We construct world-volume effective actions of exotic 5 2 2 -branes in type IIA and IIB string theories. The effective actions are given in fully space-time covariant forms with two Killing vectors associated with background isometries. The effective theories are governed by the six-dimensional N = (2, 0) tensor multiplet and N = (1, 1) vector multiplet, respectively. Performing the S-duality transformation to the 5 2 2 -brane effective action in type IIB string theory, we also work out the world-volume action of the 5 2 3 -brane. We discuss some additional issues relevant to the exotic five-branes in type I and heterotic string theories.

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We study T-duality chains of five-branes in heterotic supergravity where the first order α ′ -corrections are present. By performing the α ′ -corrected T-duality transformations of the heterotic NS5-brane solutions, we obtain the KK5-brane and the exotic 5 2 2 -brane solutions associated with the symmetric, the neutral and the gauge NS5-branes. We find that the Yang-Mills gauge field in these solutions satisfies the self-duality condition in the three-and two-dimensional transverse spaces to the brane world-volumes. The O(2, 2) monodromy structures of the 5 2 2 -brane solutions are investigated by the α ′ -corrected generalized metric. Our analysis shows that the symmetric 5 2 2 -brane solution, which satisfies the standard embedding condition, is a T-fold and it exhibits the non-geometric nature. We also find that the neutral 5 2 2 -brane solution is a T-fold at least at O(α ′ ). On the other hand, the gauge 5 2 2 -brane solution is not a T-fold but show unusual structures of space-time.

We study the world-volume effective action of an exotic five-brane, known as the M-theory 5 3 -brane (M5 3 -brane) in eleven dimensions. The supermultiplet of the world-volume theory is the N = (2, 0) tensor multiplet in six dimensions. The world-volume action contains three Killing vectorsk M I (Î = 1, 2, 3) associated with the U (1) 3 isometry. We find the effective T-duality rule for the eleven-dimensional backgrounds that transforms the M5-brane effective action to that of the M5 3 -brane. We also show that our action provides the source term for the M5 3 -brane geometry in eleven-dimensional supergravity.1 The 5 2 2 -brane has two isometries along transverse directions and its tension is proportional to g −2 s . Here gs is the string coupling constant. See [2,4,5] for the notation of 5 2 2 .