The case of the duality symmetric string is a curious one (in a recent post we began discussing this string in the context of building toward a study of duality symmetric M-theory). In this essay, which may serve as the first of a few on the topic, I want to offer an introduction to some of the characteristic features of the duality symmetric string – what I will also refer to as the doubled string – as well as discuss some of its historical connections. One thing that we will focus on at the outset is the deep connection between this extended formulation of string theory, string field theory (SFT), and the more recent development of double field theory (DFT). Such a connection is prominent not least in how we treat the string fields in constructions in which T-duality is a manifest symmetry. For the purposes of this essay, these constructions may be defined in terms of what are called double sigma models.
To help lay this out, let’s quickly review some history. In the early 1990s, a series of papers appeared by Tseytlin [1,2], Siegel [3, 4], and Duff . In these papers, the important topic of string dualities was explored, particularly the fundamental role target-space duality (T-duality) plays in string theory. T-duality is of course an old subject in string theory, and we have already spoken several times in the past about its key features. Recall, for instance, that the existence of this fundamental symmetry is a direct consequence of the existence of the string as a generalisation of point particle theory. Given how for the closed string in the presence of compact dimensions T-duality interchanges the momentum modes of a string with its winding modes around a compact cycle, one of the deep implications is that in many cases two different geometries for the extra dimensions are found to be physically equivalent.
From the space-time perspective, T-duality is a solution generating symmetry of the low energy equations of motion. However, from a world-sheet point of view, T-duality is a non-perturbative symmetry. The fact that it is an exact symmetry for closed strings suggests, firstly, that one should be able to extend the standard formulation of string theory based famously on the Polyakov action (for review, see the first chapter of Polchinski). The idea is that we may do this at the level of the world-sheet sigma-model Lagrangian density, by which I mean the motivation is to construct a manifestly T-duality invariant formulation of closed string theory on the level of the action, remembering from past discussions that we may capture T-duality transformations under the group . When we extend the theory in this way, we find that we are obliged to introduce the compact coordinates and the dual ones in the sigma model, which means we double the string coordinates in the target-space. This gives the name double string theory.
Let’s explain what this all means in clearer terms, as many of these ideas can be sketched cleanly in the context of SFT. In 1992/93, around the same time as the first duality symmetric string papers, field theory emerged as a complete gauge-invariant formulation of string dynamics [6, 7]. This led to the development of a precise spacetime action whose gauge symmetry arguably takes the most elegant possible form . What was observed, furthermore, is how the momentum and winding modes may be treated symmetrically and on equal footing. For instance, let us explicitly denote the compact coordinates and the non-compact coordinates , with . Conventionally, we define the indices such that , , and . If the string field gives component fields that depend on momentum and winding , then in position space we may assign the coordinates conjugate to the momentum and, as alluded above, new periodic dual coordinates conjugate to the winding modes.
The key point is as follows: if one attempts to write the complete field theory of closed strings in coordinate space, then as stated the full theory depends naturally on dual coordinates and . This is also to say that naturally the full phase space of the theory accompanies both the momentum and the winding modes. Or, to phrase it in a slightly different manner, for toroidal compactification there is a zero mode and , and, as the expansion of a string field provides component fields that depend on both momentum and winding, we come to the statement that the arguments of all fields in such a theory are doubled. For the doubled fields we may write the following seemingly simple action
The Lagrangian in (1) may seem straightforward, but in fact it proves incredibly complicated. One issue has to do with how the physical content of the theory becomes buried underneath unphysical and computationally inaccessible data, with the full closed string field theory comprising an infinite number of fields. This is where DFT may be motivated from first-principles; because, in response, DFT answers this problem by issuing the following simplification strategy: what if we instead choose some finite subset of string fields? An obvious choice for such a subsector of the full theory is the massless sector. In the study of DFT, we may then ask, if for the standard bosonic string the low-energy effective action is famously
what does this action become in the case of doubled coordinates on tori? Is T-duality manifest? What about for non-trivial geometries? Historically, DFT emerged with the aim to answer such questions. In fact, following Nigel Hitchin’s introduction of generalised geometry [9, 10], itself inspired by the existence of T-duality, serious efforts materialised to incorporate this mathematical insight into the study of the target-space geometry in which strings live [11, 12, 13, 14], beginning especially with the study of phase space and invariance of respective Hamiltonians. This culminated in 2009, when Hull and Zwiebach formulated such a T-duality invariant theory explicitly , formalising DFT almost two decades after the original duality symmetric string papers. What one finds is a theory constructed on the product manifold with coordinate space fields . The torus is doubled, containing the spacetime torus and the torus parameterised by the winding modes, such that are periodic on . The spectrum for the massless fields is then described in terms of the supergravity limit of string theory.
By taking this approach, DFT has presented fresh insight on T-duality in string theory, leading to the development of deeper connections between frontier theoretical physics and mathematics through the appearance and use of Courant brackets, and by gaining new insight on the deepening role generalised geometry seems to play in string theory.
Much like field theory, the doubled world-sheet theory has also been reinvigorated in the last decade or more. This follows from breakthrough work by Hull [16, 17], who established the doubled formalism to define strings in a class of non-geometric backgrounds known as T-folds. These are non-geometric manifolds where locally geometric regions are patched together such that the transition functions are T-duality transformations.
Currently, there are primarily two doubled string actions that we may consider when constructing double sigma models: Tseytlin’s first-principle construction of the duality symmetric string [1, 2, 15] and Hull’s doubled formalism [16, 17]. Both actions satisfy the requirement of T-duality appearing as a manifest symmetry, with the former possessing general non-covariance and the latter possessing general covariance.
Hull’s doubled formalism is interesting for several reasons. In this formulation we have manifest 2-dimensional Lorentz invariance from the outset, and a notable advantage is that there is a priori doubling of the string coordinates in the target space. In other words, both the Tseytlin approach and the Hull approach are formulated such that both the string coordinates and their duals are treated on equal footing. But in Hull’s formulation, invariance is effectively built in as a principle of construction. This is because for the covariant double sigma model action, the target space takes the form , in which we have a non-compact spacetime and a doubled torus. From the torus identifications we have manifest symmetry. Then, after imposing what is defined as the self-duality constraint of the theory, which contains the metric, invariance of the theory reduces directly to . In other words, while the doubled formalism starts with a covariant action that involves doubled coordinates, the invariance of this theory under is generated by imposing this self-duality constraint, which, similar to DFT, effectively halves the degrees of freedom and ensures that the remaining fields are physical.
Think of it this way: in Hull’s doubled formalism the essential motivation is to double the torus by then adding coordinates such that the fibre is ; however, typically the fields depend only on the base coordinates. Finally, the strategy is generally to proceed with a patch-wise splitting so that we have demarcated a strictly physical subspace and its dual . For a geometric background local patches are glued together with transition functions which include group valued large diffeomorphisms of the fibre. For the non-geometric case, this is approached by gluing local patches with transition functions that take values in as well as in the complete T-duality group, such that is a subgroup of large diffeomorphisms of the doubled torus.
On the other hand, Tseytlin’s first-principle formulation of the duality symmetric string and world-sheet theory for interacting chiral scalars, which presents a direct stringy extension (or stringification) of the Floreanini-Jackiw Lagrangians  for chiral fields, does not possess by principle of construction. Instead, we find that it emerges rather organically as an intrinsic characteristic of the doubled string, with the caveat being that we lose manifest Lorentz covariance on the string world-sheet. What one finds is that we must instead impose local Lorentz invariance on-shell. The equivalence of the Tseytlin and Hull actions on a classical and quantum level has been shown in [32, 33, 34]. Like DFT, both of these approaches are constructed around the generalised metric which we’ll touch on later.
It is no surprise that earlier formulations of the duality symmetric string were a primary reference in the development of DFT. In [1, 2], Tseytlin argues that the existence of the intrinsically stringy winding modes, which appear in the spectrum of the closed string compactified on a torus (created by vertex operators involving both and ), can result in 2d field theories with interactions indeed involving and . Similar models have been explored in statistical mechanics, with the key point in closed string theory being how for fully-fledged local quantum field theories we are required to treat and as independent 2d fields (dual to each other on-shell). An advantage of such an extended formulation of string theory is that we may obtain more vacua than the standard formulation. Furthermore, as one may have guessed, the notion of the duality symmetric string is based on the fact that duality symmetry becomes an off-shell symmetry of the world-sheet action. Thus, T-duality for example may be made manifest in the scattering amplitudes and on the level of the effective action.
To study the construction of the duality symmetric string, we note that directly from 2-dimensional scalar field theory constructed to be symmetric in and , Tseytlin derives the Lagrangian density
Here and are the Floreanini-Jackiw Lagrangian densities for chiral and anti-chiral fields, with and . The total Lagrangian is itself constructed so that it is manifestly invariant under the exchange of with its Hodge dual . Directly from the equations of motion one can derive chirality conditions for this theory (for a complete review see also [32, 33, 34]).
For our present purposes it is important to note that the goal for Tseytlin is to realise from 2-dimensional scalar field theory the corresponding formulation of string theory, which indeed proves general enough to incorporate the world-sheet dynamics of the winding sector. Writing the Lagrangian (3) for scalar fields and with a general background, in the Tseytlin approach we famously obtain the action
Here . We define the coordinates on such that and . The two-dimensional scalar fields depend on and they are vectors in -dimensional target space . The number of embedding coordinates is kept general, because the purpose of this action is to be as generic as possible while minimising assumptions for its construction. We also note that need not necessarily be symmetric and, from the outset, we can treat it completely generically. We also have the zweibein , where . This term appears in the definition of the covariant derivative of the scalar field , where is a flat index and is a curved index.
In its first principle construction, which occupies the earliest sections of , one can recover from this generic action (5) the standard manifestly Lorentz invariant sigma model action for strings propagating in a curved background. Furthermore, if we exclude the dilaton for simplicity we may define , where we reintroduce explicit notation for the string tension , is the metric tensor on the target space, and is the Kalb-Ramond field.
Keeping to a generic analysis with a general , after a number of steps one finds that (5) may be rewritten in the following way,
Here it is conventional to define and . The action is manifestly diffeomorphism and Weyl invariant, but it is not manifestly invariant under local Lorentz transformations. Moreover, notice that (6) must be invariant for the finite transformation of the zweibein, because the physical theory should be independent of . This means that if under such a transformation we have , where one may recognise is a Lorentz matrix dependent on , we also have an induced infinitesimal transformation of the form with . Now, substituting , we obtain
however, as stated, the action is not manifestly invariant under such transformations. The requirement of on-shell local Lorentz invariance is fundamental to the entire discussion at this point. As Tseytlin comments in a footnote , alternatively we may prefer Siegel’s manifestly Lorentz covariant formulation, but with that we obtain extra fields and gauge symmetries; whereas in extending the Floreanini-Jackiw formulation it is fairly simple to introduce interactions and, ultimately, we find that the condition in the Siegel approach that requires decoupling of the Lagrange multiplier corresponds to what we will review as the Lorentz invariance condition in the Floreanini-Jackiw approach.
For the action (6), a way to attack the requirement of on-shell Lorentz invariance is by seeing in  that it demands we satisfy the condition
The general idea is that the tree-level string vacua should be assumed to correspond to , which define the Weyl and Lorentz invariant quantum field theory. In performing the background field expansion, we may take the expansion to be near the classical solution of the equations of motion with the trace of the expectation value of the energy-momentum tensor as well as the trace vanishing on-shell. In Tseytlin’s formulation, denotes precisely this epsilon trace such that . The vanishing of shows local Lorentz invariance. So let us now vary (6) under local Lorentz transformation, which is proportional to the equations of motion
This equation for is equivalent to equation 4.3 in . In order for the variation of the action to vanish under such a transformation, we derive the condition
In other words, the condition that must be satisfied to recover local Lorentz invariance depends on the solution of the equations of motion for the zweibein. In fact, one will recognise that what is observed is completely analogous to the standard string theory formulation based on the Polyakov action, where one will recall that the equations of motion for the world-sheet metric determines the vanishing of the energy-momentum tensor .
This constraint must be imposed on a classical and quantum level.
The key point is that now we can choose the flat gauge , thanks to the invariances under diffeomorphisms, Weyl transformations, and finally local Lorentz invariance imposed on-shell. This is crucial for the formulation of the dual symmetric string in that, using the flat gauge for the zweibein, we are effectively performing the analogous procedure as when fixing the conformal gauge in standard string theory. Keeping and constant, we can compute the equations of motion for the field to give
In the flat gauge this result becomes
From (12) a now famous identity appears, where, in the flat gauge and along the equations of motion for , the following constraint on and is obtained :
One may recognise the tensor structure of (13) in terms of the action of an element. The important thing to highlight is that throughout the lengthy calculation to get to this point, and are held constant. (When and are not treated as constant, a number of interesting questions arise which extend beyond the scope of the present discussion). What is also important is that, after rotating , the matrix can always be put into diagonal form such that
It remains to be said that , which means that the constraint (13) defines the indefinite orthogonal group of matrices with in . The inner product may now be written as
in which the matrix eventually takes on the explicit definition of an invariant metric in the 2D target space . Although, admittedly, this cursory review has omitted many important and interesting details, the pertinent point in terms of this essay is as follows. The action (6) turns out to describe rather precisely a mixture of chiral and anti-chiral scalars. In demanding local Lorentz invariance and the vanishing of the Lorentz anomaly, this requires that with . In working through the complete logic of the calculation, we observe quite explicitly that inasmuch the requirement of local Lorentz invariance is imposed through the condition (10), this leads one naturally to an interpretation of the matrix as a 2D target space metric with coordinates
If we make a change of coordinates in the target space, particularly by defining a set of new chiral coordinates, the matrix takes on the off-diagonal form of the constant metric typically considered in DFT (for review, see ) and elsewhere. The chiral coordinates we define are
In this frame, the matrix is then shown to be
It follows that the condition (13) transforms into the constraint
on the symmetric matrix , which can be parametrised by a symmetric matrix and an antisymmetric matrix . Therefore, remarkably, the symmetric matrix takes the precise form of the generalised metric in which is found to be positive definite.
To conclude, in the chiral coordinates we arrive at a famous form of the Tseytlin action,
This action is manifestly invariant. When transformations are applied to (20), we obtain exactly what we would anticipate for the standard string in the sense of T-duality invariance under and for the generalized metric .
For completeness, from the action (6) in arriving at (20), it should be clear that what we are working with is a sigma model for the dual symmetric string. The generalised version of the celebrated action (20) is indeed often written as
This final action can be argued to be a very natural generalisation for the standard string on a curved background. It not only contains the generalised metric , but also another symmetric metric with signature and an antisymmetric 2-tensor . The coordinates are defined with the background fields in general depending on .
In the last decade especially, Tseytlin’s formulation has been refocused in various studies concerning the nature of the doubled string and its geometry. One notable example to which we will return in a moment, pre-dates the first primary collection of DFT papers and, in many ways, can be interpreted to give a prediction to DFT. I am refering to the 2008 paper David S. Berman, Neil B. Copland, and Daniel C. Thompson , where they investigated the background field equations for the duality symmetric string using an action equivalent to that of Tseytlin’s but constructed in the context of Hull’s doubled formalism. In recent years, a series of publications on doubled sigma models have appeared in connection [19, 20, 21, 22], where in  the double sigma model is for example directly related to DFT.
Another example refers directly to both Tseytlin and DFT from a different perspective. In the years after 2009 when Hull and Zwiebach published their important paper formalising DFT, it was recognised that while a deep connection exists between DFT and generalised geometry, with the former locally equivalent to the latter, it does not completely come into contact with its formal mathematical structures. In fact, an open research question remains motivated by the unmistakeable resemblance DFT has with generalised geometry and the formal gap that remains between them. Recent work in mathematics and physics has displayed some promise, suggesting that the use of para-Hermitian and para-Kähler manifolds may be the solution [23, 24, 25]. Related to these efforts is a recent reformulation of string theory under the heading metastring theory [24, 26, 27, 28, 29], which begins, similar to the studies on double sigma models, with a generalised version of the first-principle Tseytlin action for the duality symmetric string. The metastring is therefore a chiral T-duality invariant theory that, in many ways, wants to generalise from DFT and make direct connection with things like Born geometry , relying on the consistency of Tseytlin’s formulation.
If a direct consequence of making T-duality manifest is that the winding modes are treated on equal footing with momentum, then for DFT all of these properties are incorporated into one field theory. The result, as mentioned, is a doubled coordinate space. In metastring theory, on the other hand, the target space of the world-sheet formulation is a phase space, much like in Tseytlin’s original construction. The coordinates of this phase space are indeed doubled, but unlike in DFT they are also conjugate such that in this case the dual coordinates are related directly to energy-momentum coordinates. In other words, is now identified with . This means that, instead of a physical spacetime formulation, the goal of metastring theory is to construct a sigma model as a phase space formulation of the string and its dynamics.
The implications of metastring theory, as they have so far been conjectured, are intriguing. For example, there have been claims toward obtaining a family of models with a 3+1-dimensional de Sitter spacetime, argued to be realised in the standard tree-level low-energy limit of string theory in the case of a non-trivial anisotropic axion-dilaton background . A key statement here is that, while string theory has purely stringy degrees of freedom (from first principles consider simply the difference between the left and right-moving string modes), these are not captured by standard effective field theory approaches and their spacetime descriptions. Such approaches are usually employed when investigating de Sitter space. In the phase-space formulation of the metastring, these purely stringy degrees of freedom (generally chiral and non-commutating) are argued to be captured explictly. When it comes to the hope of obtaining an effective de Sitter background, one of the major claims in this non-commutative phase-space formalulation is how, in the doubled and generalised geometric description, the effective spacetime action translates directly into the see-saw formula for the cosmological constant. Furthermore, in this cosmic-string-like solution related to the concept of an emergent de Sitter space, it is argued that the metastring leads naturally to an expression of dark energy, represented by a positive cosmological constant to lowest order. Finally, it is argued that the intrinsic stringy non-commutativity provides a vital ingredient for an effective field theory that reproduces to lowest order the sequestering mechanism [29, 30] and thus a radiatively stable vacuum energy.
Building from the Tseytlin action (21), this world-sheet theory of chiral bosons not only takes the heterotic string to its maximal logical completion (a point to be discussed another time), the total doubled space that it sees naturally accomodates stringy non-geometries. With the development of DFT and Hull’s doubled formalism in mind, one interesting question that we can ask concerns whether the best features of all of these approaches can be put together under a more general formulation. There is already a lot in Tseytlin’s original first-principle construction, and so one idea is to generalise from this theory. This was one motivation for my MRes thesis. Another question concerns the presence of generalised geometry and finally how, given a completely generalised treatment of the duality symmetric string, how may we extend the ideas toward the study of duality symmetric M-theory, where exceptional field theory seeks to promote the U-duality group to a manifest symmetry of the spacetime action [37, 38].
These comments take us back to the work of Berman et al. , who started to point toward the same question of generalisation in their approach that combines Tseytlin’s action with Hull’s doubled formalism. It is a very interesting entry into the ideas described, and it is this paper where my own MRes thesis more or less entered the picture.
Moreover, the approach in my MRes was basically to follow the prescription first adopted by Berman et al; however, the action they used to study the doubled beta-functionals for the interacting chiral boson model was constructed in the case where the background fields depend trivially on the doubled coordinates but non-trivially on the non-compact spacetime coordinates. This means that in their approach the target-space was constructed in terms of a torus fibration over a base . One may think of this as a description of string theory in which the target space is locally a bundle, while is some generic base manifold that may be thought of simply as a base space.
While such constructions are important and deserve attention moving forward – we will certainly discuss cases in the future of more complicated bundles, for example – for my MRes the idea was to first strip everything back and generalise the result with minimal assumptions. The first step, for example, was to not demand anything about the dependence of the background fields. What we arrived at was an action of the form
where is the generalised metric and we also have a generic 2-tensor (that we continued to treat generically). In doing away with a base-fibre split (we also dropped a topological term, which isn’t so important here), what we have is the sort of action considered originally by Tseytlin. In fact, (22) is the most general doubled action we can write without manifest Lorentz invariance, because it allows us to calculate the background fields in a way in which the fields maintain arbitrary dependence on the full doubled geometry. That is to say, in taking the democratic approach in which everything becomes doubled, we’re ultimately seeking an effective spacetime theory that corresponds to completely generic non-geometric geometries. At the same time, the structure of the action is precisely the sort proposed to lead directly to DFT , and it also remains equivalent to the Polyakov action in the standard formulation of string theory.
Due to the fact that there are papers pending on these calculations and associated topics, I will leave more details for future entries and for when they more formally appear on arxiv.References
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