RIKEN Accel. Prog. Rep. 48 (2015)

Ⅰ. HIGHLIGHTS OF THE YEAR

Current Reflection and Transmission at Conformal Defects:
Applying BCFT to Transport Process†T. Kimura∗1,∗2 and M. Murata∗3
A powerful method for studying critical phenomena with conformal defects is boundary conformal
field theory (BCFT). However, it has not been completely understood how BCFT describes the reflection/transmission at conformal defects. In this report, we define the reflection/transmission coefficientfor conserved currents, as a natural generalization ofthat based on the energy-momentum tensor.1)
We consider two one-dimensional quantum systems
connected by a junction, which can be considered as
an impurity interacting with the bulk. Let us assume
that the first system is in the positive domain x > 0,
the second is in the negative x < 0, and they are connected at the origin as depicted in Fig. 1(a). Now
we shall describe the above system in terms of BCFT.
Corresponding to the two quantum systems, the BCFT
picture involves two CFTs: CFT1 and CFT2 . These
CFTs are defined in the upper and lower half planes
respectively as depicted in Fig. 1(b). The real axis,
which divides the two CFTs, stands for the world lineof the impurity, or the defect. We can reformulate thissystem to obtain CFT1 ×CFT2 in the upper half plane
thanks to the folding trick,2) as shown in Fig. 2. In thisway, the junction of the one-dimensional quantum systems can be mapped into a CFT boundary condition.

Fig. 2. From the defect to the boundary. By using thefolding trick, a system with the defect is mapped intoanother system defined on the upper half plane withthe boundary.

R-matrix based on this boundary state |B,j,bR[C]ij,ab = −0|j1i,a j 1 |B.
0|B

(1)

Since we have three constraints for this matrix, it has
only one degree of freedom. Letting ωB [C] bebdab ωB [C] = −0|K1a K 1 |B1,
k1 k2 (k1 + k2 )
0|B

the R-matrix is given by

k1
k1 k2
k2
R[C] =1
k1 + k2

1k2k1

+ ωB [C]

(2)

−1
1

1−1

,(3)

where dab is the Cartan–Killing form, ki is the levelof the Kac–Moody algebra, and Kna = k2 jn1,a − k1 jn2,a .
We can introduce the reflection and transmission coefficients based on this matrix:
Fig. 1. From the impurity to the defect. (a) Two onedimensional systems are connected through the impurity at x = 0. (b) Adding the time direction and taking
the continuum limit, that system is mapped into thetwo-dimensional system with the defect along the line
x = 0.

We assume that CFT1,2 have the same symmetrysubalgebra C, which is preserved at the conformal defect. For such a defect,the we choose following curtot,atot,arent gluing condition jn+ j −n |B = 0, wheretot,aˆ and jjntakes values in the Kac–Moody algebra C,
is the anti-holomorphic part. We then introduce the†∗1
∗2
∗3Condensed from the article in Nucl. Phys. B885 266 (2014)
RIKEN Nishina Center
Institut de Physique Théorique, CEA Saclay
Institute of Physics, ASCR

R[C] =

R11 + R22,
k1 + k2

T [C] =

R12 + R12,
k1 + k2(4)

which satisfy the current conservation R + T = 1. Applying this formula to the coset-type boundary statewith C = su(2),3) we obtain(k +k ) (k +k )S001 2 Sρ11 2
2k1 k2T [su(2)] =1 − (k +k ) (k +k ) ,(5)(k1 + k2 )2
S 1 2 S 1 2
ρ001with the modular
 S-matrix
 of SU(2)k labeled by
 two(k)integers, Sρµ =2k+2 sinπ
k+2 (2ρ + 1)(2µ + 1).

References
1) T. Quella et al., JHEP 0704 (2007) 095.
2) M. Oshikawa, I. Affleck, Phys. Rev. Lett. 77 (1996) 2604.
3) T. Quella, V. Schomerus, JHEP 0206 (2002) 028.

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