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Register a new userClassification of the BPS states in Bagger-Lambert Theory
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We classify, in a group theoretical manner, the BPS configurations in the multiple M2-brane theory recently proposed by Bagger and Lambert. We present three types of BPS equations preserving various fractions of supersymmetries: in the first type we have constant fields and the interactions are purely algebraic in nature; in the second type the equations are invariant under spatial rotation SO(2), and the fields can be time-dependent; in the third class the equations are invariant under boost SO(1,1) and provide the eleven-dimensional generalizations of the Nahm equations. The BPS equations for different number of supersymmetries exhibit the division algebra structures: octonion, quarternion or complex.
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We continue our study of BPS equations and supersymmetric configurations in the Bagger-Lambert theory. The superalgebra allows three different types of central extensions which correspond to compounds of various M-theory objects: M2-branes, M5-branes, gravity waves and Kaluza-Klein monopoles which intersect or have overlaps with the M2-branes whose dynamics is given by the Bagger-Lambert action. As elementary objects they are all 1/2-BPS, and multiple intersections of $n$-branes generically break the supersymmetry into $1/2^n$, as it is well known. But a particular composite of M-branes can preserve from 1/16 up to 3/4 of the original ${cal N}=8$ supersymmetries as previously discovered. In this paper we provide the M-theory interpretation for various BPS equations, and also present explicit solutions to some 1/2-BPS equations.
Among newly discovered M2, M5 objects in the Bagger-Lambert-Gustavsson theory, our interest is about half BPS vortices which are covariantly holomorphic curves in transverse coordinates. We restrict ourselves to the case where the global symmetry is broken to so(2) x so(2)x so(4) for the mass deformed Bagger-Lambert theory. A localized object with finite energy exists in this theory where the mass parameter supports regularity. It is time independent but carries angular momentum coming solely from the gauge potential by which the energy is bounded below.
We consider how to take an orbifold reduction for the multiple M2-brane theory recently proposed by Bagger and Lambert, and discuss its relation to Chern-Simons theories. Starting from the infinite dimensional 3-algebra realized as the Nambu bracket on a 3-torus, we first suggest an orbifolding prescription for various fields. Then we introduce a second truncation, which effectively reduces the internal space to a 2-torus. Eventually one obtains a large-N limit of Chern-Simons gauge theories coupled to matter fields. We consider an abelian orbifold C^4/Z_n, and illustrate how one can arrive at the N=6 supersymmetric theories with gauge groups U(N) x U(N) and Chern-Simons levels (k,-k), as recently constructed by Aharony, Bergman, Jafferis and Maldacena.
We show that the N=8 superconformal Bagger-Lambert theory based on the Lorentzian 3-algebra can be derived by taking a certain scaling limit of the recently proposed N=6 superconformal U(N)xU(N) Chern-Simons-matter theories at level (k, -k). The scaling limit (and Inonu-Wigner contraction) is to scale the trace part of the bifundamental fields as X_0 -> lambda^{-1} X_0 and an axial combination of the two gauge fields as B_{mu} -> lambda B_mu. Simultaneously we scale the level as k -> lambda^{-1} k and then take lambda -> 0 limit. Interestingly the same constraint equation partial^2 X_0=0 is derived by imposing finiteness of the action. In this scaling limit, M2-branes are located far from the origin of C^4/Z_k compared to their fluctuations and Z_k identification becomes a circle identification. Hence the scaled theory describes N=8 supersymmetric theory of 2-branes with dynamical coupling. The coupling constant is promoted to a space-time dependent SO(8) vector X_0^I and we show that the scaled theory has a generalized conformal symmetry as well as manifest SO(8) with the transformation of the background fields X_0^I.
We provide a semiclassical description of framed BPS states in four-dimensional N = 2 super Yang-Mills theories probed by t Hooft defects, in terms of a supersymmetric quantum mechanics on the moduli space of singular monopoles. Framed BPS states, like their ordinary counterparts in the theory without defects, are associated with the L^2 kernel of certain Dirac operators on moduli space, or equivalently with the L^2 cohomology of related Dolbeault operators. The Dirac/Dolbeault operators depend on two Cartan-valued Higgs vevs. We conjecture a map between these vevs and the Seiberg-Witten special coordinates, consistent with a one-loop analysis and checked in examples. The map incorporates all perturbative and nonperturbative corrections that are relevant for the semiclassical construction of BPS states, over a suitably defined weak coupling regime of the Coulomb branch. We use this map to translate wall crossing formulae and the no-exotics theorem to statements about the Dirac/Dolbeault operators. The no-exotics theorem, concerning the absence of nontrivial SU(2)_R representations in the BPS spectrum, implies that the kernel of the Dirac operator is chiral, and further translates into a statement that all L^2 cohomology of the Dolbeault operator is concentrated in the middle degree. Wall crossing formulae lead to detailed predictions for where the Dirac operators fail to be Fredholm and how their kernels jump. We explore these predictions in nontrivial examples. This paper explains the background and arguments behind the results announced in a short accompanying note.
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