## Infinite Analysis Seminar Tokyo

Seminar information archive ～02/15｜Next seminar｜Future seminars 02/16～

Date, time & place | Saturday 13:30 - 16:00 Room #117 (Graduate School of Math. Sci. Bldg.) |
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### 2018/02/06

15:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Sine-square deformation of one-dimensional critical systems (ENGLISH)

Modular invariant representations of the $N=2$ vertex operator superalgebra (ENGLISH)

**Hosho Katsura**(Department of Physics, Graduate School of Science, The Univeristy of Tokyo ) 15:00-16:00Sine-square deformation of one-dimensional critical systems (ENGLISH)

[ Abstract ]

Sine-square deformation (SSD) is one example of smooth boundary conditions that have significantly smaller finite-size effects than open boundary conditions. In a one-dimensional system with SSD, the interaction strength varies smoothly from the center to the edges according to the sine-square function. This means that the Hamiltonian of the system is inhomogeneous, as it lacks translational symmetry. Nevertheless, previous studies have revealed that the SSD leaves the ground state of the uniform chain with periodic boundary conditions (PBC) almost unchanged for critical systems. In particular, I showed in [1,2,3] that the correspondence is exact for critical XY and quantum Ising chains. The same correspondence between SSD and PBC holds for Dirac fermions in 1+1 dimension and a family of more general conformal field theories. If time permits, I will also introduce more recent results [4,5] and discuss the excited states of the SSD systems.

[1] H. Katsura, J. Phys. A: Math. Theor. 44, 252001 (2011).

[2] H. Katsura, J. Phys. A: Math. Theor. 45, 115003 (2012).

[3] I. Maruyama, H. Katsura, T. Hikihara, Phys. Rev. B 84, 165132 (2011).

[4] K. Okunishi and H. Katsura, J. Phys. A: Math. Theor. 48, 445208 (2015).

[5] S. Tamura and H. Katsura, Prog. Theor. Exp. Phys 2017, 113A01 (2017).

Sine-square deformation (SSD) is one example of smooth boundary conditions that have significantly smaller finite-size effects than open boundary conditions. In a one-dimensional system with SSD, the interaction strength varies smoothly from the center to the edges according to the sine-square function. This means that the Hamiltonian of the system is inhomogeneous, as it lacks translational symmetry. Nevertheless, previous studies have revealed that the SSD leaves the ground state of the uniform chain with periodic boundary conditions (PBC) almost unchanged for critical systems. In particular, I showed in [1,2,3] that the correspondence is exact for critical XY and quantum Ising chains. The same correspondence between SSD and PBC holds for Dirac fermions in 1+1 dimension and a family of more general conformal field theories. If time permits, I will also introduce more recent results [4,5] and discuss the excited states of the SSD systems.

[1] H. Katsura, J. Phys. A: Math. Theor. 44, 252001 (2011).

[2] H. Katsura, J. Phys. A: Math. Theor. 45, 115003 (2012).

[3] I. Maruyama, H. Katsura, T. Hikihara, Phys. Rev. B 84, 165132 (2011).

[4] K. Okunishi and H. Katsura, J. Phys. A: Math. Theor. 48, 445208 (2015).

[5] S. Tamura and H. Katsura, Prog. Theor. Exp. Phys 2017, 113A01 (2017).

**Ryo Sato**(Graduate School of Mathematical Sciences, The University of Tokyo) 16:30-17:30Modular invariant representations of the $N=2$ vertex operator superalgebra (ENGLISH)

[ Abstract ]

One of the most remarkable features in representation theory of a (``good'') vertex operator superalgebra (VOSA) is the modular invariance property of the characters. As an application of the property, M. Wakimoto and D. Adamovic proved that all the fusion rules for the simple $N=2$ VOSA of central charge $c_{p,1}=3(1-2/p)$ are computed from the modular $S$-matrix by the so-called Verlinde formula. In this talk, we present a new ``modular invariant'' family of irreducible highest weight modules over the simple $N=2$ VOSA of central charge $c_{p,p'}:=3(1-2p'/p)$. Here $(p,p')$ is a pair of coprime integers such that $p,p'>1$. In addition, we will discuss some generalization of the Verlinde formula in the spirit of Creutzig--Ridout.

One of the most remarkable features in representation theory of a (``good'') vertex operator superalgebra (VOSA) is the modular invariance property of the characters. As an application of the property, M. Wakimoto and D. Adamovic proved that all the fusion rules for the simple $N=2$ VOSA of central charge $c_{p,1}=3(1-2/p)$ are computed from the modular $S$-matrix by the so-called Verlinde formula. In this talk, we present a new ``modular invariant'' family of irreducible highest weight modules over the simple $N=2$ VOSA of central charge $c_{p,p'}:=3(1-2p'/p)$. Here $(p,p')$ is a pair of coprime integers such that $p,p'>1$. In addition, we will discuss some generalization of the Verlinde formula in the spirit of Creutzig--Ridout.