Vogel to
the vacua modular functor
coming from the conformal
field theory. This thus provides a geometric construc-
tion of the TQFT first proposed by Witten and con-
structed first by ReshetikhinTuraev from the quantum
group Uq(sl (N)). The proof is presented in a series of
four papers, of which the final one is published in the
top international journal Inventiones Mathematicae.
k
ω ij R
ω ik
i
k
2
2k
j 2πk(bb)x
j
πi/4
2πik(xy) 2 2πk(bb)y
e
η t,1
(T )(f
e
)(x) = e
f (y)e
i
R
ω jk
ω jk
ց
ւ
i
j
ω ij
i
j
k
k
Jørgen Ellegaard Andersen
i
k
j
Figure
5. T
Pentagon
relation
(11).
T 12 T 13
23 = T
23 T 12
i
ω ij
The
fundamental
pentagon
relation
in
i j
j theory
Quantum Chern-Simons
Complex
ρ i
(ij) ρ i ρ j
Jørgen Ellegaard Andersen
and Rinat Kashaev
from University of
i
ω ji
j i
Geneva has provided
a
mathematical
construction
of Quantum
j
= T + u(V )
V
Chern-Simons for complex
gauge
groups. The construction of
V
Figure 6. theory
Inversion
relation
(12). gauge groups goes
Quantum Chern-Simons
for
compact
1 to the famous work of Reshetikhin and Turaev, but
years
back
G β relations
(V ) + fulfill
2 the
i(2k
+ relations:
n) β(V )
u(V 25
)=
The following commutation
second
set of
G β remaining
(V )dF
2(2k
+ a n)
σ
σ world mathematical
it remained
challenge
the
τ, ρ i τ, (ρ ever
(13)
τ, τ σ , ρ for
i τ) = since
σ (i) τ ,
σ F σ 2ikdF β(V
σ
4kV
(14)
= τ, τ , ω σ (i)σ
, ) ikδ(β(V ))
τ, ω +
ij τ, (ω
(i) τ corresponding
community
to provide
a ij τ)
construction
of the
theories,
1
1
1
This isomorphisms offers the possibility of port con- j
i +
j
i
i
+ for
2k(k
n)φ(V
) j +
ikψ(V
)) groups. Recently
predicted by Witten,
the
non-compact
gauge
structions from one construction of this TQFT to other, Jørgen
Ellegaard
Andersen and Rinat Kashaev has posted a paper
(17)
τ, ω τ, ω ω τ = τ, ω τ, ω ω τ, i, j k, l = .
which surely will lead to interesting new results on both
sides. As an example of such an application, one can
mention that, that Andersen and Ueno's isomorphisms
allows one to conclude that the Hitchin connection
(15)
(16)
τ, ρ τ, ρ ρ τ = τ, ρ τ, ρ ρ τ,
τ, ρ i τ, ω jk ρ i τ = τ, ω jk τ, ρ i ω jk τ, i j, k,
ij
kl ij
kl
ij kl
on the
ArXiv, where they provide such a mathematical construction
3. The complex Weil representation via geometric quantisation
of the
complex
Chern-Simons
theory.
This now
allows the
Motivated
by the quantum
previous section,
we will now consider
the geometric
quantisa-
2
with the
tion of the space we
associated to a to
triangle
above,
(C ) from
mathematical
community
study
this e.g.
TQFT
a symplectic
rigorous mathe-
dy
dx
2
form Ω = x y , where (x, y) are coordinates on (C ) . Consider the correspond-
matical
point coordinates
of view (u,
and
several
such
investigations
are underway,
ing logarithmic
v) on
C 2 with the
underlying
real coordinates
preserves a mapping class group invariant Hermitian
in particular a number
have
initiated to this effect
u = of
u + projects
iu , v =
v + iv been
structure. Something which is yet to be constructed pu- that researchers.
we have the covering map
by such
QGM
rely by geometric means.
given by
π : C 2 (C ) 2
Semisimplicity
of = (exp(2πiu),
Hecke exp(2πiv))
and (walled)
π(u, v)
= (x, y).
Brauer algebras
EMISIMPLICITY OF HECKE AND (WALLED) BRAUER ALGEBRAS
dary values. For
An example of semisimplicity
example,
if δ p algebras
= 0, then this can
for Brauer
semisimple
(0, 0)
.
. . .
1
research on the study of tilting modules for
be illustrated as
quantum groups at arbitrary roots of unity.
s
. .
25
Henning Haahr
Andersen concentrated his
Together with QGM postdoc D. Tubben-
hauer and Professor C. Stroppel he first pro-
ved a surprisingly general connection bet-
ween tilting modules and cellular algebras.
non-semisimple
This theorem gives a way of producing
cellular bases for endomorphism algebras
of any such tilting module. As a first appli-
cation they obtain in this way the cellularity
of many important classes of finite dimensional algebras. As a next
r + s = minδ p + 1, p δ p + 1 + 2
r + s = minδ p + 1, p δ p + 1 + 1
step they established criteria for when such algebras - for instance
the much studied Brauer algebras - are semisimple. This generalises
and significantly simplifies several known results on semisimplicity.
r
ne (bottom, red) where the semisimplicity fails is illustrated above. We
passage from (r, s) to (r + 1, s + 1) provided by Corollary 6.4. Note that we
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