381. M2 branes and Chern-Simons theories

John Davey is a PhD student of Amihay Hanany at Physics Department of Imperial College, London. Dmitry.

Recently Chern-Simons theories have attracted much attention as they are thought to describe the world volume theory of the elusive M2 brane. In this short article I will attempt to outline some of the recent developments in this field and describe how brane tilings can be used to find and investigate a large class of these theories.

A Cartoon of an M2 brane and its transverse geometry

The recent interest in the topic of M2 branes was triggered by the discovery of highly supersymmetric Chern-Simons theories. Any theory that describes the world volume physics of an M2 brane should be maximally supersymmetric in 3 dimensions. Such a theory should therefore admit N=8 SUSY. Previously, it was thought that Chern-Simons theories in three dimensions can have at most N=3 SUSY, however Bagger and Lambert [1] recently found a Chern-Simons theory which admits the full N=8 symmetry and fits perfectly as a theory for M2 branes in flat space. Aharony, Bergman, Jafferis, and Maldacena later found the the world volume theory of a stack M2 branes transverse to a geometry [2].

A 2+1 dimensional Chern-Simons (CS) theory is a quantum field theory that has a non-dynamical gauge field . The action for such a theory has the form

Where we take to be a non-abelian gauge field transforming in the adjoint representation of the gauge group U(N).

In order for the theory to make sense, it must be well behaved under gauge transformations. While it is relatively easy to show invariance in the abelian case, the non-abelian case is a little more subtle. In this case

Where N is a integer related to the winding number of the gauge transformation performed. When quantizing the theory using Feynman’s path integral formalism, we insist upon being gauge invariant. This leads to the condition that . This integer k is called the Chern-Simons level for the gauge field . Typically every gauge group in the Chern-Simons theory has a level associated to it.

The Chern-Simons theory above is not at all supersymmetric. However it is possible to make the gauge field a component of an N=2 vector multiplet. Doing this necessarily introduces two scalar fields (one of them auxiliary) and a 2-component dirac spinor to the theory. Supersymmetry invariance naturally enhances the action to include many new terms, including the possibility of superpotential terms. Exactly which superpotential terms, and Chern-Simons levels we pick affects how much more additional supersymmetry the theory has. By carefully consideration of these factors, it is possible to find a theory that can admit the full N=8 SUSY (16 supercharges).

A Typical Quiver Diagram

For certain special Chern-Simons theories we can pictorally represent the Lagrangian by using a directed graph called a quiver diagram. These theories are called quiver gauge theories. Each node of the quiver corresponds to a gauge group and each edge corresponds to a chiral field. Each chiral field transforms under a bi-fundamental representation of the two gauge groups that the edge connects. To define the quiver gauge theory’s lagrangian, superpotential data must also be supplied. In this discussion we shall insist that each field should appear exactly twice in the superpotential – once in a positive and once in a negative term. This is known as the toric condition. It should be noted that, due to gauge invariance, each superpotential term corresponds to a closed loop in the quiver, although the converse is not always true. To completely specify the theory living in the world volume of a stack of M2 branes, we must also specify Chern-Simons levels for all of the gauge groups in the theory.

A Typical Brane Tiling

A brane tiling can be viewed as an extension of a quiver to include superpotential data. The tiling is a bi-partite graph that lives on , or alternatively one can view it as forming a tiling of the plane. The tiling’s dual graph is the periodic quiver and so the tiling’s faces correspond to gauge groups and its edges correspond to chiral bi-fundamental fields. A given term in the superpotential is composed of all of the fields (edges) that the corresponding node connects to. As the tiling is bi-partite we can colour its nodes, say, black and white with different coloured nodes corresponding to superpotential terms of different signs. The tiling’s superpotential then naturally satisfies the toric condition as each edge connects precisely one white and one black node. As is mentioned above, Chern-Simons levels must also be specified to fully define the theory. Faces in the tiling are given integers equal to these Chern-Simons levels.

Once a brane tiling and a set of Chern-Simons terms are chosen then it is interesting to analyse the Moduli Space of Vacua of the theory.  For CS theories with superpotentials satisfying the toric condition, this space is generically a toric Calabi-Yau 4-fold.  The first and in some ways most simple object we look at is the space of vacua when only the superpotential’s F-terms are taken into account. This space is known as the Master Space of the theory and can be given by the quotient where can be regarded as a charge matrix associated with the F-terms. The space is a space of special liner combinations of fields called perfect matchings. A perfect matching is a collection of fields such that each term in the superpotential contains exactly one of these fields. These perfect matchings can be found easily from the tiling [3] and , which encodes the relationships between these perfect matchings, is also very easy to find.  The columns of  correspond to perfect matchings and rows of  correspond to charges.

It is also possible to take into account the D-term conditions. The moduli space of vacua, taking into account both F-terms as well as D-terms is called the mesonic moduli space and can be identified with the space transverse to an M2-brane in M-theory.  Rows of the charge matrix correspond to baryonic symmetries of the theory and tell us how perfect matchings (corresponding to columns) are charged under these symmetries. The computation of   is straightforward and is described in [3].  It turns out that the total number of charges (from both and ) is equal to which means that the (complex) dimension of the mesonic moduli space is equal to 4. This is expected as there should be 8 real (or 4 complex) dimensions transverse to an M2 brane in 11 dimensional M-theory.

The mesonic moduli space, being a toric Calabi-Yau 4-fold, can be specified by a convex collection of lattice points in called a toric diagram, which is a kind of fingerprint for the manifold. This toric diagram can be found directly from the and matrices that define the theory’s mesonic moduli space. The total charge matrix can constructed by gluing together the two charge matrices:

The kernel of this total charge matrix contains the toric data of the CY 4 of interest. Specifically, a collection of 4-vectors are generated which form a convex zonotope in . We should note that the toric points live in a 3-dimensional hypersurface in this 4-dimensional space, which is a manifestation of the Calabi-Yau condition on the mesonic moduli space. This collection of toric points defines the aforementioned toric diagram. The process of finding this toric data for a given Chern-Simons theory is known as the forward algorithm and is a very efficient way of finding large numbers of Chern-Simons theories that live on M2 branes. Given a tiling and a set of Chern-Simons levels, we can easily compute details of the mesonic moduli and so find out which M2 brane geometry the Chern-Simons theory corresponds to. As a special case, we find the well known ABJM theory [2].

One interesting phenomena worthwhile mentioning is that of toric duality. This is when two different CS theories, corresponding to two different tilings with different CS levels have the same mesonic moduli space and so describe the same M2 brane physics. This phenomena is well known and is called toric duality. In Phases of M2-brane Theories [3], these dualities are tested by comparing both the spectrum of gauge invariant operators and also scaling dimensions.

References:

[1]  J. Bagger and N. Lambert, arXiv:0712.3738 [hep-th].

[2] O. Aharony, O. Bergman, D. L. Ja eris and J. Maldacena, arXiv:0806.1218 [hep-th].

[3] J. Davey, A. Hanany, N. Mekareeya and G. Torri arXiv:0903.3234 [hep-th].