Polchinski_string theory_tracingcurves blog

Reading Polchinski – My notes on string theory

When I was in the first year of my undergraduate degree, I started self-studying string theory. At that time, I set myself the task of working through Joe Polchinski’s two volume textbook, String Theory. It was honestly one of the best years of my life, despite the general uncertainty about my academic future at the time. I also began a project of sharing my own notes on this blog, as an extension of my enthusiasm.

Between the time as a first year undergraduate student and the commencement of my Masters the next year, I hadn’t much time to finish converting my notes to latex and uploading them to this blog. Things move fast, and quickly I was on to double field theory, developing my interests in M-theory, and other cool stuff with pressures of producing a thesis. Also, in that time, I had to change my web hosting service. As a result of transferring my blog, the latex broke in all of my old posts (including what string notes I had uploaded) which was a bit frustrating.

I have been intending to reupload what notes I had converted to latex and then to also continue my project for sometime. Now that I have settled into my PhD years, I feel able again to take up the task (of course, as and when the time becomes available). As part of my scholarship, I also feel obliged to participate in science communication. Sharing technical and explanatory notes in maths/physics is probably where I am most capable.

***

I consider my handwritten notes a product of rigorous examination and review, organised around the central motivation to rederive the whole of bosonic and then superstring theory from first principles, or, where appropriate, at least from as close to first principles as possible. They are a thorough companion to Polchinski’s textbook, providing a workthrough of its many pages and subtle details. They also offer a lot of comments from my own perspective, such as when making an interesting observation or offering more background work than presented in Polchinski’s textbook. I have also developed my own preferences, such as emphasising at the outset the notion of string theory as a generalisation point particle theory.

Due to the way I upload latex to my blog nowadays, what I currently plan to do is start from the very beginning and reupload everything that was already posted and then work toward uploading the rest. As the notes were originally written for myself, converting them to latex and organsing them in a presentable way takes quite a bit of time. I like the idea of uploading one batch of notes at a time as individual posts, such as in presenting one lecture note at a time (working page by page through Polchinski). Eventually I will also create a directory on my blog, where the reader may navigate by topic through the complete notes. (I will also file everything under the tag, ‘Notes on String Theory’). They may or may not prove useful for others, but, to be honest, I ultimately doing it because it is something I enjoy.

As for what is to come next: in the first entry we will start with a brief review of the non-relativistic string, before moving to a review of the relativistic free-point particle in a separate note. This will then take us to the first pages of Polchinski and the construction of the Nambu-Goto action.

Generalised supergravity and the dilaton

I spent sometime in the early autumn months thinking about the cosmological constant problem (CC). This was actually secondary, because my primary note taking focused more on S-duality and manifestly duality invariant actions, non-perturbative corrections, and the dilaton. But my supervisor, Tony, has spent a lot of time thinking about this problem, with one of his big ideas being vacuum energy sequestering, so naturally there is motivation whenever we get the chance. There has also been some renewed interest in the CC problem in the context of generalised double sigma models and double field theory. In general, there is a lot of interesting cosmology to be investigated here.

I’m currently drafting a post on the CC problem from the view of string theory. This will hopefully provide the reader with a thorough introduction. But as a passing comment in this short note, it suffices to say that the role of the CC in string theory is generally mysterious. In standard textbook analysis, one sees that the mystery starts with the massless sector contribution, with the dilaton central to the discussion; but the mysteriousness comes further into focus once the role of dual geometry is investigated and the peculiar change of the CC under duality transformation. Intuitively, I am inclined to think that a piece of the picture is missing.

One idea I find interesting to play with involves adding extra fields. Another idea people play with is redefining the dilaton. An example comes from a breakthrough paper by Tseytlin and Wulff [1].

Admittedly, I wasn’t aware of this paper until my early autumn investigations. Within it, a 30 year old problem is solved using the Green-Schwarz (GS) formulation of supergravity theory. The short version is that, in the standard GS formulation of Type IIB string theory there is a problem with the number of degrees of freedom. The space-time fermions have 32 components. An on-shell condition reduces the degrees of freedom to 16, but it needs to be 8. It was later discovered that kappa-symmetry is present in the theory, which is a non-trivial gauge symmetry, and this symmetry may be used to reduce the remaining 8 degrees of freedom. However, issues remained in proving a number of associated conjectures – that is, until Tseytlin and Wulff formulated generalised type IIB SUGRA on an arbitrary background.

The key observation is that generalised SUGRA is equivalent to standard SUGRA plus an extra vector field. Furthermore, one of the characteristics is that, under generalised T-duality, there is a modification of the dilaton such that a non-linear term is added \Phi \rightarrow \tilde{\Phi} = \Phi + I \cdot \tilde{x} [2]. I think this is quite interesting, and it is something I want to look at more deeply in the future.

Although the context of the calculation is completely different to my own investigations, it is worth noting that this generalised Type IIB theory can be obtained from double field theory. Perhaps not surprisingly, I have seen some pin their hopes that generalised SUGRA could contribute to solving the cosmological constant problem (and potentially also give de Sitter vacua). That seems premature, from my vantange; but in any case it is an interesting bit of work by Tseytlin, Wulff, and others.

References

[1] Tseytlin, A.A., Wulff, L., \textit{Kappa-symmetry of superstring sigma model and generalized 10d supergravity equations}. (2016). [arXiv:1605.04884 [hep-th]].

[2] Tseytlin, A.A., et al, Scale invariance of the $\eta$-deformed $AdS5 \times S5$ superstring, T-duality, and modified type II equations. (2016). [arXiv:1511.05795 [hep-th]].

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]].