Metastring Theory and Modular Space-time

Metastring Theory and Modular Space-time

Laurent Freidel, Robert G. Leigh and Djordje Minic

Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo ON, N2L 2Y5, Canada
Department of Physics, University of Illinois, 1110 West Green St., Urbana IL 61801, U.S.A.
Department of Physics, Virginia Tech, Blacksburg VA 24061, U.S.A.

July 14, 2019
Abstract

String theory is canonically accompanied with a space-time interpretation which determines S-matrix-like observables, and connects to the standard physics at low energies in the guise of local effective field theory. Recently, we have introduced a reformulation of string theory which does not rely on an a priori space-time interpretation or a pre-assumption of locality. This metastring theory is formulated in such a way that stringy symmetries (such as T-duality) are realized linearly. In this paper, we study metastring theory on a flat background and develop a variety of technical and interpretational ideas. These include a formulation of the moduli space of Lorentzian worldsheets, a careful study of the symplectic structure and consequently consistent closed and open boundary conditions, and the string spectrum and operator algebra. What emerges from these studies is a new quantum notion of space-time that we refer to as a quantum Lagrangian or equivalently a modular space-time. This concept embodies the standard tenets of quantum theory and implements in a precise way a notion of relative locality. The usual string backgrounds (non-compact space-time along with some toroidally compactified spatial directions) are obtained from modular space-time by a limiting procedure that can be thought of as a correspondence limit.

1 Introduction

After more than 40 years [1] the deep nature of string theory [2] remains largely hidden. In its conventional formulation, space-time is taken to be the target space of a worldsheet sigma model. It is widely taken for granted that the raison d’être for string theory is to provide local effective field theories on a (non-compact) space-time in a setting that incorporates quantum gravity. These theories are complete from this field theory point of view in the sense that they are apparently ultraviolet finite.

Whenever one pushes the theory to its limits, by looking for example at high energies or short distances, there are indications that the structure of local quantum field theory in a fixed space-time cannot be correct. Certainly the UV finiteness fits with this. More generally, presumably in any theory of quantum gravity, one expects cross-talk between short and long distances and thus some form of non-locality. This is manifested in a variety of ways. It is well-known that there are no local observables in gravity, a fact that was so crucial in the development of holographic space-times. But perhaps even more fundamentally, if one probes quantum gravity theory at very short distances, of the order of the Schwarzchild radius of some probe, then it has been suggested that some sort of ‘classicalization’ may emerge, involving large scale physics. Conceptually, this feels consistent with one of the avatars of string theory, T-duality, in which under certain conditions, short and long distance physics are swapped — a new notion of space-time emerges at short distances (at least along compactified dimensions). Presumably all of these exotic properties of string theories are tied to the fact that what we conceive of as classical geometries are fully discoverable only by particle-like probes. So if we ask any question of string theories that gets at some non-particle aspect, we are likely to lose contact with an understanding within local effective field theory. There are many examples of this sort of effect, involving either perturbative or non-perturbative string physics. A central issue going hand in hand with the emergence of space-time, is the emergence and nature of locality.

In two recent letters [3, 4] we introduced a new formulation of string theory as a quantum theory living outside of the usual space-time framework. Our motivation for developing such a theory, which we now call metastring theory, is manyfold. It is based on the same fundamental concepts as is the usual string theory, departing from it in its initial assumptions about physical space-time. In the present paper, we will explore some aspects of this theory, establishing a number of foundational principles and interpretations. Some of the structure of the theory that we construct is shared by double field theory [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] and the so-called generalized geometries [16, 17, 18]. As we move through the paper, we will be specific about the differences between our formulation and those treatments.

Classically, our starting point will be the Tseytlin action. The form of this action, at least for flat backgrounds (which we mostly confine ourselves to in this paper), can be derived directly from the Polyakov path integral. One of the main features of this formulation is that it is chiral; a second feature is that the target space of this formulation is a phase space and not space-time. The utility of this formulation is that T-duality acts linearly on the target space coordinates, which also explains its role in double field theory. As we described in [4], our interpretation is more general than just implementing T-duality, but touches on the foundations of quantum theory as it relates to string theory. In quantum gravity, there are a number of distinct ways to formulate theories, differing in what is taken as the set of fundamental objects. Are the fundamental objects the smallest (particles, strings) or the largest (space-time itself)? Making either choice means that that choice must define the other. In the worldsheet path integral formulation of string theory, the fundamental probes are strings;111Of course, string theory contain other objects that become visible at finite coupling. These are expected to play a vital role in a complete theory. in the usual formulation we regard them as probes of a given space-time theory. But another point of view is that they define what we mean by space-time, that the geometry is determined by how probes interact with one another.

In the usual formulation of string theory, all the probes agree on a notion of space-time, as space-time is the target space. This of course is ambiguous when (spatial) dimensions are compactified, but becomes unambiguous in a given limit (such as large or small radius). In the chiral phase-space formulation, T-duality gives an action on the phase-space coordinates. At least classically, a choice of a space-time can be thought of as a choice of polarization, in that we identify space-time with a (Lagrangian) submanifold222We emphasize that when we talk about phase space, we always mean the phase space of probes of space-time and momentum space, such as strings, and not of the phase space of gravitational fields, which are emergent in string theory.. In double field theory, one imposes a constraint that is equivalent to identifying a particular submanifold of this phase space as space-time.

In the absence of interactions amongst strings, it is perhaps not obvious that different strings should view the same Lagrangian submanifold as space-time. We think of this as an implementation of Born reciprocity (). This interpretation is particularly clear if we think in terms of string wave-functionals whose natural basis specifies the position in space-time of string loops. In this context, passage to other Lagrangian submanifolds is obtained by Fourier transformation. In fact, this Fourier transform implements generalized T-dualities in the compact case. In ordinary quantum mechanics we may, depending on convenience, choose a position or momentum basis of states; it is a fundamental property of quantum theory that this choice of polarization is immaterial. In quantum gravity, if all probes agree on what we mean by space-time, then we have broken Born duality — there is a preferred choice of polarization, the space-time one. Thus, we emphasize that a suitable notion of quantum gravity is not as a quantization of a space-time theory, but rather should be viewed in a broader context in which space-time is a choice of polarization. This is the structure that metastrings provides. From this point of view the fact that there is a preferred interpretation of space-time in the usual string theory implies some degree of classicality. We refer to this as absolute locality: the same space-time is shared by all probes, independently of their energy state or their history. It is worth pointing out that absolute locality is an assumption that underlies the interpretation of all cosmological observations, as well as all high energy experiments.

We distinguish absolute from relative locality [19, 20], the idea that each probe has, in a sense, its own notion of space-time. Colloquially, it is only when probes talk to one another, through interactions, that they compare their choices. One manifestation of this idea is that the dual momentum-energy space becomes curved (indeed, in quantum field theories, absolute locality is implemented by the linearity of momentum space), an idea that goes back to Max Born [21, 22]333As well, a few attempts have been made to incorporate momentum space curvature as a regulator in quantum field theory, without any definite success. The efforts of Snyder [23] and Golfand [24] are particularly noteworthy. Curved momentum space plays a central role also in 3d quantum gravity[25, 26].. Another motivation for introducing metastring theory is to implement the idea of relative locality in a theory that has a chance to be a complete theory of quantum gravity. We will see that indeed there is a notion of relative locality that emerges in the metastring.

Fixing a specific submanifold as space-time can be thought of within the process of quantizing the string as a choice of specific boundary conditions, constraining the form of string zero modes, in particular, the monodromies. This is the first primary difference between the usual string and the metastring: in the metastring, we do not impose such constraints from the outset, but merely ask the metastring to be consistent with its gauge symmetries and with worldsheet locality. Thus our first task in this paper will be to formulate the Tseytlin theory allowing for generic monodromies.

Such a formulation requires us to consider carefully the general problem of summing over worldsheets. Because the Tseytlin theory does not possess manifest worldsheet Lorentz invariance at the level of the action, we consider the formulation of Lorentzian worldsheets, extending old work of Giddings and Wolpert [27], Krichever and Novikov [28, 29], and Nakamura [30]. The Lorentzian formalism allows us to consider a more generalized notion of closed string boundary conditions, based not on the vanishing of monodromies, but on the continuity of symplectic flux.

The relaxation of the zero mode sector to allow for general monodromies cannot be implemented without restrictions. Consistent with the diffeomorphism constraint, we will in general have ‘dyonic states’ in the spectrum. Thus, the imposition of worldsheet locality on the algebra of vertex operators is a non-trivial condition. Remarkably, we find that this constraint implies that there is a unique444As we will clarify later, the uniqueness applies to a certain class of boundary conditions which do not include, for example, orbifolds. Lorentzian lattice dual to the target space.

The usual interpretation in ordinary string theory would be that this lattice is the Narain lattice of a string theory on a fully compactified Lorentzian space-time. It seems unlikely555We note that Moore [31] has previously tried to make sense of such a compactification. that such an interpretation gives rise to a sensible theory (causality for example, would seem hard to implement). Note that in such an interpretation, the space-time is a Lagrangian submanifold of the target space. By studying the quantum algebra of vertex operators, we find that in fact another interpretation comes to the forefront involving a quantum notion of Lagrangian submanifold, which we refer to as modular space-time. In fact, this interpretation fits well with ideas in ordinary quantum mechanics formulated by Aharonov and Rohrlich [32]. These authors have shown that modular observables are the ones that allow to observe quantum interferences. They have no classical analog and obey non-local equations of motions. They also argue that, remarkably, thanks to the uncertainty principle, this dynamical non-locality does not lead to a violation of causality. This dynamical non-locality is the source of some of the most striking quantum mechanical effects, such as the Aharonov-Bohm or Aharonov-Casher effects [33, 34]. We establish here that the modular space-time experienced by the metastring is colloquially obtained by replacing classical coordinates by modular coordinates which form a commutative sub-algebra of the quantum phase space algebra. The appearance of modular space-time is fundamentally non-perturbative, and even if it contains in some sense a doubling of the target it cannot be understood in terms of corrections as considered in the context of double field theory. Some of the key features of modular space-time have been already discussed on the other hand, although not in our terms, in the context of the ‘non-geometrical backgrounds’ such as monodrofolds or T-folds [35, 36, 37, 38, 39, 40].

It is of interest to consider the notions of ‘quantum’ and ‘classical’ in what we have described here. Even in the usual string theory, there are many layers to these notions; certainly, the worldsheet theory is quantum in the usual sense (being a (path-integral) quantization of a well-defined classical theory). From the space-time point of view (even if we confine attention to string perturbation theory), it is also quantum in the sense of the S-matrix interpretation in (asymptotically-) flat backgrounds, and perturbative in the corresponding expansion in powers of . Clearly, given the progress over the last 20 years, it is not enough to describe string theory as an S-matrix theory, and this is even more clear if, as in the metastring, there is no a-priori notion of space-time. The metastring is formulated as a worldsheet theory, and so there is a definite notion of quantum from the worldsheet point of view. However, it has long been known [41] that in the Polyakov string there is no direct notion of ; instead there is a length parameter that sets the scale of length on the target space. In the Tseytlin form of the action, there are actually two scales and whose product and quotient correspond to and respectively.

In fact, given our notion of modular space-time we should ask in what sense the usual string backgrounds can be recovered. In fact, as we will now summarize, they can be recovered from the metastring via ‘classical’ (for lack of a more precise term) limits. Modular space-time corresponds to a cell in phase space whose size is set by and . It reduces to the classical notion of Lagrangian submanifold in a limit, such as , in which the cell is squashed (preserving volume) in half the directions. Depending on how this squashing is done, one may obtain a theory identical to any compactification of the usual bosonic string (and presumably any superstring as well) with any number of non-compact directions. The low energy physics of such a compactification is local and causal.

Another consequence of these ideas is that they inevitably lead to a certain gravitization of quantum theory. This notion has been suggested before [42], but such discussions have always been hampered by the necessity of discussing it within (semi-)classical GR. It seems natural in unifying the geometrical nature of general relativity and the rigid algebraic structure of quantum theory that both must learn from each other. In the context of the metastring, the rigidity of the quantum theory is encoded into the flatness of the polarization metric , a metric in phase space that tells us how to define the notion of Lagrangian submanifolds. In order to make the metastring consistent on general backgrounds this metric needs to be curved and hence the rigidity of quantum mechanics will be relaxed once we show that the metastring theory is a consistent quantum theory. Trying to quantize the metastring and keep the flatness of the polarization metric leads to inconsistent truncations and presumably explains some of the tensions and difficulty inherent to double field theory, for example. Indeed we will later see that the metastring admits in its spectrum vertex operators which are the seeds of deformation of the polarization metric .

In the future we intend to develop the theory of metastrings on arbitrary backgrounds. To begin, in this paper we will consider the semi-classical structure of the simplest example, involving only a flat background. Although this is far from our ultimate goals, it is important to establish a firm foundation, based on free worldsheet field theory techniques.

The organization of this paper is thus as follows. In Section 2, we recall the derivation of the Tseytlin -model, which we interpret as a chiral theory on a -dimensional target that we call phase space. In this section, we also discuss some geometrical aspects of this target and the symmetries and constraints of the -model. In particular we show how the chiral -model necessitates the introduction of a quantum metric (also called generalized metric) and polarization metric and a phase space 2-form . The absence of worldsheet Lorentz invariance of the -model action leads us in Section 3 to consider the formulation of Lorentzian worldsheets. In Section 4, we consider the canonical analysis of the metastring. In particular we construct the symplectic structure on a strip geometry and show that there is a consistent notion of closed string boundary conditions. A more thorough analysis of the gluing of arbitrary genus Lorentzian worldsheets is reserved for a future publication. In Section 5, we briefly summarize some features of quantum amplitudes of the metastring, culminating in the derivation of the unique Lorentzian lattice as a label of the zero modes of the metastring states. In Section 6, we discuss metastring observables and their canonical bracket. We also show how the classical metastring observables are the canonical generators of phase space diffeomorphism symmetry. The imposition of mutual locality at the quantum level leads us to the realization that the classical notion of projecting to a Lagrangian submanifold must be replaced by the notion of -periodicity. In Section 7, we elaborate on this idea and argue that -periodicity can be interpreted in terms of modular variables. We finish this section with a brief discussion of ‘classical’ limits of modular space-time and how the effective description of strings can be done in terms of fields defined on a modular space-time. In an Appendix, we briefly extend our previous discussion of symplectic structure to worldsheets with time-like boundaries and thus establish a few notions of the open metastring. In Section 8, we conclude with comments on the present status of the metastring theory and future investigations.

2 Sigma Model in Phase Space

We are now ready to formulate the metastring theory. As we mentioned above, our aim is to establish a theory that is capable of describing curved space-times and momentum space simultaneously. We review here the passage to such a theory, which we obtain by deforming the usual Polyakov path integral formulation. We begin our discussion [3] by examining the Polyakov action coupled to a flat metric ,

(1)

where denote the Hodge dual and exterior derivative on the worldsheet, respectively. We generally will refer to local coordinates on as , while it is traditional to interpret as local coordinates on a target space , here with Minkowski metric . Since we are in Lorentzian signature, and . Note that has dimensions of length-squared if we take to have dimension of length, so appears in the path integral as . is the string length which is related to the slope parameter by , where is the Planck constant of the worldsheet quantum theory. With this definition has the usual coefficient in units of . In order for the Polyakov action to be well-defined, one must demand that the integrand be single-valued on . For example, on the cylinder it would be sufficient that is periodic666The most general condition would be to ask that where is a Lorentz transformation. In this work we only consider the case where . This restriction is a fundamental limitation of our analysis that excludes, in particular, orbifolds. with respect to with period . However, and this is a crucial point, this does not mean that has to be a periodic function, even if is non-compact. Instead, it means that must be a quasi-periodic function which satisfies

(2)

Here is the quasi-period, or monodromy, of . If is not zero, there is no a priori geometrical interpretation of a closed string propagating in a flat space-time – periodicity goes hand-in-hand with a space-time interpretation. Of course, if were compact and spacelike then would be interpreted as winding, and it is not in general zero [3]. However since we want ultimately to generalize the -duality to curved backgrounds, we do not want to impose the restriction that there is a space-time interpretation of the monodromies. Instead we want to find what conditions these monodromies have to satisfy. As we stressed in [3] the string can be understood more generally to propagate inside a portion of a space that we will refer to as phase space . What matters here is not that string theory possesses or not a geometrical interpretation but whether it can be defined consistently. This is no different than the usual CFT perspective, in which there are only a few conditions coming from quantization that must be imposed; a realization of a target space-time is another independent concept. It has always been clear that the concept of T-duality must change our perspective on space-time, including the cherished concept of locality, and so it is natural to seek a relaxation of the space-time assumption.

In order to present our perspective on T-duality, let us consider the dimensionless first order action777The passage from the usual Polyakov formulation to this can be performed straightforwardly in the full worldsheet path integral.

(3)

where is a momentum scale, is a length scale and is a one form with dimension of mass. If we integrate the one form we get back the space-time Polyakov action, and if we integrate we get the momentum space Polyakov action. Indeed, if we integrate out , we find and we obtain the Polyakov action

Now, the reader may come to the conclusion that are not independent scales, and this would be true within the confines of this flat non-interacting theory. However, the introduction of here is an important step conceptually [41]. In any theory of quantum gravity, we expect to find three dimensionful constants, , and . Putting aside, this implies that quantum gravity depends both on a length scale and an energy scale (here we are using the language of dimension for simplicity). As was emphasized by Veneziano long ago, the usual formulation of string theory as a theory of quantum gravity contains a puzzle: there is apparently only one dimensionful scale, (or equivalently, ) that appears directly in the quantum phase factor. In the presence of both a length scale and an energy scale , we can reconstruct

(4)

The Newton constant is proportional to the latter scale, , depending on the dimension and the details of compactification.

Of course, in the present context, these constant scales can be reabsorbed into a redefinition of the fields . The significance of the parameters are only seen when we ask questions about specific probes in the phase space target theory (e.g., we compare a probe momentum to ), or if we consider backgrounds that have their own inherent length scales (such as a curvature scale).

Now, on the other hand, if we integrate out instead, we get and so we can locally write where can be thought of as a momentum coordinate. It is in this sense that there is “one degree of freedom” in even though it is a worldsheet 1-form– on-shell, is locally equivalent to the scalar . Notice though that this is true only locally, and in order to interpret it globally we must allow to be multi-valued on the worldsheet. That is, even if we assume that is single-valued to begin with, should carry additional monodromies associated with each non-trivial cycle of . This means that the function is only quasi-periodic with periods given by the momenta

(5)

The action for (obtained by integrating out ) becomes essentially the Polyakov action, with the addition of a boundary term

(6)

Note that has the dimension of length while has the dimension of momentum. We recover the Polyakov action for the momentum variable, with playing the role of for this dual theory. The presence of the boundary term in (6) is related to the fact that the transformation corresponds to the string Fourier transformation [43]. Indeed, as we will see, for a boundary located at fixed , and are conjugate variables satisfying

(7)

where is the periodic delta function.

In order to obtain a formulation where we are left with a phase space action, a natural idea is to partially integrate out . In a local coordinate system on the worldsheet, we write the decomposition of the momentum one-form

(8)

In conformal co-ordinates the first order action then reads888Our conventions are such that in the conformal frame the 2d metric is and , and . Here means .

(9)

The equations of motion for are simply

(10)

By integrating out , we insert the equation of motion and get the action in Hamiltonian form:

(11)

Now we locally introduce a momentum space coordinate such that . Like , this coordinate is not periodic, its quasi-period is proportional to the string momentum. Using this coordinate the action becomes simply

(12)

The main point is that in this action both and are taken to be quasi-periodic. The usual Polyakov formulation is recovered if one insists that is single-valued, and the usual T-dual formulation is recovered if one insists that quasi-periods of appear only along space-like directions and have only discrete values.

It is convenient, as is often used in the double field theory formalism [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], to unify both and in one space (that we often refer to as phase space) and introduce a dimensionless coordinate on

(13)

To write the action, we introduce a constant neutral999Here neutral means that is of signature , while is of signature . metric , a constant metric and a constant symplectic form

(14)

where is the -dimensional identity matrix and is the -dimensional Lorentzian metric, denoting transpose. The presence of a symplectic structure expresses the fact that is a symplectic manifold. The space-time vectors of the form defines a subspace of . Similarly momentum space vectors of the form form defines another transversal subspace of . Moreover, we see that both the space-time subspace or momentum-space subspace are Lagrangian subspaces of of maximal dimension. That is the symplectic structure vanishes on both of them and . We can also see that both and are null subsets of with respect to . That is if and similarly for . The metric has therefore the property that its null subspaces are Lagrangian manifolds of maximal dimension. A choice of Lagrangian subspace of phase space is called a choice of polarization. We therefore refer to the metric as a polarization metric or P-metric . This metric is of signature and it is therefore neutral. The subscript refer to the fact that the metric is constant in the present discussion.

The metric already appears in the context of double field theory and generalized geometry [17] and is often referred to as the generalized metric. We feel however that this denomination misses the point that is a phase space and that this metric can be understood as descending from the quantum probability metric applied to coherent states [4, 43]. Therefore we refer to this metric as the quantum metric or Q-metric . This metric is of signature , the two negative eigenvalues corresponding to the time direction in space-time and the energy direction in energy-momentum space. When restricted onto the space-time Lagrangian subspace it provides the space-time metric .

The P-metric and the Q-metric are not independent in the present context: if we define

(15)

we see that is an involutive transformation preserving , that is,

(16)

defines a chiral structure101010Also called a para-complex structure in the mathematical literature [44]. on phase space . We also introduce the constant symplectic form:

(17)

which expresses the fact that is a symplectic manifold.

Using these definitions, the action is written as a -model on :

(18)

The term proportional to is a total derivative. However since there are monodromies, it will be relevant, as we will see, to keep track of it. One sees that the Hamiltonian is ultra-local – it depends only on the space derivatives. In view of the pioneering work [45, 46, 47], we call this expression the Tseytlin action111111See also [48]. . The Tseytlin action is such that its target is .

This space is equipped as usual with a symplectic structure, and in order to carry a string we emphasize that it contains two metrics, . The Q-metric can be thought of as being an extension to of the space-time metric, while as we will see more precisely later, the P-metric is related to the decomposition of phase space into space-time and energy-momentum . A point that will become important later is the fact that space-time can be characterized as the kernel of while energy-momentum is the kernel of . In the case at hand we also have that the momentum Lagrangian is the image of the space-time one by the chirality map . As we will see, this last property is specific to a geometry with vanishing -field.

At first one might wonder how one can double the target space dimension without doubling the degrees of freedom. This is related to the fact that the metastring is chiral: i.e., there are no terms quadratic in time derivatives. This is achieved thanks to the presence of the chiral structure and, in particular, the fact that it squares to unity. While the Polyakov string contains both left- and right-movers, the metastring contains only left- and right-movers that are chiral in the target. As we will see, the left-movers have negative chirality while the right-movers have positive chirality.

2.1 More General Backgrounds and Born Geometries

Although in this paper we will work exclusively with the flat theory described by (18), it is instructive to consider the generalizations to which we will turn our attention in future publications. One might expect that can be replaced by more general structures.

In fact, it is a simple extension of the above construction to include a curved background in the Polyakov action

(19)

We can recast this action in the first order form by introducing dual and fields by or equivalently

(20)

The first order dimensionless action reads

(21)

Following the same procedure as above, we obtain

(22)

As before, by introducing the dimensionless coordinates , we write the action as

(23)

where now

(24)

Let us finally remark that the general metric can be obtained from the trivial one by an O transformation: , where

(25)

is an matrix and is the frame field corresponding to .

Thus, the usual string theory in curved backgrounds corresponds to making the Q-metric dynamical (but not the P-metric or the symplectic structure ). Let us discuss further generalizations. Suppose that we first generalize to general structures . Furthermore, given the existence of and , there is a natural way to understand this geometrical structure from the point of view of quantum mechanics. If one takes the point of view of geometric quantization [49, 50], the construction of the Hilbert space associated with a phase space requires the introduction of a complex structure compatible with . Such a complex structure defines the notion of coherent states as holomorphic functionals and equips the phase space with a quantum-metric via the relation [3]. This structure is, in effect, what Born suggested to be part of quantum gravity in the 1930’s [21]. In the string case if the field vanishes is related to via a complex structure. This is no longer true if does not vanish. We can still define the map in this case, but the Q-metric and the symplectic structure are no longer compatible.

However, the Born proposal is not enough. As we have pointed out in [3], in the metastring theory we must take to be dynamical as well. As we have seen above, it is that governs the splitting of phase space into space and momentum space. In particular, one can think of space-time as a Lagrangian submanifold, that is a manifold of maximal dimension on which the symplectic structure vanishes. Analogously, momentum space is just another Lagrangian submanifold in this description, which is transverse to the space-time Lagrangian submanifold. Thus we end up with a bilagrangian structure on . That is a decomposition of into two transverse Lagrangian manifolds: and . What is remarkable is the fact that a bilagrangian structure is uniquely characterized by a polarization metric . This metric is characterised by the fact that and . In other words, the geometrical notion of is to provide a bilagrangian decomposition of phase space. The neutral metric that seems like a purely stringy metric is in fact a very natural object from the point of view of phase space, in that it labels its decomposition into space and momentum. In order to prove this, let us introduce a structure which is on the vectors tangent to the space-time Lagrangian and on the momentum space Lagrangian . This is a real structure which satisfies . Since and are Lagrangians also satisfies an anti-compatibility condition with : . These two properties in turn show that is a neutral metric .

We have already emphasized the importance of the endomorphism , which relates the two metrics. Its properties enforce the chirality of the -model. We thus suppose that the geometry of should be constrained by the property .

It is relative locality that suggests that both and be dynamical. In particular, in canonical quantum theory is a purely kinematical structure and , which describes the choice of polarization, can be modified by unitary dynamics. Conversely, in the context of gravitational dynamics, is a purely kinematical structure (because space-time provides the preferred basis or polarization), while , through its space-time part, can be made dynamical. According to Born, when we introduce gravity into quantum theory we have to make into a truly dynamical quantity. When we introduce quantum theory into gravity, we have to make the neutral metric dynamical, and thus in the context of quantum gravity, both and have to be dynamical.

The neutral metric is, together with the generalized phase space metric , indispensable for the definition of space-time as a maximally null subspace of with the space-time metric given by the restriction of the metric to this -null subspace [3].

The structure can also be described in terms of the two real structures and the map . We can check that the relation between these maps is given by

(26)

If, in addition, we assume that vanishes we have that is a complex structure and that . Phase space geometries that have satisfying these conditions were referred to as Born geometries in [3]. These possess para-quaternionic structure (because , but and they anticommute). Born geometry represents a natural unification of quantum and space-time and phase space geometries, and it implies a new view on the kinematical and dynamical structure of quantum gravity. This structure is natural in a quantum particle theory. In string theory it is also natural to allow for a non zero -field, in which case is no longer a complex structure121212As a side comment, note also that in the mathematical literature the Born reciprocity idea has been at the root of the invention of quantum groups. Indeed, quantum groups, originally designed by Drinfeld [51, 52] as doubles, are self-dual algebraic structures and the famous R-matrix is the kernel of the Fourier transformation. Another independent invention of a subclass of quantum groups [53, 54], the bi-crossproduct ones, directly stems from the algebraic implementation of the Born self-dualization idea, a principle at play in 3d gravity [55]. Finally let us note that the canonical quantization of curved momentum space has also been discussed in other contexts as well [56, 57, 58, 59, 60]. .

2.2 T-duality

The expression of T-duality in the Polyakov formulation of constant backgrounds appears as the worldsheet symmetry

(27)

which exchanges and in the conformal gauge. The phase space formulation on the other hand breaks the symmetry between and . The T-duality symmetry does not appear as a worldsheet symmetry, but instead appears as a target space symmetry. This manifest transfer of the symmetry property from worldsheet to target is one of the main advantages of this formulation. In order to see this let’s consider, given and , the chiral operator . It can be written explicitly as

(28)

What is remarkable about this operator is the fact that it is an transformation leaving the P-metric invariant and that, as we have mentioned above, it is a chiral structure which squares to the identity:

(29)

From its definition it can be seen that , so it also preserves :

(30)

These properties imply that the map

(31)

is a symmetry of the bulk action, and it expresses the T-duality symmetry.

Note however that does not preserve . When the field vanishes it maps into , while if the -field is non-zero it rotates non-trivially the Lagrangian subspaces. An explicit computation gives with

(32)

In the constant background case this breaking of T-duality appears only as a change of the boundary conditions via the boundary term. Another way to express this is to notice that when the -field is non-zero, the momentum Lagrangian is no longer aligned with the subspace orthogonal to with respect to . Indeed, this space is simply given by since .

2.3 Usual string viewed from phase space

It is clear from the previous analysis that the formulation (23) begs for a natural generalization where possesses arbitrary monodromy and where not only the constant Q-metric is promoted to an arbitrarily curved metric , but also the P-metric and symplectic structure are allowed to be dynamical. In the general case we promote to be functions of . Such a generalization aims to provide a string theory formulation where T-duality is manifest even in the curved context [61]. The action is given by the consequent generalization of (18) and we call such a generalization the metastring. Double field theory, on the other hand, usually considers the effective field theory based on the restricted structure , where both the symplectic structure and the P-metric are treated as background structures, while is allowed to have a specific type of dependence131313The fields are demanded to be projectable. A recent exception [15] considers a non-trivial while still keeping a flat P-metric . Another notable exceptions are in the context of beta function calculations [62, 63, 64]. .

Before doing so, it is necessary to pause for a moment and understand what specific conditions characterize the Polyakov string within the metastring. Let us start by listing the necessary and sufficient conditions that the Tseytlin string has to satisfy in order to be a Polyakov string in disguise. There are 5 conditions:

  • is an involution preserving .

  • are maximally degenerate, i.e., of rank .

  • is a closed form.

  • The fields only depend on the degenerate directions of ; that is

    (33)
  • The fields possess monodromy only in the degenerate direction of ; that is

    (34)

    where is the monodromy.

In the case where are constant and given by (24), the matrix projects to zero the energy-momentum vectors , that is, the vectors belonging to the Lagrangian . On the other hand projects out the space-time derivative . This means that the conditions (34) and (33) read respectively and . They imply that the fields depend on while monodromy is only in the momentum Lagrangian . We will analyze what happens when we relax these conditions.

The mildest condition to relax is the last, in which we allow monodromy in all directions. In the case where all the fields are constant and extra monodromies are allowed only in space-like directions, this corresponds to the torus compactification of the Polyakov string. If monodromy is allowed in the time-like direction, the usual interpretation is in terms of thermal solitons and gives rise (under Euclidean continuation) to the string free energy, etc. [65, 66]. In a later section, we will carefully consider this generalization and show that there is a consistent but non-trivial notion of closed string boundary conditions.

Next we can relax the condition (33) by allowing the fields themselves to depend on all coordinates in . This generalization is one of the most interesting and will need to be dynamically constrained in order to give admissible backgrounds. In particular, it implies considering the new possibility where is no longer a flat metric. This entails relaxing the condition that the splitting between space-time and energy-momentum is universal. That is, it relaxes the hypothesis of absolute locality and allows us to have a framework in which locality is relative, or, in colloquial terms, a framework where each string can carry a different space-time.

Another level of relaxation is to allow to not be closed. This would impede its interpretation as a symplectic form in Born geometry. Although this generalization deserves study, it is beyond the scope of our present discussion. As we will see [43, 67, 61] these three levels of relaxation are admissible both at the classical and the quantum level.

The next level of generalization would be to consider a string where is not maximally degenerate. For instance, as we will see later, if is invertible, there is no propagating open string. For simplicity, we will keep the condition of maximal degeneracy for now. We have seen that in the Polyakov case the kernel of plays the role of the space-time Lagrangian . By keeping the property of maximal degeneracy, we keep the concept of a preferred Lagrangian defined by the metastring fields. Moreover we will see that the open metastring boundary naturally propagates inside . If we want to keep the compatibility condition between open and closed string in the sense that the open string possesses half the closed string degree of freedom, we have to keep the condition of maximal degeneracy.

Finally, we are also going to see in this work that it is inconsistent to relax the first condition: we always need to be a chiral structure if we want to keep the conformal symmetry of the theory. In summary, the metastring action is given by

(35)

where the fields which correspond respectively to a neutral P-metric , a Q-metric and a 2-form, are all dynamical and depend on . We demand however that is maximally degenerate and that is a chiral structure.

2.4 Global Symmetries

We now comment on the global symmetries of the flat Tseytlin action (18). We still assume in this section that , and are constant matrices. Let us first use the fact that since is a neutral metric, we can always choose a frame where it assumes the form given in (24), that is . As we have seen in (25) we can, in this frame, trivialize by an O transformation. Without loss of generality we can therefore take for illustration in the form (14).

2.4.1 Double Lorentz symmetry

The first global symmetry of the action is the double Lorentz group , preserving both and . That is, we define

(36)

This group is isomorphic to . The component is generated by matrices of the form

(37)

where , and , i.e., . There are two types of ‘boosts’ here. First, we have the usual ones , that act in the usual way on space-time and energy-momentum space defined as Lagrangian subspaces of . Secondly, the boosts mix space-time and energy-momentum space in a non-trivial manner. This is the group of symmetries of the metastring theory, the action being invariant under . Thus the group of Lorentz transformations is generalized to its double since its Lie algebra is locally isomorphic to . This fact can be clearly seen if we look at the action of this group on the chiral components of . We find that it acts diagonally:

(38)

where is a Lorentz transformation.

The component of the symmetry group is generated by . This corresponds to the exchange of two Lagrangian subspaces.

2.4.2 Discrete symmetries

The metastring possesses three distinct discrete symmetries.141414There are also discrete elements of the double Lorentz group acting locally on , such as the inversion , corresponding to . The first one that we have already seen is the duality symmetry

(39)

We also have the PT symmetry

(40)

and the time reversal symmetry

(41)

where is a matrix such that and it also satisfies and . This is given by

(42)

It is interesting to note that this matrix anti-commutes with

(43)

This means that the combination of time reversal and duality symmetry is implemented by the map

(44)

where is a complex structure which preserves :

(45)

The found here are those of the corresponding (trivial) Born geometry. Here we have seen that they are involved in symmetries of the flat Tseytlin model that act on both worldsheet and target space.

2.5 Time translation symmetry

Another important symmetry of the Tseytlin action is the time translation symmetry. We consider the transformation, described in the local conformal frame

(46)

This corresponds to a translation along a -independent vector field. In the case where are constant, we are just describing a global translation of the flat target space. We emphasize that there is a larger symmetry here under certain conditions on . Indeed, under such a -dependent transformation the action transforms by a boundary term

(47)

where is the monodromy. We see that this variation vanishes if belongs to the kernel of . It is important to note that this necessarily implies that is null with respect to the P-metric , . In other words, belongs to the momentum space Lagrangian . We will analyze later the consequences of this extra symmetry.

2.6 Constraints

Let us now understand the nature of the Virasoro constraints in this formulation. In string theory we integrate over all worldsheet metrics, that is we integrate over all conformal structures and quotient by the action of 2d diffeomorphisms. This imposes Hamiltonian and diffeomorphism constraints on the data. For now, we focus on a given cylinder, in which the worldsheet metric is conformally equivalent to , coming back to general worldsheets later. If we change the conformal frame infinitesimally, we have to introduce a new time and space coordinate frame. A variation of the conformal structure can be encoded in two functions via

(48)

A new conformal frame is obtained by a redefinition of the local frame coordinates with

(49)

The variation of the space and time derivatives due to this local change of frame is given by

(50)

We can now determine the Hamiltonian and diffeomorphism constraints from the variations , , which in local coordinates read

(51)
(52)

Finally, it is also important to consider variations of the coordinate frames that do not change the conformal structure. These are given by the Weyl and Lorentz transformations: , and , respectively. Demanding invariance under these variations leads to the (classical) constraints

(53)

In order to see that these reduce on-shell to the usual Hamiltonian and diffeomorphism constraints of string theory, and that the Lorentz and Weyl constraints are trivially satisfied, let us first write these constraints in a slightly different form. Consider the vectors

(54)

and rewrite all the constraints in terms of and . (In the following we denote by the contraction with the metric .) The constraints are then

(55)

Note that in the flat case the constraint is identically satisfied. In this case, the Lorentz condition simply becomes . In the flat case the equation of motion implies that . This means that depends only on . The Lorentz condition means that belongs to a Lagrangian subspace , that is a null subspace of the P-metric . Choosing such that is the kernel of , we can use the time symmetry described earlier to fix the gauge where . This is the gauge in which we will now work. Notice that this gauge choice, given , fixes a relationship between chirality on the worldsheet and -chirality in the target space.

Also, in this language, the Hamiltonian and diffeomorphism constraints are given by:

(56)

where we have denoted . In terms of the phase space coordinates , the constraints read , . These reduce to the usual form

(57)

once we impose the duality equations , .

2.6.1 Energy momentum Tensor

We would like to write the phase space action in a more covariant manner in order to clarify the constraints. Indeed, so far we have heavily relied on the space-time splitting which assumes a conformal frame on the worldsheet. We now introduce a fully covariant formulation of metastring theory that does not assume a particular choice of coordinates on the worldsheet.

In order to find a covariant formulation, we introduce the co-frame field

(58)

with and the corresponding frame fields which we denote as

(59)

They are such that . Given these definitions the metric can be written as . It is convenient to write everything in terms of a chiral frame: and in which the metric reads . The Tseytlin action can be now written

(60)

This action is manifestly diffeomorphism and Weyl invariant, but not manifestly locally Lorentz invariant.

We define the energy momentum tensor as We make this definition rather than the usual one involving the variation with respect to the metric, because in the absence of Lorentz symmetry, the stress current is not automatically symmetric. We then find

(61)
(62)

The generators of Weyl and Lorentz transformation act on the frame fields as :

(63)
(64)

and the Tseytlin action transforms as where the Weyl and Lorentz generators are given by , . This gives explicitly:

(65)

where . The generators of conformal transformations are then given by and and read as follows

(66)

in agreement with our previous derivation.

The new feature of this formulation is the fact that worldsheet Lorentz invariance is not manifest; under an infinitesimal Lorentz transformation the action transforms as (assuming )

(67)

and the constraint has to be imposed, in other words, has to be null with respect to the neutral metric. It is only after the imposition of this constraint, which implies on-shell after use of the time symmetry, that we recover the usual Polyakov formulation where this symmetry is satisfied on-shell for the flat background. As we will see the non manifest Lorentz symmetry is akin to the non manifest Weyl invariance of the massive deformations of Polyakov string. It is one of the most challenging but also one of the most interesting and fruitful aspects of this new formulation. The deep quantum implications of this fact will be explored in [43, 67]. See also [62, 68, 64].

2.6.2 Euclidean form and Level Matching

The description given here may look unfamiliar since it is intrinsically Lorentzian and refer to a particular time slicing . As we will see in the next section the Lorentzian nature of the metastring is one of its key features. That is, once the Lorentzian structure and the proper time is given, it is possible to do a Wick rotation and write the previous expressions in terms of Euclidean coordinates. We do this here for the reader’s convenience in order to connect to the more usual notation. To do so, we switch to Euclidean worldsheet coordinates , , and . With this convention we can replace and . In general, the frame field can be decomposed in terms of a conformal factor and imaginary internal rotation parameter and a Beltrami differential : . We denote by the components of the inverse frame field and its conjugate, which is explicitly given, in this parameterization, by . It is illuminating to write down the constraints in terms of the Euclidean variables. The first quantity to consider is the equation of motion . Since this equation imposes a soldering between the worldsheet chirality determined by the choice of holomorphic coordinates and the target space chirality determined by and it implies that

(68)

These equations relate the worldsheet notion of chirality (LHS) with the target space notion as eigenspaces of value of . Note that the RHS does not contain reference to the worldsheet chiral structure. This is the essential soldering phenomenon happening in the metastring that allows us to promote the worldsheet notion of T-duality and to a linear target space operation and this will eventually allow us to promote T-duality to a symmetry valid in general backgrounds.

Once we assume the chiral soldering to be in place, we can easily write the constraints in the usual form