# M-Brane Models and Loop Spaces

Abstract: I review an extension of the ADHMN construction of monopoles to M-brane models. This extended construction gives a map from solutions to the Basu-Harvey equation to solutions to the self-dual string equation transgressed to loop space. Loop spaces appear in fact quite naturally in M-brane models. This is demonstrated by translating a recently proposed M5-brane model to loop space. Finally, I comment on some recent developments related to the loop space approach to M-brane models.

Topics: Loop space (64%), Loop (topology) (63%), Brane (55%)

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Abstract: We construct the duality-symmetric actions for a large class of six-dimensional models describing hierarchies of non-Abelian scalar, vector and tensor fields related to one another by first-order (self-)duality equations that follow from these actions. In particular, this construction provides a Lorentz invariant action for non-Abelian self-dual tensor fields. The class of models includes the bosonic sectors of the $6d$ (1,0) superconformal models of interacting non-Abelian self-dual tensor, vector, and hypermultiplets.

40 citations

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Abstract: We construct an alternative form of the M5-brane action in which the sixdimensional worldvolume is subject to a covariant split into 3+3 directions by a triplet of auxiliary fields. We consider the relation of this action to the original form of the M5-brane action and to a Nambu-Poisson 5-brane action based on the Bagger-Lambert-Gustavsson model with the gauge symmetry of volume preserving diffeomorphisms.

37 citations

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Abstract: We consider the non-abelian theory [1] for an arbitrary number N
5 of five-branes and construct self-dual string solution with an arbitrary N
2 unit of self-dual charges. This generalizes the previous solution of non-abelian self-dual string [2] of N
5 = 2, N
2 = 1. The radius-transverse distance relation describing the M2-branes spike, particularly its dependence on N
2 and N
5, is obtained and is found to agree precisely with the supergravity description of an intersecting M2-M5 branes system.

25 citations

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Abstract: We consider the non-abelian self-dual two-form theory arXiv:1203.4224 and find new exact solutions. Our solutions are supported by Yang-Mills (anti)instantons in 4-dimensions and describe wave moving in null directions. We argue and provide evidence that these instanton string solutions correspond to M-wave (MW) on the worldvolume of multiple M5-branes. When dimensionally reduced on a circle, the MW/M5 system is reduced to the D0/D4 system with the D0-branes represented by the Yang-Mills instanton of the D4-branes Yang-Mills gauge theory. We show that this picture is precisely reproduced by the dimensional reduction of our instanton string solutions.

14 citations

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Abstract: We consider the non-abelian theory (http://arxiv.org/abs/arXiv:1203.4224) for an arbitrary number $N_5$ of five-branes and construct self-dual string solution with an arbitrary $N_2$ unit of self-dual charges. This generalizes the previous solution of non-abelian self-dual string (http://arxiv.org/abs/arXiv:1207.1095) of $N_5=2, N_2=1$. The radius-transverse distance relation describing the M2-branes spike, particularly its dependence on $N_2$ and $N_5$, is obtained and is found to agree precisely with the supergravity description of an intersecting M2-M5 branes system.

4 citations

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Abstract: In previous work we proposed a field theory model for multiple M2-branes based on an algebra with a totally antisymmetric triple product. In this paper we gauge a symmetry that arises from the algebra's triple product. We then construct a supersymmetric theory with no free parameters that is consistent with all the continuous symmetries expected of a multiple M2-brane theory: 16 supersymmetries, conformal invariance, and an SO(8) R-symmetry that acts on the eight transverse scalars. The gauge field is not dynamical. The result is a new type of maximally supersymmetric gauge theory in three dimensions.

1,578 citations

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Abstract: In the course of closing supersymmetry on parallel M2 branes up to a gauge transformation, following the suggestion in [J. Bagger, N. Lambert, Phys. Rev. D 75 (2007) 045020, hep-th/0611108 ] of incorporating a gauge field which only has topological degrees of freedom, we are led to assume a certain algebraic structure for the low energy theory supposedly living on parallel M2 branes.

1,422 citations

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01 Jan 1993

Abstract: This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Recent developments in mathematical physics (e.g., in knot theory, gauge theory and topological quantum field theory) have led mathematicians and physicists to look for new geometric structures on manifolds and to seek a synthesis of ideas from geometry, topology and category theory. In this spirit this book develops the differential geometry associated to the topology and obstruction theory of certain fibre bundles (more precisely, associated to gerbes). The new theory is a 3-dimensional analogue of the familiar Kostant-Weil theory of line bundles. In particular the curvature now becomes a 3-form. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kaehler geometry of the space of knots, Cheeger-Chern-Simons secondary characteristic classes, and group cohomology. Finally, the last chapter deals with the Dirac monopole and Dirac's quantization of the electrical charge. The book will be of interest to topologists, geometers, Lie theorists and mathematical physicists, as well as to operator algebraists. It is written for graduate students and researchers, and will be an excellent textbook. It has a self-contained introduction to the theory of sheaves and their cohomology, line bundles and geometric prequantization a la Kostant-Souriau.

936 citations

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Abstract: The partition function of $${\mathcal{N}=6}$$
supersymmetric Chern–Simons-matter theory (known as ABJM theory) on $${\mathbb{S}^3}$$
, as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super–matrix model is closely related to a matrix model describing topological Chern–Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on $${{\rm AdS}_4\times\mathbb{C}\mathbb{P}^3}$$
and gives the correct N
3/2 scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in $${\mathbb{C}\mathbb{P}^3}$$
. We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi–Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi–Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two ’t Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi–Yau, and leads to an expansion around topological Chern–Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.

608 citations

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Abstract: Recently a Nahm transform has been discovered for magnetic bags, which are conjectured to arise in the large n limit of magnetic monopoles with charge n We interpret these ideas using string theory and present some partial proofs of this conjecture We then extend the notion of bags and their Nahm transform to higher gauge theories and arbitrary domains Bags in four dimensions conjecturally describe the large n limit of n self-dual strings We show that the corresponding Basu-Harvey equation is the large n limit of an equation describing n M2-branes, and that it has a natural interpretation in loop space We also formulate our Nahm equations using strong homotopy Lie algebras

544 citations