Starting from the Nambu-Goto bosonic membrane action, we develop a geometric description suitable for -brane backgrounds. With tools of generalized geometry we derive the pertinent generalization of the string open-closed relations to the -brane case. Nambu-Poisson structures are used in this context to generalize the concept of semiclassical noncommutativity of -branes governed by a Poisson tensor. We find a natural description of the correspondence of recently proposed commutative and noncommutative versions of an effective action for -branes ending on a -brane. We calculate the power series expansion of the action in background independent gauge. Leading terms in the double scaling limit are given by a generalization of a (semi-classical) matrix model.
July 14, 2019
Extended generalized geometry and a DBI-type effective action for branes ending on branes
Branislav Jurčo, Peter Schupp, Jan Vysoký
Charles University in Prague, Faculty of Mathematics and Physics, Mathematical Institute
Prague 186 75, Czech Republic, email@example.com
Jacobs University Bremen
28759 Bremen, Germany, firstname.lastname@example.org
Czech Technical University in Prague
Faculty of Nuclear Sciences and Physical Engineering
Prague 115 19, Czech Republic, email@example.com
Keywords: Sigma Models, p-Branes, M-Theory, Bosonic Strings, Nambu-Poisson Structures, Courant-Dorfman Brackets, Generalized Geometry, Noncomutative Gauge Theory.
Dedicated to the memory of Julius Wess and Bruno Zumino
Among the most intriguing features of fundamental theories of extended objects are novel types of symmetries and concomitant generalized notions of geometry. Particularly interesting examples of these symmetries are T-duality in closed string theory and the equivalence of commutative/noncommutative descriptions in open string theory. These symmetries have their natural settings in generalized geometry and noncommutative geometry. Low energy effective theories link the fundamental theories to potentially observable phenomena in (target) spacetime. Interestingly, the spacetime remnants of the stringy symmetries can fix these effective theories essentially uniquely without the need of actual string computations: “string theory with no strings attached.”
The main objective of this paper is to study this interplay of symmetry and geometry in the case of higher dimensional extended objects (branes). More precisely, we intended to extend, clarify and further develop the construction outlined in  that tackles the quest to find an all-order effective action for a system of multiple -branes ending on a -brane. The result for the case of open strings ending on a single D-brane is well known: The Dirac-Born-Infeld action provides an effective description to all orders in [2, 3, 4]. The way that this effective action has originally been derived from first principles in string theory is rather indirect: The effective action is determined by requiring that its equations of motion double as consistency conditions for an anomaly free world sheet quantization of the fundamental string. A more direct target space approach can be based on T-duality arguments. Moreover, there is are equivalent commutative and non-commutative descriptions , where the equivalency condition fixes the action essentially uniquely [6, 7]. This “commutative-noncommutative duality” has been used also to study the non-abelian DBI action [8, 6]. In the context of the M2/M5 brane system a generalization has been proposed in.
In this paper, we focus only on the bosonic part of the action. The main idea of , inspired by , was to introduce open-closed membrane relations, and a Nambu-Poisson map which can be used to relate ordinary higher gauge theory to a new Nambu gauge theory [10, 11, 12, 13]. See also the work of P.-M. Ho et al. [14, 15, 16, 17] and K. Furuuchi et al. [18, 19] on relation of M2/M5 to Nambu-Poisson structures. It turns out that the requirement of “commutative-noncommutative duality” determines the bosonic part of the effective action essentially uniquely. Interesting open problems are to determine, in the case of a M5-brane, the form of the full supersymmetric action and to check consistency with -symmetry and (nonlinear) selfduality.
Nambu-Poisson structures were first considered by Y. Nambu already in 1973 , and generalized and axiomatized more then 20 years later by L. Takhtajan . The axioms of Nambu-Poisson structures, although they seem to be a direct generalization of Poisson structures, are in fact very restrictive. This was already conjectured in the pioneering paper  and proved three years later in [22, 23]. For a modern treatment of Nambu-Poisson structures see [24, 25, 26].
Matrix-model like actions using Nambu-Poisson structures are a current focus of research (see e.g. [27, 28, 29, 30]) motivated by the works of [31, 32, 33, 34, 35] and others. See also [36, 37] for further reference. Among the early approaches, the one closest to ours is the one of [38, 39], which uses -symmetry as a guiding principle and features a non-linear self-duality condition. It avoids the use of an auxiliary chiral scalar  with its covariance problems following a suggestion of . For these and alternative formulations, e.g., those of , based on superspace embedding and -symmetry, we refer to the reviews [43, 44].
Generalized geometry was introduced by N. Hitchin in [45, 46, 47]. It was further elaborated in . Although Hitchin certainly recognized the possible importance for string backgrounds, and commented on it in , this direction is not pursued there. Recently, a focus of applications of generalized geometry, is superstring theory and supergravity. Here we mention closely related work [49, 50]. The role of generalized geometries in M-theory was previously examined by C.M. Hull in . A further focus is the construction of the field theories based on objects of generalized geometry. This is mainly pursued in [52, 53] and in , see also . Generalized geometry (mostly Courant algebroid brackets) was also used in relation to worldsheet algebras and non-geometric backgrounds. See, for example, [56, 57, 58] and [59, 60]. One should also mention the use of generalized geometry in the description of T-duality, see, or the lecture notes . An outline of the relation of T-duality with generalized geometry can be found in . Finally, there is an interesting interpretation of D-branes in string theory as Dirac structures of generalized geometry in [64, 65]. Finally, in , we have used generalized geometry to describe the relation between string theory and non-commutative geometry.
This paper is organized as follows:
In section 3, we review basic facts concerning classical membrane actions. In particular, we recall how gauge fixing can be used to find a convenient form of the action. We show that the corresponding Hamiltonian density is a fiberwise metric on a certain vector bundle. We present background field redefinitions, generalizing the well-known open-closed relations of Seiberg and Witten.
In section 4, we describe the sigma model dual to the membrane action. It is a straightforward generalization of the non-topological Poisson sigma model of the case.
Section 5 sets up the geometrical framework for the field redefinitions of the previous sections. An extension of generalized geometry is used to describe open-closed relations as an orthogonal transformation of the generalized metric on the vector bundle . Compared to the string case, we find the need for a second “doubling” of the geometry. The split in and has its origin in gauge fixing of the auxiliary metric on the -dimensional brane world volume and the two parts are related to the temporal and spatial worldvolume directions. To the best of our knowledge, this particular structure with has not been considered in the context of M-theory before.
In section 6, we introduce the -form gauge field as a fluctuation of the original membrane background. We show that this can be viewed as an orthogonal transformation of the generalized metric describing the membrane backgrounds. On the other hand, the original background can equivalently be described in terms of open variables and this description can be extended to include fluctuations. Algebraic manipulations are used to identify the pertinent background fields. The construction requires the introduction of a target manifold diffeomorphism, which generalizes the (semi-classical) Seiberg-Witten map from the string to the brane case.
This map is explicitly constructed in section 7 using a generalization of Moser’s lemma. The key ingredient is the fact that , which appears in the open-closed relations, can be chosen to be a Nambu-Poisson tensor. Attention is paid to a correct mathematical formulation of the analogue of a symplectic volume form for Nambu-Poisson structures.
Based on the results of the previous sections, we prove in section 8 the equivalence of a commutative and semiclassically noncommutative DBI action. We present various forms of the same action using determinant identities of block matrices. Finally, we compare our action to existing proposals for the M5-brane action.
In section 9, we show that the Nambu-Poisson structure can be chosen to be the pseudoinverse of the -form background field . In analogy with the case, we call this choice “background independent gauge”. However, for we have to consider both algebraic and geometric properties of in order to obtain a well defined Nambu-Poisson tensor . The generalized geometry formalism developed in section 5 is used to derive the results in a way that looks formally identical to the much easier case. (This is a nice example of the power of generalized geometry.)
In section 10, we introduce a convenient splitting of the tangent bundle and rewrite all membrane backgrounds in coordinates adapted to this splitting using a block matrix formalism. We introduce an appropriate generalization of the double scaling limit of  to cut off the series expansion of the effective action.
In the final section 11 of the paper, we use background independent gauge, double scaling limit, and coordinates adapted to the non-commutative directions to expand the DBI action up to first order in the scaling parameter. It turns out that this double scaling limit cuts off the infinite series in a physically meaningful way. We identify a possible candidate for the generalization of a matrix model. For a discussion of the underlying Nambu-Poisson gauge theory we refer to .
Thorough the paper, is a fixed positive integer. Furthermore, we assume that we are given a -dimensional compact orientable worldvolume with local coordinates . We may interpret as a time parameter. Integration over all coordinates is indicated by , whereas the integration over space coordinates is indicted as . Indices corresponding to the worldvolume coordinates are denoted by Greek characters , etc. As usual, . We assume that the -dimensional target manifold is equipped with a set of local coordinates . We denote the corresponding indices by lower case Latin characters , etc. Upper case Latin characters , etc. will denote strictly ordered -tuples of indices corresponding to coordinates, e.g., with . We use the shorthand notation and . The degree -parts of the exterior algebras of vector fields and forms are denoted by and , respectively.
Where-ever a metric on is introduced, we assume that it is positive definite, i.e., is a Riemannian manifold. With this choice we will find a natural interpretation of membrane backgrounds in terms of generalized geometry. For any metric tensor , we denote, as usually, by the components of the inverse contravariant tensor.
We use the following convention to handle -tensors on . Let be a -form on . We define the corresponding vector bundle map as , where . We do not distinguish between vector bundle morphisms and the induced -linear maps of smooth sections. We will usually use the letter also for the matrix of in the local basis of and of , that is . Similarly, let ; the induced map is defined as for . We use the letter also for the matrix of , that is . Clearly, with these conventions and .
Let be a smooth map. We use the notation , and correspondingly . Similarly, . We reserve the symbol for spatial components of the -form , that is, . We define the generalized Kronecker delta to be whenever the top -index constitutes an even permutation of the bottom one, if for the odd permutation, and otherwise. In other words, . We use the convention . Thus, in this notation we have .
3 Membrane actions
The most straightforward generalization of the relativistic string action to higher dimensional world volumes is the Nambu-Goto -brane action, simply measuring the volume of the -brane:
where are components of the positive definite target space metric , and is the -tuple of scalar fields describing the -brane. In a similar manner as for the string action, one can introduce an auxiliary Riemannian metric on and find the classically equivalent Polyakov action of the -brane:
where can be chosen arbitrarily (but fixed), and . Using the equations of motion for ’s:
where , in , one gets back to (1). In the rest of the paper, we will choose . Using reparametrization invariance, one can always (at least locally) choose coordinates such that , , where denotes the space-like components of the metric. In this gauge, the first term in action (2) splits into two parts, one of them containing only the spatial derivatives of and the spatial components of the metric . Using now the equations of motion for , one gets the gauge fixed Polyakov action111The gauge constraints on , and imply an energy-momentum tensor with vanishing components and . These constraints must be considered along with the equations of motion of the action (4), to ensure equivalence with the actions (1) and (2). As discussed in , the subgroup of the diffeomorphism symmetries that remains after gauge fixing is a symmetry of the gauge-fixed p-brane action (4) and also transforms the pertinent components of the energy-momentum tensor into one another (even if they are not set equal to zero). The constraints can thus be consistently imposed at the level of states.
The second term can be rewritten in a more convenient form once we define
Using this notation, one can write
From now on, assume that is a positive definite metric on . Note that from the symmetry of it follows that . We can view as a fibrewise bilinear form on the vector bundle . Moreover, at any , one can define the basis of as , where is the orthonormal basis for the quadratic form at . In this basis one has , which shows that is a positive definite fibrewise metric on .
For any , we can add the following coupling term to the action:
The resulting gauge fixed Polyakov action has the form
This can be written in the compact matrix form by defining an ()-row vector
The action then has the block matrix form
From now on, unless explicitly mentioned, we may assume that is not necessarily of the form (5), i.e., can be any positive definite fibrewise metric on . Any further discussions will, of course, be valid also for the special case (5). Since is non-degenerate, we can pass from the Lagrangian to the Hamiltonian formalism and vice versa. The corresponding Hamiltonian has the form
The expression in the Hamiltonian and a similar expression play the role of “open membrane metrics” and first appeared in the work of Duff and Lu  already in 1990. Hamilton densities for membranes have also been discussed around that time, see e.g. .222We believe that the Hamiltonian (10) has been known, in this or a similar form, to experts for a long time but we were not able to trace it in even older literature, cf.  for the string case. More recently, the Hamiltonian as well as the open membrane metrics appeared, e.g., in . We thank D. Berman for bringing this paper to our attention. The block matrix in the Hamiltonian can be viewed as positive definite fibrewise metric on defined on sections as
for all and . For and , one gets exactly the inverse of the generalized metric corresponding to a Riemannian metric and a -form . Note that, analogously to the case, can be written as a product of block lower triangular, diagonal and upper triangular matrices:
Before we proceed with our discussion of the corresponding Nambu sigma models, let us introduce another parametrization of the background fields and . In analogy with the case, we shall refer to and as to the closed background fields. Let denote the matrix in the action (9), that is,
This matrix is always invertible, explicitly:
Further, let us assume an arbitrary but fixed -vector and consider a matrix of the form
such that the equality , i.e.,
holds. This generalization was introduced and used in . Again, in analogy with the case , we will refer to and as to the open backgrounds. More explicitly, we have the following set of open-closed relations:
For fixed , given there exist unique such that the above relations are fulfilled, and vice versa. The explicit expressions are most directly seen from the equality , again using the formula for the inverse of the block matrix . In particular,
and the explicit expression for can be found straightforwardly. Obviously, the inverse relations are obtained simply by interchanging , , , and . Using these relations, we can write the action (9) equivalently in terms of the open backgrounds , and the (so far auxiliary) -vector .
In terms of the corresponding Hamiltonian (10), the above open-closed relations give just another factorization of the matrix . This time we have
In the sequel it will be convenient to distinguish the respective expressions of above introduced matrices and in the closed and open variables. For the former we we shall use and and for the latter we introduce and , respectively. Hence the open-closed relations can be expressed either way: or . Note, that the latter form is just equivalent to the statement about the decomposability of a 2x2 block matrix with the invertible upper left block as a product of lower triangular, diagonal, and upper triangular block matrices, the triangular ones having unit matrices on the diagonal. Note that for and , the open-closed relations (see ) are usually written simply as
To conclude this section, note that taking the determinant of the matrix , we may prove the useful identity:
To show this, just note that can be decomposed in two different ways, either
Taking the determinant of both expressions and comparing them yields (25).
4 Nambu sigma model
In analogy with the case, we may ask whether there is a Nambu sigma model classically equivalent to the action (9). To see this, introduce new auxiliary fields and , which transform according to their index structure under a change of coordinates on . Define an -row vector . The corresponding (non-topological) Nambu sigma model then has the form:
where can be either of and , supposing that the open-closed relations hold. Using the equations of motion for , one gets back the Polyakov action (9). For the detailed treatment of Nambu sigma models see .
Yet another parametrization of – using new background fields , which we refer to as Nambu background fields333Here, instead of fixing and finding open variables in terms of closed ones, we fix to be zero and find, again using the open-closed relations, unique as functions of and , or vice versa. – can be introduced
We will denote as the matrix expressed with help of Nambu background fields . Using (14), one gets the correspondence between closed and Nambu sigma background fields:
Clearly, is a Riemannian metric on and is a fibrewise positive definite metric on . It is important to note that in general, for , is not necessarily induced by a -vector on . This also means that it is not in general a Nambu-Poisson tensor. However; for , it is easy to show that is a bivector.
Also note that even if is a skew-symmetrized tensor product of ’s (5), is not in general the skew-symmetrized tensor product of ’s.
The converse relations are:
Again, it is instructive to pass to the corresponding Hamiltonians. First, find the canonical Hamiltonian to (26), that is
Second, use the equations of motion to get rid of . In analogy with the case, one expects that resulting Hamiltonian coincides with (10), that is
Indeed, we get
when using the Nambu background fields, in which case we shall introduce the notation for it. This shows that to any one can uniquely find and vice versa, since they both come from the respective unique decompositions of the matrix .
We will refer to the Poisson sigma model, when expressed – using – in open variables as to augmented Poisson sigma model.
5 Geometry of the open-closed brane relations
For , the open-closed relations (24) can naturally be explained using the language of generalized geometry. We have developed this point of view in . One expects that similar observations apply also for case. In the previous section we have already mentioned the possibility to define the generalized metric on the vector bundle by the inverse of the matrix (12). Here we discuss an another approach to a generalization of the generalized geometry starting from equation (16). Denote .
The main goal of this section is to show that we can without any additional labor adapt the whole formalism of  to the vector bundle .
Define the maps , using block matrices as
for all . Next, define the map as
We see that they have exactly the same form as (24) for . The purpose of this section is to obtain these relations from the geometry of the vector bundle .
We define an inner product on to be the natural pairing between and , that is:
for all , , , and . Note that this pairing has the signature .
Now, let be a vector bundle endomorphism squaring to identity, that is, . We say that is a generalized metric on , if the fibrewise bilinear form
defined for all , is a positive definite fibrewise metric on . It follows from definition that is orthogonal and symmetric with respect to the inner product . Moreover, it defines two eigenbundles , corresponding to eigenvalues of . It follows immediately from the properties of , that they are both of rank , orthogonal to each other, and thus
Moreover, and form the positive definite and negative definite subbundles of , respectively. From the positive definiteness of it follows that has zero intersection both with and , and is thus a graph of a unique vector bundle isomorphism . The map can be written as a sum of a symmetric and a skew-symmetric part with respect to : . From the positive definiteness of , it follows that is a positive definite fibrewise metric on . From the orthogonality of and we finally obtain that:
The map , or equivalently the fibrewise metric can be reconstructed using the data and to get
Note that the above block matrix can be decomposed as a product
The maps can be parametrized as
where is a symmetric covariant -tensor on , are vector bundle morphisms, , and and are symmetric and skew-symmetric fibrewise bilinear forms on , respectively. The fields are not arbitrary, since has to be a positive definite fibrewise metric on . One immediately gets that have to be positive definite. The conditions imposed on can be seen from the equalities
We see that there are two equivalent conditions on : the fibrewise bilinear form , or -tensor have to be positive definite. Inspecting the action (9), we see that only the case when is relevant for our purpose.
Now, let us turn our attention to the explanation of the open-closed relations. For this, consider the vector bundle automorphism , orthogonal with respect to the inner product , that is,
for all . Given a generalized metric , we can define a new map . It can be easily checked that is again a generalized metric. Obviously, the respective eigenbundles are related using , namely:
We have also proved that every generalized metric corresponds to two unique fields and . This means that to given and , and an orthogonal vector bundle isomorphism , there exists a unique pair , corresponding to . We will show that open-closed relations are a special case of this correspondence. Also, note that and are related as
Now, consider an arbitrary skew-symmetric morphism , that is
for all , and . It can easily be seen that the vector bundle isomorphism , defined as
for all , is orthogonal with respect to the inner product . Its inverse is simply . Let be the generalized metric corresponding to . Note that can be expressed as
Using the relation (40), we obtain that
We see that the vector bundle morphism corresponding to satisfies
Using the decomposition of the matrices, we can write this also as
Comparing both expressions, we get the explicit form of open-closed relations:
We have proved that for given and any , and can be found uniquely. Inverse relations can be obtained by interchanging , and . Note that, actually, the last equation follows from the first two. Now let us turn our attention to the case of in the form (37). One has