Doubled diffeomorphisms and the generalised Ricci curvature

I was asked a question the other week about the idea of doubled diffeomorphisms, such as those found in double field theory. A nice way to approach the concept is to start with dualised linearised gravity [1]. That is to say, we start with a theory considering only the field h_{ij}(x^{\mu}, x^a, \tilde{x}_a)  . This field transforms under normal linearised diffeomorphism as

\delta h_{ij} = \partial_i \epsilon_j + \partial_j \epsilon_i \ \ (1)

and, under the dual diffeomorphism as

\tilde{\delta} h_{ij} = \tilde{\partial}_i \tilde{\epsilon}_j + \tilde{\partial}_j \tilde{\epsilon}_i. \ \ (2)

Now, take the basic Einstein-Hilbert action

S_{EH} = \frac{1}{2k^2} \int \ \sqrt{-g} \ R, \ \ (3)

and expand to quadratic order in the fluctuation field h_{ij}(x) = g_{ij} - \eta_{ij}  . Just think of standard linearised gravity with the following familiar quadratic action

S^2_{EH} = \frac{1}{2k^2} \int \ dx \ [\frac{1}{4} h^{ij}\partial^2 h_{ij} - \frac{1}{4} h \partial^2 h + \frac{1}{2} (\partial^i h_{ij})^2 + \frac{1}{2} h \partial_i \partial_j h^{ij}]. \ \ (4)

This is the Feirz-Pauli action and it is of course invariant under (1). But we want a dualised theory. The naive thing to do, for the field h(x, \tilde{x}) , is to add a second collection of tilde dependant terms. In comparison with (4), we also update the integration measure to give

S^2_{EH} =  \frac{1}{2k^2} \int \ dx d\tilde{x} \ [\frac{1}{4} h^{ij}\partial^2 h_{ij} - \frac{1}{4} h \partial^2 h \\ + \frac{1}{2} (\partial^i h_{ij})^2 + \frac{1}{2} h \partial_i \partial_j h^{ij} + \\ \frac{1}{4} h^{ij} \tilde{\partial}^2 h_{ij} - \frac{1}{4} h \tilde{\partial}^2 h \\ + \frac{1}{2} (\tilde{\partial}^i h_{ij})^2 + \frac{1}{2} h \tilde{\partial}_i \tilde{\partial}_j h^{ij}]. \ \ (5)

If you decompose x, \tilde{x} such that h_{ij} (x) no longer depends on \tilde{x} , then this action simply reduces to linearised Einstein gravity on the coordinate space x^a.  Similarly, for the dual theory.

When the doubled action (5) is varied under \tilde{\delta} , the second line is invariant under (2). However, the first line gives

\tilde{\delta} S = \int [dx d\tilde{x}] [h^{ij} \partial^2 \tilde{\partial}_i \tilde{\epsilon}_j + \partial_i h^{ij} (\partial^k \tilde{\partial}_{k})\tilde{\epsilon}_j \\ - h \partial^2 \tilde{\partial} \tilde{\epsilon} + h(\partial_i \tilde{\partial}^i)\partial_j \tilde{\epsilon}^j \\ + \partial_i h^{ij} \partial^k \tilde{\partial}_j \tilde{\epsilon}_k + (\partial_j \partial_j h^{ij})\tilde{\partial} \tilde{\epsilon}. \ \ (6)

As one can see, the terms on each line would cancel if the tilde derivatives were replaced by ordinary derivatives. Rearranging and grouping like terms, and then relabelling some indices we find

\tilde{\delta} S = \int [dx d\tilde{x}] \ [(\tilde{\partial}_j h^{ij})\partial^k (\partial_i \tilde{\epsilon}_k - \partial_k \tilde{\epsilon}_i) \\ + (\partial_i \partial_j h^{ij} - \partial^2h) \tilde{\partial} \tilde{\epsilon} \\ + (\partial^i h_{ij} - \partial_j h)(\partial \tilde{\partial})\tilde{\epsilon}^j. \ \ (7)

For this to be invariant under the transformation \tilde{\delta} we have to cancel each of the terms. In order to cancel the variation, new fields with new gauge transformations are required. For the first term, a hint comes from the structure of derivatives, namely the fact we have a mixture of tilde and non-tilde derivatives. The Kalb-Ramond b-field mixes derivatives in this way, and, indeed, for the first term to cancel we may add b_{ij} . We denote this inclusion to the action as S_b

S_b = \int [dx d\tilde{x}] \ (\tilde{\partial}_j h^{ij})\partial^k b_{ik}, \\ with \ \ \tilde{\delta}b_{ij} = - (\partial_i \tilde{\epsilon}_j - \partial_j \tilde{\epsilon}_i). \ \ (8)

The second term can similarly be killed upon introduction of the dilaton \phi . It takes the form

S_{\phi} = [dx d\tilde{x}] (-2) (\partial_i \partial_j h^{ij} - \partial^2 h) \phi, \ \ \text{with} \ \ \tilde{\delta}\phi = \frac{1}{2}\tilde{\partial} \tilde{\epsilon}. \ \ (9)

This is quite nice, if you think about it. It is not the full story, because in the complete picture of double field theory we need to add more terms and their are several subtlties. In the naive case of dualised linearised gravity, we find in any case that linearised dual diffeomorphisms for the field h_{ij} requires, naturally and perhaps serendipitously, a Kalb-Ramond gauge field and a dilaton – i.e., the closed string fields for the NS-NS sector.

We are now only left with one term, which is the one with curious structure on the third line in (7). To kill this term, we can observe that the gauge parameter \tilde{\epsilon} satisfies the constraint \partial \cdot \tilde{\partial} = 0 derived from the level matching condition. This constraint says that fields and gauge parameters must be annihilated by \partial \tilde{\partial} , and it is fairly easy to find in an analysis of the spectrum in closed string field theory.

So that is one way to attack the remaining term. But what is also interesting, I think, is that it is possible to accomplish the same goal by adding more fields to the theory. This is a non-trivial endeavour, to be sure, as the added fields would need to be invariant under \delta and \tilde{\delta} transformations. Ideally, one would likely want to be able to generalise the added fields to the formal case of the duality invariant theory. But it presents an interesting question.

***

From the perspective of string field theory, double field theory wants to describe a manifestly T-duality invariant theory (we talked about this in a number of past posts). The strategy is to look at the full closed string field theory comprising an infinite number of fields, and instead select to focus on a finite subset of those fields, namely the massless NS-NS sector. So DFT is, at present, very much a truncation of the string spectrum.

As a slight update to notation to match convention, for the massless fields of the NS-NS sector let’s now write the metric g_{ij}  , with the b-field b_{ij}  and dilaton \phi  the same as before. The effective action of this sector is famously

\displaystyle S_{NS} = \int dX \sqrt{-g} e^{-2\phi} [R + 4(\partial \phi)^2 - \frac{1}{12}H^2] + \text{higher derivative terms}. \ \ (10)

As one can review in any string textbook, this action is invariant under local gauge transformations: diffeomorphisms and a two-form gauge transformation. The NS-NS field content transforms as

\displaystyle \delta g_{ij} = L_{\lambda} g_{ij} = \lambda^{k} \partial_k g_{ij} + g_{kj}\partial_i \lambda^k + g_{ik}\partial_i \lambda^k,

\displaystyle \delta b_{ij} = L_{\lambda} b_{ij} = \lambda^k \partial_k b_{ij} + b_{kj}\partial_i \lambda^k + b_{ik}\partial_i\lambda^k,

\displaystyle \delta \phi = L_{\lambda} \phi = \lambda^k \partial_k \phi. \ \ (11)

We define the Lie derivative L_{\lambda}  along the vector field \lambda^i  on an arbitrary vector field V^i  such that the Lie bracket takes the form

\displaystyle L_{\lambda} V^i = [\lambda, V]^i = \lambda^j \partial_j V^i - V^j \partial_j \lambda^i. \ \ (12)

For the Kalb-Ramond two-form b_{ij}  , the gauge transformation is generated by a one-form field \tilde{\lambda}_i

\displaystyle \delta b_{ij} = \partial_i \tilde{\lambda}_j - \partial_j \tilde{\lambda}_i.  \ \ (13)

One way to motivate a discussion on doubled or generalised diffeomorphisms in DFT is to understand that what one wants to do is essentially generalise the action (10). This means that at any time we should be able to recover it. The generalised theory should therefore possess all the same symmetries (with added requirement of manifest invariance under T-duality), including diffeomorphism invariance.

In the generalised metric formulation [2] the DFT action reads

\displaystyle S_{DFT} = \int d^{2D} X e^{-2d} \mathcal{R}, \ \ (14)

where

\displaystyle \mathcal{R} \equiv 4\mathcal{H}^{MN}\partial_M \partial_N d - \partial_M \partial_N \mathcal{H}^{MN} \\ - 4\mathcal{H}^{MN}\partial_{M}d\partial_N d + 4\partial_M \mathcal{H}^{MN} \partial_N d \\ + \frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL} - \frac{1}{2} \mathcal{H}^{MN}\partial_{N}\mathcal{H}^{KL}\partial_{L}\mathcal{H}_{MK}. \ \ (15)

This action is constructed [2] precisely in such a way that it captures the same dynamics as (10). Here \mathcal{H} is the generalised metric, which combines the metric and b-field into an O(D,D) valued symmetric tensor such that

\displaystyle \mathcal{H}^{MN}\eta_{ML}\mathcal{H}^{LK} = \eta^{NK}, \ \ (16)

where \eta  is the O(D,D) metric. We spoke quite a bit about the generalised metric and the role of O(D,D) in a past post (see this link also for further definitions, recalling for instance the T-duality transformation group is O(D,D; \mathbb{R})  , which is discretised to O(D,D; \mathbb{Z}) . If O(D,D) is broken to the discrete O(D,D;\mathbb{Z}) , then one can interepret the transformation as acting on the background torus on which DFT has been defined). Also note that in (15) d is the generalised dilaton. In the background independent formulation of DFT [5], e^{-2d} is shown to be a generalised density such that the dilaton \phi  with the determinant of the undoubled metric g = \det g_{ij}  on the whole space is combined into an O(D,D)  singlet d establishing the identity \sqrt{-g}e^{-2\phi} = e^{-2d} . We’ll talk a bit more about this later.

There are a number of important characteristics built into the definition of the generalised Ricci (15). Firstly, it is contructed to be an O(D,D)  scalar. One can show that the action (14) possesses manifest global O(D,D) symmetry

\displaystyle \mathcal{H}^{MN} \rightarrow \mathcal{H}^{LK}M_{L}^{M}M_{K}^{N} \ \ \text{and} \ \ X^{M} \rightarrow X^{N}M_{N}^{M}, \ (17)

where M_{L}^{K} is a constant tensor which leaves \eta^{MN} invariant such that

\displaystyle \eta^{LK} M_{L}^{M} M_{K}^{N} = \eta^{MN}. \ \ (18)

Importantly, given O(D,D) extends to a global symmetry, we may define this under the notion of generalised diffeomorphisms. Unlike with the supergravity action (10), which is invariant under the gauge transformations (11) and (12), in DFT the metric and b-field are combined into a single object \mathcal{H}  . So the obvious task, then, is to find a way to combine the diffeomorphisms and two-form gauge transformation in the form of some generalised gauge transformation. This is really the thrust of the entire story.

To see how this works, as a brief review, we define some doubled space \mathbb{R}^{2D}.  To give a description of this doubled space, all we need to start is some notion of a differential manifold with the condition that we have a linear transformation of the coordinates X^{\prime} = hX  , where h \in O(D,D) (similar to the transformation we defined in the post linked above). We will include the generalised dilaton d  and we also include the generalised metric \mathcal{H} , although we can keep this generic in definition should we like. For \mathcal{H} we require only that it satisfies the O(D,D) constraint \mathcal{H}^{-1} = \eta \mathcal{H} \eta , where, from past discussion, one will recall \eta is the 0(D,D) metric. It transforms \mathcal{H}^{\prime}(X^{\prime}) = h^{t}\mathcal{H}(X)h  . We now have everything we need.

Definition 1. A doubled space \mathbb{R}^{2D}(\mathcal{H},d) is a space equipped with the following:

1) A positive symmetric 2D \times 2D-\text{matrix} field \mathcal{H} , which is the generalized metric. This metric must satisfy the above conditions and transform covariantly under O(D,D).

2) A generalised dilaton scalar d , which is a 2D scalar density such that d = \phi - \frac{1}{2} \ln \det h (we’ll show this in a moment).

a) The generalised dilaton is related to the standard dilaton as already described above.

With this definition, we can then advance to define the notion of an O(D,D)  module, generalised vectors and vector fields, and so on. To keep our discussion short, the point is that in defining an O(D,D)  vector we may combine from before the vector \lambda^i and one-form \tilde{\lambda}_i as generalised gauge parameters

\displaystyle \xi^M = (\tilde{\lambda}_i, \lambda^i). \ \ (19)

One can see how this is done in [2,3]. In short, the combination of the gauge transformations into the general gauge transformation with parameter \xi^M  is defined under the action of a generalised Lie derivative. The result is simply given here as

\displaystyle \mathcal{L}_{\xi}A_M \equiv \xi^P \partial_P A_M + (\partial_M \xi^P - \partial^P \xi_M)A_p,

\displaystyle \mathcal{L}_{\xi}B^M \equiv \xi^P \partial_P B^M + (\partial^M \xi_P - \partial_P \xi^M)B^p. \ \ (20)

From this definition, where, it should be said, A  and B  are generalised vectors, we can eventually write the generalised Lie derivative of \mathcal{H} and d  .

\displaystyle \mathcal{L}_{\xi} \mathcal{H}_{MN} = \xi^P \partial_P \mathcal{H}_{MN} + (\partial_M \xi^P - \partial^P \xi_M)\mathcal{H}_{PN} + (\partial_N \xi^P -  \partial^P \xi_N)\mathcal{H}_{MP},

\displaystyle \mathcal{L}_{\xi}(e^{-2d}) = \partial_M(\xi^M e^{-2d}). \ \ (21)

What we see is that, indeed, the generalised dilaton, which we may think of as an O(D,D)  singlet, transforms as a density. This means we may think of it as a generalised density. It can also be shown that the Lie derivative of the O(D,D)  metric \eta  vanishes and therefore the metric is preserved.

What we want, for the purposes of this post, is the generalised Lie derivative of the generalised scalar curvature (15). What we find is that, indeed, it transforms as a scalar provided that the definition of (15) includes the full combination of terms.

\displaystyle \mathcal{L}_{\xi} \mathcal{R} = \xi^M \partial_M \mathcal{R}.  (22)

Or, looking at the action (14) as a whole, the subtlety is that the generalised dilaton forms part of the integration measure. The action does not possess manifest generalised diffeomorphism invariance in the typical sense that we might think about it, but it is constructed precisely in such a way that

\displaystyle \mathcal{L}_{\xi}(e^{-2 d})\mathcal{R} = \partial_I (\xi^{I} e^{-2d}\mathcal{R}) \ \ (26)

vanishes in the action integral (due to being a total derivative). So we find (14) does indeed remain invariant.

As a brief aside, from the transformations of the generalised metric and the dilaton, we can define an algebra [4]

\displaystyle [\mathcal{L}_{\xi_1}, \mathcal{L}_{\xi_2}] = \mathcal{L}_{\xi_1} \mathcal{L}_{\xi_2} - \mathcal{L}_{\xi_2} \mathcal{L}_{\xi_1} = \mathcal{L}_{[\mathcal{L}_{\xi_1} \mathcal{L}_{\xi_2}]_C}, \ \ (23)

where we find first glimpse at the presence of the Courant bracket. Provided the strong O(D,D) constraint of DFT is imposed

\displaystyle \partial_N A_I \partial^{N} A^J = 0 \ \forall \ i,j, \ \ (24)

then the Courant bracket governs this algebra such that

\displaystyle [\mathcal{L}_{\xi_1}, \mathcal{L}_{\xi_2}]^{M}_{C} = \xi_{1}^{N}\partial_{N}\xi_{2}^{M} - \frac{1}{2}\xi_{1N}\partial^{M}\xi_{2}^{N} - (\xi_1 \leftrightarrow \xi_2). \\ (25)

An important caveat or subtlety about this algebra is that it does not satisfy the Jacobi identity. This means that the generalised diffeomorphisms do not form a Lie algebroid. But nothing fatal comes from this fact for the reason that, whilst we may like to satisfy the Jacobi identity, the gauge transformation leaves all the fields invariant that fulfil the strong O(D,D) constraint.

In closing, recall that DFT starts with the low-energy effective theory as a motivation. It is good, then, that a solution of (24) is to set \tilde{\partial} = 0 giving (10). The Ricci scalar is the only diffeomorphism invariant object in Riemannian geometry that can be constructed only from the metric with no more than two derivatives. In DFT, we have an action constructed only from the generalised metric and doubled dilaton with their derivatives.

References

[1] Hull, C.M., and Zweibach, B., Double field theory. (2009). [arXiv:0904.4664 [hep-th]].

[2] Hohm, O., Hull C.M., and Zwiebach, B., Generalized metric formulationof double field theory. JHEP, 08:008, 2010. [arXiv:1006.4823 [hep-th]].

[3] Zwiebach, B., Double field theory, T-duality, and Courant brackets. [arXiv:1109.1782 [hep-th]].

[4] Hull, C.M., and Zwiebach, B., The gauge algebra of double field theory and courant brackets. Journal of High Energy Physics, 2009(09):090–090, Sep 2009. [arXiv:0908.1792 [hep-th]].

[5] Hohm, H., Hull, C.M., and Zwiebach, B., Background independent actionfor double field theory. Journal of High Energy Physics, 2010(7), Jul 2010. [arXiv:1003.5027 [hep-th]].

Stringification as categorisation

In quantum field theory one is typically taught to use perturbation theory when the equations of motion for the fields are nonlinear and weakly interacting. For example, in \phi^4  theory one can use a formal series as described by Rosly and Selivanov [1]. Perturbative theory is about mastering series expansions. The basic idea, upon constructing some correlation function in the full nonlinear model, is to expand in powers of \alpha  , namely the interaction strength. In the language of perturbative physics, Feynman diagrams give a representation of each term in the expansion such that we use them to illustrate linear operators. This ultimately enables us to obtain a good approximation to the exact solution. Needless to say, there is a real power and usefulness about perturbative methods and the sum of Feynman diagrams.

When computing amplitudes with Feynman diagrams, the amplitudes depend on various topological properties (i.e., vertices, loops, and so on). Although not always made explicit in the perturbative view, from the Fenynman diagrams of 0-dimensional points with 1-dimensional graphs (to use the language of p-branes, which we’ll get to in a moment), we have topologies that describe linear operators: i.e., what Feynman diagrams start to make explicit is the deeper role of topology in physics [2]. This was summarised wonderfully in a lovely article by Atiyah, Dijkgraaf, and Hitchin [3]. Mathematically, and from the perspective of geometry, the main idea is that a linear operator behaves very much like an n-dimensional manifold going between manifolds of one dimension less, which we may define as a cobordism (i.e., think of a stringy ‘trousers’ diagram) [2,4].

Now, consider the story of p-branes, in particular the perspective as we pass from standard quantum field theory to string theory. The language of p-branes as first described by Duff et al [5] may be reviewed in any introductory string theory textbook. We can, from first-principles, motivate string theory thusly: in a special, if not unique way, we may generalise the point-like 0-dimensional particle to the 1-dimensional string, which is made explicit when we generalise the action for a relativistic particle to the Nambu-Goto action for the relativistic string. In the language of p-branes, which are p-dimensional objects moving through a D(D \geq p)  dimensional space-time, a 0-brane is a (0-dimensional) point particle that that traces out a (0+1)-dimensional worldline. The generalisation of the point particle action S_0 = -m \int ds to a p-brane action in a D(\geq p) -dimensional space-time background is given by S_p = -T_p \int d\mu_p . Here T_p is the p-brane tension with units mass/vol, and d\mu_p is the (p + 1)-dimensional volume element. For the special case where p=1 such that we have 1-brane, we obtain the string action which sweeps out a (1+1)-dimensional surface that is the string worldsheet propagating through space-time. We can also go on to speak of higher-dimensional objects, such as those that govern M-theory. For instance, a 2-brane is a membrane. Historically, these were considered as 2-dimensional particles. There are also 3-branes, 4-branes, and so on.

This generalising process, if we can describe it that way, is what I like to think of as stringification. For the case where p=1 , Feynman diagrams of ordinary quantum field theory with 2-dimensional cobordisms represent world-sheets traced out by strings. The generalising picture, or stringification, show these 2-dimensional cobordisms equipped with extra structure give a powerful mathematical language (describing the relation between physics and topology, as string diagrams enable us to sum over the various topologies and provide a valuable mathematical tool for thinking about composition). But of course this picture can still be extended. Not only does the important analogy between operators and cobordisms come directly into focus, it is also, in some sense, where stringification meets categorification. That is, from the maths side, we arrive at the logic of higher-dimensional algebra and the arrows of monoidal and higher categories. In each, physical processes are describe by morphisms or functors (functors are like morphisms between categories). This generalising picture toward higher geometry, higher algebra, and, indeed, higher structures is called ‘categorifying’ or ‘homotopifying’ (my notes on which I have started to upload to this blog). In this post, I want to think a bit about this idea of stringification as categorification.

***

There is a view of M-theory, and I suppose of fundamental physics as whole, that I find fascinating and compelling: stringification as the categorisation of physics. The notion of stringification is not formal, but captures if nothing else an intuition about a certain generalising process or abstract story, or at least that is how I presently see it. It is a term I have picked up that used to float around in different contexts a couple of decades ago. As described through the language of p-branes, the story begins with the generalisation or stringification of point particle theory (and all that it implies) toward the existence of the string and eventually other extended objects in fundamental physics. Meanwhile, the notion of categorification is certainly formal, signalling, at its origin, the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories. This process, when iterated, gives definition to the notion of n-category theory, where we also replace functions with functors, and equations between functions by natural isomorphisms between functors [6]. As Schreiber pointed out in 2004, there is a sort of harmony between these two processes – stringification and categorification – which has certainly started to clarify over the last decade or more.

As one example, the observation that Schreiber describes in the linked post refers to boundaries of membranes attached to stacks of 5-branes, which conceptually appear as a higher-dimensional generalisation of how boundaries of strings appear.

To understand this think, firstly, of the simple example of the existence of D-branes (Dirichlet membranes) and how the endpoints of open strings can end on these extended objects. In fact, an introductory string textbook will guide one to see why the equations of motion of string theory require that the endpoints of an open string satisfy one of two types of boundary conditions (Dirichlet or Neumann) ending on a brane. If the endpoint is confined to the condition that it may move within some p-dimensional hyperplane, one then obtains a first description of Dp-branes. (I think this was one of the first things I calculated when learning strings!). For the sake of saving space I won’t go into the arrangement of D-branes or other related topics. The main point that I am driving at, the technicalities of which we could review in another post, is how these branes are dynamic and as such they may influence the dynamics of a string (i.e., how an open string might move and vibrate). Thus, the arrangement of branes (e.g., we can have parallel branes or ‘stacks’) will also impact or control the types of particles in our theory. It is truly a beautiful picture.

In p-brane language, if you take the Nambu–Goto action and for the quantum theory study the spectrum of particles, you will see that it exhibits what we may describe as the photon, which of course is the fundamental quantum of the electromagnetic field. Now, what is nice about this is that, the resemblance of the photon is actually a p-dimensional version of the electromagnetic field, so it is in fact a p-dimensional analogue of Maxwell’s equations.

What Schreiber is highlighting in his post is not just that in string theory, the points of the string ending on a Dp-brane give rise to ordinary gauge theory. (One could even take the view that string theory predicts electromagnitism such that string theory predicts the existence of D-branes. It is by their nature that these extended objects all carry an electromagnetic field on their volume, i.e., what we call the brane volume). The point made is that, given there is reason to extend the picture further – the picture of stringification so to speak – to higher-dimensional generalisations, we can then replace strings with membranes, and so on. From the maths side, it was realised that from the perspective of categories, something analogous is happening: replacing points with arrows (i.e., morphisms) one finds the gauge string may be described by the structure of nonabelian gerbes (a gerbe is just a generalised analogue of a fibre bundle), and so on.

When I first learned strings, the picture of stringification was in my mind but I didn’t yet have a word for it. I also didn’t possess category theoretic language at the time; it was really only a vague sense of a picture, perhaps emphasised in the way I learned string theory. So when I discovered and read last year about the idea of stringification as categorisation [7] in Schreiber’s thesis, I was excited.

A nice illustration comes from the first pages of this work. Take some ordinary point-particle, which traces out a worldline over time t . The thrust of the idea is that, given some charge, there is a connection in some bundle (yet unspecified) such that, locally, a group element g \in G is associated to the path. Diagrammatically this may be represented as,

Now consider some time t^{\prime} , where t^{\prime} > t . The particle has travelled a bit further,

We can of course compose these paths. The composition is associative and the operation is multiplication. In fact, what we’re doing is multiplying the group elements. We can also define an inverse g^{-1} . The punchline is that, from the theory of fibre bundles with connection, we can consider how this local picture may fit globally. If g is an element in a non-abelian group, the particle we are generalising is non-abelian. Generalise from a point-particle to a string, and the diagrammatic representation of the world-sheet takes the form

Ultimately, we can continue to play this game and develop the theory of non-abelian strings (and on to higher-dimensional branes), which, it turns out, corresponds with a 2-category theory [7,8]. Sparing details, in n-category theory a 2-category is a special type of category wherein, besides morphisms between objects, it possesses morphisms between morphisms. What is interesting about this example is how we can go on to show the idea of SUSY quantum mechanics on loop space relates to ideas in higher gauge theory, particularly in the sense of categorifying standard gauge theory. For example, John Baez’s paper on higher Yang-Mills [9]. But even before all of that, from the view of perturbative string theory being the categorification of supersymmetric quantum mechanics, we can play the same game such that the generalisation of the membranes of M-theory are a categorification of the supersymmetric string, and so on. The intriguing and, perhaps, grand idea, is that this process of stringification as categorification can be utilised to describe the whole of physics, or, so, it is suspected.

***

I’ve been thinking about this picture quite a bit recently, perhaps spurred by all of my ongoing studies in M-theory. The view to be encircled, as the notion of categorisation enters the stringy picture, also marks for me the beginning of the story about higher structures in fundamental physics (in terms of the view of category theory and higher category theory). In a sense, as much as I currently understand it (as I am very much in the process of studying and forming my thoughts on the matter) we are encircling not much more than an abstract story; but it is one in which many tantalising hints exist about a potentially foundational view.

The history of this higher structure view is rich with examples [10, 11], and, for many reasons, it leads us directly to a study of the plausible existence of M-theory. From the use of braided monoidal categories in the context of string diagrams through to knot theory (See Witten’s many famous lectures); the notion of quantum groups; Segal’s famous work on the axioms of conformal field theory (described in terms of monoidal functors and the category 2Cob_{\mathbb{C}} whose morphisms are string world-sheets such that we can compose the morphisms, and so on); and of course the work of Atiyah in topological quantum field theory (TQFT) followed by Dijkgraaf’s thesis on 2d TQFTs in terms of Frobenius algebras – the list is far to big to summarise in a single paragraph. All of this indicates, in some general sense, a very abstract story from basic quantum mechanics through to string theory and, I would say, as a natural consequence M-theory.

It is a fascinating perspective. There is so much to be said about this developing view, including why higher geometry and algebra seem to hold the important clues of M-theory as a fundamental theory of physics. What is also interesting, as I am beginning to understand, is that in the higher structure picture, a striking consequence from a geometric persective is that the geometry of fundamental physics (higher geometry and supergeometry) may not be described by spaces with sets of points. And, in fact, we start to see this for each value of p  . Instead of a traditional notion space associated with the definition of topological spaces or differentiable manifolds, the geometric observation is that what we’re dealing with is functorial geometry of the sort described by Grothendieck, or synthetic differential geometry of the sort described by Lawvere, or a variation of them both.

Anyway, this is just a short note of me thinking aloud.

References

[1] Rosly, A.A., and Selivanov, K.G., On amplitudes in self-dual sector of Yang-Mills theory. [arXiv:9611101 [hep-th]].

[2] Baez, J., and Stay, M., Physics, Topology, Logic and Computation: A Rosetta Stone. [arXiv:0903.0340 [quant-ph]].

[3] Atiyah, M., Dijkgraaf, R., and Hitchin, N., Geometry and physics. Phil. Trans. R. Soc., (2010), A.368, 913–926. [http://doi.org/10.1098/rsta.2009.0227].

[4] Baez, J., and Lauda, A., A Prehistory of n-Categorical Physics. [https://math.ucr.edu/home/baez/history.pdf.].

[5] M. J. Duff, T. Inami, C. N. Pope, E. Sezgin [de], and K. S. Stelle, Semiclassical quantization of the supermembrane. Nucl. Phys. B297 (1988), 515.

[6] Baez, J., and Dolan, J., Categorification. (1998). [arXiv: 9802029 [math.QA]].

[7] Schreiber, U., From Loop Space Mechanics to Nonabelian Strings [thesis]. (2005). [hep-th/0509163].

[8] Baez, J. et al., Categorified Symplectic Geometry and the Classical String. (2008). [math-ph/0808.0246v1].

[9] Baez, J., \textit{Higher Yang–Mills theory}. (2002). [hep-th/0206130].

[10] Baez, J., and Lauda, A., A Prehistory of n-Categorical Physics. [https://math.ucr.edu/home/baez/history.pdf.]

[11] Jurco, B. et al., \textit{Higher structures in M-theory}. (2019). [arXiv:1903.02807v2].

Start of new semester, thinking about double field theory cosmology

I haven’t added much to my blog in the past weeks. With university kicking off again, and with Tony and I having our first work sessions of the semester, it has been quite busy. I’ve also been adjusting to being back at university after summer holiday, and with being back on campus for the first time since lock down due to the pandemic. So I’ve been finding my feet again with new daily structure and routine.

I’ve also been working on a number of projects, some short-term and some long-term, which have kept me quite occupied. It is the battle of constantly balancing enticing questions and ideas that define the day. It’s what makes life exciting and keeps me coming back to physics, I suppose.

In the last week or so we’ve been talking more about double field theory cosmology, mainly from the perspective of how matter couples. As a developing area of research there are many interesting questions one can ask. It’s quite interesting stuff, to be honest, and I’m looking forward to potentially pursuing a few side projects in this area. As it relates, I’m interested in higher {\alpha^{\prime}} corrections, non-perturbative solutions, and {\alpha^{\prime}} deformed geometric structures.

To share a bit more, one thing that is quite neat about DFT cosmology is how, under a cosmological ansatz [1,2], the equations coupled to matter take the form

\displaystyle 4d^{\prime \prime} - 4(d^{\prime})^2 - (D-1)\tilde{H}^2 + 4\ddot{d} - 4 \dot{d}^2 - (D - 1)H^2 = 0

\displaystyle (D - 1)\tilde{H}^2 - 2 d^{\prime \prime} - (D - 1)H^2 + 2\ddot{d} = \frac{1}{2}e^{2d} E

\displaystyle  \tilde{H}^{\prime} - 2\tilde{H}d^{\prime} + \dot{H} - 2h\dot{d} = \frac{1}{2} e^{2d}P. \\ (1)

Here {E} and {P} denote energy density and pressure, respectively. These equations are duality invariant provided {E \leftrightarrow -E} and {P \leftrightarrow -P} . The approaches that make use of these equations are typically restricted to dilaton gravity. That is to say, the B-field is switched off. From what I presently understand the reason for this is because it is generally unknown how proceed with the full massless string sector explicit in the theory.

For a homogeneous and isotropic cosmology the metric takes the form

\displaystyle  dS^2 = -dt^2 + \mathcal{H}_{MN} dx^M dx^N

\displaystyle  = -dt^2 + a^2(t) dx^2 + a^{-2}(t) d\tilde{x}, \ \ (2)

where {t} is physical time, {a(t)} is the cosmological scale factor, {x} denote are co-moving spatial coordinates. In general, the basic fields reduce to the cosmological scale factor {a(t, \tilde{t})} and the dilaton {\phi(t, \tilde{t})} .

Most pertinently, as we are dealing with a manifestly T-duality invariant theory, what one finds is that T-duality results in scale factor duality. In some ways, this is expected. With the B-field off, the background fields transform

\displaystyle  a(t, \tilde{t}) \rightarrow \frac{1}{a(\tilde{t},t)},

\displaystyle \phi(t, \tilde{t}) \rightarrow \phi(\tilde{t}, t). \ \ (3)

The T-duality invariant combination of the scale factor and the dilaton is

\displaystyle  \phi \equiv \phi - d\ln a, \ \ (4)

where {d = D-1} is the number of spatial dimensions with D space-time dimensions.

It will be interesting to read more about the work that has so far been done in this area. One thing that is very clear, the approaches to DFT cosmology that I have so far looked at ultimately go back to Tseytlin and Vafa [3], and, also, of course, to efforts in string gas cosmology.

The main thing about these types of approaches behind (1) is that, rather than using T-duality variables, they leverage T-duality frames. The assumption, again, is the use of the section condition (conventional in DFT), which states the fields only depend on a D-dimensional subset of the space-time variables. We’ve talked about this in the past on this blog. There are different, often arbitrary choices, of this condition – what we call frames – and these different frames are related by T-duality.

The most basic example is the supergravity frame with standard coordinates transformed to the winding frame with dual coordinates. And so, what one can do, is calculate supergravity and winding frame solutions of the cosmological equations (1), with these solutions being T-dual to each other [4].

In review of ongoing efforts, it will be interesting to see what ideas might arise in the coming weeks.

References

[1] H. Wu and H. Yang, Double Field Theory Inspired Cosmology. JCAP 1407, 024 (2014) doi:10.1088/1475- 7516/2014/07/024 [arXiv:1307.0159 [hep-th]].

[2] R. Brandenberger, R. Costa, G. Franzmann and A. Welt- man, T-dual cosmological solutions in double field theory. [arXiv:1809.03482 [hep-th]].

[3] A. A. Tseytlin and C. Vafa, Elements of string cosmol- ogy. Nucl. Phys. B 372, 443 (1992) doi:10.1016/0550- 3213(92)90327-8 [hep-th/9109048].

[4] H. Bernardo, R. Brandenberger, G. Franzmann, T-Dual Cosmological Solutions of Double Field Theory II. [ arXiv:1901.01209v1 [hep-th]].

Conference: Higher structures in quantum field theory and string theory

This week I am attending a conference on higher structures in quantum field theory and string theory. It’s an event that I have been excited about since the new year. So far there have been some very nice talks, with interesting ideas and calculations presented.

There is the expression about going down a rabbit hole. In the world of mathematical concepts and fundamental physics, it is easy to get excited about an especially stimulating talk and follow down several rabbit holes. I’m trying to stay especially focused on presentations that are more directly related with my current research, but sometimes the excitment and sense of interest in the discussion topic becomes too strong! This afternoon I am looking forward to Bob Knighton speak on an exact AdS/CFT correspondence and Fiona Seibold talk about integrable deformations of superstrings. The rest of the week should also be a lot of fun.

Meanwhile, in the background I’ve been working on my PhD research (even though I don’t formally start until 1 August) and some double sigma model stuff. I’m hoping to also have my next post on categorical products, duality, and universality finished, which, as it is currently drafted, also talks a bit more about M-theory motivations but I may save this part for a detailed entry of its own.

O(D,D) and Double Field Theory

1. Introduction

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 {O(d,d)} 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 {O(d,d)} ?

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 [5]. The main thing to remember, or take note of, is how T-duality is encoded in the transformations R \leftrightarrow \frac{l_s}{R} , p \leftrightarrow w  , which describe an equivalence between radius and inverse radius, with the exchange of momentum modes {p} and the intrinsically stringy winding modes {w} 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 S^1  for example, as momentum and winding are exchanged, the coordinates {x} on {S^1} are exchanged with the dual {S^1} coordinates \tilde{x}  .

When T-duality is explicit we have for the mass operator,

\displaystyle  M^2 = (N + \tilde{N} - 2) + p^2 \frac{l_s^2}{R^2} + \tilde{w}^2 \frac{R^2}{l_s^2}, \ (1)

where the dual radius is {\frac{R^2}{l_s} \leftrightarrow \frac{\tilde{R}^2}{l_s} = \frac{l_s}{R^2}} with {p \leftrightarrow \tilde{w}} . Here {l_s} 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 {\mathbb{Z}_2} , but for a d-dimensional torus the duality group is the indefinite orthogonal group {O(d,d; \mathbb{Z})} , with {d} the number of compact dimensions.

And, actually, since we’re here one can motivate the idea another way [6]. 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 {S^1} compactification and set {R = \sqrt{2}} , one finds four additional massless gauge bosons that correspond to {pw = \pm 1} , {N + \tilde{N} = 1} . One can combine these states with the two {U(1)} gauge fields to enlarge the {U(1)^2} gauge symmetry in the form

\displaystyle  U(1) \times U(1) \rightarrow SU(2) \times SU(2). \ (2)

If we want to generalise from the example of an {S^1} 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 {G = U(1)^{2n}} and the toroidal moduli {g_{ij}, b_{ij}} , parameterising a moduli space of inequivalent vacua. This moduli space is {n^2} -dimensional coset space

\displaystyle  \mathcal{M}^{n} = \frac{O(n,n)}{O(n) \times O(n)} / \Gamma_T, \ (3)

where {\Gamma_T = O(n,n; \mathbb{Z})} . In other words, it is the T-duality group relating equivalent string vacua. (In my proceeding notes I sometimes use O(d,d) and O(n,n) interchangably).

But the example I really want to get to comes from the classical bosonic string sigma model and its Hamiltonian formulation [7]. 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

\displaystyle  G_{ab} (\partial_{\tau} X^{a} \partial_{\tau} X^b + \partial_{\sigma} X^a \partial_{\sigma} X^b) = 0

and

\displaystyle  G_{ab}\partial_{\tau}X^a \partial_{\sigma} X^b = 0, \ (4)

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

\displaystyle H(X; G,B) = -\frac{1}{4 \pi \alpha^{\prime}} \begin{pmatrix}\partial_{\sigma} X \\ 2 \pi \alpha^{\prime} P \end{pmatrix}^T \mathcal{H}(G, B) \begin{pmatrix} \partial_{\sigma} X \\ 2\pi \alpha^{\prime} P \end{pmatrix}

\displaystyle = -\frac{1}{4\pi \alpha^{\prime}} \begin{pmatrix}  \partial_{\tau} X \\ -2\pi \alpha^{\prime} W \end{pmatrix}^T \mathcal{H}(G, B) \begin{pmatrix}  \partial_{\tau}X \\ -2\pi \alpha^{\prime} P \end{pmatrix} \ (5).

This {\mathcal{H}(G,B)} is what we will eventually come to define as the generalised metric. Keeping to the Hamiltonian formulation of the standard string, the appearance of {O(d,d)} follows. We first may define generalised vectors given some generalised geometry {TM \oplus T \star M} , in which the tangent bundle {TM} of a manifold {M} is doubled in the sum of the tangent and co-tangent bundle. The vectors read:

\displaystyle  A_{P}(X) = \partial_{\sigma} X^a \frac{\partial}{\partial x^a} + 2\pi \alpha^{\prime}P_a dx^a

and

\displaystyle  A_W(X) = \partial_{\tau} X^a \frac{\partial}{\partial x^a} - 2\pi \alpha^{\prime}W_a dx^a. \ (6)

Now, in this set-up, {O(d, d)} 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

\displaystyle  A^T_{P} \mathcal{H} A_P = 0 \ \ \text{and} \ \ A^T_P L A_P = 0. \ (7)

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 {L} defines the group {O(d,d)} . Moreover, a {d \times d} matrix {Z} is an element of {O(d,d)} if and only if

\displaystyle  Z^T L Z = L \ (8),

where

\displaystyle  L = \begin{pmatrix} 0 & \mathbb{I} \\ \mathbb{I} & 0 \\ \end{pmatrix}. \ (9)

The moral of the story here is that the generalised vectors solving the constraint in (7) are related by an {O(d,d)} 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 {O(D,D,\mathbb{Z})}

From a field theory perspective, there is a lot to unearth about the presence of {O(d,d)} , 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 {g_{IJ}} with Riemann curvature {R(g)} , the Kalb-Ramond field {b_{IJ}} with the conventional definition for the field strength {H=db} , and a dilaton scalar field {\phi} – 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 {S^1} . Generalising to a d-dimensional compactification, we of course have {O(d,d)} and for the double internal space we may write the coordinates {X^i = (x^i, \tilde{x}_i)} , where {i = 1,...,d} . But what we really want to do is to double the entire space such that {D = d + n} , with {I = 1,..., 2D} , and then see what happens. Consider the standard formulation of DFT known as the generalised metric formulation (for a review of the fundamentals see [8]). The effort begins with the NS-NS supergravity action

\displaystyle  S_{SUGRA} = \int dX \sqrt{-g} \ e^{-2\phi} [R + 4(\partial \phi)^2 - \frac{1}{12}H^2] \ + \ \text{higher derivative terms}. \ (10)

In the case of toroidal compactification defined by {D} -dimensional non-compact coordinates and {d} -dimensional compact directions, the target space manifold can be defined as a product between {d} -dimensional Minkowski space-time and an {n} -torus, such that {\mathbb{R}^{d-1,1} \times T^{n}} where, as mentioned a moment ago, {D = n + d} . We have for the full undoubled coordinates {X^{I} = (X^{a}, X^{\mu})} with {X^{a} = X^{a} + 2\pi} being the internal coordinates on the torus. The background fields are {d \times d} matrices taken conventionally to be constant with the properties:

\displaystyle G_{IJ} = \begin{pmatrix} \hat{G}_{ab} & 0 \\ 0 & \eta_{\mu \nu} \\ \end{pmatrix}, \ \ B_{IJ} = \begin{pmatrix} \hat{B}_{ab} & 0 \\ 0 & 0 \\ \end{pmatrix}, \ \ \text{and} \ \ G^{IJ}G_{JK} = \delta^{I}_K. \ (11)

We define {\hat{G}_{ab}} as a flat metric on the torus and {\eta_{\mu \nu}} is simply the Minkowski metric on the {d} -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 {G_{IJ}} and {B_{IJ}} come together in a single generalised geometric entity, which we begin to construct with the internal metric denoted as

\displaystyle E_{IJ} = G_{IJ} + B_{IJ} = \begin{pmatrix} \hat{E}_{ab} & 0 \\ 0 & \eta_{\mu \nu} \\ \end{pmatrix} \ (12)

for the closed string background fields, with {\hat{E}_{ab} = \hat{G}_{ab} + \hat{B}_{ab}} as first formulated by Narain et al [9]. It is important to note that the canonical momentum of the theory is {2\pi P_{I} =  G_{IJ}\dot{X}^{J} + B_{IJ} X^{\prime J}} , where, in the standard way, {\dot{X}} denotes a {\tau} derivative and {X^{\prime}} denotes a {\sigma} derivative. Famously, the Hamiltonian of the theory may then also be constructed from the expansion of the string modes for coordinate {X^{I}} , the canonical momentum, and from the Hamiltonian density to take the following form

\displaystyle  H = \frac{1}{2} Z^{T} \mathcal{H}(E) Z + (N + \bar{N} - 2). \ (13)

Or, to write it in terms of the mass operator,

\displaystyle  M^{2} = Z^{T} \mathcal{H}(E) Z + (N + \bar{N} - 2). \ (14)

The structure of the first terms in (14) should look familiar. In summary, in an {n} -dimensional toroidal compactification, the momentum {p^{I}} and winding modes {w_{I}} become {n} -dimensional objects. So the momentum and the winding are combined in a single object known as the generalised momentum Z = \begin{pmatrix} w_{I} \\ p^{I} \\ \end{pmatrix}  . This generalised momentum Z is defined as a 2D  -dimensional column vector, and we will return to a discussion of its transformation symmetry in a moment. Meanwhile, in (13) and (14) N  and \bar{N}  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 \mathcal{H}(E) , which is a 2D \times 2D  symmetric matrix constructed from G_{IJ}  and B_{IJ}  with E = E_{IJ} = G_{IJ} + B_{IJ}  . We will discuss the generalised metric in just a few moments.

As is fundamental to closed string theory there is the Virasoro constraint {L_{0} - \bar{L}_{0} = 0} , where {L_{0}} and {\bar{L}_{0}} are the Virasoro operators. This fundamental constraint remains true in the case of DFT. Except in DFT this condition on the spectrum gives {N - \bar{N} = p_{I}w^{I}} or, equivalently,

\displaystyle  N - \bar{N} = \frac{1}{2} Z^{T} L Z, \ (15)

where

\displaystyle L = \begin{pmatrix} 0 & \mathbb{I} \\ \mathbb{I} & 0 \\ \end{pmatrix}. \ (16)

This is, indeed, the same {L} 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 {L} somewhat vaguely and simply consider it as a constant matrix. We denote {\mathbb{I}} as a {D \times D} 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 [10] for T-duality transformations with the standard sigma model [11,12]. That is to say, {\mathcal{H}} takes a form in which there is clear mixing of the background fields. It is defined as follows,

\displaystyle \mathcal{H}(E) = \begin{pmatrix} G - BG^{-1}B & BG^{-1} \\ -G^{-1}B & G^{-1} \\ \end{pmatrix}. \ (17)

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 {G_{ij}} and the Kalb-Ramond field {B_{ij}} , in DFT these then must assume the form of an {O(d,d)} 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 {\mathcal{M}} may be similarly written as (3).

The overall dimension of the moduli space is {n^2} which follows from the parameters of the background matrix {E_{ij}} , with {n(n+1)/2} for {G_{ij}} plus {n(n-1)/2} for {B_{ij}} . The zero mode momenta of the theory define the Narain lattice {\Gamma_{n,n} \subset \mathbb{R}^{2n}} , and it can be proven that {\Gamma_{n,n}} 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 {O(n, \mathbb{R}) \times O(n, \mathbb{R})} . The Hamiltonian (13) remains invariant from separate {O(n, \mathbb{R})} 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 {O(n,n) / O(n) \times O(n)} coset form of the {n^2} moduli fields.

4. {O(n,n,\mathbb{Z})}

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 {Z} as it appears in (14). If we shuffle the quantum numbers {w,p} , which means we exchange {w} for {p} and vice versa, the transformation symmetry of {Z} is well known to be

\displaystyle  Z \rightarrow Z = h^{T}Z^{\prime}. \ (18)

For now, {h} $ is considered generally as a {2D \times 2D} invertible transformation matrix with integer entries, which mixes {p^{I}} and {w_{I}} after operating on the generalized momentum. It follows that {h^{-1}} 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 {Z \rightarrow Z^{\prime}} as a one-to-one correspondence, the level-matching condition (15) with the above symmetry transformation (18) gives

N - \bar{N} = \frac{1}{2} Z^{T}LZ = \frac{1}{2} Z^{T \prime}L Z^{\prime}

\displaystyle  = \frac{1}{2} Z^{T \prime} h L h^{T} Z^{\prime}. \ (19)

For this result to be true, it is necessary as a logical consequence that the transformation matrix {h} must preserve the constant matrix {L} . This means it is required that

\displaystyle  h L h^{T} = L, \ (20)

which also implies

\displaystyle  h^{T} L h = L. \ (21)

These last two statements can be proven, producing several equations that give conditions on the elements of {h} . 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 {a, b, c, d} be {D \times D} matrices, such that {h} may be represented in terms of these matrices

\displaystyle h = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}. \ (22)

The condition in which {h} preserves {L} demands that the elements {a, b, c, d} satisfy in the case of (20)

\displaystyle  a^{T}c + c^{T}a = 0, \ b^{T}d + d^{T}b = 0,

and

\displaystyle  a^{T}d + c^{T}b = 1. \ (23)

Likewise, similar conditions are found for the case (21), for which altogether it is proven that {h^{-1}} has invertible entries. What this ultimately means is that although we previously considered {h} vaguely as some transformation matrix, it is in fact an element of {O(D,D, \mathbb{R})} and {L} is an {O(D,D, \mathbb{R})} invariant metric. Formally, an element {h \in O(D,D, \mathbb{R})} is a {2D \times 2D} matrix that preserves, by its nature, the {O(D,D, \mathbb{R})} invariant metric {L} (16) such that

\displaystyle  O(D,D,\mathbb{R}) = \bigg \{h \in GL(2D, \mathbb{R}) \ : \ h^{T}Lh = L \bigg \}. \ (24)

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 {O(D,D)} then we must have the following transformation property in the case {Z^{T} \mathcal{H}(E) Z \rightarrow Z^{\prime T} \mathcal{H}(E^{\prime}) Z^{\prime}} :

\displaystyle  Z^{\prime T}\mathcal{H}(E^{\prime}) Z^{\prime} = Z^{T}\mathcal{H}(E)Z

\displaystyle  = Z^{\prime T} h \mathcal{H}(E)h^{T} Z^{\prime}. \ (25)

By definition, given the principle requirement of (25) it is therefore also required that the generalised metric transforms as

\displaystyle  \mathcal{H}(E^{\prime}) =  h\mathcal{H}(E)h^{T}. \ (26)

The primary claim here is that for the transformation of {E} we find

\displaystyle (E^{\prime}) = h(E) =  \begin{pmatrix}  a & b \\ c & d \\ \end{pmatrix}(E) \equiv (aE + b)(cE + d)^{-1}. \ (27)

One should note that this is not matrix multiplication, and {h(E)} 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 {\mathcal{H}} is acting on the background {E} , what we end up with is the following

\displaystyle (E^{\prime T}) = \begin{pmatrix} a & -b \\ -c & d \\ \end{pmatrix}(E^{T}) \equiv (aE^{T} - b)(d - cE^{T})^{-1}, \ (28)

where in the full derivation of this definition it is shown (E^{\prime T}) = \begin{pmatrix} a & -b \\ -c & d \\ \end{pmatrix} E^T.

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 {E} from the identity background {E^{\prime} = \mathbb{I}} , where conventionally {E = G + B} and {G = AA^{T}}. Recall, also, the definition for the generalised metric metric (17). Then for {E = h_{E}(\mathbb{I})} , what is {h_{E} \in O(D,D, \mathbb{R})} ? To answer this, suppose we know some {A} such that

\displaystyle h_{E} =  \begin{pmatrix} A & B(A^{T})^{-1} \\ 0 & (A^{T})^{-1} \\ \end{pmatrix}. \ (29)

It then follows

\displaystyle  h_{E}(I) = (A \cdot \mathbb{I} + B(A^{T})^{-1})(0 \cdot \mathbb{I} + (A^{T})^{-1})^{-1}

\displaystyle  = (A + B(A^{T})^{-1}) A^{T} = AA^{T} + B = E = G + B. \ (30)

This means that the {O(D,D)} transformation creates a {G + B} background from the identity. Additionally, the transformation {h_E} is ambiguous because it is always possible to substitute {h_E} with {h_E \cdot g} , where we define {g(\mathbb{I}) = \mathbb{I}} for {g \in O(D,D, \mathbb{R})} . In fact, it is known that {g} defines a {O(D) \times O(D)} subgroup of {O(D,D)} {g^{T}g = gg^{T} = I} .

In conclusion, one can show that {\mathcal{H}} transforms appropriately, given that up to this point {h_{E}} was constructed in such a way that the metric {G} is split into the product {A} and {A^{T}} , with the outcome that only {A} is entered into {h_{E}} . To find {G} we simply now consider the product {h_{E}h_{E}^{T}} ,

\displaystyle h_{E}h_{E}^{T} = \begin{pmatrix} A & B(A^{T})^{-1} \\  0 & (A^{T})^{-1} \\ \end{pmatrix} \begin{pmatrix}  A^{T} & 0 \\ -A^{-1}B & A^{-1} \\ \end{pmatrix}

\displaystyle  = \begin{pmatrix} G - BG^{-1}B & BG^{-1} \\ -G^{-1}B & G^{-1} \\ \end{pmatrix} = \mathcal{H}(E). \ (31)

If we now suppose naturally {E^{\prime}} is a transformation of {E} by {h} , such that {E^{\prime} = h(E) = hh_{E}(\mathbb{I})} , we also have {E^{\prime} = h_{E^{\prime}}(\mathbb{I})} . Notice that this implies {h_{E^{\prime}} = hh_{Eg}} up to some ambiguous and so far undefined {O(D,D,\mathbb{R})} subgroup defined by {g} . Putting everything together, we obtain the rather beautiful result

\displaystyle \mathcal{H}(E^{\prime}) = h_{E^{\prime}}h^{T}_{E^{\prime}} = hh_{Eg}(hh_{Eg})^{T} = hh_{E}h^{T}_{E}h^{T} = h\mathcal{H}(E)h^{T}. \ (31)

\Box

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 {O(D,D,\mathbb{R})} .

In conclusion, and to summarise, in DFT there is an explicit restriction on the winding modes {w_{I}} and the momenta {p^{I}} 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 {n} -dimensional toroidal space, so that in the quantum theory the symmetry group is restricted to {O(n,n,\mathbb{Z})} subgroup to {O(D,D,\mathbb{R})} . The group {O(n,n,\mathbb{Z})} is as a matter of fact the T-duality symmetry group in string theory. It is conventional to represent the transformation matrix {h \in O(n,n,\mathbb{Z})} in terms of {O(D,D,\mathbb{R})} such that

\displaystyle h = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}

with,

\displaystyle a = \begin{pmatrix} \tilde{a} & 0 \\ 0 & 1 \\ \end{pmatrix},

\displaystyle b = \begin{pmatrix} \tilde{b} & 0 \\ 0 & 0 \\ \end{pmatrix},

\displaystyle c = \begin{pmatrix} \tilde{c} & 0 \\ 0 & 0 \\ \end{pmatrix}

and

\displaystyle d = \begin{pmatrix} \tilde{d} & 0 \\ 0 & 1 \\ \end{pmatrix}. \ (32)

Each of {\tilde{a}, \tilde{b}, \tilde{c}, \tilde{d}} are {n \times n} matrices. They can be arranged in terms of {\tilde{h} \in O(n,n,\mathbb{Z})} as

\displaystyle \tilde{h} = \begin{pmatrix} \tilde{a} & \tilde{b} \\ \tilde{c} & \tilde{d} \\ \end{pmatrix}. \ (33)

Invariance under the {O(D,D,\mathbb{Z})} group of transformations is generated by the following transformations. To simplify matters, let us define generally the action of an {O(D,D)} element as

\displaystyle \mathcal{O} = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} = \mathcal{O}^{T}L\mathcal{O}. \ (34)

Residual diffeomorphisms: If {A \in GL(D, \mathbb{Z})} , then one can change the basis for the compactification lattice {\Gamma} by {A \Gamma A^{T}} . The action on the generalised metric is

\displaystyle \mathcal{O}_{A} = \begin{pmatrix} A^{T} & 0 \\ 0 & A^{-1} \\ \end{pmatrix}, \ \ A \in GL(D, \mathbb{Z}), \ \ \det A = \pm 1. \ (35)

B-field shifts: If we define {\Theta} to be an antisymmetric matrix with integer entries, one can use {\Theta} to shift the B-field producing no change in the path integral. For compact d-dimensions, this amounts to {B_{IJ} \rightarrow B_{IJ} + \Omega_{IJ}} . It follows that the {O(D,D)} transformation acts on the generalised metric,

\displaystyle \mathcal{O}_{\Omega} = \begin{pmatrix} 1 & \Omega \\ 0 & 1 \\ \end{pmatrix}, \ \  \Omega_{IJ} = - \Omega_{JI} \in \mathbb{Z}. \ (36)

Factorised dualities: We define a factorised duality as a {\mathbb{Z}_2} duality corresponding to the {R \rightarrow \frac{1}{R}} transformation for a single circular direction (i.e., radial inversion). It acts on the generalised metric as follows

\displaystyle \mathcal{O}_{T} = \begin{pmatrix} 1-e_{i} & e_{i} \\ e_i & 1-e_{i} \\ \end{pmatrix}, \ (37)

where {e} is a {D \times D} matrix with 1 in the {(i, i)} -th entry, and zeroes elsewhere {(e_{i})_{jk} =  \delta_{ij}\delta_{ik}} . Altogether, these three essential transformations define the T-duality group {O(D,D,\mathbb{Z})} , as first established in [14,15]. To calculate a T-dual geometry one simply performs the action (26) or (28) using an {O(D,D,\mathbb{R})} 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.

References

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[9] 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.

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[11] Mark Bugden. A tour of t-duality: Geometric and topological aspects of t-dualities, 2019.

[12] T.H. Buscher. Path Integral Derivation of Quantum Duality in Nonlinear SigmaModels.Phys. Lett. B, 201:466–472, 1988.

[13] Daniel C. Thompson. T-duality invariant approaches to string theory, 2010.[14] Alfred D. Shapere and Frank Wilczek. Selfdual Models with Theta Terms.Nucl.Phys. B, 320:669–695, 1989.[15] Amit Giveon, Eliezer Rabinovici, and Gabriele Veneziano. Duality in StringBackground Space.Nucl. Phys. B, 322:167–184, 1989.

[14] Alfred D. Shapere and Frank Wilczek. Selfdual Models with Theta Terms. Nucl. Phys. B, 320:669–695, 1989.

[15] Amit Giveon, Eliezer Rabinovici, and Gabriele Veneziano. Duality in String Background Space. Nucl. Phys. B, 322:167–184, 1989.

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