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The Calabi-Yau manifold is a stable complex 3D (or real 6D) manifold on the geometry of which information about strings can be stored as fibre bundles of tensors, more or less like the electromagnetic four-vector can be seen as a vector on the small circle in Kaluza-Klein theory (corresponding to $\operatorname{U}(1)$ symmetry), represented in the ecxtended 5d metric tensor as eight extra symmetric off-diagonal terms.

The Calabi-Yau manifold incorporates the all three gauge symmetries in particle physics, i.e. $\operatorname{SU}(3)\operatorname{SU}(2)_{l}\operatorname{U}(1)_{Y}$. These gauge symmetries correspond to Lie groups and gauge fields. The gauge transformations leave spacetime untouched, unlike the the gauge transformations used to derive general relativity (corresponding to the Poincaré Poincaré group) which only touch upon spacetime itself (the symmetry demand being obviously that spacetime transformations leave the GR defined Lagrangian unchanged), so not an a particle field in spacetime.

How is this symmetry incorporated in the Calabi-Yau manifold, and how does that introduce graviton modes?

Samuel Adrian Antz
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Gerald
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    Why would it need to be "incorporated into the Calabi-Yau manifold"? The 10d theory you're compactifying is already a theory of (super)gravity. – ACuriousMind Jul 14 '22 at 17:14
  • @ACuriousMind The extra compactified dimensions can be associated with the three basic forces. Say a certain string mode corresponds to electric charge only, so we have an electron. The string vibrates in this way because of the shape of the manifold. How is the charge "mass" induced by the shape of the CY manifold? – Gerald Jul 14 '22 at 17:34
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    It isn't, and very little of what you are saying is compatible with my own technical understanding of string theory: For instance, non-Abelian Yang-Mills type gauge theories come from (coincident) D-branes, not from a string "vibrating because of the shape of the manifold", and for non-Abelian gauge enhancements we typically want singularities (so no smooth CY). – ACuriousMind Jul 14 '22 at 17:43
  • @ACuriousMind You mean intersecting branes between which strings can be attached? I'm a bit confused how a vibration causes mass. – Gerald Jul 14 '22 at 17:48
  • In general relativity on a spacetime manifold, the gauge group for gravity comes from the structure constants $f_{ab}^c$ the commutators $£{\xi_a}\xi_b=f{ab}^c\xi_c$ for a basis ${\xi_d|d\in I}$ ($I$ an index set) of the manifold's Killing vector fields. I'd be interested if e.g. @ACuriousMind shows how this relates to the situation in string theory. – J.G. Jul 14 '22 at 17:49
  • No, I'm saying that's where the "non-mass" gauge theories like $\mathrm{SU}(3)$ come from. Again, mass/gravity is already there in the 10-dimensional theory and does not need to arise from the compactification, which is why I don't understand the premise of this question at all. Can you give any reference that claims that gravity has to arise from the compactification? – ACuriousMind Jul 14 '22 at 17:50
  • @ACuriousMind I recently read a question here about a stack of N branes of which M are rotated through the same angle. The U(N) symmetry gets broken into U(N-M)×U(M) and the strings are attached between them. This would be similar to Higgs. Does this mean mass is already inherent to the CY? – Gerald Jul 14 '22 at 17:57
  • @ACuriousMind But if gravity is already there what does it mean that string theory automatically generates gravity? – Gerald Jul 14 '22 at 18:00
  • @ACuriousMind Ah. It's already there in the non-compactified 10d space? – Gerald Jul 14 '22 at 18:50
  • Extra dimensions in string theory aren't used in a Kaluza-Klein way. More precisely, Kaluza-Klein mechanism can occur in string theory, but it isn't used to explain the known forces. In Kaluza-Klein, symmetries (isometries) of the compact space become gauge symmetries. Calabi-Yau doesn't have such symmetries - main phenomenological significance of the Calabi-Yau is its topology - and the gauge forces come from open strings on D-branes, or from gauge boson states of the closed heterotic string. – Mitchell Porter Jul 17 '22 at 19:13
  • @MitchellPorter Is a brane made of the same "stuff" as the string or is it a space brane? – Gerald Jul 18 '22 at 00:54
  • Branes can carry some extra charges but are just as physical as strings. – Mitchell Porter Jul 20 '22 at 04:40
  • @MitchellPorter Charges like electric or color charge? – Gerald Jul 30 '22 at 00:07
  • Generalizations of these called p-forms (p-dimensional differential forms) – Mitchell Porter Jul 30 '22 at 01:09
  • @MitchellPorter Which means new gauge fields? Are branes like space or do the exist in space, like strings? – Gerald Aug 01 '22 at 08:52
  • At our current level of understanding, you may as well think of them as existing in space. But in string theory, space itself is not necessarily fundamental (e.g. emergent dimensions of space in AdS/CFT or "M(atrix) theory"), so the ultimate nature of strings and branes may be something other than "existing in space". – Mitchell Porter Aug 02 '22 at 03:49
  • @MitchellPorter Which means that space is "build" from strings or branes? – Gerald Aug 02 '22 at 07:41
  • Or maybe strings, branes, and space are all built from something else! Maybe see "matrix models" – Mitchell Porter Aug 05 '22 at 02:51
  • @MitchellPorter Maybe space is constituted by non-local hidden variables... Who knows... – Gerald Aug 05 '22 at 04:23

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  1. The Yang-Mills type gauge groups of an effective theory in string theory do not arise solely from the compactification, but from "wrapping" D-branes (to which open strings can be attached) around non-trivial cycles in homology - that's the connection to the geometry of the compactification manifold. Non-Abelian gauge groups arise from coincident D-branes (for a discussion of why, see this question and its answers, and more generally this phenomenon is known as gauge enhancement), which in the geometry are typically modelled by taking some limit in which the compactification manifold becomes singular so that some of the distinct cycles "merge" at the singular point and the branes become coincident. Blowing up the singularity (i.e. returning to the non-singular version) corresponds to symmetry breaking (this is mostly an M-theory viewpoint).

  2. In contrast, gravity - which as you say is not gauge theory like the others, for an exact discussion of what is "gauge" about gravity see this answer of mine - is not generated by D-branes or singularities or anything like that: The 10-dimensional uncompactified low-energy effective description of string theory/M-theory, i.e. the theory that is being compactified when we talk about Calabi-Yau manifolds and whatnot, is already a theory of supergravity, because the spectrum of states of the superstring already contains a massless spin-2 particle, and by arguments very similar to how massless spin-1 bosons are always associated with Yang-Mills type gauge theories (see this answer of mine and Weinberg's book for the full argument for spin-1 and this answer and its links for the spin-2 argument), such a massless spin-2 boson is always the graviton of a theory that looks like gravity.

So there is no need to get gravitons from the compactification - they are always there from the outset (and this is arguably the reason people started thinking about string theory as a potential theory of quantum gravity in the first place after it was initially designed to explain QCD).

ACuriousMind
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    But how can there already be a spectrum if states in a non-compactified 10d space, in which there are no string states corresponding to other particles yet? Because it's a closed string state, which the graviton is? Or because of the branes in it? – Gerald Jul 14 '22 at 18:43
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    Is the connection with QCD the elastic strings that were envisioned between quarks? Surely these are no gravitons yet. – Gerald Jul 14 '22 at 19:00
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    @Gerald If you think that there is no spectrum in non-compactified 10d space, you're simply wrong. Any introductory text on string theory should derive both the non-compactified open and closed string spectrum. Can you give any references for the claims you are making? – ACuriousMind Jul 14 '22 at 20:15
  • "Any introductory text on string theory should derive both the non-compactified open and closed string spectrum" So in 10d non-compactified space there is a spectrum? – Gerald Jul 15 '22 at 08:10