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A Case for 14-Dimensions

  1. Dec 19, 2013 #1
    While looking back at some gradings for exceptional Lie algebras, I re-discovered an old jewel of a paper (hep-th/9704054) by Itzhak Bars, known for his work on supergravity and the 13-dimensional S-theory. Essentially, Bars argues that a 14-dimensional theory could lie behind non-perturbative string theory and its dualities. He shows how to embed type A,B,C, heterotic and type-I superalgebras covariantly in the framework of 14 dimensions with signature (11,3) and SO(11,3) symmetry and that in lower dimensions one can embed three sets of 32 supercharges as different projections of 64 supercharges, which form three distinct superalgebras.

    The SO(11,3) symmetry has also been used by Nesti and Percacci in arXiv:0909.4537 in a GraviGUT model, that has ties with Lisi's E8 theory, as can be seen from the grading of E8(-24)'s algebra: 14*+64*+so(11,3)+R+64+14. The gradings clearly shows the 64 supercharges, (11,3) signature symmetry and some extra G2 symmetry. A supersymmetric form of Lisi's theory might very well be equivalent to Bars' 14-dimensional theory that would contain M-theory and F-theory.

    Lisi has also expressed interest in the other non-compact form of E8, namely E8(8) which admits the algebraic grading: 14*+64*+so(7,7)+R+64+14. Itzhak Bars hasn't explored this (7,7) signature theory yet, but in light of E8(C) with grading: 14*+64*+so(14,C)+C+64+14, it should be dual to the SO(11,3) theory in the sense that E8(8) and E8(-24) are just real non-compact forms of E8(C). The complex E8(C) theory, in this sense, would be more fundamental.

    Other gradings of E8 have been used extensively in the construction of 57-dimensional extremal black hole charge spaces in D=3 (e.g. E8(C)=1*+56*+E7(C)+C+56+1) (hep-th/0008063). There, the E8(8) and E8(-24) gradings amount to a choice of split-octonion or octonion variables inside the algebra of complexified octonions. The construction of a Jordan C*-algebra and its extended Freudenthal triple system leads to the full E8(C) symmetry, that is ultimately necessary to accommodate solutions where the E7 quartic invariant takes negative values and leads to non-real values for the 57D conformal invariant.

    Does anybody have any thoughts on such a 14-dimensional theory?
     
    Last edited: Dec 19, 2013
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  3. Dec 19, 2013 #2

    MathematicalPhysicist

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    The final theory will have [tex]\infty[/tex] dimensions anyway...
     
  4. Dec 19, 2013 #3

    tom.stoer

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    What are the physical reasons to go beyond 10 dimensions?
     
  5. Dec 20, 2013 #4

    MathematicalPhysicist

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    We shouldn't go beyond 1+3, but mathematically we can do as we wish, and with no empirical evidence who can stop us?

    In PDEs we also deal with equations with two time variables and even more, like in the ultrahyperbolic pde.
     
  6. Dec 20, 2013 #5

    tom.stoer

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    Of course we can. The question is whether it's physics or pure math.
     
  7. Dec 20, 2013 #6

    MathematicalPhysicist

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    Well, history has it that most pure math is being used in theoretical physics.

    So the distinction between the two in my case is quite blur anyway.
     
  8. Dec 20, 2013 #7

    tom.stoer

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    In history math was related to physics and reality (GR, QM, QFT, ...). This changed with string theory ;-)
     
  9. Dec 20, 2013 #8
    Arguments for M-theory in D=11 (e.g. nonperturbative string dualities, D=11 SUGRA, etc.) are well understood. Vafa's proposal for F-theory in D=12 relies on the observation that the IIB D-string with a U(1) super-Maxwell gauge field on its worldsheet has a critical dimension of 12, as one has to introduce additional ghosts of spin(0,1) which shifts the central charge by -2 giving a total space with signature (10,2).

    Sezgin was able to formulate a super Yang-Mills theory in (11,3) but going past 14-dimensions (e.g. signature (12,4)) yields unwanted contributions to the Yang-Mills field equation and fails to be supersymmetric. Note: Sezgin considered spacetime superalgebras containing a single Majorana-Weyl spinor generator with 2^{n+3} real components and all possible bosonic generators that can occur in their anticommutation relation. The (11,3) superalgebra was found to be maximal in the series: (8,0), (9,1), (10,2), (11,3).

    What makes the story more amusing is that the structure of (8,0), (9,1), (10,2) and (11,3) signature theories (symmetry and supercharges) is already encoded in non-compact forms of E6, E7 and E8. Namely, one has the Lie algebraic gradings (which can be lifted to subgroups):

    E6(-26)=8*+16*+so(8)+R+R+16+8

    E6(-26)=M1,2(O)*+so(1,9)+R+M1,2(O)

    E7(-25)=1*+32*+so(10,2)+R+32+1

    E8(-24)=14*+64*+so(11,3)+R+64+14.

    Usually, these non-compact forms of E6, E7 and E8 are interpreted as U-duality groups for magic supergravities in D=5,4 and 3 dimensions, respectively. However, the U-duality interpretation makes use of different gradings (e.g. in D=3, one uses E8(-24)=1*+56*+E7(-25)+R+56+1 where E8(-24) acts on a 57 dimensional charge-entropy space).
     
  10. Dec 20, 2013 #9

    tom.stoer

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    Yes. I agree.

    But the experimental, physical support is zero.
     
  11. Dec 20, 2013 #10
    Eric Weinstein's mystery theory (hyped in the Guardian by Marcus du Sautoy earlier this year, and later presented in a talk at Oxford) is supposed to involve 14 dimensions. Peter Woit wrote: "The metric tensor is a symmetric bilinear form, so 10 components in 4 d. So, you could make a bundle over your 4d spacetime, with 10d fibers given by the symmetric bilinear forms on the tangent space." My theory about his theory was that it was a 64-dimensional spinor coupled to an SO(14) GraviGUT, with the first two generations coming from the spinor and the third generation from somewhere else (like Lisi, Weinstein has to handwave to get three generations).

    If Bars' 14-dimensional theory has any reality, you could look for a 3-brane in its spectrum, and then try to obtain a Weinstein-like theory as the worldvolume theory of the 3-brane... But I have a question for kneemo. The conventional stringy wisdom regarding the idea of gravity as a gauge theory (a la LQG, etc) is that it only works in the special case of 2+1 dimensions. Do you have some special take on GraviGUTs which is supposed to be consistent with M-theory?
     
  12. Dec 20, 2013 #11
    Sezgin suggested there may be a superbrane with (3,3) dimensional worldvolume propagating in (11,3) dimensions. Vafa's (2,2) worldvolume in F-theory would be recovered from this. In F-theory the (10,2) is connected to the usual (9, 1) reformulation via wrapping of a (1,1) part of the (2,2) worldvolume about a compact (1,1) space, leaving a (1,1) string in 10 dimensions - the type IIB string. With this, one can use T-duality to go from IIB to IIA and S-duality to get to M-theory.

    Bars argued that the 32 supercharges of IIB and IIA are distinct, as projections of the 64 supercharges coming from the Weyl spinor of SO(11,3), coming from a BPS equation. Each projection branch admits compactifications down to 4D with SO(3, 1) Lorentz symmetry and inherited internal symmetries. The kicker is that in D=4 Bars extends the SO(3,1) symmetry to SO(3,1)xSO(1,1) (or another variant, as one is coming down from (10,2) or (11,3) signature) and gets a novel Kaluza-Klein family generation mechanism from the extra (1,1) coordinates. This mechanism might be useful in the GraviGUT approaches.
     
    Last edited: Dec 20, 2013
  13. Dec 20, 2013 #12

    arivero

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    Happy Xmas.

    SO(10) works as the symmetry group of the 9 sphere, so classical Kaluza-Klein for GUT group SO(10) needs only 13 dimensions. I am not sure how many dimensions should a Kaluza Klein with GUT group E6 (which is the greatest group with complex representations as needed for the fermions) have.
     
  14. Dec 20, 2013 #13

    arivero

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    Happy Xmas.

    SO(10) works as the symmetry group of the 9 sphere, so classical Kaluza-Klein for GUT group SO(10) needs only 13 dimensions. I am not sure how many dimensions should a Kaluza Klein with GUT group E6 (which is the greatest group with complex representations as needed for the fermions) have.
     
  15. Dec 20, 2013 #14
    You might be right. It was said that Weinstein's theory is a type of 14D Yang-Mills theory. He may very well be using SO(14,C) and a complex 64-dimensional spinor. This is the structure found in the full complexified form of E8. SO(11,3) and SO(7,7) Yang-Mills theories with real 64-dimensional spinors would be special cases of this.

    As for relating GraviGUTs to M-theory, for now it seems the D=14 SO(11,3) super Yang-Mills theory with real 64 spinor is the closest match.
     
    Last edited: Dec 20, 2013
  16. Dec 21, 2013 #15
    E6 contains all the symmetries of the Cayley plane, so a 16+4=20-dimensional classic Kaluza-Klein construction should suffice.

    A more modern approach would have E6(C) as a structure group of a nonassociative C*-algebraic bundle. The E6(C) gauge degrees of freedom are then interpreted as inner fluctuations of a nonassociative geometry.
     
  17. Dec 22, 2013 #16

    tom.stoer

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    I don't want to be impolite, but I am still interested in physical reasons to consider these models. Which physical problems are solved by these models? And what is the physical reason to go beyond 10 or 11 dimensions?
     
  18. Dec 22, 2013 #17
    When reduced to a 4+2 version of the Standard Model, Bars claims the strong CP violation problem of QCD is resolved (without an axion) and the model predicts a dilaton driven electroweak spontaneous symmetry breaking, which has implications for string theory vacuum selection, dark matter composition and inflation.
     
    Last edited: Dec 22, 2013
  19. Dec 23, 2013 #18

    tom.stoer

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    What does "4+2" mean? Is it a two-time formalism?

    How is a "4+2" version of the standard model related to the "3+1" version which has been verified experimentally?(within the accessable energy range)

    Why do we need a dilaton-driven symmetry breaking instead of the Higgs-mechanism?

    As far as I unterstand you correctly, another benefit of this huge mathematical structure is to get rid of the axion, correct?

    What about experimentally testable predictions beyond the standard model? Which particles and interactions do we have to find in order to prove the model? Which experiments can disprove the model?
     
    Last edited: Dec 23, 2013
  20. Dec 23, 2013 #19
    Yes, it's a two-time formalism with an extra Sp(2,R) symmetry that can be gauge fixed down to 3+1 dimensions without leaving behind any Kaluza-Klein type modes or extra components of vector and spinor fields in the extra 1+1 dimensions. An action in d+2 dimensions requires a dilaton (as a singlet of SO(d,2)) in order to achieve two-time gauge symmetry. (Note: The 4+2-dimensional SM has SO(4,2) symmetry, which interestingly is also the isometry group of AdS_5.). The dilaton couples to the Higgs in a purely quartic potential where electroweak symmetry breaking by the Higgs is driven by the vev of the dilaton.

    The higher structure coming from 4+2 dimensions prevents the problematic F ∗ F term in QCD from appearing in 3+1 dimensions, thus resolving the strong CP problem without the axion.

    Bars' 4+2 SM predicts a Higgs-dilaton mixing. There are regions of parameter space where the mixing is large enough for the rates to diphoton and ZZ to receive observable corrections. Measurements ruling out Higgs-dilaton mixing can disprove Bars' proposed 4+2 model.
     
  21. Dec 23, 2013 #20

    tom.stoer

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    Thanks. For me this is too much speculative input, but now I can make the relation to established physics.
     
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