Program (last update May, 17)
June 14 (Mon)
9:30 -- 10:30 Tetsuji Miwa (Kyoto)
Omega Trick in XXZ model
10:40 -- 11:40 Christian Korff
(Glasgow)
Quantum
Integrable Systems and Enumerative Geometry I:
a free
fermion formulation of quantum cohomology
13:30 -- 14:30 Hidetoshi Awata (Nagoya)
Five-dimensional AGT Relation,
q-W Algebra and Deformed $\beta$-ensemble
14:40 -- 15:40 Yasushi Komori (Rikkyo)
Multiple Bernoulli polynomials and multiple
$L$-functions of root
systems
16:00 -- 17:00 Masato Okado (Osaka)
Stable rigged configurations and X=M for sufficiently
large rank
9:30 -- 10:30 Christian Korff (Glasgow)
Quantum Integrable Systems and Enumerative Geometry
II:
crystal
limit and the WZNW fusion ring
10:40 -- 11:40 Rinat Kashaev (Geneve)
Introduction to quantum Teichmuller theory
13:30 -- 14:30 Tomoki Nakanishi (Nagoya)
Dilogarithm identities in conformal field theory and cluster
algebras
14:40 -- 15:40 Roberto Tateo (Torino)
The ODE/IM correspondence and its
applications
16:00 -- 17:00 Tomohiro Sasamoto (Chiba)
Height distributions of 1D Kardar-Parisi-Zhang equation
June 16 (Wed)
9:30 -- 10:30 Roberto Tateo (Torino)
Thermodynamic
Bethe Ansatz and the AdS/CFT correspondence
10:40 -- 11:40 Yuji Satoh (Tsukuba)
Gauge/string
duality and thermodynamic Bethe ansatz equations
13:30 -- 14:30 Rinat Kashaev (Geneve)
Discrete
Liouville equation and quantum Teichmuller theory
14:40 -- 15:40 Kentaro Nagao (Nagoya)
Instanton counting with adjoint matters via affine Lie algebras (changed)
16:00 -- 17:00 Masahito Yamazaki (IPMU)
Dimer, crystal, free fermion and
wall-crossing
Christian Korff (Glasgow)
(1) Quantum Integrable Systems and Enumerative Geometry I:
a free
fermion formulation of quantum cohomology
I will present a brief introduction to the quantum cohomology ring
of the Grassmannian. It first appeared in works by Gepner, Vafa,
Intriligator and Witten and a particular specialisation of it can be
identified with the fusion ring of the gauged $\widehat{gl}(n)$-WZNW
model. The talk will focus on how one derives known (geometric) results
about the quantum cohomology ring in a simple combinatorial setting
using well-known techniques from quantum integrable systems. For
instance, performing a Jordan-Wigner transformation one derives the
Vafa-Intriligator formula for Gromov-Witten invariants. The free fermion
formalism also allows one to derive new results such as recursion
relations for Gromov-Witten invariants and a fermion product formula.
(2) Quantum Integrable Systems and Enumerative Geometry II:
crystal limit and the WZNW fusion ring
The second talk will focus on a closely related ring, the fusion
ring of the $\hat{sl}(n)$-WZNW model. It will be discussed how this ring
arises from the crystal limit of the $U_q\hat{sl(2)}$-vertex model with
"infinite" spin and $n$ lattice sites. Its transfer matrix can be
interpreted as the generating function of complete symmetric
polynomials in a noncommutative alphabet, the generators of the affine
plactic algebra. (The latter is an extension of the finite plactic
algebra first introduced by Lascoux and Sch\"utzenberger.) Exploiting
the Jacobi-Trudy formula one introduces noncommutative Schur polynomials
and defines the fusion product in a purely combinatorial manner. In
close analogy to the discussion of the quantum cohomology ring one
derives the Verlinde formula for the fusion coefficients (the structure
constants of the fusion ring) via the algebraic Bethe ansatz. I shall
conclude by stating the precise relationship between the quantum
cohomology and fusion ring in terms of a simple projection formula which directly relates
Gromov-Witten invariants and fusion coefficients. The former count
rational curves of finite degree, while the latter are dimensions of
spaces of generalized theta functions over the Riemann sphere with three
punctures.