In continuation of a past entry, this week I was intending to write more about double sigma models. I wanted to offer several further remarks on the intrinsic aspects of the doubled world-sheet formalism, and also give the reader a sense of direction when it comes to interesting questions about the geometry of the doubled string.
However, I realised that I have yet to share on this blog many of my notes on Double Field Theory (DFT). We’ve talked a bit about the Courant Bracket and the strong constraint and, in a recent post, we covered a review of Tseytlin’s formulation of the duality symmetric string for interacting chiral bosons that relates to the formulation of DFT. But, as a whole, it would be useful to discuss more about the latter before we continue with the study of double sigma models. There is a wonderfully deep connection between two, with a lot of the notation and concepts employed in the former utilised in the latter, and eventually a lot of concepts become quite interrelated.
We’ll start with some basics about DFT, focusing particularly on the T-duality group and the generalised metric formulation. In a later entry, we’ll deepen the discussion with gauge transformations of the generalised metric; generalised Lie derivatives; Courant brackets, generalised Lie brackets, and Dorfman brackets; among other things. The endgame for my notes primarily focuses on the generalised Ricci and the question of DFT’s geometric constitution, which we will also discuss another time.
For the engaged reader interested in working through the seminal papers of Zwiebach, Hull, and Hohm, see [1,2,3,4].
2. What is ?
As we’ve discussed in other places, DFT was formulated with the purpose of incorporating target space duality (T-duality) in way that is manifest on the level of the action. One will recall that, in our review of the duality symmetric string, the same motivation was present from the outset. I won’t discuss T-duality in much depth here, instead see past posts or review Chapter 8 in Polchinski . The main thing to remember, or take note of, is how T-duality is encoded in the transformations , , which describe an equivalence between radius and inverse radius, with the exchange of momentum modes and the intrinsically stringy winding modes in closed string theory, or in the case of the open string an exchange of Dirichlet and Neumann boundary conditions. More technically, we have an automorphism of conformal field theory. In the case of compactifying on for example, as momentum and winding are exchanged, the coordinates on are exchanged with the dual coordinates .
When T-duality is explicit we have for the mass operator,
where the dual radius is with . Here is the string scale. One may recognise the first terms as the number operators of left and right moving oscillator excitations. The last two terms are proportional to the quantised momentum and winding. Compactified on a circle, the spectrum is invariant under , but for a d-dimensional torus the duality group is the indefinite orthogonal group , with the number of compact dimensions.
And, actually, since we’re here one can motivate the idea another way . A generic aspect of string compactifications is that there exist subspaces of the moduli space which feature enhanced gauge symmetry. The story goes back to Kaluza-Klein. Take an compactification and set , one finds four additional massless gauge bosons that correspond to , . One can combine these states with the two gauge fields to enlarge the gauge symmetry in the form
If we want to generalise from the example of an compactification to higher-dimensional toroidal compactifications, we can do so such that the massless states at a generic point in the moduli space include Kaluza-Klein gauge bosons of the group and the toroidal moduli , parameterising a moduli space of inequivalent vacua. This moduli space is -dimensional coset space
where . In other words, it is the T-duality group relating equivalent string vacua. (In my proceeding notes I sometimes use and interchangably).
But the example I really want to get to comes from the classical bosonic string sigma model and its Hamiltonian formulation . It is fairly straightforward to work through. Along with the equations of motion, constraints in the conformal gauge are found to be of the form
which determine the dynamics of the theory. Then in the Hamiltonian description, one can calculate the Hamiltonian density from the standard Lagrangian density. After some calculation, which includes obtaining the canonical momentum and winding, the Hamiltonian density is found to take the form
This is what we will eventually come to define as the generalised metric. Keeping to the Hamiltonian formulation of the standard string, the appearance of follows. We first may define generalised vectors given some generalised geometry , in which the tangent bundle of a manifold is doubled in the sum of the tangent and co-tangent bundle. The vectors read:
Now, in this set-up, naturally appears in the classical theory ; because we take the generalised vector (6) with the constraint (4) and, in short, find that the energy-momentum tensor can be written as
The two constraints in (7) tell quite a bit: we have the Hamiltonian density set to zero with the second constraint being quite key. It will become all the more clear as we advance in our discussion that this defines the group . Moreover, a matrix is an element of if and only if
The moral of the story here is that the generalised vectors solving the constraint in (7) are related by an transformation. This transformation is, in fact, T-duality. But to formalise this last example, let us do so finally in the study of DFT and its construction.
3. Target Space Duality, Double Field Theory, and
From a field theory perspective, there is a lot to unearth about the presence of , especially given the motivating idea to make T-duality manifest. What we want to do is write everything in terms of T-duality representations. So all objects in our theory should have well-defined transformations.
We can then ask the interesting question about the field content. What one will find is that for the NS-NS sector of closed strings – i.e., gravitational fields with Riemann curvature , the Kalb-Ramond field with the conventional definition for the field strength , and a dilaton scalar field – these form a multiplet of T-duality. From a geometric viewpoint, this suggests some sort of unifying geometric description, which, as discussed elsewhere on this blog, may be formalised under the concept of generalised geometry (i.e., geometry generalised beyond the Riemannian formalism).
Earlier, in arriving at (1), we talked about compactification on . Generalising to a d-dimensional compactification, we of course have and for the double internal space we may write the coordinates , where . But what we really want to do is to double the entire space such that , with , and then see what happens. Consider the standard formulation of DFT known as the generalised metric formulation (for a review of the fundamentals see ). The effort begins with the NS-NS supergravity action
In the case of toroidal compactification defined by -dimensional non-compact coordinates and -dimensional compact directions, the target space manifold can be defined as a product between -dimensional Minkowski space-time and an -torus, such that where, as mentioned a moment ago, . We have for the full undoubled coordinates with being the internal coordinates on the torus. The background fields are matrices taken conventionally to be constant with the properties:
We define as a flat metric on the torus and is simply the Minkowski metric on the -dimensional spacetime. As usual, the inverse metric is defined with upper indices. In (11) we also have the antisymmetric Kalb-Ramond field. Finally, for purposes of simplicity, we have dropped the dilaton. Of course one must include the dilaton at some point so as to obtain the correct form of the NS-NS supergravity action, but for now it may be dropped because the motivation here is primarily to study the way in which and come together in a single generalised geometric entity, which we begin to construct with the internal metric denoted as
for the closed string background fields, with as first formulated by Narain et al . It is important to note that the canonical momentum of the theory is , where, in the standard way, denotes a derivative and denotes a derivative. Famously, the Hamiltonian of the theory may then also be constructed from the expansion of the string modes for coordinate , the canonical momentum, and from the Hamiltonian density to take the following form
Or, to write it in terms of the mass operator,
The structure of the first terms in (14) should look familiar. In summary, in an -dimensional toroidal compactification, the momentum and winding modes become -dimensional objects. So the momentum and the winding are combined in a single object known as the generalised momentum . This generalised momentum is defined as a -dimensional column vector, and we will return to a discussion of its transformation symmetry in a moment. Meanwhile, in (13) and (14) and are the usual number operators counting the excitations familiar in the standard bosonic string theory. One typically derives these when obtaining the Virasoro operators. We also see the first appearance of the generalised metric , which is a symmetric matrix constructed from and with . We will discuss the generalised metric in just a few moments.
As is fundamental to closed string theory there is the Virasoro constraint , where and are the Virasoro operators. This fundamental constraint remains true in the case of DFT. Except in DFT this condition on the spectrum gives or, equivalently,
This is, indeed, the same we defined before. Given some state and some oscillators, the fundamental constraint (15) must be satisfied, with the energy of such states computed using (13). For the time being, we treat somewhat vaguely and simply consider it as a constant matrix. We denote as a identity matrix.
Continuing with basic definitions, the generalised metric that appears in (13) and (14) is similar to what one finds using the Buscher rules  for T-duality transformations with the standard sigma model [11,12]. That is to say, takes a form in which there is clear mixing of the background fields. It is defined as follows,
One inuitive motivation for the appearance of the generalised metric is simply based on the fact that, if we decompose the supergravity fields into the metric and the Kalb-Ramond field , in DFT these then must assume the form of an tensor. The generalised metric, constructed from the standard spacetime metric and the antisymmetric two-form serves this purpose. On the other hand, the appearance of the generalised metric can be approached from a more general perspective that offers a deeper view on toroidal compactifications. In (13) what we have is in fact an expression that serves to illustrate the underlying moduli space structure of toroidal compactifications [9,13], which, as we have discussed, for a general manifold may be similarly written as (3).
The overall dimension of the moduli space is which follows from the parameters of the background matrix , with for plus for . The zero mode momenta of the theory define the Narain lattice , and it can be proven that is even and also self-dual. These properties ensure that, in the study of 1-loop partition functions, the theory is modular invariant with the description enabling a complete classification of all possible toroidal compactifications (for free world-sheet theories). The feature of self-duality contributes . The Hamiltonian (13) remains invariant from separate rotations of the left and right-moving modes that then gives the quotient terms. As for the generalised metric, we may in fact define it as the coset form of the moduli fields.
In a lightning review of certain particulars of DFT, we may deepen our discussion of the T-duality group by returning first to the generalised momentum as it appears in (14). If we shuffle the quantum numbers , which means we exchange for and vice versa, the transformation symmetry of is well known to be
For now, $ is considered generally as a invertible transformation matrix with integer entries, which mixes and after operating on the generalized momentum. It follows that should also have invertible entries, this will be shown to be true later on. Importantly, if we have a symmetry for the theory, this means a transformation in which we may take a set of states and, upon reshuffling the labels, we should obtain the same physics. Famously, it is indeed found that the level-matching condition and the Hamiltonian are preserved. If we take as a one-to-one correspondence, the level-matching condition (15) with the above symmetry transformation (18) gives
For this result to be true, it is necessary as a logical consequence that the transformation matrix must preserve the constant matrix . This means it is required that
which also implies
These last two statements can be proven, producing several equations that give conditions on the elements of . The full derivation will not be provided due to limited space (complete review of all items can again be found in [1,2,3,4,8]); however, to illustrate the logic, let be matrices, such that may be represented in terms of these matrices
The condition in which preserves demands that the elements satisfy in the case of (20)
Likewise, similar conditions are found for the case (21), for which altogether it is proven that has invertible entries. What this ultimately means is that although we previously considered vaguely as some transformation matrix, it is in fact an element of and is an invariant metric. Formally, an element is a matrix that preserves, by its nature, the invariant metric (16) such that
Finally, if the aim of DFT at this point is to completely fulfil the demand for the invariance of the massless string spectrum, it is required from (13) for the energy that, if the first term is invariant under then we must have the following transformation property in the case :
By definition, given the principle requirement of (25) it is therefore also required that the generalised metric transforms as
The primary claim here is that for the transformation of we find
One should note that this is not matrix multiplication, and is not a linear map. What we find in (27) is actually a well known transformation in string theory that appears often in different contexts, typically taking on the appearance of a modular transformation. Given the notational convention that is acting on the background , what we end up with is the following
where in the full derivation of this definition it is shown
Proof: To work out the full proposition with a proof of (26), we may also demonstrate the rather deep relation between (26) and (28). The basic idea is as follows: imagine creating from the identity background , where conventionally and . Recall, also, the definition for the generalised metric metric (17). Then for , what is ? To answer this, suppose we know some such that
It then follows
This means that the transformation creates a background from the identity. Additionally, the transformation is ambiguous because it is always possible to substitute with , where we define for . In fact, it is known that defines a subgroup of .
In conclusion, one can show that transforms appropriately, given that up to this point was constructed in such a way that the metric is split into the product and , with the outcome that only is entered into . To find we simply now consider the product ,
If we now suppose naturally is a transformation of by , such that , we also have . Notice that this implies up to some ambiguous and so far undefined subgroup defined by . Putting everything together, we obtain the rather beautiful result
Thus ends the proof of (26). A number of other useful results can be obtained and proven in the formalism, including the fact that the number operators are invariant which gives complete proof of the invariance of the full spectrum under .
In conclusion, and to summarise, in DFT there is an explicit restriction on the winding modes and the momenta to take only discrete values and hence their reference up to this point as quantum numbers. The reason has to do with the boundary conditions of -dimensional toroidal space, so that in the quantum theory the symmetry group is restricted to subgroup to . The group is as a matter of fact the T-duality symmetry group in string theory. It is conventional to represent the transformation matrix in terms of such that
Each of are matrices. They can be arranged in terms of as
Invariance under the group of transformations is generated by the following transformations. To simplify matters, let us define generally the action of an element as
Residual diffeomorphisms: If , then one can change the basis for the compactification lattice by . The action on the generalised metric is
B-field shifts: If we define to be an antisymmetric matrix with integer entries, one can use to shift the B-field producing no change in the path integral. For compact d-dimensions, this amounts to . It follows that the transformation acts on the generalised metric,
Factorised dualities: We define a factorised duality as a duality corresponding to the transformation for a single circular direction (i.e., radial inversion). It acts on the generalised metric as follows
where is a matrix with 1 in the -th entry, and zeroes elsewhere . Altogether, these three essential transformations define the T-duality group , as first established in [14,15]. To calculate a T-dual geometry one simply performs the action (26) or (28) using an transformation and, in general, one may view the formalism with the complete T-duality group as a canonical transformation on the phase space of a given system.
 Chris Hull and Barton Zwiebach. Double field theory.Journal of High EnergyPhysics, 2009(09):099–099, Sep 2009.
 Chris Hull and Barton Zwiebach. The gauge algebra of double field theory andcourant brackets.Journal of High Energy Physics, 2009(09):090–090, Sep 2009.
 Olaf Hohm, Chris Hull, and Barton Zwiebach. Generalized metric formulationof double field theory.JHEP, 08:008, 2010.
 Olaf Hohm, Chris Hull, and Barton Zwiebach. Background independent actionfor double field theory.Journal of High Energy Physics, 2010(7), Jul 2010.
 Joseph Polchinski.String theory. Vol. 1: An introduction to the bosonic string.Cambridge Monographs on Mathematical Physics. Cambridge University Press,12 2007.
 Stefan F ̈orste and Jan Louis. Duality in string theory.Nuclear Physics B -Proceedings Supplements, 61(1-2):3–22, Feb 1998.
 Felix Rennecke. O(d,d)-duality in string theory.Journal of High Energy Physics,2014(10), Oct 2014.
 Barton Zwiebach. Double Field Theory, T-Duality, and Courant Brackets.Lect.Notes Phys., 851:265–291, 2012.
 K.S. Narain, M.H. Sarmadi, and Edward Witten. A Note on Toroidal Compact-ification of Heterotic String Theory.Nucl. Phys. B, 279:369–379, 1987.
 T.H. Buscher. A Symmetry of the String Background Field Equations.Phys.Lett. B, 194:59–62, 1987.
 Mark Bugden. A tour of t-duality: Geometric and topological aspects of t-dualities, 2019.
 T.H. Buscher. Path Integral Derivation of Quantum Duality in Nonlinear SigmaModels.Phys. Lett. B, 201:466–472, 1988.
 Daniel C. Thompson. T-duality invariant approaches to string theory, 2010. Alfred D. Shapere and Frank Wilczek. Selfdual Models with Theta Terms.Nucl.Phys. B, 320:669–695, 1989. Amit Giveon, Eliezer Rabinovici, and Gabriele Veneziano. Duality in StringBackground Space.Nucl. Phys. B, 322:167–184, 1989.
 Alfred D. Shapere and Frank Wilczek. Selfdual Models with Theta Terms. Nucl. Phys. B, 320:669–695, 1989.
 Amit Giveon, Eliezer Rabinovici, and Gabriele Veneziano. Duality in String Background Space. Nucl. Phys. B, 322:167–184, 1989.*Cover image: ‘Homology cycles on a torus’. Wikipedia, Creative Commons. *Edit for spelling, grammar, and syntax.