A note on singular D-branes in group manifolds

Abstract

After reviewing D-branes as conjugacy classes and various charge quantizations (modulo k) in WZW model, we develop the classification and systematic construction of all possible untwisted D-branes in Lie groups of A-D-E series. D-branes are classified according to their positions in the maximal torus. The moduli space of D-branes is naturally identified with a unit cell in the weight space which is exponentiated to be the maximal torus. However, for the D-brane classification, one may consider only the fundamental Weyl domain that is surrounded by the hyperplanes defined by Weyl reflections. We construct all the D-branes by the method of iterative deletion in the Dynkin diagram. The dimension of a D-brane always becomes an even number and it reduces as we go from a generic point of the fundamental domain to its higher co-dimensional boundaries. Quantum mechanical stability requires that only D-branes at discrete positions are allowed.

Classification and quantum moduli space of D-branes in group manifolds

“…from which one can see that if ψ 1 and ψ 2 /2 are integers, h ≡ e 2πiX becomes the identity matrix to enhance the centralizer Z(h) to SO (5). The points specified by both ψ 1 ∈ Z and ψ 2 ∈ 2Z are indeed the points on the central lattice of SO(5) as discussed before.…”

“…The symmetry enhancement is therefore completely fixed by the position of the D-brane in the weight space ( ψ). For any simple Lie group G, the classification of the D-branes can be described in terms of the Dynkin diagram as follows [5]. Suppose the D-brane location ψ is belonging to the intersection of k hyperplanes.…”

“…The points specified by both ψ 1 ∈ Z and ψ 2 ∈ 2Z are indeed the points on the central lattice of SO(5) as discussed before. 2 Note also that l 1 = 1 and l 2 = 2 in SO (5). In order to determine Weyl domains in weight space, one has to know all the possible enhancement lines given by the coordinates ψ i 's.…”

“…Denote an n × n submatrix of a parent 5 × 5 matrix by (i 1 , · · · , i n ) indicating the subspace (x i 1 , · · · , x in ) on which the submatrix is acting. As shown in appendix B.3 in [5], SU(2) α 1 for a long root is generated by two 2's of SU(2) specified by (1, 2), (4, 5) blocks, whereas SU(2) α 2 for a short root is generated by a 3 of SU(2) given by (1,3,5) block. Similarly, SU(2) α 3 is generated by two 2's of (1, 4), (2, 5) blocks, while SU(2) α 4 is generated by a 3 of (2, 3, 4) block.…”

“…For B r -series, the discussion of D-brane moduli space involves the global structure of the groups that comes from the difference between the integral lattice and the co-root lattice. For example, SO (5) and Sp (2) share the same fundamental domain but they are different in the period of integral lattice, resulting in different topological structures.…”