Stringy Things

Thinking About the Strong Constraint in Double Field Theory

I’ve been thinking a lot lately about the strong (or section) constraint in Double Field Theory. In this post, I want to talk a bit about this constraint.

Before doing so, perhaps a lightning review of some other aspects of DFT might be beneficial, particularly in contextualising why the condition appears in the process of developing the formalism.

One of the important facets of DFT is the unification of B-field gauge transformations and diffeomorphisms acting on the spacetime manifold {M}. The result is a generalisation of diffeomorphisms acting on the doubled space {P} [1]. This doubled space is not too difficult to conceptualise from the outset. Think, for instance, how from the perspective of a fully constructed closed string theory, the closed string field theory on a torus is naturally doubled. But in DFT, as we advance the formalism, things of course become more complicated.

Simply put, from a first principle construction of DFT, two motivations are present from the start: 1) to make T-duality manifest and 2), extend the spacetime action for massless fields. For 2), the low-energy effective action that we want to extend is famously,

\displaystyle  S_{MS} =  \int d^{D}x \sqrt{-g}e^{-2\phi}[R + 4(\partial \phi)^{2} - \frac{1}{12}H^{ijk}H_{ijk} + \frac{1}{4} \alpha^{\prime} R^{ijkl}R_{ijkl} + ...] \ (1)

In reformulating the low-energy effective action, the tools we use begin with the fact that the coordinates in DFT are doubled such that {X^{M} = (\tilde{x}_{i}, x^{i})}. Given that the full closed string theory is rather complicated – i.e., the field arguments are doubled and we would have infinite fields, so Lagrangian wouldn’t be trivial – this motivates from the start to restrict our focus to a subset of fields. Naturally, we choose the massless sector, with the motivation to obtain a description of spacetime where the gravity-field {g_{ij}}, Kalb-Ramond field {b_{ij}} and the dilaton {\phi} are manifest. We also work with a generalised metric which rediscovers the Buscher rules such that we write, mixing {g} and {b}-fields,

\displaystyle \mathcal{H}_{MN}(E) = \begin{pmatrix} g^{ij} & -g^{ik}b_{kj} \\ b_{ik}g^{kj} & g_{ij}-b_{ik}g^{kl}b_{lj} \\ \end{pmatrix} \in O(D,D) \ (2)

Here we define {E = g_{ij} + b_{ij}}.

Most importantly, to find the analogue for {S_{MS}} formulated in an O(D,D) covariant fashion (O(D,D) is the T-duality group), a lot of the problems we need to solve and the issues we generally face are deeply suggestive of constructions in generalised geometry. We will talk a little bit about generalised geometry later.

1. The Strong Constraint

Where does the strong constraint enter into this picture? From the cursory introduction provided above, one of the quickest and most direct ways to reaching a discussion about the strong constraint follows [1]. Relative to the discussion in this paper, beginning with the standard sigma model action, where the general background metric {g_{ij}} and the Kalb-Ramond field {b_{ij}} are manifest, our entry point proceeds from the author’s review of the first quantised theory of the string and the obtaining of the oscillator expansion for the zero modes. Important for this post is what turns out to be the definition of the derivatives from the oscillator expansions,

\displaystyle  D_{i} = \partial_{i} - E_{ik}\bar{\partial}^{k}, \\ D \equiv g^{ij}D_{j}

\displaystyle  \bar{D}_{i} = \partial_{i} + E_{ki}\bar{\partial}^{k}, \\ \bar{D} \equiv g^{ij}\bar{D}_{j} \ (3)

Famously, as can be reviewed in any standard string theory textbook (for instance, see [2]), in the first quantised theory we find the Virasoro operators with zero mode quantum numbers,

\displaystyle  L_{0} = \frac{1}{2}\alpha^{i}_{0}g_{ij} \alpha^{j}_{0} + (N-1)

\displaystyle \bar{L}_{0} = \frac{1}{2}\bar{\alpha}^{i}_{0}g_{ij}\bar{\alpha}^{j}_{0} + (\bar{N} - 1) \ (4)

In (2) {N} and {\bar{N}} are the number operators. Importantly, in string theory, from the Virasoro operators we come to find the level matching constraint that matches left and right-moving excitations. This is an unavoidable constraint in closed string theory that demands the following,

\displaystyle  L_{0} - \bar{L}_{0} = 0 \ (5)

As this is one of the fundamental constraints of string theory, it follows that all states in the closed string spectrum must satisfy the condition defined in (5). The complete derivation can be found in [2].

So far everything discussed can be reviewed from the view of standard string theory. What we now want to do is use the definition of the derivatives in (3) and express (5) as,

\displaystyle  L_{0} - \bar{L}_{0} = N - \bar{N} - \frac{1}{4}(D^{i}G_{ij}D^{j} - \bar{D}^{i}G_{ij}\bar{D}^{j})

\displaystyle  = N - \bar{N} - \frac{1}{4}(D^{i}D_{i} - \bar{D}^{i}\bar{D}_{i}) \ (6)

After some working, using in particular the derivative definitions and the definitions of the background fields (see an earlier discussion in [1]), one can show that

\displaystyle  \frac{1}{2}(D^{i}D_{i} - \bar{D}^{i}\bar{D}_{i}) = -2\partial^{i}\tilde{\partial}_{i} \ (7)

In (7) we use the convention as established in [1] to denote ~ as relating to the dual coordinates. Notice, then, that what remains is a relatively simple contraction between normal and dual derivatives. What is significant about (7) is that we can now express the fundamental string theory constraint (5) as a constraint on the number operators. Since {L_{0} - \bar{L}_{0} = 0} for all states of the theory we find,

\displaystyle  N - \bar{N} = \frac{1}{2}(-2\partial_{i}\tilde{\partial}^{i} = -\partial^{i}\tilde{\partial}_{i} \equiv \partial \cdot \bar{\partial} = p_{i}\omega^{i} \ (8)

So we see that from the number operators we have constraints involving differential operators. But what is this telling us? In short, it basically depends on the fields we use for the closed string theory. The fields that are arguably most natural to include are of a first quantised state expressed in the sum,

\displaystyle  \sum_{p,\omega} e_{ij}(p,\omega) \alpha^{i}_{-1}\bar{\alpha}^{j}_{-1}c_{1}\bar{c}_{1} |p, \omega \rangle

\displaystyle  \sum_{p,\omega} d (p,\omega) (c_{1}C_{-1} - \bar{c}_{1}\bar{c}_{-1}) |p, \omega \rangle \ (9)

Where we have momentum space wavefunctions {e_{ij}} and {d (p,\omega)}. Furthermore, in the first line, {e_{ij}} denotes the fluctuating field {h_{ij} + b_{ij}}. The {c} terms are ghosts. So what we observe in (9) is matter and ghost fields acting on a a vacuum with momentum and winding.

Here comes the crucial part: given {N = \bar{N} = 0} it follows that the fields, which, to make explicit depend on normal and dual coordinates, {e_{ij}(x, \tilde{x})} and {d(x,\tilde{x})} are required to satisfy,

\displaystyle  \partial \cdot \tilde{\partial} e_{ij} (x, \tilde{x}) = \partial \cdot \tilde{\partial} d(x, \tilde{x}) = 0 \ (10)

This is weak version of the strong or section constraint, a fundamental constraint in DFT for which we can go on to define an action. What it says is that every field of the massless sector must be annihilated by the differential operator {\partial \cdot \tilde{\partial}}.

This constraint (10), when developed further, turns out to actually be very strong. When we proceed to further generalise in our first principle construction of DFT, first with the study of {O(D,D)} transformations and then eventually the construction of {O(D,D)} invariant actions, we come to see that not only all fields and gauge parameters must satisfy the constraint {\partial \cdot \tilde{\partial}}. But this constraint is deepened, in a sense, to an even stronger version that includes the product of two fields.

The argument is detailed and something we’ll discuss in length another time, with the updated definition that {\partial \cdot \tilde{\partial}} annihilates all fields and all products of fields. That is, if we let {A_{i}(x, \tilde{x})} be in general fields or gauge parameters annihilated by the constraint {\partial^{M}\partial_{M}}, we now require all products {A_{i}A_{j}} are killed such that,

\displaystyle  \partial_{M}A_{i}\partial^{M}A_{j} = 0, \forall i,j \ (11)

Here {\partial_{M}A_{i}\partial^{M}} is an {O(D,D)} scalar. Formally, the result (11) is the strong {O(D,D)} constraint. What, finally, makes this condition so strong is that, from one perspective, it kills half of the fields of the theory and we in fact lose a lot of physics! In full string theory the doubled coordinates are physical. Effectively, however, the above statement ultimately implies that our fields only depend on the real space-time coordinates, due to a theorem in {O(D,D)} in which there is always some duality frame {(\tilde{x}^{\prime}_{i},\tilde{x}^{\prime})} in which the fields do not depend on {\tilde{x}^{\prime}_{i}}. So we only have dependence on half of the coordinates.

There is maybe another way to understand or motivate these statements. In the standard formulation of DFT, what we come to find is the appearance of the generalised Lie derivative. It is essentially unavoidable. The basic reason has to do with how, in pursuing the construction of the {O(D,D)} invariant action as highlighted at start, which includes the generalised metric {\mathcal{H}}, we find that the conventional Lie derivative is not applicable. It is not applicable in this set-up because, even when using trivial gauge parameters, we find that it simply does not vanish. In other words, as can be reviewed in [1], {\mathcal{L}_{\xi} \neq 0}. So the definition of the Lie derivative becomes modified using what we define as the neutral metric {\eta}. Why this is relevant has to do with how, interestingly, from the generalised Lie derivative (or Dorfman bracket) we may then define an infinitesimal transformation that, in general, does not integrate to a group action [3-5]. This means that it does not generate closed transformations.

The convention, as one may anticipate, is to place quite a strong restriction on the space of vector fields and tensors. Indeed, from the fact that DFT is formulated by way of doubling the underlying manifold, we have to use constraints on the manifold to ensure a consistent physical theory. But this restriction, perhaps as it can be viewed more deeply, ultimately demands satisfaction of what we have discussed as the strong constraint or section condition. So it is again, to word it another way, the idea that we have to restrict the space of vectors and tensors for consistency in our formulation that perhaps makes (11) more intuitive.

There is, of course, a lot more to the strong constraint and what it means [5], but as a gentle introduction we have captured some of its most basic implications.

2. Some Nuances and Subtleties

Given a very brief review of the strong constraint, there are some nuances and caveats that we might begin to think about. The first thing to note is that the strong constraint can be relaxed to some degree, and people have started researching weakly constrained versions of DFT. I’m not yet entirely familiar with these attempts and the issues faced, but an obvious example would be the full closed string field theory on a torus, because this is properly doubled from the outset and subject only to the weak level-matching constraint {\tilde{\partial} \cdot \partial = 0}.

I think a more important nuance or caveat worth mentioning is that, as discussed in [3], the strong constraint does not offer a unique solution. That is to say, from what I currently understand, there is no geometrical information that describes the remaining coordinates on which the fields depend. This contributes to, in a sense, an arbitrariness in construction because there is a freedom to choose which submanifold {P} is the base {M} for the generalised geometry {TM}.

In a future post, we’ll discuss more about this lack of uniqueness and other complexities, as well as detail more thorough considerations of the strong constraint. As related to simplified discussion above, the issue is that we can solve the basic consistency constraints that govern the theory by imposing the strong constraint (11). This is what leads to the implied view that DFT is in fact a highly constrained theory despite doubled coordinates, etc. In this approach, we have restriction on coordinate dependence such that, technically, the fields and gauge parameters may only depend on the undoubled slice of the doubled space. We haven’t discussed the technicalities of the doubled space in this post, but that can be laid out another time. The main point being that this solution is controlled. But there are also other solutions, of which I have not yet studied, but where it is understood that the coordinate dependence is no longer restricted (thus truly doubled) at the cost of the shape of geometric structure. At the heart of the matter, some argue [3] that when it comes to this problem of uniqueness the deeper issue is a lack of a bridge between DFT and generalised geometry. This is also a very interesting topic that will be saved for another time.

References

[1] B. Zwiebach, ‘Double Field Theory, T-Duality, and Courant Brackets’ [lecture notes]. 2010. Available from [arXiv:1109.1782v1 [hep-th]].

[2] J. Polchinski, ‘String Theory: An Introduction to the Bosonic String’, Vol. 1. 2005.

[3] L. Freidel, F. J. Rudolph, D. Svoboda, ‘Generalised Kinematics for Double Field Theory’. 2017. [arXiv:1706.07089 [hep-th]].

[4] B. Zwiebach O. Hohm. Towards an invariant geometry of double field theory. 2013. [arXiv:1212.1736v2 [hep-th]].

[5] B. Zwiebach O. Hohm, D. Lust. The Spacetime of Double Field Theory : Review, Remarks and Outlook. 2014. [arXiv:1309.2977v3 [hep-th]].

[6] K. van der Veen, ‘On the Geometry of String Theory’ [thesis]. 2018. Retrieved from [https://pdfs.semanticscholar.org/8c17/53af2fce4d174ca63a955c39ee2fedf37556.pdf]

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