Twisted D-branes of the WZW model and level-rank duality

Abstract

We analyze the level-rank duality of ω c -twisted D-branes of su(N ) K (when N and K > 2). When N or K is even, the duality map involves Z Z 2 -cominimal equivalence classes of twisted D-branes. We prove the duality of the spectrum of an open string stretched between ω c -twisted D-branes, and ascertain the relation between the charges of level-rank-dual ω c -twisted D-branes.

Group dualities, T‐dualities, and twisted K‐theory

“…To conclude this section, we finally mention level‐rank dualities, which are certain mysterious relations between WZW theories that involve swapping a parameter for the group with one for the Virasoro level (see, for example, .) This is quite a deep subject and does seem to be connected with twisted ‐theory calculations, though not in a totally straightforward way.…”

Level-rank duality of untwisted and twisted D-branes of the WZW model

“…We defined a one-to-one map α → α between the spinor ε-twisted D-branes of so(2n) 2k and the spinor ε-twisted D-branes of so(2k) 2n . We then showed that the multiplicity n β In both the su(N) K and sp(n) k WZW theories, the charges of level-rank-dual untwisted D-branes are equal (modulo sign) [5,6], with a slightly more complicated relationship holding between the charges of twisted D-branes [7]. In the case of so(N) K , however, the charges of the D-branes do not exhibit any simple relationship under level-rank duality.…”

“…In both the su(N) K and sp(n) k WZW theories, the charges of level-rank-dual untwisted D-branes are equal (modulo sign) [5,6], with a slightly more complicated relationship holding between the charges of twisted D-branes [7]. In the case of so(N) K , however, the charges of the D-branes do not exhibit any simple relationship under level-rank duality.…”

“…(6.10) of ref. [7]), where S ′ and S′ are the modular transformation matrices of so(2n − 1) 2k+1 and so(2k − 1) 2n+1 respectively. Since by eq.…”

“…More recently, it has been shown [5,6,7] that the untwisted and twisted D-branes in the boundary su(N) K WZW model [8]- [25] respect level-rank duality; that is, there exists a one-to-one map between the (equivalence classes of) D-branes of su(N) K and those of su(K) N . The open-string spectra associated with level-rank-dual D-branes are isomorphic, and the charges of level-rank-dual untwisted D-branes are equal (modulo sign), with a slightly more complicated relationship holding between the charges of twisted D-branes.…”