## Infinite Analysis Seminar Tokyo

Seminar information archive ～04/01｜Next seminar｜Future seminars 04/02～

Date, time & place | Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
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### 2008/06/14

13:30-16:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Confluent KZ equations for $sl_2$ and quantization of monodromy preserving deformation

Exact Solution and Physical Combinatorics of Critical Dense

Polymers

**孫 娟娟**(東大数理) 13:30-14:30Confluent KZ equations for $sl_2$ and quantization of monodromy preserving deformation

[ Abstract ]

We obtain a system of confluent Knizhnik-Zamolodchikov (KZ) equations which generalizes that of KZ equations for $sl_2$,

and give integral solutions of the system. We also study the relation between the system and monodromy preserving deformation theory,

and recover quantizations of Painlev\\'e equations P_I-P_V with affine Weyl group symmetry which are introduced by H.Nagoya.

We obtain a system of confluent Knizhnik-Zamolodchikov (KZ) equations which generalizes that of KZ equations for $sl_2$,

and give integral solutions of the system. We also study the relation between the system and monodromy preserving deformation theory,

and recover quantizations of Painlev\\'e equations P_I-P_V with affine Weyl group symmetry which are introduced by H.Nagoya.

**Paul A. Pearce**(Univ. of Melbourne) 15:00-16:00Exact Solution and Physical Combinatorics of Critical Dense

Polymers

[ Abstract ]

A Yang-Baxter integrable model of critical dense polymers on the

square lattice

is introduced corresponding to the first member ${\\cal LM}(1,2)$ of a

family of logarithmic

minimal models. The model has no local degrees of freedom, only non-

local degrees

of freedom in the form of extended polymers. The model is built

diagrammatically using the

planar Temperley-Lieb algebra and solved exactly on finite width

strips using transfer matrix

techniques. The bulk and boundary free energies and finite-size

corrections are

obtained from the Euler-Maclaurin formula. The spectra are classified

by selection rules and

the physical combinatorics of the eigenvalue patterns of zeros in the

complex

spectral-parameter plane. This yields explicit finitized conformal

characters.

In particular, in the scaling limit, we confirm the central charge

$c=-2$ and conformal weights

$\\Delta_{1,s}=\\frac{(2-s)^2-1}{8}$ for $s=1,2,3,\\ldots$ where $s-1$ is

the number

of defects.

A Yang-Baxter integrable model of critical dense polymers on the

square lattice

is introduced corresponding to the first member ${\\cal LM}(1,2)$ of a

family of logarithmic

minimal models. The model has no local degrees of freedom, only non-

local degrees

of freedom in the form of extended polymers. The model is built

diagrammatically using the

planar Temperley-Lieb algebra and solved exactly on finite width

strips using transfer matrix

techniques. The bulk and boundary free energies and finite-size

corrections are

obtained from the Euler-Maclaurin formula. The spectra are classified

by selection rules and

the physical combinatorics of the eigenvalue patterns of zeros in the

complex

spectral-parameter plane. This yields explicit finitized conformal

characters.

In particular, in the scaling limit, we confirm the central charge

$c=-2$ and conformal weights

$\\Delta_{1,s}=\\frac{(2-s)^2-1}{8}$ for $s=1,2,3,\\ldots$ where $s-1$ is

the number

of defects.