Classical worldvolumes as generalised geodesics
aa r X i v : . [ h e p - t h ] J a n Classical worldvolumes as generalised geodesics
Charles Strickland-Constable
Department of Physics, Astronomy and Mathematics, University of Hertfordshire,College Lane, Hatfield, AL10 9AB, UK
E-mail: [email protected]
Abstract:
It is a standard result that the integral curves of an auto-parallel vector field aregeodesics which, for null and timelike vectors, are the paths of freely-falling particles in generalrelativity. We introduce a definition of an “auto-parallel” generalised vector field and showthat it gives the analogous statements for the classical worldvolumes of strings and branes inarbitrary background field configurations. This appears to give a unified description of theworldvolume equations of strings and branes, similar to the way that generalised geometryprovides a unified description of maximal supergravity theories. We present details of thecases of string worldsheets in O (10 ,
10) generalised geometry and M2 branes restricted to thefour dimensions of SL (5 , R ) × R + generalised geometry. A key quantity is the infinitesimalflow of the conjugate momentum along the generalised tangent vector, which is equated tothe gradient of the Hamiltonian, viewed as a function on spacetime. ontents O (10 ,
10) generalised geometry 52.3 Classical strings as generalised geodesics 1: generalised connection approach 8 SL (5 , R ) × R + generalised geometry with Lorentz group SO (3 ,
2) 164.3 Equations of motion from generalised geodesics 18
In ordinary differential geometry, a vector field X defines a congruence of (its integral) curves.In standard presentations, it is stated that if the vector field is auto-parallel in the Levi-Civitaconnection (i.e. it is parallel-transported along its integral curves) ∇ X X = 0 , (1.1)then these curves are (affinely-parameterised) geodesics for the corresponding metric. Ingeneral relativity (GR), the metric has Lorentzian signature and if these curves are timelikeor null, then they describe the trajectories of massive or massless particles freely-falling in thegravitational field encoded by the metric (see e.g. [1]). In the timelike case, these geodesicpaths minimise the action functional given by the invariant relativistic length of the path,given the boundary conditions. An action which describes also null geodesics can be foundby introducing a worldline metric. Using worldline reparameterisation invariance, this canbe set equal to one, leaving the Lagrangian as the square of the invariant interval. We willconsider this gauge-fixed action and its generalisations in this work.In string theory, the analogue of classical freely-falling particles in GR are classical strings(see e.g. [2–4]). Their dynamics are described by a mathematically similar action to the– 1 –oint particle, but generalised to the case where the worldline has become a two-dimensionalworldsheet. In addition to worldsheet reparameterisation invariance, this is invariant underlocal rescalings of the worldsheet metric, which enables us to write it in a gauge-fixed formvery similar to that of the particle. There is also a coupling to a two-form potential B in thetarget space, given simply as the integral of the pull-back of B to the worldsheet. Similarstatements can be made about other branes arising in string theories, albeit with additionalcomplications due to non-linearities and the absence of conformal symmetry.In classical GR, the study of geodesics is fundamental to understanding the structureof the theory and forms a key component of all aspects from orbits of planets to singularitytheorems. In string theory, historically the focus has largely been centred on questions ofquantisation, but understanding classical string (and brane) solutions has been of significantinterest, for example, in semi-classical analysis of the AdS/CFT correspondence [5–17].Given the similarities (at least in spirit) between the mathematical descriptions of all ofthese objects, it is natural to wonder if there is an analogue of the tangent vector X for,say, strings. Further, one can also consider what might be the analogue of equation (1.1) inthe case that one extends the tangent vector to be a field over spacetime. Ideally, we wouldlike an equation such that the tangent vector object defines a foliation of spacetime by stringworldsheets if it satisfies the differential equation.At first glance, one would say that a two-dimensional surface will have two ordinarytangent vectors, and the equations of motion become some differential equations relating themto the background metric and field strength H = d B . One could leave it at that. However,it has been noted long ago [18] that these equations can be viewed as the preservation ofthe tangent vectors in the worldsheet-null directions by the connections with torsion ∇ ( ± ) ∼∇ ± H , suggesting that they have more geometric structure. In this work we will show thatthese equations have considerably more structure still, which generalises also to other branes.As the target space ingredients of the string equations are the metric and B field, oneimmediately suspects that generalised geometry [19, 20], which combines these objects into ageneralised metric, will be a suitable framework in which to look for such additional structure.While this is indeed the framework used in this paper, we should note that there is already awide literature studying how generalised geometry, and similar constructions using a doubledspacetime, describe supergravity, string worldsheets and non-geometric backgrounds [21–37].As far as worldsheet statements are concerned, many of these works have focused on eitherthe geometry of the target space of sigma-models with supersymmetry or the constructionof actions for strings and branes, looking to make the duality symmetries manifest in theformulation and to quantise the systems in those terms.In this work, we will encounter several ideas which have inevitably appeared before inthis literature. However, we will focus only on the classical equations of motion of the ob-jects, rather than action principles. We see how the two ordinary tangent vectors v = ∂∂τ and ˜ v = ∂∂σ of a string worldsheet are combined into a generalised vector V = v + g · ˜ v .The Virasoro constraints are then the vanishing bilinears of this vector in the (Lorentziansignature) generalised metric G and the O (10 ,
10) metric on the generalised tangent space.– 2 –e then imagine that this is extended off the worldsheet of a single string to a vector fieldon spacetime, in at least some open set containing the worldsheet, similarly to how one canmove from the tangent vector of a curve to a local vector field X . This is so that we candefine derivatives of the vector field in all directions in spacetime, rather than purely alongthe worldsheet, even though in the end these must cancel out from the equations. In this way,we will formulate our discussion in terms of a generalised vector field on the target space.Our key results will concern the formulation of the equations of motion in terms ofthis generalised vector V . These equations, together with the Virasoro conditions, thenencapsulate the full system in our language. If they are satisfied on a patch of the targetspacetime, then we have a foliation of spacetime by classical string worldsheet solutions.(Other results on the existence of such foliations have appeared in [38, 39]). However, onecould also simply require them to be solved on a single two-dimensional worldsheet (suchthat V is the generalised tangent vector to it) and this would then give an isolated stringworldsheet solution. As the derivatives which are not along the worldsheet cancel from theequations, the manner in which the generalised tangent vector is extended to a local fielddoes not affect them.As a first pass, we write the equation of motion as two equations, each resembling (1.1),but using generalised Levi-Civita connections. These are simply the equations in terms of ∇ ( ± ) from [18] re-branded as generalised geometry objects.However, we then reformulate the ordinary geodesic equation (1.1) in a form utilising theLie derivative, such that no connection appears explicitly. We later interpret this in terms ofthe conjugate momentum and Hamiltonian viewed as a function on spacetime. Remarkably,writing the exact same equation using the Dorfman derivative (or generalised Lie derivative)and generalised metric we recover the equations of motion for the string. We thus propose thisas a plausible definition of the analogue of the auto-parallel condition for a generalised vector,even though no notion of generalised parallel transport itself is developed here. The objectsin this equation have the same interpretation as for the ordinary geodesic equation, but fora generalised covector conjugate momentum, similar to those which have been discussed inprevious studies (see e.g. [32, 34]). It is also noteworthy that our condition reproduces theequation of motion without using the quadratic constraints on the generalised tangent vector,suggesting that it could be a meaningful definition also in the absence of these constraints(as (1.1) is also for non-null vector fields).Further to the above, we show that the exactly analogous equations written in a differ-ent form of generalised geometry, with generalised structure group SL (5 , R ) × R + , describethe equations for the M2 brane restricted to the four dimensions (out of eleven) which areincluded in this generalised geometry. This is the four-dimensional case of exceptional gen-eralised geometry [40, 41] which describes the dimensional restrictions of eleven-dimensionalsupergravity (and type II supergravity via different decompositions) [42, 43], and forms thebasis for the internal part of exceptional field theory [44–46]. As for the string, we use a gauge-fixed form of the M2 brane theory, with the gauge fixing constraints now corresponding to thequadratic constraints on our generalised tangent vector. In the process we encounter objects– 3 –eminiscent of those in previous studies of membrane worldvolume theories in the context ofextended geometry [47–52]. Our construction then proceeds in exactly the same way as in ourdiscussion of the string. This suggests that our definition of the auto-parallel condition andformulation of the equations of motion will be universal across generalised geometries. Wealso stress that no assumptions are made about the nature of the background fields, whichneed not solve the supergravity equations of motion.The structure of the paper is as follows. In section 2 we review the action and equationsof motion for the string in a background metric and B -field, introduce the necessary elementsof O (10 ,
10) generalised geometry and show how to formulate the equations of motion usinga generalised Levi-Civita connection. Next, in section 3 we reformulate the ordinary auto-parallel condition without explicit use of a connection and give our definition of an auto-parallel generalised vector field. This is then shown to reproduce the equations of section 2for the string in O (10 ,
10) generalised geometry. In section 4 we provide the correspondingstatements for the M2 brane in SL (5 , R ) × R + generalised geometry. We end with somediscussion of our results in section 5. The action of the classical string in a background field configuration ( g, B ), gauge fixed toconformal gauge on the worldsheet, is S = − Z d s (cid:16) η αβ g mn + ǫ αβ B mn (cid:17) ∂x m ∂s α ∂x n ∂s β (2.1)The resulting classical equations of motion can be written as η αβ h ∂ x m ∂s α ∂s β + Γ pmq ∂x p ∂s α ∂x q ∂s β i − ǫ αβ H mpq ∂x p ∂s α ∂x q ∂s β = 0 (2.2)Writing this out explicitly using η = diag( − , +1) and ( s α ) = ( τ, σ ) and denoting v m = ∂x m ∂τ ˜ v m = ∂x m ∂σ (2.3)we have − Dv m Dτ + D ˜ v m Dσ − H mpq v p ˜ v q = 0 (2.4)where for any vector w ∈ Γ( T M | worldsheet ) Dw m Dτ = ∂w m ∂τ + v p Γ pmn w n Dw m Dσ = ∂w m ∂σ + ˜ v p Γ pmn w n (2.5)are the target space covariant derivatives along the τ and σ directions. Here we take ǫ = +1. This matches the sign conventions of the B -field in the generalised geometryconstruction. – 4 –n addition to these equations of motion, one must also impose the Virasoro constraints,which are the equations of motion of the worldsheet metric written in the conformal gaugein which we wrote our action (2.1). These are the vanishing of the energy momentum tensor T αβ = g mn ∂ α x m ∂ β x n − η αβ η γδ g mn ∂ γ x m ∂ δ x n (2.6)Let us now extend to the analogue of a congruence of curves, and consider a coordinatesystem on spacetime for which τ and σ are the first two coordinates (often called “static gauge”in the literature). We then promote v m and ˜ v m to vector fields on a patch of spacetime, ratherthan just on a two-dimensional embedded worldsheet. If these satisfy the relevant equationof motion they will then define a foliation of the patch of spacetime by string worldsheets.In that setup, the above equation of motion becomes − v p ∇ p v m + ˜ v p ∇ p ˜ v m − H mpq v p ˜ v q = 0 (2.7)while the Virasoro constraints are g ( v, v ) + g (˜ v, ˜ v ) = 0 g ( v, ˜ v ) = 0 (2.8)Clearly, it is possible to satisfy the first of (2.8) with both v and ˜ v non-zero only if the metric g has indefinite signature.In section 2.3 we will recover this system from generalised geometry via generalised con-nections. In section 3.1 we will see a more universal generalised geometry formulation. O (10 , generalised geometry In this section, we briefly recall some of the features of generalised geometry needed to describethe string. We mostly follow the presentation of [24], to which the reader can refer for fulldetails of these constructions.Firstly, our spacetime is equipped with a B -field, which may not be globally defined, sowe introduce patches of the spacetime such that between the patches the B -field transformsas B ′ = B − dΛ (2.9)A generalised vector field V = v + λ is a vector field v together with a one-form λ defined oneach of the patches as above. Between the patches, the one-form part transforms as λ ′ = λ − i v dΛ (2.10)As these one-forms are explicitly twisted by the gauge transformations in this way, we willrefer to this as the “twisted picture” representation of a generalised vector. The action ofthese gauge transformations preserves the O (10 ,
10) inner product η h V, V i = η ( V, V ) = i v λ (2.11)– 5 –nd so, including also the GL (10 , R ) action of diffeomorphisms, we can think of the structuregroup of the generalised tangent space to be O (10 ,
10) (even though in fact it lies only in aparabolic subgroup).One can also discuss the “untwisted picture” representation of generalised vectors. Thegeneralised tangent bundle E is isomorphic to the direct sum T ⊕ T ∗ , and the isomorphismcan be made explicit using the B -field. The one-form˜ λ = λ − i v B (2.12)is invariant under the gauge transformations between patches, and so v + ˜ λ is a well-definedsection of T ⊕ T ∗ over our spacetime.In [24], this isomorphism is presented in terms of the components of generalised vectorswith respect to certain split frames for the generalised tangent space. Further discussion ofthe twisted vs untwisted pictures can be found in [53]. In this paper, we will do most of ourcalculations working with the untwisted picture representation, so that generalised vectorswill simply be the sum of a vector field and a one-form field (and we will drop the tilde onthe one-form from the notation).One of the key structures in generalised geometry is the Dorfman derivative (or gener-alised Lie derivative) which generates the action of infinitesimal generalised diffeomorphisms(i.e. combined diffeomorphisms and B -field gauge transformations). With respect to thetwisted picture components, we have L V V ′ = [ v, v ′ ] + L v λ ′ − i v d λ (2.13)Introducing O (10 ,
10) indices
M, N via (cid:0) V M (cid:1) = v m λ m ! (cid:0) ∂ M (cid:1) = ∂ m ! (cid:0) η MN (cid:1) = 12 ! (2.14)and raising and lowering these indices with η and its inverse, we can write the Dorfmanderivative as ( L V V ′ ) M = V N ∂ N V ′ M + ( ∂ M V N − ∂ N V M ) V ′ N (2.15)Its action on an section W of E ∗ , written with a lower index, is then( L V W ) M = V N ∂ N W M + ( ∂ M V N − ∂ N V M ) W N (2.16)In the untwisted picture, this takes the form L V V ′ = [ v, v ′ ] + L v λ ′ − i v d λ − i v i v ′ H (2.17)where L v denotes the ordinary Lie derivative and H = d B is the field strength of B . Thissatisfies the Leibniz identity, giving E the structure of a Leibniz algebroid [54], though it is– 6 –sually referred to as a Courant algebroid [55], as it has more structure still. It is not a Liealgebroid as the Dorfman derivative is not anti-symmetric, but the symmetric part is exact L V V ′ + L V ′ V = 2d h V, V ′ i (2.18)Another important object for us will be the generalised metric G , which gives anotherinner product on E . In the untwisted picture, the generalised metric can be written simplyin terms of a metric g on the spacetime via G ( V, V ) = h g ( v, v ) + g − ( λ, λ ) i (2.19)In this paper, we will take the spacetime metric g to have signature (9 ,
1) so that G is stabilisedby O (9 , × O (9 , ⊂ O (10 , η and G . Given two orthonormal frames ˆ e + a and ˆ e − ¯ a for the tangent bundle (with duals e + a and e − ¯ a ) these can be defined (in the untwisted picture) byˆ E + a = ˆ e + a + e + a ˆ E − ¯ a = ˆ e − ¯ a − e − ¯ a (2.20)Clearly, there is an O (9 , × O (9 ,
1) family of these frames rotating the a and ¯ a indicesseparately, reflecting the O (9 , × O (9 , ⊂ O (10 ,
10) structure defined by the generalisedmetric. A generalised vector written with respect to these frames as V = V + a ˆ E + a + V − ¯ a ˆ E − ¯ a has G ( V, V ) = V + a V + a + V − ¯ a V − ¯ a η ( V, V ) = V + a V + a − V − ¯ a V − ¯ a (2.21)Note that we raise and lower a, b, c, . . . with g ab , g ab and ¯ a, ¯ b, ¯ c, . . . with g ¯ a ¯ b , g ¯ a ¯ b . One can also define generalised connections acting on generalised tensor bundles Q to belinear differential operators D : Q −→ E ∗ ⊗ Q (2.22)with a natural notion of generalised torsion L ( D ) V − L V = T ( V ) · (2.23)defined as the change in the Dorfman derivative when one inserts the generalised connectionin place of the partial derivative.In [24], there is a lengthy discussion of how one can construct torsion-free generalisedconnections and in particular those which preserve the generalised metric. In fact, this involvesintroducing an auxiliary line bundle with structure group R + , so that the generalised structure Note again that when writing O (10 ,
10) indices, we will use the O (10 ,
10) inner product η to raise andlower them. This can lead to clashes with signs when decomposing O (10 ,
10) indices. We must be careful to define that V ¯ a = V A =¯ a with an upper index and D ¯ a = D A =¯ a with a lower index. – 7 –roup is enhanced to O (10 , × R + . The generalised Levi-Civita connections are thenthose which are compatible with an SO (9 , × SO (9 ,
1) subgroup of this, with the additionalcompatibility requirements introducing the dilaton field into the structure. There is no uniquechoice for any given generalised metric, but rather a family of such connections, though theundetermined parts drop out of all physical operators which are built from them. We will notrecount the full construction here, but merely note the result that such generalised connectionsact on generalised vectors according to D a V b + = ∇ a V b + − H abc V c + − (cid:0) δ ab ∂ c φ − η ac ∂ b φ (cid:1) V c + + A + a bc V c + ,D ¯ a V b + = ∇ ¯ a V b + − H ¯ abc V c + ,D a V ¯ b − = ∇ a V ¯ b − + H a ¯ b ¯ c V ¯ c − ,D ¯ a V ¯ b − = ∇ ¯ a V ¯ b − + H ¯ a ¯ b ¯ c V ¯ c − − (cid:0) δ ¯ a ¯ b ∂ ¯ c φ − η ¯ a ¯ c ∂ ¯ b φ (cid:1) V ¯ c − + A − ¯ a ¯ b ¯ c V ¯ c − , (2.24)where A ± are undetermined tensors which have the symmetries A + abc = − A + acb , A +[ abc ] = 0 , A + a ab = 0 ,A − ¯ a ¯ b ¯ c = − A − ¯ a ¯ c ¯ b , A − [¯ a ¯ b ¯ c ] = 0 , A − ¯ a ¯ a ¯ b = 0 , (2.25) We form a generalised vector, using the C ± bases ˆ E + a and ˆ E − ¯ a as V + a = ( v a + ˜ v a ) V − ¯ a = ( v ¯ a − ˜ v ¯ a ) (2.26)where for now, v and ˜ v are arbitrary vector fields. In what follows though, v ± ˜ v will be seento be the null directions in the string worldsheet metric, making concrete the notion that the V ± correspond to the left and right moving directions in the string, as remarked in previousworks (e.g. [24]).We claim that the system of section 2 is encapsulated in the equations V + a D a V − ¯ a = 0 V − ¯ a D ¯ a V + a = 0 (2.27)To see this, first we expand the equations using the explicit formulae for the generalisedLevi-Civita connection (2.24) V + c D c V − ¯ a = V + c (cid:16) ∇ c V − ¯ a + H c ¯ a ¯ b V − ¯ b (cid:17) V − ¯ c D ¯ c V + a = V − ¯ c (cid:16) ∇ ¯ c V + a − H ¯ cab V + b (cid:17) (2.28)Now we use the components (2.26) and align the frames ˆ e + a = ˆ e − a = ˆ e a so as to effectivelydecompose under the Lorentz group of the tangent bundle which sits diagonally inside the– 8 –eneralised structure group SO (9 , × SO (9 , V + c (cid:16) ∇ c V − a + H cab V − b (cid:17) = [ ∇ v v − ∇ ˜ v ˜ v ] a + [ ∇ ˜ v v − ∇ v ˜ v ] a + H cab ( v c v b − ˜ v c ˜ v b + ˜ v c v b − v c ˜ v b )= (cid:16) [ ∇ v v − ∇ ˜ v ˜ v ] a + H cab ˜ v c v b (cid:17) + [ ∇ ˜ v v − ∇ v ˜ v ] a V − ¯ c (cid:16) ∇ ¯ c V + a − H ¯ cab V + b (cid:17) = [ ∇ v v − ∇ ˜ v ˜ v ] a − [ ∇ ˜ v v − ∇ v ˜ v ] a − H cab ( v c v b − ˜ v c ˜ v b − ˜ v c v b + v c ˜ v b )= (cid:16) [ ∇ v v − ∇ ˜ v ˜ v ] a + H cab ˜ v c v b (cid:17) − [ ∇ ˜ v v − ∇ v ˜ v ] a (2.29)Thus we see that equations (2.27) are equivalent to the pair of equations[ ∇ v v − ∇ ˜ v ˜ v ] a + H cab ˜ v c v b = 0 ∇ ˜ v v − ∇ v ˜ v = 0 (2.30)The first of these equations is the string equation of motion (2.7). The second equationlooks like an unwanted additional condition. However, because the Levi-Civita connection istorsion-free this equation can be written as[ v, ˜ v ] = 0 (2.31)which is the condition that there are coordinates τ and σ for which v = ∂∂τ ˜ v = ∂∂σ (2.32)Thus equations (2.27) on a generalised vector V firstly impose that the constituent vectors v and ˜ v commute and thus define a foliation of spacetime by string worldsheets, and then alsoimpose that those worldsheets are solutions of the classical equations of motion for the stringin the background generalised metric G .The Virasoro constraints (2.8) are also neatly encapsulated in terms of the generalisedmetric and the O (10 ,
10) metric. In particular, G ( V, V ) = V + c V + c + V − ¯ c V − ¯ c = (cid:0) g ( v, v ) + g (˜ v, ˜ v ) (cid:1) η ( V, V ) = V + c V + c − V − ¯ c V − ¯ c = g ( v, ˜ v ) (2.33)are the two quantities which are set to zero by these. We thus conclude that a generalisedvector field with non-vanishing v and ˜ v , satisfying (2.27), with vanishing quantities (2.33),encodes a foliation of spacetime by classical string worldsheets. This is analogous to a nullgeodesic congruence in general relativity. We will come back to the issue of whether theworldsheet is non-singular, in the sense that both v and ˜ v are non-zero, in the discussion inthe final section.If one considers V ± a ˆ e a as ordinary vector fields, the quantities appearing in equation (2.28)are of course the ordinary connections with torsion usually denoted by ∇ ± in the literature.– 9 –t has long been known [18] that the equations of motion of the string could be written as thevanishing of the ordinary covariant derivatives in (2.28), though the statement that given suchvector fields V ± one obtains a foliation of spacetime by string worldsheets (at least locally)has not been greatly emphasised.The conditions (2.27) at first appear slightly ad hoc, and are not obviously recognisableas the analogue of the usual auto-parallel condition ∇ X X = 0. In the next section, we willsee that if one reformulates the ordinary auto-parallel condition suitably, equations (2.27) arein fact precisely the analogous conditions in generalised geometry. We begin this section by writing a connection-free expression for the quantity ∇ X X , as thisexpression is the one which we will generalise in our key definition. By substituting in theLevi-Civita connection (using its torsion-free property so that L = L ∇ and d = d ∇ ), one caneasily verify the identity ∇ X X = g − · h L X ( g · X ) − d[ g ( X, X )] i (3.1)for any vector field X , where we employ the notation that given a tangent vector X , g · X isthe one-form obtained via ( g · X ) m = g mn X n . Correspondingly, we define that a generalised vector field V is auto-parallel if P V V := G − · h L V ( G · V ) − d[ G ( V, V )] i = 0 (3.2)where G is the generalised metric and ( G · V ) M = G MN V N gives a section of E ∗ . We claimthat the vanishing of this operator is precisely the conditions (2.27). We first demonstratethat this recovers equations (2.27) by evaluating the expression in terms of generalised Levi-Civita connections. We then do an alternative calculation which does not involve generalisedconnections to recover equation (2.7) directly. We note again that we have not defined anotion of parallel transport here, but choose the label auto-parallel as our expression preciselymirrors (3.1). This is similar to the philosophy by which integrable generalised G -structureswere said to have generalised special holonomy in [56].Into the definition we can then insert any generalised Levi-Civita connection, since theseare generalised torsion-free, and decompose into O (9 , × O (9 ,
1) objects. We have, for W = G · V , using (2.16)( L V ( G · V )) a = V B D B W a + ( D a V B ) W B − η aa ′ η BB ′ ( D B ′ V a ′ ) W B = V b D b W a + V ¯ b D ¯ b W a + ( D a V b ) W b + ( D a V ¯ b ) W ¯ b − W b D b V a + W ¯ b D ¯ b V a = 2 V ¯ a D ¯ a V a + V c D a V c + V ¯ a D a V ¯ a = 2 V ¯ a D ¯ a V a + D a (cid:0) G ( V, V ) (cid:1) (3.3)– 10 –ext, viewing d G ( V, V ) as a section of E ∗ , so that its inner product with a generalised vector V ′ = v ′ + λ ′ is v ′ m ∂ m G ( V, V ), we have[d G ( V, V )] a = ∂ a ( V c V c + V ¯ c V ¯ c ) = ∂ a (cid:0) G ( V, V ) (cid:1) (3.4)so that ( P V V ) a = h G − · (cid:2) L V ( G · V ) − d[ G ( V, V )] (cid:3)i a = 2 V ¯ a D ¯ a V a (3.5)is (twice) the operator in the first of equations (2.27). Similarly, keeping careful track ofconventional factors, we find( P V V ) ¯ a = h G − · (cid:2) L V ( G · V ) − d[ G ( V, V )] (cid:3)i ¯ a = 2 V a D a V ¯ a (3.6)Thus we see that equations (2.27) are the analogue of the auto-parallel condition, givenby (3.2).Let us also note, that we can recover (2.7) directly, without explicit use of generalisedconnections. As E ≃ E ∗ , the generalised covector G · V can also be viewed as a generalisedvector, with untwisted picture components η − · G · V = ˜ v + g · v (3.7)We thus have that L V ( η − · G · V ) = [ v, ˜ v ] + L v ( g · v ) − i ˜ v d( g · ˜ v ) − i v i ˜ v H = [ v, ˜ v ] + h v p ∇ p v m + ( ∇ m v p ) v p − ˜ v p ( ∇ p ˜ v m − ∇ m ˜ v p ) + H mnp v n ˜ v p i d x m = [ v, ˜ v ] + h v p ∇ p v m − ˜ v p ∇ p ˜ v m + H mnp v n ˜ v p i d x m + ∂ m ( v p v p + ˜ v p ˜ v p )d x m (3.8)Again, with careful consideration of the numerical factors, we find η − · h L V ( G · V ) − d G ( V, V ) i = [ v, ˜ v ] + h v p ∇ p v m − ˜ v p ∇ p ˜ v m + H mnp v n ˜ v p i d x m = 0 (3.9)becomes the equation of motion (2.7) together with the vanishing of the commutator [ v, ˜ v ]. To understand why (3.1) is a physically natural formulation of the auto-parallel condition,we consider the action of a particle in general relativity: S = Z d λ (cid:18) g mn ∂x m ∂λ ∂x n ∂λ (cid:19) (3.10) As the identification of E with E ∗ is via η MN , and (2.14) contains a factor 1 /
2, there are many awkwardfactors of 2 in this section. We could simply ignore the distinction between E and E ∗ here, but as we cannotdo this in other generalised geometries, we maintain it. – 11 –riting v m = ∂x m ∂λ and imagining that we have a congruence of trajectories as above, theconjugate momentum to the coordinate x m and the Hamiltonian are thus p m = g mn v m H = p m v m − L = g mn v m v n = g ( v, v ) = g − ( p, p ) (3.11)so that the auto-parallel condition, formulated as (3.1), is the statement that L v p = d H (3.12)This equation is reminiscent of one of Hamiltons equations (usually written in textbooks as˙ p = − ∂ H ∂q ) but with the simple time derivative replaced by the Lie derivative along the flowgenerated by the vector field v . This is natural, as the vector field v defines a Hamiltonian“time” coordinate λ on the target spacetime. If we work in a coordinate system with λ as the first coordinate, in which v = ∂∂λ , then the Lie derivative expression reduces to thepartial derivative with respect to this coordinate on the components of tensor fields. TheLie derivative thus provides a covariantisation of the derivative with respect to Hamiltonian“time” λ . Put another way, the Lie derivative is a (covariant) time derivative which isnatural if we match the gauge for spacetime diffeomorphisms to the gauge for worldlinediffeomorphisms. This is often referred to as static gauge in the literature.However, the interpretation is slightly different to the usual Hamiltonian formalism inother ways. The Hamiltonian is usually thought of as a function on the phase space of thesystem, expressed in terms of the coordinates and conjugate momenta. In our case, theHamiltonian depends on the coordinates only through the inverse metric, and is quadratic inthe momenta. However, in (3.12), the Hamiltonian is thought of simply as a function on thespacetime manifold, albeit a function which is expressed in terms of the value of the vectorfield v . One could thus also compare this to a simple potential force law. The left side couldbe thought of as the force (rate of change of momentum), while the right side is the gradientof the potential energy for that force. In this case the potential and the momentum are bothwritten in terms of the same vector field v , together with the metric.We also note that given a Killing vector k , the usual statement of the conservation lawcan also be seen easily from (3.12), without introducing a connection. We have: ∂∂λ (cid:2) g ( v, k ) (cid:3) = L v h p, k i = h d H , k i + h p, L v k i = L k H + h p, L v k i = g ( v, L k v ) + h p, L v k i = h p, L k v + L v k i = 0 (3.13)We will now show that all of this works in the same way for the string, but with a gen-eralised notion of momentum. First, we examine the conjugate momentum to the coordinate– 12 – m for the string action (2.1). This is given by ρ m = g mn v m − B mn ˜ v n (3.14)Here we immediately encounter an apparent difference from the situation above: this quan-tity is not gauge-invariant with respect to B -field gauge transformations. This is typical ofconjugate momenta in the presence of gauge fields, and one encounters the same for a particlecoupled to an electromagnetic vector potential A m . As we change gauge via B ′ = B + dΛ,we have that ρ transforms as ρ ′ = ρ − i ˜ v dΛ (3.15)and thus, the one-form ρ transforms as the one-form part of a generalised vector in the twistedpicture. The vector part of this generalised vector is ˜ v , the tangent vector in the spacelikedirection along the string. We can think of this as being a momentum dual to the “winding”or charge of the string, though here we do not assume any circle directions or isometries in thespacetime. This motivates the definition of the momentum generalised vector (in the twistedpicture) as: P = ˜ v + ρ = ˜ v + g · v − i ˜ v B (3.16)(This object was previously identified in the discussion of [32].) If we move to the untwistedpicture representation this becomes simply P = ˜ v + ρ = ˜ v + g · v (3.17)which is the generalised vector V with v and ˜ v interchanged. This can be written in O (10 , P M = η MN G NP V P (3.18)so that identifying E ≃ E ∗ (i.e. using the O (10 ,
10) metric to raise and lower indices) wehave the generalised covector P = G · V (3.19)Noting that the Hamiltonian of the string is H = ρ m v m − L = g mn ( v m v n + ˜ v m ˜ v n ) = G ( V, V ) = G − ( P, P ) (3.20)we now see that the auto-parallel condition for the generalised tangent vector V for the stringfrom (3.2) becomes (viewing d H as a section of E ∗ as above (3.4)) L V P = d H (3.21)which is clearly the precise analogue of equation (3.12) for the generalised geometry system.Thus, we have the statement that the infinitesimal flow of the generalised momentum along The unsightly factor of 1 / O (10 ,
10) metric η . It isremoved on viewing P and d H instead as sections of E as below. – 13 –he generalised diffeomorphism generated by V is equal to the gradient of the Hamiltonian.If we fix the gauge for spacetime diffeomorphisms, i.e. our local coordinates, to be such thatthe worldsheet coordinates τ and σ are the first two coordinates, thus matching the gauge onthe worldsheet, then one could expect that the Dorfman derivative will involve only simplepartial derivatives along those directions. In fact this is only true if one also imposes theVirasoro constraint g ( v, ˜ v ) = 0.We can also view P naturally as a section of E as we originally defined it in (3.17), andalso d H as a section of E with normalisation η ( V, d H ) = i v d H . Doing so, (3.21) becomes L V P = d H (3.22)From this, we also see that, just as the equations of motion are symmetric in the interchangeof the worldsheet coordinates τ and σ , this equation is symmetric in V interchanged with P .In particular, L P V = 2d η ( P, V ) − L V P = 2d G ( V, V ) − d G ( V, V ) = d G ( V, V ) = d H (3.23)so that in fact the equation can be written as the vanishing of the Courant bracket (i.e. theanti-symmetric part of the Dorfman derivative)[ V, P ] = 0 (3.24)Naively, the exchange of σ and τ appears to resemble the ingredients of a T-dualitytransformation. However, this should not be confused with T-duality, as we do not changethe background metric and B -field. It is simply the exchange of τ and σ on the worldsheet.Let us also examine the possible analogue of the conservation law for a generalised Killingvector K with L K G = 0 [57, 58]. We have ∂∂τ (cid:2) G ( V, K ) (cid:3) = L V h P, K i = h d H , K i + h P, L V K i = L K H + h P, L V K i = G ( V, L K V ) + h P, L V K i = h P, L K V + L V K i = 2 h P, d η ( K, V ) i = L P h V, K i = ∂∂σ h V, K i (3.25)Up until the last steps, this is identical to the calculation for the ordinary geodesic above, butunlike the Lie bracket, the Dorfman derivative is not anti-symmetric. This reflects that thegeneralised tangent space is in general a Leibnitz algebroid [54], rather than a Lie algebroid,and is related to the tensor hierarchy of gauge transformations of the B -field [59–63] and– 14 –ts associated L ∞ structure [64–69]. We thus do not get automatic conservation of the innerproduct of K with the generalised tangent vector V , but we do have conservation if η ( V, K ) =0 (for example if K ∝ V given that V satisfies the Virasoro constraints).Finally, let us note that, similarly to in ordinary geometry where a null Killing vector isautomatically auto-parallel, a generalised Killing vector satisfying η ( K, K ) = G ( K, K ) = 0is also auto-parallel in our generalised sense. Thus, such generalised Killing vectors may giverise to string worldsheet solutions. As we discuss in the conclusion, whether we get a stringor not depends on whether both the vector and one-form components of K are non-vanishing.The condition for this is simply that g ( k, k ) = 0 where k is the vector component of K . In this section we provide the corresponding construction for the M2 brane. We find that theexact same conditions on a generalised vector field in a different version of generalised geom-etry, with generalised structure group SL (5 , R ) × R + , reproduce the equations of motion for agauge-fixed formulation of the worldvolume of the M2 brane, restricted to a four-dimensionalsector of a dimensional split (which is necessary for the formulation of the exceptional gener-alised geometry). In contrast to most constructions of exceptional generalised geometry, weinclude the time direction in this four-dimensional sector, such that the generalised metricdefines an SO (3 ,
2) subgroup of SL (5 , R ) × R + . The bosonic part of the M2 brane action is [70] S = Z d σ h − √− γ (cid:16) γ αβ g mn ∂ α x m ∂ β x n − (cid:17) + ǫ αβγ A mnp ∂ α x m ∂ β x n ∂ γ x p i (4.1)which gives rise to the equations of motion for the field x m ∇ α ∂ α x m + Γ pmq γ αβ ∂ α x p ∂ β x q − ǫ αβγ F mnpq ∂ α x n ∂ β x p ∂ γ x q = 0 (4.2)where ∇ α is the worldvolume Levi-Civita connection, and γ αβ = g mn ∂ α x m ∂ β x n (4.3)for the worldvolume metric. Note that the equation of motion for the worldvolume metric setsit equal to the pullback of the target spacetime metric to the worldvolume. This complicatesthe theory substantially.Here, we use the formulation of the M2 brane in which the worldvolume diffeomorphismsare gauge-fixed (as in [17]) so that the metric γ αβ has γ i = 0 γ + det[ γ ij ] = 0 (4.4)– 15 –or indices α = (0 , i ) and i = 1 ,
2. This results in the action S = Z d σ h (cid:16) g mn ∂ x m ∂ x n − g mn g m ′ n ′ λ mm ′ λ nn ′ (cid:17) + ǫ αβγ A mnp ∂ α x m ∂ β x n ∂ γ x p i (4.5)where, anticipating what is to come, we use the shorthand λ mn = ǫ ij ∂ i x m ∂ j x m with theconvention that ǫ = +1.The conjugate momentum for the coordinate x m is then the target space one-form p m = g mn ∂ x n − A mnp λ np (4.6)which, as for the string, is not gauge invariant under A ′ = A + dΛ. The Hamiltonian is thengiven by H = g ( v , v ) + g mn g m ′ n ′ λ mm ′ λ nn ′ (4.7)where we have written v m = ∂ x m . The equation of motion for the field x m takes the form v p ∇ p v m + λ np ∇ n λ pm + F mnpq v n λ pq = 0 (4.8)where we have extended v and λ to fields on a patch of spacetime, as for equation (2.7). SL (5 , R ) × R + generalised geometry with Lorentz group SO (3 , s = e δ µν d y µ d y ν + g mn d x m d x n (4.9)with m, n = 0 , , , A (3) to have components A mnp along the internal directions, and imposethat all fields depend only on the internal coordinates x m . This is a typical ansatz for awarped compactification to seven-dimensional Minkowski space, but here we take the internalmetric g mn to have Lorentzian signature, while the external factor is warped Euclidean. Forsimplicity, in this paper we will set the warp factor to zero.The theory of eleven-dimensional supergravity restricted to the four-dimensional ansatzoutlined above admits a description in terms of a generalised geometry [40–42, 48] withstructure group E ≃ SL (5 , R ) × R + and generalised tangent space E ≃ T ⊕ Λ T ∗ (4.10)A generalised vector field V ∈ Γ( E ) is given by a vector field v together with a collection oftwo-forms which transform under the gauge transformations A ′ = A + d λ via λ ′ = λ + i v dΛ (4.11) This is derived using the convention that on the Lorentzian worldsheet ǫ = +1 and ǫ = − – 16 –he key objects which we need here are the Dorfman derivative (or generalised Lie derivative)which for generalised vectors V = v + λ and V ′ = v ′ + λ ′ is given by L V V ′ = L v v ′ + ( L v λ ′ − i v ′ d λ ) (4.12)and the generalised metric G ( V, V ) = ( g mn + A pqm A pqn ) v m v n − A pqm v m λ pq + λ pq λ pq (4.13)where indices are raised and lowered with the ordinary metric g mn . As here we take g mn tobe Lorentzian, the generalised metric here is stabilised by SO (3 , ⊂ SL (5 , R ).We will also be interested in sections W = ζ + β of E ∗ ≃ T ∗ ⊕ Λ T . The one-form parts ζ of these transform under (4.11) as ζ ′ = ζ − β y dΛ (4.14)which can be verified as this leaves the natural inner product h W, V i = v y ζ + β y λ (4.15)invariant.As for the discussion of O (10 ,
10) generalised geometry in section 2.2, there are twodescriptions of generalised vectors which are equally good. The description above, with two-forms transforming under gauge transformations, is the twisted picture of this geometry.Given such a generalised vector V = v + λ , we can also define V Untwisted = v + λ − i v A (4.16)which is a global section of T ⊕ Λ T ∗ , thus realising the isomorphism E ≃ T ⊕ Λ T ∗ . In [42],this is described in terms of the components with respect to particular frames called splitframes. In the untwisted picture, the generalised metric and Dorfman derivative are given bydifferent explicit formulae to those above. We have instead L V V ′ = L v v ′ + ( L v λ ′ − i v ′ d λ ) − i v i v ′ F (4.17)and the generalised metric becomes simply G ( V, V ) = g mn v m v n + λ pq λ pq (4.18)In what follows, we will also be interested in an expression for the Dorfman derivative actingon a section of E ∗ . Written in terms of the untwisted objects V = v + λ and W = ζ + β , thisis given by L V W = h L v ζ + β y d λ − v y ( β y F ) i + L v β (4.19) The symbol y denotes the contraction of a multivector into a form with the same conventions as in [42]. – 17 –here is also another bundle, denoted by N , which contains the parameters for the gauge-transformations of the gauge-transformations in the supergravity [42]. For the SL (5 , R ) × R + generalised geometry, its fibre transforms in the ′ + representation of SL (5 , R ) × R + [71],where the generalised tangent space transforms in the + . The formula, (2.18) for thesymmetric part of the Dorfman derivative generalises to the statement that L V V ′ + L V ′ V = ∂ × E ( V × N V ′ ) (4.20)where the symbol × X means that one takes the tensor product and then projects (covariantly)onto the part in the bundle X . In constructions of extended geometry, where one looks to enlarge the spacetime byadding additional coordinates corresponding to the non-vector directions in the generalisedtangent space, the physical spacetimes (or duality frames) are often defined by restricting thedependence of fields to a set of directions which are mutually null in the section condition.In other words, the derivatives satisfy ∂X × N ∗ ∂Y = 0 for all fields and parameters X and Y . This is referred to as the strong section condition, and it is needed for the closure of thealgebra of generalised diffeomorphisms in these constructions [72]. We claim that the same generalised geodesic equation as before, L V P = d H (4.21)encodes the equations of motion for integral surfaces of the generalised vector V which matchthe gauge-fixed M2 brane theory.To make this claim, first we need to understand how the generalised vector encodes threedirections in the four-dimensional space. To see this, let us impose the condition that V isnull in the projection to N , i.e. V × N V = 0 (4.22)This is the analogue of the Virasoro condition η ( V, V ) = 0 for the string. In terms of thevector and two-form components of V in the untwisted picture, it says that i v λ = 0 λ ∧ λ = 0 (4.23)The second condition implies that λ is rank one and thus λ = λ ∧ λ for two one-forms λ and λ . Via the four dimensional metric these give rise to two vectors v = g − ( λ , · )and v = g − ( λ , · ). These vectors must be linearly independent for a non-zero two-form λ .Further, the other condition in (4.23) says that the vector v is orthogonal to v and v in themetric, matching the first gauge fixing condition for the worldvolume metric (4.4). Assumingthat v is timelike, v and v are then spacelike and, without loss of generality, orthogonal. For O ( d, d ) generalised geometry, N is the trivial real line bundle so that V × N V ′ is simply the scalarfunction 2 h V, V ′ i . – 18 –hus, the generalised vector satisfying V × N V = 0 becomes equivalent to three vectorswhich are orthogonal to each other in the four-dimensional metric. These will become thethree directions along the worldvolume of the M2 brane. Imposing also that V is null in thegeneralised metric G ( V, V ) = 0 as for the string, we find that g mn v m v n + g mn g m ′ n ′ λ mm ′ λ nn ′ = 0 (4.24)which matches the second gauge condition for the worldvolume metric (4.4). This is solvedif v is timelike and v and v are spacelike with appropriately related magnitudes. As forthe string, we assume that the components of our generalised vector fit this pattern andare labelled “non-degenerate” as such. Though there are other configurations which wouldsolve the same constraints, and we cannot impose this type of non-degeneracy at the level of SL (5 , R ) covariant conditions, we leave this issue for the discussion.Next, we must examine the conjugate momentum as above. As for the string, the naturalone-form (4.6) has the correct gauge transformation under A ′ = A + dΛ to be the one-formcomponent of a local section of E ∗ ≃ T ∗ ⊕ Λ T in the twisted picture. In the untwistedpicture, this section is given by the global relation P = g · v + v ∧ v = G · V (4.25)In terms of these objects, it is clear that the Hamiltonian (4.7) takes the form H = G ( V, V ) = G − ( P, P ) (4.26)However, note that the symmetry between V and P that we found in section 3.2 is special tothe case of the string and does not have any analogue here. As for our discussion of the string,similar objects to (4.25) and (4.26) have appeared in previous works looking at membranesigma models in extended geometries [50–52].It thus remains to show that (4.21) indeed encapsulates the equations of motion. Us-ing (4.19) we have that as a section of E ∗ , in the untwisted picture:[ L V ( G · V )] m = v p ( ∇ p v m + ∇ m v p ) + λ pq (d λ ) pqm + F mnpq v n λ pq [ L V ( G · V )] mn = ( L v β ) mn (4.27)where β mn = g mp g nq λ pq are the components of the bivector part of G · V . Equation (4.21)then becomes v p ( ∇ p v m + ∇ m v p ) + λ pq (d λ ) pqm + F mnpq v n λ pq = 12 ∇ m (cid:0) v p v p + λ pq λ pq (cid:1) L v β = 0 (4.28)Via some simple manipulations, the first of these equations becomes equivalent to (4.8).Setting v m = v m and β mn = 2 v [ m v n ]2 as suggested above, the second equation becomes[ v , v ] ∧ v + v ∧ [ v , v ] = 0 (4.29)– 19 –hich is solved as [ v , v ] = [ v , v ] = 0 if the vectors v , v , v can be assigned coordinates, aswe require for our worldvolume. This completes the demonstration that (4.21) encapsulatesthe equations of motion of the M2 brane in this formulation of generalised geometry.We also briefly note that the manipulations (3.25) in this case lead to ∂∂τ (cid:2) G ( V, K ) (cid:3) = L V h P, K i = h P, ∂ × E ( K × N V ) i (4.30)so that we also have a conservation law here, provided that our generalised Killing vector K has K × N V = 0. Any generalised Killing vector which is null in the generalised metric andhas K × N K = 0 will automatically be auto-parallel in this case too.Also, while equation (4.21) naively appears to involve derivatives in directions other thanthose along the worldvolume, when expanded we find that this is not the case. This could bemade manifest by expanding the equation in terms of a generalised Levi-Civita connection [42]for the SO (3 ,
2) generalised metric, as we did in section 3.1 for the string.
A summary of our main result is as follows. The classical worldvolume of a string or branein a background generalised metric G has an associated generalised tangent vector V whichis null in the generalised metric and section condition G ( V, V ) = 0 V × N V = 0 (5.1)When extended to a generalised vector field on an open set containing the worldvolume, itsolves the equation of motion L V P = d H (5.2)on the worldvolume, where P = G · V is the generalised conjugate momentum covector and H = G ( V, V ) = G − ( P, P ) is the Hamiltonian/energy function on the spacetime. Equa-tion (5.2) can be thought of as the analogue of the auto-parallel condition for the generalisedvector field V . The extension of V to a local vector field is technically necessary for theDorfman derivative to be defined, though the equations need only be solved on the world-volume. If the equations are solved everywhere, then we obtain a foliation of spacetime byworldvolume solutions, at least locally.In the case of the string, we can argue a converse result: given a generalised vector field V satisfying these equations which is non-singular in an appropriate sense to be discussed,one obtains a foliation of spacetime by classical worldsheet solutions. We have not firmlyestablished this converse result for the membrane, though one could expect that it will beconfirmed by further analysis.This converse follows from considering the solutions to equations (5.1) and (5.2) fora generalised vector field V . Let us suppose that both vectors v and ˜ v derived from the– 20 –eneralised vector are non-vanishing on some local patch of spacetime. Since [ v, ˜ v ] = 0, theydefine a folitation into two-dimensional sheets, and we can choose coordinates on the sheetssuch that they are coordinate induced vectors. Since η ( V, V ) = 0 ⇒ g ( v, ˜ v ) = 0, they areorthogonal and G ( V, V ) = g ( v, v ) + g (˜ v, ˜ v ) = 0 implies that they are either both null or one isspacelike and the other timelike. If they are both null, then orthogonality implies that theyare proportional, and thus the worldsheet degenerates to be one-dimensional. Assuming thatthis does not happen, i.e. that g ( v, v ) = 0, we thus recover that the worldsheet coordinates τ and σ are the coordinates inducing v and ˜ v .Throughout the paper, we have avoided the question of how, in addition to the con-straints (5.1), one could specify that the generalised vector field associated to the foliationis non-singular in this way, and so gives a two-dimensional sheet in the string case. In fact,this question seems to be rich with possibility. Consider, for example, the type IIA decom-position of SL (5 , R ) × R + generalised geometry (see [53] for a full presentation of type IIAdecompositions). There the generalised tangent space decomposes as E ≃ T ⊕ T ∗ ⊕ R ⊕ Λ T ∗ V = v + λ + ω + ω (5.3)and the condition V × N V = 0 becomes i v λ = 0 ω ∧ ω = 0 i v ω + ω λ = 0 (5.4)while G ( V, V ) = 0 becomes G ( V, V ) = v p v p + λ p λ p + ( ω ) + ( ω ) pq ( ω ) pq = 0 (5.5)This gives rise to many possibilities. For example, we could have only v and λ non-zero andbe left with a system like our treatment of the string above (but including background RRfluxes). Alternatively, we could have only v and ω non-zero, which would give a system forthe D2 brane, similar to our picture for the M2 brane above. As the D2 (and the type IIAdecomposition of exceptional geometry) is the straightforward dimensional reduction of theunwrapped M2 system, the equation of motion should match. Another possibility would haveonly v and ω , giving the D0 brane. The case where only v itself is non-zero (and thus a nullvector) would presumably correspond to an infinite boost limit of any of these objects, andthus a pure momentum state. This last statement seems universal.One could thus hope that these conditions will give all possible objects described by therelevant duality group, including objects such as the M5 and NS5 branes for the higher rankexceptional groups, and that (5.2) will provide all of their equations of motion. One potentialcomplication could be the inclusion of the gauge fields that appear on the worldvolumes ofthese objects. However, there is reason to be hopeful there also: the generalised tangentvectors in those cases include more degrees of freedom than purely the directions along theworldvolumes. For example, if one considers D-branes in type II theories, the generalisedvector contains a one-form which could encode the one-form gauge field. Further, for the– 21 –ype IIB NS5 brane there is an additional one-form and for the M5 and type IIA NS5 there isa two-form. Thus, the generalised vectors in question appear to have the degrees of freedomto include the relevant gauge fields. The corresponding charges for these components of thegeneralised vectors correspond to the objects which end on the relevant brane, e.g. the stringcharge in the case of D-branes and the M2 charge in the case of M5 branes, as one wouldexpect.Note that our description has advantages over previous formulations of membrane sigmamodels [50–52] in this respect. Dualities change the dimensions of the worldvolumes of branes.For example, T duality on a circle shifts the dimension of a D-brane up or down by onedepending on whether the worldvolume wraps the circle or not. This makes it problematicto formulate a “duality invariant” action, if the dimension of the worldvolume one integratesover is fixed. However, our generalised vector field description does not suffer from thisproblem, as the dimension of the brane is determined by the decomposition of the generalisedvector in the duality frame in question. One could also hope that the equations could includethe possibility of type-changing solutions, which degenerate along different loci and containcomponents with different dimensions, at least in limiting cases. This might then describeintersections of different objects, such as strings ending on branes, all as the solution to asingle set of equations.We should note that some of the above observations concerning the possible solutions ofthe algebraic equations (5.1) are fairly well known in the literature (see [73, 74] and referencestherein): they can be seen as the BPS condition and the -BPS condition. There is also a clearlink here to the recent works [74–77] in which the target space supergravity solutions of flatbranes are seen to correspond to plane waves in some duality frame. The condition V × N V = 0is usually referred to as the section condition in extended geometries, and subspaces of vectorswhich mutually satisfy it correspond to the spacetime directions of the various duality frames(sometimes also called polarisations) in the extended space. In [74–77] it is shown thatthe different duality frame perspectives on a null wave in the extended spacetime give thesupergravity solutions corresponding to the various branes whose charges are included in thegeneralised tangent space. The duality frame in which the solution corresponds to a wave isone in which the generalised vector V is of the pure vector type (and thus manifestly satisfies V × N V = 0).Our result here could be thought of as a generalisation of this statement from the per-spective of the worldvolume theories and within the realm of generalised geometry, wherethe anchor map π : E → T fixes the duality frame. However, whereas the solutions of [74–77] describe only the flat worldvolumes corresponding to the standard supersymmetric branesolutions in flat space, our result contains arbitrary solutions of the worldvolume theoriesin arbitrary backgrounds (restricted to the dimensions included in the generalised geometryin the exceptional case). For a generalised vector with only a null vector component, ourequations describe ordinary null geodesics, corresponding to null waves. In a sense, our resultcould be viewed simply as writing that system in generalised geometry covariant language.Having done this, it then seems inevitable by symmetry that the other solutions to (5.1) will– 22 –orrespond to the objects carrying the other charges.It would also be interesting to link our results with supersymmetry. Supersymmetricbranes are encoded in the spacetime geometry (with fluxes) by (generalised) calibrations [20,78–81]. In [82], it was shown that the calibration forms are related to the pull-backs to thebranes of the generalised Killing vectors which arise as the commutators of supersymmetrieson the backgrounds (see [58, 83] for details of these superalgebras in generalised geometrylanguage), thus confirming a conjecture from [84]. Thus, in our language it could be thatthese supersymmetric worldvolumes correspond precisely to those generalised Killing vectors.Another point which goes beyond the scope of the present work concerns global aspectsof our worldvolumes. We have worked entirely within a gauge-fixed framework i.e. someparticular choice of coordinates on the worldvolume, which may not be available globally.Though our expressions (5.1) and (5.2) are manifestly coordinate-free on the target space, wemust wonder how to patch together our generalised vectors on the overlaps of these patchesof the worldvolume. We leave this issue for future consideration.An intriguing but speculative possible extension of these ideas concerns “higher” geome-try. While an ordinary vector defines a one-parameter family of diffeomorphisms along whichthe particles flow as they propagate, the generalised vector is more complicated. For example,in the string case, it encodes a diffeomorphism and a gauge transformation of the two-formfield B which together are a general bosonic symmetry of the background fields. However, V should be thought of as an element of an L algebra rather than a Lie algebra. Thus, it is notclear what a generalised vector “integrates to”. Previous efforts to exponentiate generalisedLie derivatives in the context of Double Field Theory [85] have led authors to consider exoticforms of geometry for the doubled or extended space [86–91]. Correctly understanding inwhat sense the generalised vector field presented here may describe a flow, along which ourcanonical momentum is preserved, may well involve appealing to these constructions. Acknowledgments
We would like to thank Dan Waldram for useful discussions.
References [1] R. M. Wald, “General Relativity,” University of Chicago Press, 1984[2] M. B. Green, J. H. Schwarz and E. Witten, “SUPERSTRING THEORY. VOL. 1:INTRODUCTION,” Cambridge University Press, 1987[3] J. Polchinski, “String theory. Vol. 1: An introduction to the bosonic string,” CambridgeUniversity Press, 1998[4] K. Becker, M. Becker and J. H. Schwarz, “String theory and M-theory: A modernintroduction,” Cambridge University Press, 2006[5] D. E. Berenstein, J. M. Maldacena and H. S. Nastase, “Strings in flat space and pp waves fromN=4 superYang-Mills,” JHEP , 013 (2002) [arXiv:hep-th/0202021 [hep-th]]. – 23 –
6] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “A Semiclassical limit of the gauge / stringcorrespondence,” Nucl. Phys. B , 99-114 (2002) [arXiv:hep-th/0204051 [hep-th]].[7] J. G. Russo, “Anomalous dimensions in gauge theories from rotating strings in AdS(5) x S**5,”JHEP , 038 (2002) [arXiv:hep-th/0205244 [hep-th]].[8] E. Sezgin and P. Sundell, “Massless higher spins and holography,” Nucl. Phys. B , 303-370(2002) [erratum: Nucl. Phys. B , 403-403 (2003)] [arXiv:hep-th/0205131 [hep-th]].[9] G. Mandal, N. V. Suryanarayana and S. R. Wadia, “Aspects of semiclassical strings in AdS(5),”Phys. Lett. B , 81-88 (2002) [arXiv:hep-th/0206103 [hep-th]].[10] M. Alishahiha and M. Ghasemkhani, “Orbiting membranes in M theory on AdS(7) x S**4background,” JHEP , 046 (2002) [arXiv:hep-th/0206237 [hep-th]].[11] J. A. Minahan, “Circular semiclassical string solutions on AdS(5) x S(5),” Nucl. Phys. B ,203-214 (2003) [arXiv:hep-th/0209047 [hep-th]].[12] M. Alishahiha and A. E. Mosaffa, “Circular semiclassical string solutions on confining AdS /CFT backgrounds,” JHEP , 060 (2002) [arXiv:hep-th/0210122 [hep-th]].[13] S. Frolov and A. A. Tseytlin, “Multispin string solutions in AdS(5) x S**5,” Nucl. Phys. B ,77-110 (2003) [arXiv:hep-th/0304255 [hep-th]].[14] S. Frolov and A. A. Tseytlin, “Rotating string solutions: AdS / CFT duality innonsupersymmetric sectors,” Phys. Lett. B , 96-104 (2003) [arXiv:hep-th/0306143 [hep-th]].[15] A. A. Tseytlin, “Spinning strings and AdS / CFT duality,” [arXiv:hep-th/0311139 [hep-th]].[16] N. P. Bobev, H. Dimov and R. C. Rashkov, “Semiclassical strings in Lunin-Maldacenabackground,” Bulg. J. Phys. , 274-285 (2008) [arXiv:hep-th/0506063 [hep-th]].[17] S. A. Hartnoll and C. Nunez, “Rotating membranes on G(2) manifolds, logarithmic anomalousdimensions and N=1 duality,” JHEP , 049 (2003) [arXiv:hep-th/0210218 [hep-th]].[18] S. J. Gates, Jr., C. M. Hull and M. Rocek, “Twisted Multiplets and New SupersymmetricNonlinear Sigma Models,” Nucl. Phys. B , 157-186 (1984)[19] N. Hitchin, “Generalized Calabi-Yau manifolds,” Quart. J. Math. , 281-308 (2003)[arXiv:math/0209099 [math.DG]].[20] M. Gualtieri, “Generalized complex geometry,” [arXiv:math/0401221 [math.DG]].[21] U. Lindstrom, “Generalized N = (2,2) supersymmetric nonlinear sigma models,” Phys. Lett. B , 216-224 (2004) [arXiv:hep-th/0401100 [hep-th]].[22] U. Lindstrom, R. Minasian, A. Tomasiello and M. Zabzine, “Generalized complex manifolds andsupersymmetry,” Commun. Math. Phys. , 235-256 (2005) [arXiv:hep-th/0405085 [hep-th]].[23] U. Lindstrom, M. Rocek, R. von Unge and M. Zabzine, “Generalized Kahler manifolds andoff-shell supersymmetry,” Commun. Math. Phys. , 833-849 (2007) [arXiv:hep-th/0512164[hep-th]].[24] A. Coimbra, C. Strickland-Constable and D. Waldram, “Supergravity as Generalised GeometryI: Type II Theories,” JHEP , 091 (2011) [arXiv:1107.1733 [hep-th]].[25] M. J. Duff, “Duality Rotations in String Theory,” Nucl. Phys. B , 610 (1990) – 24 –
26] A. A. Tseytlin, “Duality Symmetric Formulation of String World Sheet Dynamics,” Phys. Lett.B , 163-174 (1990)[27] A. A. Tseytlin, “Duality symmetric closed string theory and interacting chiral scalars,” Nucl.Phys. B , 395-440 (1991)[28] C. M. Hull, “A Geometry for non-geometric string backgrounds,” JHEP , 065 (2005)[arXiv:hep-th/0406102 [hep-th]].[29] C. M. Hull, “Doubled Geometry and T-Folds,” JHEP , 080 (2007) [arXiv:hep-th/0605149[hep-th]].[30] D. S. Berman, N. B. Copland and D. C. Thompson, “Background Field Equations for theDuality Symmetric String,” Nucl. Phys. B , 175-191 (2008) [arXiv:0708.2267 [hep-th]].[31] O. Hohm, W. Siegel and B. Zwiebach, “Doubled α ′ -geometry,” JHEP , 065 (2014)doi:10.1007/JHEP02(2014)065 [arXiv:1306.2970 [hep-th]].[32] C. D. A. Blair, E. Malek and A. J. Routh, “An O ( D, D ) invariant Hamiltonian action for thesuperstring,” Class. Quant. Grav. , no.20, 205011 (2014) [arXiv:1308.4829 [hep-th]].[33] A. S. Arvanitakis and C. D. A. Blair, “The Exceptional Sigma Model,” JHEP , 064 (2018)[arXiv:1802.00442 [hep-th]].[34] R. Bonezzi, F. Diaz-Jaramillo and O. Hohm, “Old Dualities and New Anomalies,” Phys. Rev. D , no.12, 126002 (2020) [arXiv:2008.06420 [hep-th]].[35] W. Siegel, “Superspace duality in low-energy superstrings,” Phys. Rev. D , 2826-2837 (1993)[arXiv:hep-th/9305073 [hep-th]].[36] C. Hull and B. Zwiebach, “Double Field Theory,” JHEP , 099 (2009) [arXiv:0904.4664[hep-th]].[37] O. Hohm, C. Hull and B. Zwiebach, “Generalized metric formulation of double field theory,”JHEP , 008 (2010) [arXiv:1006.4823 [hep-th]].[38] P. ˇSevera and T. Strobl, “Transverse generalized metrics and 2d sigma models,” J. Geom. Phys. , 103509 (2019) [arXiv:1901.08904 [math.DG]].[39] A. Chatzistavrakidis, A. Deser, L. Jonke and T. Strobl, “Strings in Singular Space-Times andtheir Universal Gauge Theory,” Annales Henri Poincare , no.8, 2641-2692 (2017)[arXiv:1608.03250 [math-ph]].[40] C. M. Hull, “Generalised Geometry for M-Theory,” JHEP , 079 (2007)[arXiv:hep-th/0701203].[41] P. P. Pacheco, D. Waldram, “M-theory, exceptional generalised geometry and superpotentials,”JHEP , 123 (2008). [arXiv:0804.1362 [hep-th]].[42] A. Coimbra, C. Strickland-Constable and D. Waldram, “ E d ( d ) × R + Generalised Geometry,Connections and M theory,” arXiv:1112.3989 [hep-th].[43] A. Coimbra, C. Strickland-Constable and D. Waldram, “Supergravity as Generalised GeometryII: E d ( d ) × R + and M theory,” JHEP , 019 (2014) [arXiv:1212.1586 [hep-th],arXiv:1212.1586]. – 25 –
44] O. Hohm and H. Samtleben, “Exceptional Field Theory I: E covariant Form of M-Theoryand Type IIB,” Phys. Rev. D , no.6, 066016 (2014) [arXiv:1312.0614 [hep-th]].[45] O. Hohm and H. Samtleben, “Exceptional field theory. II. E ,” Phys. Rev. D , 066017(2014) [arXiv:1312.4542 [hep-th]].[46] H. Godazgar, M. Godazgar, O. Hohm, H. Nicolai and H. Samtleben, “Supersymmetric E Exceptional Field Theory,” JHEP , 044 (2014) [arXiv:1406.3235 [hep-th]].[47] M. J. Duff and J. X. Lu, “Duality Rotations in Membrane Theory,” Nucl. Phys. B , 394-419(1990)[48] D. S. Berman and M. J. Perry, “Generalized Geometry and M theory,” JHEP , 074 (2011)[arXiv:1008.1763 [hep-th]].[49] M. J. Duff, J. X. Lu, R. Percacci, C. N. Pope, H. Samtleben and E. Sezgin, “Membrane DualityRevisited,” Nucl. Phys. B , 1-21 (2015) [arXiv:1509.02915 [hep-th]].[50] M. Hatsuda and K. Kamimura, “SL(5) duality from canonical M2-brane,” JHEP , 001 (2012)[arXiv:1208.1232 [hep-th]].[51] Y. Sakatani and S. Uehara, “Branes in Extended Spacetime: Brane Worldvolume Theory Basedon Duality Symmetry,” Phys. Rev. Lett. , no.19, 191601 (2016) [arXiv:1607.04265 [hep-th]].[52] Y. Sakatani and S. Uehara, “Exceptional M-brane sigma models and η -symbols,” PTEP ,no.3, 033B05 (2018) [arXiv:1712.10316 [hep-th]].[53] D. Cassani, O. de Felice, M. Petrini, C. Strickland-Constable and D. Waldram, “Exceptionalgeneralised geometry for massive IIA and consistent reductions,” JHEP , 074 (2016)[arXiv:1605.00563 [hep-th]].[54] D. Baraglia, “Leibniz algebroids, twistings and exceptional generalized geometry,” J. Geom.Phys. , 903-934 (2012) [arXiv:1101.0856 [math.DG]].[55] D. Roytenberg, “Courant algebroids, derived brackets and even symplectic supermanifolds”,Ph.D. Thesis, U.C. Berkeley, arXiv:math/9910078 .[56] A. Coimbra, C. Strickland-Constable, D. Waldram, “Supersymmetric Backgrounds andGeneralised Special Holonomy,” [arXiv:1411.5721 [hep-th]][57] M. Grana, R. Minasian, M. Petrini and D. Waldram, “T-duality, Generalized Geometry andNon-Geometric Backgrounds,” JHEP , 075 (2009) [arXiv:0807.4527 [hep-th]].[58] A. Coimbra and C. Strickland-Constable, “Supersymmetric Backgrounds, the KillingSuperalgebra, and Generalised Special Holonomy,” JHEP , 063 (2016) [arXiv:1606.09304[hep-th]].[59] O. Hohm and H. Samtleben, “Gauge theory of Kaluza-Klein and winding modes,” Phys. Rev. D , 085005 (2013) [arXiv:1307.0039 [hep-th]].[60] B. de Wit and H. Samtleben, “Gauged maximal supergravities and hierarchies of nonAbelianvector-tensor systems,” Fortsch. Phys. , 442-449 (2005) [arXiv:hep-th/0501243 [hep-th]].[61] B. de Wit, H. Nicolai and H. Samtleben, “Gauged Supergravities, Tensor Hierarchies, andM-Theory,” JHEP , 044 (2008) [arXiv:0801.1294 [hep-th]]. – 26 –
62] S. Lavau, “Tensor hierarchies and Leibniz algebras,” J. Geom. Phys. , 147-189 (2019)[arXiv:1708.07068 [hep-th]].[63] A. Kotov and T. Strobl, “The Embedding Tensor, Leibniz–Loday Algebras, and Their HigherGauge Theories,” Commun. Math. Phys. , no.1, 235-258 (2019) [arXiv:1812.08611 [hep-th]].[64] D. Roytenberg and A. Weinstein, “Courant Algebroids and Strongly Homotopy Lie Algebras,”[arXiv:math/9802118 [math.QA]].[65] O. Hohm and B. Zwiebach, “ L ∞ Algebras and Field Theory,” Fortsch. Phys. , no.3-4,1700014 (2017) [arXiv:1701.08824 [hep-th]].[66] B. Jurˇco, L. Raspollini, C. S¨amann and M. Wolf, “ L ∞ -Algebras of Classical Field Theories andthe Batalin-Vilkovisky Formalism,” Fortsch. Phys. , no.7, 1900025 (2019) [arXiv:1809.09899[hep-th]].[67] M. Cederwall and J. Palmkvist, “ L ∞ Algebras for Extended Geometry from BorcherdsSuperalgebras,” Commun. Math. Phys. , no.2, 721-760 (2019) [arXiv:1804.04377 [hep-th]].[68] R. Bonezzi and O. Hohm, “Leibniz Gauge Theories and Infinity Structures,” Commun. Math.Phys. , no.3, 2027-2077 (2020) [arXiv:1904.11036 [hep-th]].[69] S. Lavau and J. Stasheff, “ L ∞ -algebra extensions of Leibniz algebras,” [arXiv:2003.07838[math-ph]].[70] E. Bergshoeff, E. Sezgin and P. K. Townsend, “Properties of the Eleven-Dimensional SuperMembrane Theory,” Annals Phys. , 330 (1988)[71] D. S. Berman, H. Godazgar, M. Godazgar and M. J. Perry, “The Local symmetries of M-theoryand their formulation in generalised geometry,” JHEP , 012 (2012) [arXiv:1110.3930 [hep-th]].[72] D. S. Berman, M. Cederwall, A. Kleinschmidt and D. C. Thompson, “The gauge structure ofgeneralised diffeomorphisms,” JHEP , 064 (2013) [arXiv:1208.5884 [hep-th]].[73] N. A. Obers and B. Pioline, “U duality and M theory,” Phys. Rept. , 113-225 (1999)[arXiv:hep-th/9809039 [hep-th]].[74] D. S. Berman and C. D. A. Blair, “The Geometry, Branes and Applications of ExceptionalField Theory,” Int. J. Mod. Phys. A , no.30, 2030014 (2020) [arXiv:2006.09777 [hep-th]].[75] J. Berkeley, D. S. Berman and F. J. Rudolph, “Strings and Branes are Waves,” JHEP ,006 (2014) [arXiv:1403.7198 [hep-th]].[76] D. S. Berman and F. J. Rudolph, “Branes are Waves and Monopoles,” JHEP , 015 (2015)[arXiv:1409.6314 [hep-th]].[77] D. S. Berman and F. J. Rudolph, “Strings, Branes and the Self-dual Solutions of ExceptionalField Theory,” JHEP , 130 (2015) [arXiv:1412.2768 [hep-th]].[78] R. Harvey and H. B. Lawson, Jr., “Calibrated geometries,” Acta Math. , 47 (1982)[79] K. Becker, M. Becker and A. Strominger, “Five-branes, membranes and nonperturbative stringtheory,” Nucl. Phys. B , 130-152 (1995) [arXiv:hep-th/9507158 [hep-th]].[80] J. Gutowski, G. Papadopoulos and P. K. Townsend, “Supersymmetry and generalizedcalibrations,” Phys. Rev. D , 106006 (1999) [arXiv:hep-th/9905156 [hep-th]]. – 27 –
81] L. Martucci and P. Smyth, “Supersymmetric D-branes and calibrations on general N=1backgrounds,” JHEP , 048 (2005) [arXiv:hep-th/0507099 [hep-th]].[82] O. de Felice and J. Geipel, “Generalised Calibrations in AdS backgrounds from ExceptionalSasaki-Einstein Structures,” [arXiv:1704.05949 [hep-th]].[83] A. Coimbra and C. Strickland-Constable, “Supersymmetric AdS backgrounds and weakgeneralised holonomy,” [arXiv:1710.04156 [hep-th]].[84] A. Ashmore, M. Petrini and D. Waldram, “The exceptional generalised geometry ofsupersymmetric AdS flux backgrounds,” JHEP , 146 (2016) [arXiv:1602.02158 [hep-th]].[85] O. Hohm and B. Zwiebach, “Large Gauge Transformations in Double Field Theory,” JHEP ,075 (2013) [arXiv:1207.4198 [hep-th]].[86] R. Blumenhagen, A. Deser, D. Lust, E. Plauschinn and F. Rennecke, “Non-geometric Fluxes,Asymmetric Strings and Nonassociative Geometry,” J. Phys. A , 385401 (2011)[arXiv:1106.0316 [hep-th]].[87] C. Condeescu, I. Florakis and D. Lust, “Asymmetric Orbifolds, Non-Geometric Fluxes andNon-Commutativity in Closed String Theory,” JHEP , 121 (2012) [arXiv:1202.6366 [hep-th]].[88] D. Mylonas, P. Schupp and R. J. Szabo, “Membrane Sigma-Models and Quantization ofNon-Geometric Flux Backgrounds,” JHEP , 012 (2012) [arXiv:1207.0926 [hep-th]].[89] A. Deser and J. Stasheff, “Even symplectic supermanifolds and double field theory,” Commun.Math. Phys. , no.3, 1003-1020 (2015) [arXiv:1406.3601 [math-ph]].[90] A. Deser and C. S¨amann, “Extended Riemannian Geometry I: Local Double Field Theory,”[arXiv:1611.02772 [hep-th]].[91] L. Alfonsi, “Global Double Field Theory is Higher Kaluza-Klein Theory,” [arXiv:1912.07089[hep-th]]., no.3, 1003-1020 (2015) [arXiv:1406.3601 [math-ph]].[90] A. Deser and C. S¨amann, “Extended Riemannian Geometry I: Local Double Field Theory,”[arXiv:1611.02772 [hep-th]].[91] L. Alfonsi, “Global Double Field Theory is Higher Kaluza-Klein Theory,” [arXiv:1912.07089[hep-th]].