Thread by @Hassaan_PHY

This is a brief post about nongeometric backgrounds in string theory. We will define these terms precisely in the following but naively, these can be thought of as "spaces" on which a string moves that don't have a "geometric" interpretation. 1/n
#Physics #scicomm

A basic understanding of field theory is required. I'll try to make it as accessible as I can (without distorting the results).
In QFT, the word "background" usually means a set of fields that are present in the theory but whose dynamics aren't part of the theory. 2/n

For example, we can talk about a fermion (described by the Dirac lagrangian), moving in a background electromagnetic field (I'll call it EM field from now). We can also add interactions of this fermion with this background field. 3/n

However, in this theory, we don't include the dynamics of the EM field itself (which is done by including the kinetic terms for this field in the Lagrangian). In contrast, in QED, the EM field isn't a background field because its dynamics are part of the theory. 4/n

In string theory, by "background", we mean a set of 3 fields & these 3 fields are the metric tensor (called G), a field called the Kalb-Ramond field (it's an antisymmetric tensor field and is referred to as B field) and a scalar field called the dilaton field (called Φ). 5/n

So, a background is defined as the trio (G,B,Φ).
For the readers who want details, the B field can be thought of as the electromagnetic field for the strings except that instead of being a vector field, it is a second-rank tensor field. 6/n

It turns out that it has to be the case because the string is an extended object. The electromagnetic field for an extended object with p spatial dimensions will be a (p+1)-rank tensor (totally antisymmetric). 7/n

Two side comments here. Firstly, these background fields can be thought of as produced by strings themselves. So, these aren't something that we put in by hand. 8/n

Secondly, in superstring theory, there are other fields that come into play called the RR fields. We won't talk about them here. (G,B,Φ) define a background in the absence of RR fields. 9/n

Before describing the conditions under which such a background is "non-geometric", we have to talk about string dualities. If a transformation keeps the action of the theory invariant, we call it a symmetry of the theory 10/n

However, a transformation that changes the action but "describes the same physics" is a duality transformation. Describing the same physics naively means that the spectrum (the set of states of the theory) isn't changed. 11/n

In string theory, there are a lot of dualities but the duality that we will talk about is T duality. Suppose there is a compact direction in the space on which a string theory is defined (i.e. the space contains a circle) with a radius R. 12/n

T duality relates physics described in such a space with physics described in a space where this radius is 1/R.
Now, in literature, there are different meanings of a background being non-geometric. 13/n

Some say that a non-geometric background is a background that can't be described as a manifold with a well-defined metric (i.e. not describable in terms of Riemannian geometry). Let's call this definition d1. 14/n

Some say that a nongeometric background is a background where T duality transformations are required to make the background well-defined (we will mention what this means later). Let's call this definition d2. Some people employ a stricter definition as follows. 15/n

When we apply T duality to a background (G,B,Φ), it changes in a non-trivial way to another background (G',B',Φ') called the T-dual background of (G,B,Φ). 16/n

Now, some say that a background is nongeometric if T duality transformations are required to make the background well defined but it is not T-dual to any background that can be described in terms of Riemannian geometry. Let's call this definition d3. 17/n

We now mention what we mean by a "well-defined" background. When we study curved spaces, we don't label the whole space with a single coordinate system. Instead, we employ patches to cover this space and in each patch, we have one coordinate system. 18/n

But these patches can overlap and thus, a single point can be a part of multiple patches and thus, have multiple coordinates.
How are these coordinates related? They are related by something called transition functions. 19/n

For geometric objects, these transition functions are ordinary continuous transformations called diffeomorphisms.
A similar reasoning takes place when in addition to coordinates, we define fields on a curved space. 20/n

They are also defined in patches and at a single point, we might have more than one patch and thus, more than one value of a field. These values are also related via transition functions. 21/n

Now, consider the background (G,B,Φ) defined over a curved space. If the transition functions for these fields are given by T duality transformations, we say that T-duality transformations are required to make the background "well-defined". 22/n

So, are there any examples of such non-geometric backgrounds? I will present just two related examples. A well-known example is a 3-dimensional torus (called 3-torus or T3) with a constant H field and its T-duals. 23/n

H field is the field strength of B field just like F is the field strength of the ordinary EM field. H field is also called H flux. T3 with a constant H field is a geometric background. It is easy to see that T3 has three circles in it. 24/n

We can do a T-duality transformation along one of these circles. Then we get something called a twisted torus and the flux that we get on it is called the f flux. This is also a geometric background. 25/n

The f flux is called a "geometric flux" as it has a geometric interpretation. For interested readers, f flux can be seen as structure constants of an algebra. Now, we can perform a T-duality transformation along another circle and this time, we get something called a T-fold. 26/n

T-fold is non-geometric according to definition d2 above(because we need T-duality transformations to make (G,B,Φ) well defined here) but not according to d3 (as it is T-dual to twisted torus, which is geometric). 27/n

The flux that a T-fold has is called the Q flux and it is a non-geometric flux because it lacks a geometric interpretation.
Now, requiring T-duality as transition functions is a thing that happens when two patches overlap. 28/n

But what about the description within a patch, where there are no overlaps with other patches? There is nothing non-geometric about T-folds within a patch. We say that T-folds are "locally geometric". 29/n

Can we do a T-duality along the last remaining circle? Well, not in the usual way. This is because to perform T-duality along a circle in the usual way, translations along that circle should be an isometry (i.e. the metric should not change under those translations). 30/n

However, the metric G for the T-fold isn't invariant in such translations.
This last T-duality can still nevertheless be performed in an unconventional way but I won't explain that procedure here. 31/n

I will only present some features of the resulting space. The resulting space is called the R-space and it has a non-geometric flux called R flux. 32/n

R space is even more non-geometric than the T-fold because it isn't geometric even locally. Arguments for this statement exist but I will present a simpler (but weaker) argument for this. 33/n

Suppose that there is a D3 brane (which is an extended object in 3-spatial dimensions) that wraps around the original 3-torus. There is a result from the late 90s called the Freed-Witten anomaly which says that this configuration can't exist. 34/n

Now, when we perform T-duality on a brane in a direction parallel to the brane, the brane loses that direction. For example, a D3 brane will become a D2 brane, and so on. 35/n

If we perform T-dualities along all directions of this D3 brane (and thus, all directions of the 3-torus, as the brane wraps the 3-torus) it will become a D0 brane, which is just a point particle. 36/n

It means that this D0 configuration is also not allowed as it is T-dual to a forbidden case. Now, recall that R space is obtained after T-dualizing all directions of the 3-torus (dualizing 1 direction gave a twisted torus, dualizing one more... 37/n

... gave us the T-fold, and dualizing the last one gave us R space). So, effectively what we can say is that D0 branes in R space are forbidden. 38/n

This argument suggests (but doesn't prove) that the notion of a "point" is forbidden (and probably meaningless) for an R space. So, R space is not a geometric thing even locally. 39/n

Interested readers can read arXiv:1811.11203 for a review of non-geometric backgrounds. Original papers include (along with many others) papers by Atish Dabholkar and Chris Hull. 40/n

Their arxiv numbers are arXiv:hep-th/0210209, arXiv:hep-th/0406102, and arXiv:hep-th/0512005. Cheers.

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