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The Poincare group represents the isometries of Minkowski spacetime and is a ten-dimensional manifold. String theories predict that the universe is a ten-dimensional manifold.

Question: Is this a coincidence? Or is it the main idea why physicists began to consider the possibility that the universe might be ten-dimensional?

No rigorous justification of your answer is necessary for me, since I don't think I have sufficient background to understand it. Answers linking to papers or citing references might be better received by other people who view this question and actually understand the answers, however.

Also I know that M theory is considered more up-to-date than string theory and it predicts an 11-dimensional universe (or at least one dimension higher), but my understanding was that M theory is a unifying framework for competing string theories, so basically one needs to understand string theory to understand M theory, I don't even understand string theory, so I don't want to worry about that right now. A yes/no answer to the ten-dimensional coincidence would be more than adequate for me right now. Perhaps keep in mind that other viewers will have different criteria.


Background/context: I was thinking about how the fact that O(3) has two path components actually corresponds to real-life geometric observations about rotations, reflections, and orientation. Then I thought "too bad it is a subset of 9-dimensional space and thus doesn't have any physical manifestation". Which led to the thought -- maybe it does, since string theory predicts that we live in more than three dimensions.

(It turns out this line of thinking was dumb anyway though since apparently SO(3) and presumably also O(3) is a three-dimensional manifold, so it actually wouldn't that be far-out to imagine that we lived in the tangent space of it, but whatever.)

Anyway I remembered that SO(3) wouldn't be the appropriate model to consider, since physical theories postulate that we actually live in four-dimensional spacetime, rather than three-dimensional spacetime, so I should look for the dimension of the analogous matrix group for "rotations" or "symmetries" for spacetime. I thought it was the Lorentz group (which doesn't work because it only has six dimensions) but Wikipedia corrected me and it turns out it is the Poincare group, which apparently is ten-dimensional -- just like string theories postulate for the universe.

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    $\begingroup$ Two possible sources : Google modular functions string theory and wikiversity string theory, which goes into the math a bit. As far as I have read, I never seen a connection between the Poincaré group and string theory dimensions. Bosonic string theory has 26 dimensions, as you probably know. $\endgroup$ – user108787 Aug 7 '16 at 2:44
  • $\begingroup$ Yeah I just saw that (26 dimension number) on some linked Physics.SE questions. Those resources look interesting -- I'll have to look into them further. Thank you for the recommendations! $\endgroup$ – Chill2Macht Aug 7 '16 at 2:55
  • $\begingroup$ Glad to help, it's Ramanujan's (who else :) modular functions $\endgroup$ – user108787 Aug 7 '16 at 3:05
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The quickest way to see the origin of the various dimensions that appear in string theory is to use the Green-Schwarz action functionals for super $p$-branes, including the Green-Schwarz superstring ($p = 1$):

these are the sigma-model action functionals for $p$-branes with manifest spacetime supersymmetry. They are secretly (see here for references) WZW-type models (or rather: higher dimensional analogs) for the spacetime supersymmetry algebra $\mathrm{Iso}(\mathbb{R}^{d-1,1\vert N})$ (the super Poincaré Lie algebra), and as such they exist only if a Lorentz-invariant $(p+2)$-cocycle exists on this super Lie algebra (the term responsible for $\kappa$-symmetry).

These cocycles may be classified, and they exist only for a finite number of combinations $(d,N,p)$ of

  • spacetime dimension $d$;

  • number of supersymmetries $N$ (i.e. choice of real spin representation of $\mathrm{Spin}(d-1,1)$);

  • brane worldvolume dimension $p+1$.

For $p$-branes without tensor multiples on their worldvolume (i.e. without gauge fields on their worldvolume) this classification is called the brane scan due to

Anna Achúcarro, Jonathan Evans, Paul Townsend, David Wiltshire, "Super $p$-Branes", Phys. Lett. B 198 (1987) 441 (spire)

and rigorously established in Brandt 12-13.

For fixed $N = 1$ it looks as follows (taken from Michael Duff, "Supermembranes: the first fifteen weeks" CERN-TH.4797/87 (1987) (spire)):

You see from this diagram how the critical spacetime dimensions of superstring theory arise: $d = 10$ is the maximal dimension in which there exists a spacetime-supersymmetric Green-Schwarz superstring, while $d = 11$ is the maximum dimension in which there exists a spacetime-supersymmetric Green-Schwarz supermembrane.

Incidentally, while this gives the spacetime dimensions, as a scan of all branes this scan is incomplete: it misses the D-branes and the M5-brane (which are the fundamental branes with gauge fields on their worldvolume: the D-branes carry the Chan-Paton gauge fields induced from the open strings ending on them, while the M5-brane carries a higher gauge field induced from the open membranes ending on it.)

In order to also see these branes with vector/tensor multiplet, one has to classify not just supersymmetry Lie algebras, but supersymmetry Lie $n$-algebras (Lie $n$-algebras/$L_\infty$-algebras in the sense of Stasheff, not $n$-Lie algebra in the sense of Filippov). This is discussed in

Domenico Fiorenza, Hisham Sati, Urs Schreiber, "Super Lie $n$-algebra extensions, higher WZW models and super $p$-branes with tensor multiplet fields", International Journal of Geometric Methods in Modern Physics Volume 12, Issue 02 (2015) 1550018 (arXiv:1308.5264).

The resulting classification then looks like a tree, the "Brane bouquet"

Each item in this bouquet represents one super Lie $n$-algebra, and each edge represents a higher extension by a super Lie $n$-algebra $p+2$ cocycle. The name of the super Lie $n$-algebra that the edge a starts at is that whose Green-Schwarz WZW-term is the cocycle represented by the edge, and the edge ends on the brane species which may end on the former. For instance the edgre from the super Lie $n$-algebras of the $D$-branes to those of the type II string means that type II strings end on D-branes, while the edge from the super Lie $n$-algebra of the M5-brane to that of the M2-brane means that M2-branes may end on M5-branes. Notice that this is the physical interpretation, the diagram itself arises simply from classfifying super Lie $n$-algebra cohomology.

It is not only the spacetime and brane worldvolume dimensions which are exlained this way by super Lie $n$ algebra theory, also much of the core structure of string/M-theory is govered by these algebraic structures. For instance the duality between M-theory and type IIA strings may be read off Lie $n$-algebraically, this is discussed in

Domenico Fiorenza, Hisham Sati, Urs Schreiber, "Rational sphere valued supercocycles in M-theory and type IIA string theory" (arXiv:1606.03206)

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The answer is most likely no: although string theory requires some sort of critical dimensions, they usually come in by requiring some quantum anomalies to vanish. For example, for bosonic strings propagating in flat space, the vanishing of Weyl anomaly, which is defined as the expectation value of the trace of energy momentum tensor, suggests that the dimension should be 26, although it could be modified if one considers strings propagating in curved backgrounds.

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    $\begingroup$ The question is – why do we require the conformal anomaly to vanish? This isn't a question of mathematical consistency – CFTs with central change $c \neq 0$ are consistent. With string theory, anomaly vanishing is required due to physical reasons – massless states in noncritical dimension don't form Poincare (or rather the little group) multiplets. $\endgroup$ – Prof. Legolasov May 4 at 5:57
  • $\begingroup$ @SolenodonParadoxus Indeed you can have $c\neq 0$ CFT but string theory is not just a 2d CFT. It's a world sheet quantum gravity. So by world sheet diffeomorphism invariance you need $T_{\mu\nu}=0$ classically and $\langle T_{\mu\nu}\rangle=0$ quantum mechanically. In particular the Weyl anomaly must vanish. $\endgroup$ – user110373 May 6 at 2:47
  • $\begingroup$ Please elaborate. First, I assume that you mean the matter+gravity stress-energy tensor, because otherwise your claim is wrong (usually by "stress-energy tensor" we mean only the matter part of it, which is not equal to $0$ – this is the standard terminology in General Relativity). Is that correct? Second, please explain why $\left< T_{\mu \nu} \right> = 0$ implies $c = 0$. I think that's incorrect. As a counterexample, consider noncritical string theory. Its path integral also includes gravity, and it is well defined, rendering a Liuville + matter CFT with $c \neq 0$. $\endgroup$ – Prof. Legolasov May 6 at 3:25
  • $\begingroup$ @SolenodonParadoxus In 2d $G_{\mu\nu}=0$ so classically you must have $T_{\mu\nu}=0$. This is nothing but saying gravity is topological in 2d. The quantum version can be seen by changing $g_{\mu}\to g_{\mu\nu}+\delta g_{\mu\nu}$ and consider its effect on $\langle 0|0\rangle.$ In non-critical string the total central charge $c_{Liouville}+c_{matter}+c_{ghost}$ vanishes. $\endgroup$ – user110373 May 6 at 3:53
  • $\begingroup$ You're probably right, because I've never heard of this before. Why is $G_{\mu \nu}$ identically zero in 2 dimensions? $\endgroup$ – Prof. Legolasov May 6 at 5:59
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It might. The Poincaré group can be generalized in the Bacry Lévy‐Leblond classification (https://aip.scitation.org/doi/10.1063/1.1664490) -- a 3 parameter family of symmetry groups -- that includes the Galilei/Bargmann groups, [anti-]Newton-Hooke, [anti-]deSitter, and many others. To consistently unify this with the central extension (Bargmann) of Galilei requires also adding in an 11th dimension. At this point, you begin to get deja vu.

With this extension, the Poisson-Lie manifolds of each member of the family can all be unified into a single Poisson-Lie manifold that contains a 3 parameter layering of the submanifolds. Within each layer is a 11D quadratic form (that for special cases produces quadratic forms of 3, 4 or 5 dimensions) and induces an reduced 10D quadratic form in a natural way.

The Pauli-Lubanski 4-vector of Poincaré generalizes to a 5-vector that is also associated with a quadratic form, whose signature may vary between 5, 4+1, 3+2 (the anti-deSitter case) as well as singular cases 3+1(+1), 3+0(+2) and others.

One of the parameters controls the c → ∞ (in which time becomes absolute), another controls the c → 0 (in which space becomes absolute) -- both can actually be enacted together (the results include the Static Group which has the same central extension as the Carroll group, one of the c = 0 cases). The third parameter controls the curvature of the spatial geometry and/or temporal geometry (only the latter, in the cases of [anti-]Newton-Hooke). Setting it to 0 produces the 5 "flat space" groups (Poincaré, Galilei/Bargmann, Carroll, Euclid-4D, Static).

The Cartan-Maurer equations yield a system of equations for 11 1-forms (10 in the reduced setting). In the flat space limit this yield equations for 5 "frame" 1 forms and 6 "connection" 1-forms. In the reduced 10D cases, the 5 frame 1-forms reduce to 4, which can be identified as the frame 1-forms of a Riemann-Cartan geometry.

In other replies, we've already seen an independent account for how the 10 and 11 dimensions arise. This does not rule out the possibility that a correspondence may exist between this and what I just described.

I suspect there is a correspondence here awaiting discovery. If I knew more about string theory, I would look for it, myself; and draft some kind of Rosetta stone to link and systematically line up the two descriptions side-by-side.

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