?WHAT IS... a Brane? Gregory W. Moore

The term “brane” has come to mean many things 1-form of electromagnetism. Heuristically, these to many people. Broadly speaking, it refers to a may be thought of as differential form-valued fields physical object appearing in field theories of grav- on spacetime, although a proper description turns ity and strings. It can refer to widely diverse notions, out to require notions of K-theory and differential from solitonic solutions of (super)gravity and (su- cohomology theories. Charged branes are sources per)string theories, to local boundary conditions in for these generalized gauge potentials. two-dimensional conformal field theory, to objects Let us translate some of these physical in certain categories associated with sheaves on notions into mathematics. The action principle algebraic varieties. governing a point particle of mass m and The essential physical intuition underlying the electric charge e, moving through a spacetime M, notion of a “brane” may be captured by a few sim- with metric g and Maxwell connection A , is ∗ ple examples. A p-brane is simply any object of Sparticle = W mds+ W eφ (A), where ds is the p-dimensional spatial extent. Thus a 0-brane is a induced line element on the worldline. When point particle, whereas a string is a 1-brane. The added to the standard action for g and A, namely, 1 1 etymological root of “brane” is “membrane”, the S = M vol(g)R− M 2 F ∧∗F (where bulk 16πGN 2e case p =2. The surface of the earth’s ocean may GN is Newton’s constant, R is the scalar curvature be viewed as a 2-brane wrapping the earth and of g, and F = dA is the Maxwell fieldstrength), propagating in the (3 + 1)-dimensional spacetime the action Sparticle represents a source term in of our solar system. The history of a p-brane may the Einstein-Maxwell equations of motion. Thus be described mathematically by a map φ : W→M, the brane may be studied as a solution in field where W is some reference (p +1)-dimensional theories of gravity with localized energy and manifold, while M represents a “spacetime” charge density. The generalization of the brane through which the brane propagates. M is also re- action to p-branes is of the form ferred to as the “target space”, while φW is referred ∗ ∗ (1) Sbrane = T vol(φ (g)) + εφ (C) to as the “worldvolume”. Sometimes a brane can W W have thickness, provided this is small on the scale where C is a differential form gauge potential, and of the spatial extent in p-dimensions. Thus, a rope ε is the “charge” (which may itself be represented is effectively a 1-brane and the earth’s ocean is ef- by a differential form). The generalization of Sbulk fectively a 2-brane. is the action principle of a (super)gravity or Branes play an important role in theories of (super)string theory on M. The typical supergrav- gravity, so a key physical attribute is the tension ity brane solution is a soliton—its stability is guar- T, the energy per unit volume of the brane. The ten- anteed by topological considerations, which are sion of a 0-brane is its mass. Branes can have other often intimately connected with supersymmetry. attributes, such as “charge”. The supergravity and A central point is that a brane has dynamics: it superstring theories in which branes play promi- can wiggle and bend. The oscillations are sections nent roles are generalizations of Einstein-Maxwell of the normal bundle to φ(W) ⊂Mand hence are gauge theories. In addition to the gravitational described by a (p +1)-dimensional scalar field the- field, string theories typically include a collection ory on the brane. For the earth’s ocean, the scalar of gauge potentials generalizing the connection field would represent the height of the waves. Mathematically, these degrees of freedom arise be- Gregory W. Moore is professor of physics at Rutgers cause the soliton solutions come in families. University. His email address is gmoore@physics. Physicists consider W and M of different sig- rutgers.edu. natures. They add various structures to both the

214 NOTICES OF THE AMS VOLUME 52, NUMBER 2 target space and worldvolume, endowing them the brane’s dynamics. The normal bundle scalars with gauge bundles and tensor fields, and gener- become N × N hermitian matrices. The connection alizing them to supermanifolds. In some cases they on a line bundle becomes a nonabelian gauge field, propose to discuss the quantum behavior of branes i.e, a connection on a rank N vector bundle over −1/S by integrating e brane over the space of all maps W. This fundamental phenomenon has ultimately φ; they even boldly contemplate summing over led to many startling new insights into gauge the- W topologies of . One instance in which these ory. Just one example of such an insight is the dreams can be realized, with some degree of rigor, AdS/CFT correspondence, a vast generalization of is the case in which the 1-branes are the funda- the famous relation between three-dimensional mental strings in a supersymmetric string theory. Chern-Simons gauge theory and two-dimensional One distinguished class of branes are the “D- (rational) CFT. The replacement of the normal bun- branes” of string theory. For these one can introduce dle scalars by N × N matrices leads to connections a fundamentally different viewpoint on the ques- tion: “What is a brane?” String theory describes a between D-branes and noncommutative geome- profound relation between a quantum conformal try. Using these insights in the framework of branes field theory (CFT) on a two-dimensional worldsheet within branes leads to new perspectives on hy- W and a corresponding quantum field theory on the perkähler quotient constructions and the ADHM target space M. The spacetime field theory includes construction of instantons. gravity. In this context, D-branes correspond to CFTs We began by describing a brane as an object on Riemann surfaces W with boundary. For exam- propagating through a spacetime. This puts the ple, suppose W =[0,π] × R so that φ describes the spacetime on a primary, and the brane on a sec- propagation of an open string through spacetime. We ondary footing. However, a common theme in the now select a submanifold S⊂Mand impose the study of D-branes has been the idea that in fact, boundary condition that φ : ∂W→S. For certain the (string) field theory on the brane is the primary submanifolds, the associated two-dimensional field concept, whereas the spacetime itself is a sec- theory will be conformal. (Typical examples of such ondary, derived, concept. This notion has been submanifolds include holomorphic subvarieties of given some degree of precision in the so-called complex manifolds and special Lagrangian subvari- Matrix theory formulation of M-theory. A rough eties of symplectic manifolds.) The “D” in D-brane analogy of what physicists expect may be described refers to the fact that some of the coordinate direc- in the context of purely topological branes, where tions in φ thus carry Dirichlet boundary conditions. the field theory on a brane is described in terms One may recover the notion of branes as solitons in of a noncommutative Frobenius algebra, and the supergravity via a semiclassical approximation to string field theory. “spacetime” in which it propagates is derived from Now, purely in the context of CFT, a D-brane may the Hochschild cohomology of that algebra. These be defined to be a local boundary condition preserv- ideas might ultimately lead to a profound revision ing conformal invariance. Conformal field theories of the way we regard spacetime. on Riemann surfaces with boundary can be described The recognition of the importance of branes in axiomatically as a functor from a geometric category string theory has been a central development, one to an algebraic category. Simple considerations of that is still undergoing vigorous evolution. We have gluing show that the boundary conditions should be focused above on D-branes, but there are other regarded as objects in an additive category. It is via important, but less well-understood, branes. For this route that D-branes are identified with objects example, a deeper understanding of the “solitonic in certain categories. Moreover, some CFTs carry a 5-branes” will lead to constructions of quantum CFTs special type of supersymmetry, known as N =2, and string theories in six-dimensional spacetimes. which allows a “twisting” or association with a related Further development of the theory is likely to have topological field theory. If the target space is a a wide variety of important mathematical applica- Calabi-Yau manifold, then some of the branes in the tions. CFT can be interpreted as objects in the derived cat- egory of coherent sheaves on the target. This in turn References has beautiful applications in the theory of mirror [1] Virtually all relevant research and reviews on this symmetry. material can be found on the e-print archive, http:// One more crucial point is that the dynamics on www.arxiv.org. An example of a recent review is the D-brane worldvolume is a gauge theory. In C. V. Johnson, D-brane primer, http://www.arxiv. addition to the scalar field describing fluctuations org/abs/hep-th/0007170. of the brane in the normal directions, there is a line [2] Mirror Symmetry, K. Hori et al., AMS/Clay Mathemat- bundle with connection on W. When N “elementary” ics Institute, 2003. branes are placed on top of each other, new non- [3] Topology, Geometry and Quantum Field Theory, abelian degrees of freedom are needed to describe U. Tillmann, ed. Cambridge Univ. Press, 2004.

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