Stringy Things

Literature: Duality Symmetric String and the Doubled Formalism

When it comes to a T-duality invariant formulation of string theory, there are two primary actions that are useful to study as a point of entry. The first is Tseytlin’s non-covariant action. It is found in his formulation of the duality symmetric string, which presents a stringy extension of the Floreanini-Jackiw Lagrangians for chiral fields. In fact, for the sigma model action in this formulation, one can directly reproduce the Floreanini-Jackiw Lagrangians for antichiral and chiral scalar fields. The caveat is that, although we have explicit O(D,D) invariance, which is important because ultimately we want T-duality to be a manifest symmetry, we lose manifest Lorentz covariance on the string worldsheet. What one finds is that we must impose local Lorentz invariance on-shell, and from this there are some interesting things to observe about the constraints imposed at the operator level.

The main papers to study are Tseytlin’s 1990/91 works listed below. Unfortunately there is no pre-print available, so these now classic string papers remain buried behind a paywall:
1) Tseytlin, ‘Duality Symmetric Formulation of String World Sheet Dynamics
2) Tseytlin, ‘Duality Symmetric Closed String Theory and Interacting Chiral Scalars

For Hull’s doubled formalism, on the other hand, we have manifest 2-dimensional invariance. In both cases the worldsheet action is formulated such that both the string coordinates and their duals are on equal footing, hence one thinks of the coordinates being doubled. However, one advantage in Hull’s formulation is that there is a priori doubling of the string coordinates in the target space. Here, O(D,D) invariance is effectively built in as a principle of construction. This is because for the covariant double sigma model action, the target space may be written as R^{1, d-1} \otimes T^{2D}, in which we have a non-compact spacetime and a doubled torus. From the torus identifications we have manifest GL(2D; Z) symmetry. Then after imposing what we define as the self-duality constraint of the theory, which contains an O(D,D) metric, invariance of the theory reduces directly to O(D,D; \mathbb{Z}).

    1. Hull, ‘Doubled Geometry and T-Folds
    2. Hull, ‘Geometry for Non-geometric Backgrounds
    3. Hull and Reid-Edwards, ‘Non-geometric backgrounds, doubled geometry and generalised T-duality

What is neat about the two formulations is that, turning off interactions, they are found to be equivalent on a classical and quantum level. It is quite fun to work through them both and prove their equivalence, as it comes down to the constraints we must impose in both formulations.

I think the doubled formalism (following Hull) for sigma models is most interesting on a general level. I’m still not comfortable with different subtleties in the construction, for example the doubled torus fibration background or choice of polarisation from T-duality. The latter is especially curious. But, in the course of the last two weeks, things are finally beginning to clarify and I look forward to writing more about it in time.

Related to the above, I thought I’d share three other supplementary papers that I’ve found to be generally helpful:

1) Berman, Blair, Malek, and Perry, ‘O(D,D) Geometry of String Theory
2) Berman and Thompson, ‘Duality Symmetric String and M-theory
3) Thompson, ‘T-duality Invariant Approaches to String Theory

There are of course many other papers, including stuff I’ve been studying on general double sigma models and relatedly the Pasti, Sorokin and Tonin method. But those listed above should be a good start for anyone with an itch of curiosity.

Standard
Stringy Things

Thinking About the Strong Constraint in Double Field Theory

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

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

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

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

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

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

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

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

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

1. The Strong Constraint

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2. Some Nuances and Subtleties

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

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

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

References

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

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

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

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

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

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

Standard
Stringy Things

Double Field Theory: The Courant Bracket

1. Introduction

In this post we are going to briefly and somewhat schematically discuss the appearance of the Courant bracket in Double Field Theory (DFT), following [1]. The point here is mainly to set the stage, so we jump straight into motivating the Courant bracket. In the next post, we will then study the B-transformations from the maths side and the C-bracket, following and expanding from [1] and others, with an emphasis in the end on how all of this relates to T-duality and strings.

What follows is primarily based on a larger collection of study notes, which I will upload in time.

2. Motivating the Courant Bracket

To understand the appearance of the Courant bracket in DFT, one way to start is by considering some general theory with a metric {g_{ij}(X)} and a Kalb-Ramond field (i.e., an antisymmetric tensor field) {b_{ij}(X)}, where {X \in M}. The symmetries and diffeomorphisms of {g_{ij}(X)} are generated by vector fields {V^{i}(X)}, where {V \in T(M)} with {T(M)} being the tangent bundle. As for {b_{ij}(X)}, the transformations are generated by one-forms {\xi_{i}(X)}, where {\xi^{i}(X) \subset T^{\star}(M)} with {T^{\star}(M)} being the cotangent bundle. We may combine {V^{i}(X)} and {\xi^{i}(X)} as a sum of bundles, such that (dropping indices) {V + \xi \in T(M) \oplus T^{\star}(M)}.

With these definitions, the opening question now is to ask, ‘what are the gauge transformations?’ To make sense of this, consider the following gauge parameters,

\displaystyle \delta_{V + \xi} g = \mathcal{L}_{V} g

\displaystyle \delta_{V + \xi} b = \mathcal{L}_{V} b + d\xi \ \ \ (1)

Here {\mathcal{L}} is the Lie derivative. Furthermore, note given that {V} generates diffeomorphisms, in (1) we get the Lie derivative in the direction of {V}. Also notice that {\xi} does not enter the gauge transformation of {g}; however, for the gauge transformation of {b}, we do have a one-form {\xi} and so we can take the exterior derivative. We should also note the following important properties of {\mathcal{L}}. For instance, when acting on forms the Lie derivative is,

\displaystyle \mathcal{L}_{V} = i_{V}d + di_{V} \ \ \ (2)

Where {iV} is a contraction with {V}. We’re just following the principle of a contraction with a vector times the exterior derivative. It is also worth pointing out that {\mathcal{L}} and the exterior derivative commutate such that,

\displaystyle \mathcal{L}_{V}d = d\mathcal{L}_{V} \ \ \ (3)

There are also some other useful identities that we are going to need. For instance, for the Lie algebra,

\displaystyle  [\mathcal{L}_{V_1}, \mathcal{L}_{V_2}] = \mathcal{L}_{[V_1,V_2]} \ \ (4)

Where {[V_1,V_2]} is just another vector such that {[V_1,V_2]^{k} = V_{1}^{p}\partial_{p}V_{2}^{k} - (1,2)}.

And, finally, we have,

\displaystyle  [\mathcal{L}_{X}, i_{Y}] = i_{[X,Y]} \ \ (5)

Now follows the fun part. Given the transformation laws provided in (1), we want to determine the gauge algebra. To do this, we must compute in reverse order the gauge transformations on the metric {g} and the {b}-field. For the metric we evaluate the bracket,

\displaystyle  [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]g = \delta_{V_2 + \xi_2} \mathcal{L}_{V_1}g - (1,2)

\displaystyle = \mathcal{L}_{V_1}\mathcal{L}_{V_2}g - (1,2)

Using the identity (4) we find,

\displaystyle  [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]g = \mathcal{L}_{[V_1, V_2]} g \ \ \ (6)

For the {b}-field we have,

\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}]b = \delta_{V_2 + \xi_2} (\mathcal{L}_{V_1}b + d\xi_{1}) - (1,2)

\displaystyle = \mathcal{L}_{V_1}(\mathcal{L}_{V_2}b + d\xi_2) - (1,2)

\displaystyle = \mathcal{L}_{[V_1, V_2]} + d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}V_{2}\xi_{1} \ \ \ (7)

It turns out that this bracket satisfies the Jacobi identity, although it is not without its problems because, as we will see, there is a naive assumption present in the above calculations. In the meantime, putting this aside until later, the idea now is to compare the above with (1) and see what ‘pops out’. Notice that we find,

\displaystyle [\delta_{V_2 + \xi_2}, \delta_{V_1 + \xi_1}] = \delta_{[V_1,V_2]} + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} \ \ \ (8)

In which we have discovered a bracket defined on {T(M) \oplus T^{\star}(M)},

\displaystyle  [V_{1} + \xi_{1}, V_{2} + \xi_{2}] = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2} \xi_{1} \ \ \ (9)

On the right-hand side of the equality we see a vector field in the first term and a one-form given by the final two terms. This Lie bracket is reasonable and, on inspection, we seem to have a definite gauge algebra. Here comes the problem allude a moment ago: there is a deep ambiguity in (9) in that we cannot, however much we try, determine unique parameters in our theory. Notice,

\displaystyle \delta_{V + \xi}b = \mathcal{L}_{V}b + d\xi

\displaystyle = \mathcal{L}_{V + (\xi + d \sigma)} \ \ \ (10)

The point being that the ambiguity of the one-form {\xi} is so up to some exact {d\sigma}. To put it another way, if we change {\xi} by {d\sigma}, we’re not actually changing anything at all. We would just get {\mathcal{L}_{V}b + d(\xi + d\sigma)} where, when the exterior derivative hits {d\sigma} we simply get nothing. So, given that {\xi} is ambiguous up to some exact {d\sigma}, in a sense what we have is a symmetry for a symmetry. In other words, the present construction is not sufficient.

What we can do to correct the situation is analyse the mistake in (7). Let us, for instance, look at the right-hand side of the summation sign in this equation,

\displaystyle d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}V_{2}\xi_{1}) = d(di_{V_1}\xi_{2} + i_{V_1}d\xi_{2}) - (1,2)

The logic follows that the first term {di_{V_1}\xi_{2}} is killed by {d}. It doesn’t make any contribution and its coefficient is just {1}. The trick then is to see, without loss of generality, that we may change the implicit coefficient {1} in front of {di_{V_1}\xi_{2}}. It turns out, the coefficient that we can use is {1 - \frac{\beta}{2}},

\displaystyle  = d((1-\frac{\beta}{2} di_{V_1}\xi_{2} + i_{V_1}d\xi_{2}) - (1,2)

\displaystyle = d(\mathcal{L}_{V_1}\xi_{2} - \frac{1}{2}\beta di_{V_1}\xi_{2}) - (1,2)

\displaystyle = d(\mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2}\beta d [i_{V_1}\xi_{2} - i_{V_2}\xi_{1}]) \ \ \ (11)

What we end up achieving is the construction of a much more general bracket,

\displaystyle  [V_{1} + \xi_{1}, V_{2} + \xi_{2}]_{\beta} = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2}\beta d(iV_{1}\xi_{2} - iV_{2}\xi_{2}) \ \ \ (12)

What is so lovely about this result might at first seem counterintuitive. It turns out, as one can verify, for {\beta \neq 0}, we do not satisfy the Jacobi identity. So at first (12) may not seem lovely at all! But it makes perfect sense to consider cases of non-vanishing {\beta}. In mathematics, the case for {\beta = 1} was introduced by Theodore James Courant in his 1990 doctoral dissertation [5], where he studied the bridge between Poisson geometry and pre-symplectic geometry. The idea here is to forget about the Jacobi identity – consider its loss an artefact of field theory with anti-symmetric tensors and gravity – and impose {\beta = 1}. When we do this what we obtain is indeed the famous Courant bracket. That is, given {\beta = 1}, the case of maximal symmetry is described by,

\displaystyle  [V_{1} + \xi_{1}, V_{2} + \xi_{2}]_{\beta=1} = [V_1, V_2] + \mathcal{L}_{V_1}\xi_{2} - \mathcal{L}_{V_2}\xi_{1} - \frac{1}{2} d(iV_{1}\xi_{2} - iV_{2}\xi_{2} \ \ \ (13)

Although the Jacobi identity does not hold, one can show that for {Z_{i} = V_{i} + \xi_{i}, i = 1,2,3}, the Jacobiator assumes the form,

\displaystyle [Z_1, [Z_2,Z_3]] + \text{cyclic} = dN(Z_1, Z_2, Z_3)

Which is an exact one-form. This gives us a first hint that the unsatisfied Jacobi identity does not provide inconsistencies, because exact one-forms do not generate gauge transformations.

But why {\beta = 1}? Courant argued that the correct value of {\beta} is in fact {1} because, as he discovered, there is an automorphism of the bracket. This means that if do an operation on the elements, it respects the bracket. This automorphism is, moreover, an extra symmetry known in mathematics as a B-transformation. What follows from this is, I think, actually quite special. Given the Courant bracket is a generalisation of the Lie bracket, particularly in terms of an operation on the tangent bundle {T(M)} to an operation on the direct sum of {T(M)} and the p-forms of the vector bundle, what we will discuss is how the B-transformation in mathematics relates in a deep way to what in physics, especially string theory, we call T-duality (target- space duality). This is actually one of the finer points where mathematics and physics intersect so wonderfully in DFT.

In the next post we’ll carry on with a discussion of the B-transformation and then also the C-bracket, finally showing how everything relates.

References

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

[2] Hohm, O., Hull, C., and Zwiebach, B. (2010). ‘Generalized metric formulation of double field theory’. Available from [arXiv:1006.4823v2 [hep-th]].

[3] Hull, C. and Zwiebach, B. (2009). ‘Double Field Theory’. Available from [arXiv:0904.4664v2 [hep-th]].

[4] Hull, C. and Zwiebach, B. (2009). ‘The Gauge Algebra of Double Field Theory and Courant Brackets’. Available from [arXiv:0908.1792v1 [hep-th]].

[5] Courant, T. (1990). ‘Dirac manifolds’. Trans. Amer. Math. Soc. 319: 631–661. Available from [https://www.ams.org/journals/tran/1990-319-02/S0002-9947-1990-0998124-1/].

Standard
Physics Diary

Metastable de Sitter Solutions in 2-dimensions

To obtain stable de Sitter solutions in string theory that also avoid the Swampland is no easy task. It is difficult for many reasons. To give one example, from 10-dimensional Type IIA* and Type IIB* theories, one finds de Sitter related solutions but they come with ghosts that give a negative cosmological constant. Full de Sitter spacetime is a Lorentzian manifold that is the coset space of Lorent groups, dS^d \simeq O(d,1)/O(d-1,1), the cosmology of which, as many readers will likely be familiar, is based on a positive cosmological constant. Hence, in this example, we obtain the wrong sign. Many other examples can be cited in reference to the difficulty we currently face.

To speak generally, it has gotten to the point historically that one possibility being considered is that string theory may simply conspire against de Sitter space – that is, there is some deep incompatibility between our leading theory of quantum gravity and de Sitter vacua. The No-de Sitter Conjecture is an example of an attempt to formalise such a logical possibility, motivating beyond other things the need to understand better the structure of vacua in string theory. This conjecture is by no means rigorous, but it is supported by the fact that historically de Sitter solutions have been elusive. Having said that, one must be cautious about any claims regarding de Sitter constructions, either from the perspective of obtaining de Sitter solutions or from arguments claiming de Sitter belongs to the Swampland. Our present theory is still very much incomplete and there is a lot of work to be done, and I think it is fair to suggest that the fate of de Sitter in string theory is still uncertain. As an expression of opinion, I would currently say that it is more likely de Sitter will come from something much more fundamental than some of the current strategies being proposed. It should also be kept in mind that there may be a larger issue here. Quantum gravity in de Sitter is fundamentally difficult beyond pure string theory reasons. Indeed, it is possible that de Sitter is simply unstable in quantum gravity – that it is a space that simply cannot exist quantum mechanically. For instance, think of complications in QFT in curved space. Indeed, there is a lengthy list of authoritative papers that one may cite when it comes purely to perturbative quantum gravity (see, for example, this page for further references).

Image: de Sitter geometry courtesy of ncatlab.org

Given the sociology and history, when a new paper appears claiming to have found de Sitter solutions, one takes notice. That is precisely what happened last week when Miguel Montero, Thomas Van Riet, and Gerben Venken uploaded to the archive their recent work claiming to have found metastable de Sitter solutions in lower dimensions. More precisely, these parametrically controlled solutions appear when compactifying from 4-dimensions to 2-dimensions, particularly as a result of some clever work that invokes abelian p-1-form gauge fields to stabilise the runaway potential giving in general dS_{d-p} \times S^{p} solutions. However, not all solutions are stable with controlled saddle points. Indeed, the authors find D - p > 2 solutions to be generically unstable. On the other hand, the instability in the homogeneous mode disappears when D - p = 2. In this 2-dimensional case, the solutions relate to the near horizon dS_2 \times S^2 geometry of Nariai black holes. It is also worth pointing out that these solutions are not in string theory, but the work highlights some interesting implications and raises some important questions in quantum gravity more generally, including also when it comes to the Swampland.

If I am not mistaken, I think I remember seeing similar kinds of ideas in studies of dS / CFT where people look into cases of rotating Nariai black holes maximally large at dS_2 \times S^2. The geometry is generally interesting because in that some Nariai constructions with quintessence generalise quasi-de Sitter solutions, utilising for example decompositions from 4-dimemensions to things like 2-dimensional dilaton gravity theory (this may make one recall SYK models), one will often find descriptions of the geometric space as ds^2 = \Gamma(\theta) [-d\tau^2 +\cosh^2 \frac{\tau}{l} d\psi^2 + \alpha(\theta) d\theta^{2}] + \gamma(\theta) (d\phi + k \tanh \frac{\tau}{l} d\psi)^2 which we can think of in terms of an S^2 fibered over a dS_2 base space. I kind of see it as a bit of a playground useful to experiment with and probe. In section 2 of the de Sitter paper, one can read a short overview of Nariai black hole solutions of Einstein-Maxwell theory as a precursor to the main study.

I’ve only had time to skim through the Montero et al. paper, so I still need to give it careful attention. But on quick glance I noticed a number of interesting calculations, not least the derived inequality (p -1) \mid \frac{V^{\prime}}{V} \mid \leq \mid \frac{f^{\prime}}{f} \mid that states how, in order for a dS_{d-p} \times S^{p} saddle point to exist, this constraint that relates the gauge kinetic function of the p−1-form field and the potential must be satisfied. This is an example of one of a handful of points to which I would like to go back and think about more deeply. Additionally, it seems an open question whether, if we can obtain controlled lower-dimensional de Sitter solutions from runaway potentials, do such approaches and constructions fully escape the Swampland? It is worth reading their paper with that in mind.

As alluded at the outset, one should always approach any claims about de Sitter with a healthy dose of scepticism. At the same time, van Riet is a physicist who I admire, because he is one of a number championing the need for greater mathematical rigour in string theory and quantum gravity moving forward. So, for me, this paper immediately comes with some weight and authority. If one thing is certain today, at least from my current vantage, it is the need for thoughtful caution and careful mathematical analysis; and it appears that on a few occasions the authors also stress this point in their analysis.

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Physics Diary

Generalised Geometry, Non-commutativity, and Emergence

The project I am working on for my dissertation has to do with the notion of emergent de Sitter space. Of course an ongoing problem in string theory concerns whether asymptotic de Sitter spacetime can exist as a solution, and needless to say this question serves as one motivation for the research. With what appears to be the collapse of KKLT (this is something I will write about, as from my current perspective, the list of complaints against KKLT have not yet seemed to be satisfactorily answered), this academic year I wanted to start picking at the question of perturbative string de Sitter vacua from a different line of attack (or at least explore the possibility). Often, for instance, we approach de Sitter constructions by way of a classical supergravity approach with fluxes, non-geometry, or KKLT-like constructions which add quantum effects to stablise the moduli. One could also look at an alternative to compactification altogether and invoke the braneworld formalism. But, as it is, I’ve not been entirely satisfied with existing programmes and attempts. So the question over the autumn months, as we approached the winter break, concerned whether there was anything else clever that we can think of or take inspiration from. I’m not comfortable in divulging too much at this time, not least until we have something solid. Having said that, in this post let’s talk about some of the cool and fun frontier mathematical tools relevant to my current research.

For my project the focus is on a number of important concepts, including generalised and non-commutative geometry. Within this, we may also ask questions like whether spacetime – and therefore geometry – is emergent. Sometimes in popular talks, one will hear the question framed another way: ‘is spacetime dead?’ But before getting ahead of ourselves, we may start with a very well known and familiar concept in string theory, namely T-duality. Indeed, one motivation to study generalised geometry relates to T-duality, particularly as T-duality expresses how a string experiences geometry. For example, one will likely be familiar with how, in string theory, if we consider the propagation of a string in spacetime in which one spatial dimension is curled up into a circle, the idea is that when we compactify a dimension (in this case on a circle) we modify the string mass spectrum. Less abstractly, take some 10-dimensional string theory and then compactify on a circle $S^{1}$ of radius $R$. The string moves along the circle with the momenta quantised such that p = n / R (n \in \mathbb{Z}). When compactifying the 10th dimension we obtain for the compactified direction, \displaystyle X_{(s)}^{d} (\tau, \sigma + 2\pi) = X_{(s)}^{d}(\tau, \sigma) + 2 \pi \omega R, where we now have winding modes. This is because, as one will learn from any string theory textbook, the string winds around the circle with coordinate X. We can thus write the statement \delta X = 2\pi R m (m \in \mathbb{Z}). In this basic example T-duality is the statement R \rightarrow \frac{\alpha^{\prime}}{R} with n \longleftrightarrow m. The winding modes that appear are of course a deeply stringy phenomenon. And what is interesting is the question of the generalisation of T-duality. Moreover, how might we think of string geometry in such a way that T-duality is a natural symmetry? Generalised geometry was largely motivated by this duality property, such as in the work by Nigel Hitchin. The basic mathematical statement is that the tangent bundle TM of a manifold M is doubled in the sum of the tangent and co-tangent bundle TM \oplus T \star M. In this formalism we also replace the Lie bracket with a Courant bracket, which we may write as something of the form [X + \xi, Y + \eta]_{C} = [X, Y] + L_{X} \eta - L_{Y}\xi - \frac{1}{2} d(i_{X} \eta - i_{Y}\xi) such that X \xi, Y + \eta \in \Gamma (TM \oplus T \star M). In physics, there is also motivation to ask about the geometry of spacetime in which strings propagate. For instance, the existence of winding modes and the nature in which T-duality connects these winding modes to momentum hints that perhaps the fundamental geometry of spacetime should be doubled. This idea serves as one motivation for the study and development of Double Field Theory, which is something the great Barton Zwiebach has been working on in recent years and which uses the SO(d,d) invariant formalism (see his lecture notes).

Additionally, in these areas of thinking, one will often also come across notions like non-geometry or fuzzy geometry. Sometimes these words seem used interchangably, but we should be careful about their meaning. For instance, non-geometry possess a number of characteristics that contribute to its formal definition, one being that it refers strictly to non-Riemannian geometry. Furthermore, we are also speaking of non-geometry as non-commutative geometry [X_{i}, X_{J}] \approx \mathcal{O}(l_{s}) as well as non-associative geometry [X_{i}, X_{J}] X_{k} \approx (l_{s}). One of many possible ways to approach the concept in this regard is to think quantum mechanically. If General Relativity is a very good approximation at long distances, in which we may think of smooth and continuous manifolds; at the smallest scale – such as the string scale – there are important hints that our typical understanding of geometry breaks down.

We will spend a lot of time on this blog discussing technicalities. For now, I just want to highlight some of the different formalisms and tools. In taking a larger view, one thing that is interesting is how there are many similarities between non-commutative and non-associative algebra and generalised geometry, fuzzy geometry, and finally ideas of emergence and a generalised quantum mechanics, although a precise formulation of their relation remains lacking. But this is the arena, if you will, which I think we might be able to make some progress.

As for my research, the main point of this post is to note that these are the sorts of formalisms and tools that I am currently learning. The thing about string theory is that it allows for is no sharp distinction between matter and geometry. Then to think about emergent space – that spacetime is an emergent phenomena – this infers the idea of emergent geometry, and so now we are also starting to slowly challenge present comforts about such established concepts as locality. When we think about emergent geometry we might also think of the structure of perturbative string vacua and ultimately about de Sitter space as a solution that escapes the Swampland. There is a long way to go, but right now I think in general there is an interesting line of attack.

For the engaged reader, although dated the opening article by Michael Douglas in this set of notes from the 2002 summer school at the Clay Mathematics Institute may serve an engaging introduction or overview. A basic introduction to some of the topics described in this post can also be found for instance in this set of notes by Erik Plauschinn on non-geometric backgrounds. Non-commutative (non-associative) geometry is covered as well as things like doubled geometry / field theory. Likewise, I think this paper on non-associative gravity in string theory by Plauschinn and Ralph Blumenhagen offers a fairly good entry to some key ideas. Dieter Lüst also has some fairly accessible lecture notes that offer a glance at strings and (non)-geometry, while Mariana Graña’s lecture notes on generalised geometry are a bit more detailed but serve as a basic entry. Then there are Harold Steinacker’s notes on emergent geometry from matrix models and on non-commutative geometry in relation to matrix models. Finally, there are these lecture notes by Maxim Zabzine on generalised complex geometry and supersymmetry. This is by no means comprehensive, but these links should at least help one get their feet wet.

Maybe in one of the next posts I will spend some time with a thorough discussion on non-commutativity or why it is a motivation of Double Field Theory to make T-duality manifest (and its importance).

**Cover Image: Study of Curve Folding [http://pr2014.aaschool.ac.uk/EMERGENT-TECHNOLOGIES/Curved-Folding-Workshop]

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Philosophy and General Reading

Review: Bertrand Russell’s ‘In Praise of Idleness’

In Praise of Idleness and Other EssaysIn Praise of Idleness and Other Essays by Bertrand Russell
My rating: 4 of 5 stars

View all my reviews

To some, or perhaps to many, it may seem a radical idea: idleness. But for the great British logician, mathematician, and Nobel laureate Bertrand Russell, idleness is seen as a historically rooted concept which ties intimately together the bonds of labour, leisure, and the prospect of human rationality. Or, at least that is my reading of his famously titled composition, ‘In Praise of Idleness’.

So, what does Russell mean by ‘idleness’? In some sense, it infers a socially organised definition of time that is economically independent of professional labour, in which one may instead expend their energy to fulfil personally meaningful pursuits. This could be, for example, a time for a person to explore painting or to explore a scientific pursuit or any number of interests. In some bodies of literature, such projects are called ‘existential projects’ to convey the personalisation of their meaning in one’s life. One may also call them ‘special interests’. In this sense, one can think of idleness simply as being the economically independent pursuit of a subject, activity, or quality for no other reason than it evokes the state of personal interest. Study for study sake, or a painter to paint without the pressure of starving – these are the sorts of examples that Russell evokes.

Russell_In Praise of Idleness

Idleness should thus not infer or be confused with one’s being disinclined to work or with simplistic views pertaining to individual laziness. Idleness should also not be seen as ‘the root of all evil’, as the idiom would have it. If we are to follow Russell’s arguments, idleness has substantial roots in positive human traits, such as curiosity, exploration, and invention. We also read how the notion of idleness is based on ethical, moral and empirical economic arguments. For Russell, social consumption can mean something very different, both existentially and socially, and thus humanistically. He also speaks of economic production and the way in which work and leisure cycles could generally mean something altogether more philosophically transformed in conception, particularly in terms of the meaning of leisure and its tradition and practical cultural configuration.

It is interesting to consider, on that note, how for thousands of years human beings have established traditions of celebrating different sorts of festivals – Judeo-Christian, Pagan, and so on. Think, for instance, of midwinter festivals based on the solstice or on religious themes. With these traditions follows also a deep historical relation between festivity and work. The festival represents, to frame it in terms of economic history, an interruption of daily labour cycles, with its concept rooted primarily in principles of free time for enjoyment [1]. Thinking of this, it is also interesting to recall that, using Christmas as an example, it was during the Victorian era that a formal socioeconomic relation developed between festivity, worker rights, and the commercial profit motive, particularly as middle-class families were afforded time off work with the financial means for surplus consumption. But if festivity and leisure – or idleness – are intricately related with labour by their very definition, and thus with economics, Russell’s account would seem to carry a certain diametric opposition to work patterns that exhaust the possibility of what he describes as energetic leisure.

In this sense, I read Russell’s essay as having some classical enlightenment motivation. Thoughtfulness – indeed, the time to practice thought and to explore intellectually – this seems a theme to Russell’s social philosophical view of which an advanced and aspiring rational society should strive to achieve. In other words, if idleness is a positive human experience, one which supports or fosters the individual subject to flourish rationally and, perhaps, self-actualise existentially, Russell ties this concept with the possibility of continued self-education and self-betterment, among other things. At the same time, while he celebrates the concept and experience of idleness, he also laments the loss of its broader social-economic and cultural realisation. It is argued that leisure time is expunged of idleness much as in the present-day example of Christmas, which is hyper-commercialised and seemingly increasingly filled with passive entertainments, as active energies are instead exhausted by work, intensely driven consumer cycles, and various other contemporary social behavioural patterns rigidified in such a way to maintain systemic mores. Russell’s arguments are based on traditional views of social-economic class structures, and he seems to suggest that the logic of social economy has been skewed; contemporary societies have in some ways lost sight of the meaningful idea of social production and the social purpose of consumption that may foster a more enlightened and rational society.

For these reasons, we read how with more energetic and thoughtful leisure one would then be better able to enjoy pleasures in which it was possible to take an active part. The central thrust of Russell’s argument in this regard is not so different than in present-day concepts of economic democracy and automation, in which in advanced technological society it is argued individuals should be increasingly afforded the freedom from necessary labour in order to pursue the many positive possibilities that life has to offer, including education and learning.

Reading his essay, I was reminded of a few historical examples. Think, for instance, about the development and evolution of writing and of our early mathematical ideas – a history that is intimately entwined with the genesis of civilisation. A good example comes from the ancient Babylonians. To Russell’s larger point, the early development of mathematics, much like writing, can be seen to be owed to the economic development of agriculture; because with agriculture one result was increased freedom from the precariousness of sustenance living in which people were then allowed more free time, with greater access to resources. As new technologies were conceived, and human pursuit was increasingly freed from the limits of basic survival to expand beyond that which was unavailable to hunter-gathers, the time available to explore, experiment with, and create things like writing became possible. The study of mathematics could also be pursued and formalised.

Indeed, to offer another example, the entire history of physics is riddled with such stories, like Michael Faraday playing with his magnets on a park bench in London or Issac Newton watching apples fall from trees, contemplating the nature of gravity. To the point of anthropologists and biologists who study human play, as another example that we may interpret in the frame of idleness, there is an argument to be made that what Russell is describing is in fact a fundamental biological and cognitive feature of universal human experience that is very much tied to inventiveness.

At this point, we may enter into various complex social, economic, and political arguments. Instead, as there are already many terrific reviews of Russell’s essay, both positive and critical, to close this discussion I instead want to focus on two things that struck me when recently rereading ‘In Praise of Idleness’. One playful thought was the potentially interesting applications in relation to a physics of society and of human beings, particularly regarding energetics. This has to do with the study of energy under transformation, and one may think of such transformation particularly between the individual and their labour under the fairly universal economic notion of the work-leisure trade-off. For the author, he argues that there is a sort of fetishisation of labour, especially manual labour, and he seems to want to argue that how we use labour energy is not efficient or optimised in the best ways. From the standpoint of a physics of humans and of society, it would be fascinating to see if some of his ideas are quantitatively grounded.

There are also many interesting economic points of consideration. First, it is worth noting that the contents of ‘In Praise of Idleness’ remain quite relevant today, given the resurgence of the idea of a shortened work week, especially in the UK and Europe. Some would argue that there is empirical evidence and many qualitative arguments about why the current configuration of work hours is not optimised for the benefit of both productivity and well-being [3, 4], supporting his view. Take a quantitative and qualitative view: work hours, commuting time, modern pressures of digital communication in which it is well studied that people also now routinely answer work emails in their leisure time – all of this and more matches data that substantiates the claim of an emerging culture of longer working hours [5]. Are the effects, psychological or otherwise, just as Russell observed or predicted?

On the other hand, inasmuch that the philosophical idea of idleness is tied with the economic argument of a shortened work week, how economically substantiated and viable is his argument? Some examples are as follows. If as a general rule of labour economics working more hours correlates with higher hourly rates of pay, and if as a general rule from a behavioural perspective higher rates of pay are one motivation for people in their social and economic life, then one may ask whether an economic conception of idleness is realistic. For instance, if the introduction of a shortened work week were to correspond to a cut in pay, would people be dissuaded to pursue the possibility of increased free time for the benefit of obtaining greater earnings? As this is a question about human behaviour and behavioural regulators, and hence agency, it is not so easy to model. Having said that, we observed major strikes by German steel workers in 2018 that saw them secure the right to work less at the cost of a drop in weekly earnings – although this also came with flexibility where workers may work longer hours if they choose. Perhaps agency and choice matter in this discussion.

Another point one may consider is that some economists argue that a shortened work week will likely result in an increase in earnings differentials and inequality. If, in general, those who work longer hours have higher hourly earnings than those who work shorter hours, then one would expect increased disparity in the earnings structure. Additionally, in a UK study of the public sector, a shortened work week was approximated to cost upwards of £45 billion, depending on some modelling assumptions including no increase in productivity [6].

For these reasons, when it comes to recent debates in the UK, should a shortened work week be considered some studies have shown that this reduction in time would need to be matched with an increase in productivity during work hours. There are some empirical examples where businesses that trialled shortened work hours saw productivity remain as it was or effectively increase. Although the sample is small, the argument here is that work hours – maximal output of energy during those hours – is better optimised and maintained when shortened and focused. This ties into arguments about the inefficiency of work hours within the current model – that, in the sense of Russell’s energy economics, maximum productivity and the maximum time of energetic labour – i.e., maximum labour hours – do not contradict an increase in leisure. This is partly why I think a physical theory would be interesting, if we could even construct the appropriate Hamiltonian. In empirical sociology, observations of phenomena like ’empty labour’ may also serve as an illustration of what some interpret as the outdated nature of present economic values and of modern conceptions of work [7]. Do these types of studies offer clues or evidence as to how and why economy may be reconfigured in ways in which Russell seems to indicate? It would furthermore be interesting to learn, in using separation theorem or something similar in the study of labour economics [8], whether energetic leisure serves as a positive argument in the utility function of the individual.

The problem when it comes to these sorts of economic ideas and debates is that, in many cases, we require much more accurate modelling. Current mainstream economics is quite inept at understanding the reality of human behaviour. If one considers the likes of Paul Romer’s contentions on macroeconomics (as well as notable research by many other contemporary economists), it is not controversial to say that the current economic model and its established ideas may be challenged quantitatively and qualitatively [9]. From what I can see at the present time, some arguments are emerging about the need for an interdisciplinary theory. Much like a physics of society, in which it has been suggested that a physical theory of society will not achieve systematic and objective clarity without an interdisciplinary form of research [2], in economics agent-based models are issuing similar demands. If the challenge of an objective economics is to look for the cause of instabilities inside the system, some argue that this means that what one inevitably comes up against are the details of human decision making, which, in principle, drives one toward the randomisation of decisions based on both rational and irrational processes. But it also seems more than that: it’s about thinking systemically – not just about economic models in the abstract sense but also the incentive structure and the problem that economics faces in terms of an orientation of ethics. A trivial example is as follows: if a model fosters the pathology of a simplified self-preservation worldview, and if I am one of the only two bakers in town, am I not incentivised in some way to run the other baker out of town by whatever means justified by that very principle of my own preservation? The point to be drilled into is that in social-economic modelling, simplified arguments and narratives about agents engaging in free or purely voluntary trade can, and often do, end up moralising what are otherwise deeply systemic issues. I think, in certain respects, this takes us some way toward the message in Russell’s essay about realistic economic models.

Given the transformation of the incentive structure, perhaps energetic and thoughtful leisure would be realised as an important feature of a healthy system. In terms of Russell’s arguments, framed in a systems way, the benefits would be in reducing the social deficit of reason by maximising the subject’s energetic capabilities to reason, in which education may then be ‘carried farther than it usually is at present’, fostering the provision of ‘tastes which would enable a man to use leisure intelligently’. As I read it, his argument implies the enlightenment ideal that the individual would be better scientifically informed (eg., against myths); they would potentially be better politically informed about policies and more engaged when fulfilling their democratic duties; they would make thoughtful economic decisions; and, perhaps ideally, they would approach social debates with greater consideration and in greater awareness of their own biases.

References

[1] Josef Pieper, 1999, ‘In tune with the world‘. St. Augustines Press.

[2] Guido Caldarelli, Sarah Wolf, Yamir Moreno, ‘Physics of humans, physics for society’. Nature Physics Volume 14, p. 870. DOI:10.1038/s41567-018-0266-x.

[3] Will Stronge and Aidan Harper (ed.), ‘Report: The Shorter Work Week’ [http://autonomy.work/wp-content/uploads/2019/01/Shorter-working-week-final.pdf]

[4] Lord Skidelsky, ‘Report: How to achieve shorter working hours’ [https://progressiveeconomyforum.com/wp-content/uploads/2019/08/PEF_Skidelsky_How_to_achieve_shorter_working_hours.pdf]

[5] Peter Kuhn and Fernando Lozano, ‘The Expanding Workweek? Understanding Trends in Long Work Hours among U.S. Men, 1979-2006’. Journal of Labor Economics, 26 (2) April 2008: 311-43.

[6] Centre for Policy Studies, ‘The Costs of a Four-Day Week to the Public Sector’ [https://www.cps.org.uk/research/the-costs-of-a-four-day-week-to-the-public-sector]

[7] Roland Paulsen, 2014, ‘Empty Labor: Idleness and Workplace Resistance’. Cambridge University Press.

[8] Daron Acemoglu and David Autor, ‘Lectures in Labour Economics’ [https://economics.mit.edu/files/4689]

[9] Paul Romer, 2016, ‘The Trouble with Macroeconomics’. [paulromer.net/the-trouble-with-macro/].

**Cover image: ‘Woman Reading in a Landscape’ by Jean-Baptiste-Camille Corot.

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Physics Diary

A Year in Review

Hello everyone! Today’s post will be different than the usual string theory focused engagements. Normally I would be planning to write a new entry explaining a piece of computation, uploading a string note from one of my notebooks, or organising an essay on an important physics topic. However, I have been so busy with my research that there just has not been enough hours in the day to maintain a constant flow of posts. This should change soon, and I am happy that I already have a lot written and waiting to be edited. The real difficulty has to do with the fact that I don’t like clickbait articles and I have no interest in providing watered down popular guides. The goal is to contribute to making complex subjects like quantum gravity accessible without losing conceptual and technical detail, given that accessibility here also implies an engaged reader wanting to study and understand the subject at hand. The thing about string theory is that it demands one’s full attention. If I am to maintain a research and general string theory blog I would prefer that every entry, whether based on textbook content or frontier research questions, is reasonably substantial and certainly thorough so that it may be beneficial to the reader. I’ve had some great feedback on my articles and notes so far, which I have found both affirming and motivating. I’ve also received some nice feedback about the odd personal post. Slowly over time the process of sharing more personal updates and keeping a personal physics diary is something with which I have become more comfortable. I thought that in this post it might be nice at the turn of the calendar to write about the last year – a year in review of sorts.

It is actually fitting that I would write such a post on this of all days. It so happens that this morning I received a formal PhD offer! Moving on to a PhD is a reality to which I have been orientating myself for some time. But no matter how many times I have thought about it and have tried to prepare for it, especially in terms of my current research focus, when I received my offer this morning it still felt as though everything was happen very quickly. Objectively, I suppose a lot has happened at a relatively rapid pace. Just last year I was an undergraduate being academically accelerated to a full-time research year. I am now only a few months into that research year, planning my MRes dissertation and celebrating the fact that I have been formally offered a PhD position in quantum gravity.

In the time between first arriving at the University of Nottingham for the start of the 2018/19 academic year and the present, I couldn’t possibly list all of the things that I have studied. It has been intense. It started with a complete and comprehensive review of string theory, rederiving the whole of bosonic string theory for my own notes before moving on to superstring theory. In just that time I also taught myself conformal field theory, I had to brush up on quantum field theory, and I had to learn an assortment of important tools ranging from BRST quantisation and the Faddeev–Popov approach to computing string scattering amplitudes, learning about string compactifications, and then trying to cram everything I could about orbifolds and D-branes. Then, as we pushed further into 2019 I moved to superstrings and string geometry while also learning long lists of other physical concepts and mathematical tools in addition to continually working to sharpen my existing knowledge. But what stands out the most about the last year is the Swampland – in fact, I think for me it was the year of the Swampland. It is an absolutely fascinating space of research and I very much enjoyed my time in Spain attending a PhD summer school reviewing things like the Distance and Weak Gravity Conjectures. What also stands out from that first arrival at university to the present time is learning Calabi-Yau manifolds and related geometry; pure spinors, which I started studies while at the summer school in Spain; and then last autumn having to catch up on advanced gravity theory and the braneworld formalism. And now here we are with all of this stuff and more as I work to learn non-commutative geometry and contemplate the nuances of string de Sitter solutions.

Although this really only scratches the surface of an entire year, these descriptions provide some sense of range. I am by no means an expert in any of it, to be frank. Going back to some old calculations I often need to remind myself of certain first principle definitions, like when recently uploading my CFT notes. Typically, it seems like a day is equivalent to a week, as there is just so much to learn. Of the material covered so far and of all the concepts, tools, and theorems studied I can say that one thing I’ve learned is that, in terms of a definitive and coherent theme to research in string / M-theory, in this field every day there seems a new mountain that needs to be climbed. A few days ago it was more to do with gauge-gravity duality and matrix models, which I am learning. Today, it is ‘axilaton’ models. No matter how many mountains one seems to climb, the nature of frontier physics research seems to guarantee tomorrow there will be many new summits.

The last year has also been a momentous one for me personally. Having Asperger’s (Autism Spectrum Condition), which has been described clinically as severe, I experience many challenges in basic day to day life. This also includes assistance with functioning and needs. Now being a formal member of university also presents many additional challenges. What I can that also stands out about 2019 is that I am also ever so grateful to be at a school with tremendous support staff. Being able to participate in a formal academic institution, thanks to the support I receive, has opened up so many new opportunities that would otherwise not have been possible. It has probably been one of my greatest years. Growing up with little support, perhaps this story will serve as an example of how important proper support is for people like myself – or anyone for that matter – to succeed. It reminds me that in the future I would like to write more about living with Asperger’s. It is something I have tried to write about in the past, however successfully or unsuccessfully. But I think the message is also more general – everyone needs support to be themselves, to pursue their interests, and ultimately to self-actualised in a healthy and positive way.

Thinking about the future is something I find difficult. In moving from the past to the present, I’m not quite sure how to project forward in time. What I know for certain is that the next year should be a productive one, given the current trajectory. At present I am planning my MRes dissertation and thinking about possible PhD projects, with the troubles of string de Sitter vacua very much an interest. As I have written before, non-perturbative theory seems to be my point of entry into string research, instead of computing scattering amplitudes or focusing on SCFTs for example. I am thankful to be working under my brilliant professor, Tony Padilla, who is encouraging in this regard and also with my other interests, such as for instance exploring non-geometric backgrounds and matrix models. Every discussion we have is a stimulating, and I enjoy going to his office every week with new ideas to share. Non-geometry will be a lasting topic. One motivation for it, of course, has to do with the no-go theorems for supergravity, which, in turn, relates to questions about the sigma-model prescription that gives geometric vacua. Moreover, there are a number of suggestions in string / M-theory that a perturbative string vacua will not be geometric in the typical sense. Instead, it will be non-geometric. What this means, and to explore some of the mathematical/physical intuition as to why we might think about non-geometric vacua, I think this would make for a terrific future post. Additionally, if a further consequence of strings is that geometry and gravity may even be emergent concepts, and that there is some hint at possibly the idea of non-commutative (non-associative) theory of gravity, then I think another principle of direction is to try and investigate how these are related. We could also ask about how non-perturbative vacua and non-geometric vacua are related, if in fact a formal relation may be defined.

These sorts of topics and questions I suspect will define much of my research year in 2020. But, then again, with each new mountain there sometimes also appears an exotic new valley waiting to be explored. I think I shall take it day by day.

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