Tony's Home

The Structure of theMacroSpace ofMany-Worlds

 has both Geometricand AlgebraicAspects,

and Sum-Over-HistoriesPaths are related to PrimeNumbers.

Here is the relevance of theMcKay Correspondence.

The MacroSpace of Many-Worlds is related to Jordan-likeAlgebras and

Bosonic ClosedStrings

Hilbert Space, Octonions,and Cl(8) Clifford Algebra

von Neumann algebras

 What is MacroSpace and Where does it Come From?

Physics models, including the D4-D5-E6-E7-E8VoDou Physics model, have two parts:


In the case of the D4-D5-E6-E7-E8 VoDouPhysics model, the basic ingredients of M are:

The quantum theory MacroSpace Q is then made up of all thepossible M states.

The structure of Q is determined by how those M states (Worldsof the Many-Worlds) fit together.

The structure of Q  has both Geometricand AlgebraicAspects:

The Geometric Structure of Q comes from a chain of Lie Algebras whose groups are the related to Automorphism Groups of Jordan-like Algebras describing the Algebraic Structure of Q:
Groups ---- of -------------------- Jordan-like AlgebrasD4 ---------------- Gauge Bosons --------- Chevalley Algebra Chev3(O)D5 ------ SpaceTime + Internal Symmetry -- Freudenthal Algebra Fr2(O)E6 ---------------- Fermions ------------- Freudenthal Algebra Fr3(O)E7 --------------- Many-Worlds ----------------- Brown Algebra Br3(O)----------------------- E8 ------------------------------------------

248-dim E8 is made up of 133-dim E7 plus 112-dim Br3(O) plusQuaternionic SU(2).

The lowest dimensional non-trivial representation of E8 is its248-dim Adjoint representation, and the algebra E8 corresponds to itsE8 group as Automorphism Group. Therefore

E8 isSelf-Automorphic, and the sequence ends with E8,

which contains both of the

Geometric andAlgebraicdescriptions of the MacroSpace of Many-Worlds


Click on the following links to see about:


For a pdf version of the following material, go to physics/0102042at, or click HERE.

 [[ In this html version, materialadded after the latest version of physics/0102042has been enclosed by [[ ]] double brackets.]]


26-dim Bosonic StringTheory 

The 26 dimensions of Closed Unoriented Bosonic String Theory are interpreted as the 26 dimensions of the traceless Jordan algebra J3(O)o of 3x3 Octonionic matrices, with each of the 3 Octonionic dimenisons of J3(O)o having the following physical interpretation: 4-dimensional physical spacetime plus 4-dimensional internal symmetry space; 8 first-generation fermion particles; 8 first-generation fermion anti-particles. This interpretation is consistent with interpreting the strings as World Lines of the Worlds of Many-Worlds Quantum Theory
[[ The strings are closed because they represent the world-lines of fermion particle-antiparticle pairs ( the pair of fermions acting as a boson so that the entire string is bosonic ) from the time of their creation to their eventual mutual annihilation,
   *    / \   ... /   \ //     |         (The illustrated closed string is red.\     |          It interacts with a partially shown gray string.) \   / \...  \ /    *  
perhaps with lots of interactions with lots of other particles/antiparticles of other world-lines in the meantime, so that part of each string might represent a photon or other particle of any type formed by interaction of one of the particle/antiparticle pair.

Note that since our Universe began with a Big Bang, all its particles originate from pair creation since then. For pairs that do not appear to reconnect for mutual annihilation within the volume of 26-dimensional spacetime being considered in working with the String Theory,

****************    \    ...   /   \      \ /    \      |         (The illustrated string is red.     \     |          It interacts with a partially shown gray string.      \   / \...      A perfect absorber in the future         \ /            is indicated by ******* ).        *  
the string is closed by considering the 26-dimensional spacetime to be a compactified 25+1 dimensional Minkowski spacetime due to considering the Universe to "... be a perfect absorber in the future ...[as in]... the Wheeler-Feynman ... absorber theory of radiation ..." described by Narlikar in his book Introduction to Cosmology (Cambridge 1997) (Section 8.8.1) and related to the Collective Electrodynamics of Carver Mead. For most of the matter in our Galactic Cluster, such an absorber could be a Black Hole of the Black Hole Era. Such a compactification is also similar to the conformally compactified 3+1 dimensional Minkowski spacetime M# used by Penrose and Rindler in their book Spinors and Space-Time, Volume 2 (Cambridge 1986) (particularly Chapter 9). ]]
and the 26 dimensions as the degrees of freedom of the Worlds of the Many-Worlds. 

Note that:

Details about some material mentioned on the above chart can beseen on these web pages:


The following sections are about:

To see some interesting connections among such things as the24-dimensional Leech Lattice and the 256-dimensional Cl(8) CliffordAlgebra, go to click here.

The Lie algebra E6 of the D4-D5-E6-E7-E8VoDou Physics model can be represented in terms of 3 copies ofthe 26-dimensional traceless subalgebra J3(O)o of the 27-dimensionalJordan algebra J3(O) by using the fibrationE6 / F4 of 78-dimensional E6 over 52-dimensional F4 and thestructure of F4 as doubled J3(O)obased on the 26-dimensional representation of F4.In this view, the 26-dimensional traceless subalgebra J3(O)o is arepresentation of

the 26-dim Theory ofUnoriented Closed Bosonic Strings produces a BohmQuantum Theory with geometry of E6/ F4.

In such an interpretation:


Click here to see how the Bosonic StringTheory of 26-dim J3(O)o is related to an M-theory based on the full27-dimensional J3(O).

Click here to see how the Bosonic StringTheory of 26-dim J3(O)o is related to an F-theory based on the28-dimensional J4(Q).


Closed Unoriented BosonicStrings:

[[ In the D4-D5-E6-E7-E8VoDou Physics model, closed strings represent the world-lines offermion particle-antiparticle pairs ( the pair offermions acting as a boson so that the entire string is bosonic) from the time of their creation to their eventual mutualannihilation (perhaps with lots of interactions withlots of other particles/antiparticles of other world-lines in themeantime). ]]

Michio Kaku, in his books, Introduction to Superstrings andM-Theory (2nd ed) (Springer-Verlag 1999) and Strings, ConformalFields, and M-Theory (2nd ed) (Springer-Verlag 2000) diagrams theUnoriented Closed Bosonic String spectrum:

Joseph Polchinski, in his books String Theory vols. I and II(Cambridge 1998), says: "... [In] the simplest case of 26 flatdimensions ... the closed bosonic string ... theory has the maximal26-dimensional Poincare invariance ... [and] ... is theunique theory with this symmetry ... It is possible to have aconsistent theory with only closed strings ... masslessspectra, with Guv representing the graviton [and] ... PHIthe dilaton ... [and also] ... the tachyon ... [forthe] Closed unoriented bosonic string [are]...":

What about the size/scale of each ofthe 26 dimensions of Closed Unoriented Bosonic String Theory?

Represent the size/scale of each dimension as a radius R, with R =infinity representing a flat large-scale dimension. Let Lpl denotethe Planck length, the size of the lattice spacing in the HyperDiamondLattice version of the D4-D5-E6-E7-E8VoDou Physics model. Joseph Polchinski says "... as R ->infinity winding states become infinitely massive, while the compactmomenta go over to a continuous spectrum. ... at the opposite limit R-> 0 ... the states with compact momentum become infinitelymassive, but the spectrum of winding states ... approaches acontinuum ... it does not cost much energy to wrap a string around asmall circle. Thus as the radius goes to zero the spectrum againseems to approach that of a noncompact dimension. ... In fact, theR-> 0 and R-> infinity limits are physically identical. Thespectrum is invariant under ...[

R -> R' = (Lpl)^2 / R

]... This equivalence is known as T-duality. ... The space ofinequivalent theories is the half-line [ R > Lpl ]. We could take instead the range [ 0 < R <Lpl ] but it is more natural to think in terms of the larger ofthe two equivalent radii ... in particular questions of locality areclearer in the larger-R picture. Thus [from the larger-R point ofview], there is no radius smaller than the self-dual radius [Rself-dual = Lpl ]. ...". T-duality structures are is similar toPlanck Pivot Vortex structures.


Consider a (purple) world-lineString of one World of the MacroSpace of Many-Worlds and itsinteractions with another (gold)world-line World String, from the point of view of one point of the(purple) World String, seen so close-upthat you don't see in the diagram that the(purple) and(gold) World Strings are both reallyclosed strings when seen at very large scale:

From the given point (diagram origin) of the(purple) World String:

In the D4-D5-E6-E7-E8VoDou Physics model gravitation in the 26-dimensionalBosonic StringTheory MacroSpace of the Many-Worlds justifies theHameroff/Penrose idea:

Superposition Separationis the separation/displacement of a mass separated from itssuperposed self. The picture is spacetime geometry separating fromitself.

[ Note that Gravity may not propagate in the26 dimensions of the MacroSpace of the Many-Worlds in exactly thesame way as it propagates in our 4-dimensional physical SpaceTime.]


Gravitation from nearby World Strings might account for atleast some Dark Matter that isindirectly observed in our World String, an ideasimilar to one described (in the context of a superstringmodel that is in many ways very different from the D4-D5-E6-E7-E8VoDou Physics model) by Nima Arkani-Hamed,Savas Dimopoulos, Gia Dvali, Nemanja Kaloper in their paperManyfoldUniverse, hep-ph/9911386, and also in anarticle by the first three authors in the August 2000 issue ofScientificAmerican.
[[ In the D4-D5-E6-E7-E8 VoDou Physics model, closed strings represent the world-lines of fermion particle-antiparticle pairs from the time of their creation to their eventual mutual annihilation,
   *    / \   ... /   \ //     |         (The illustrated closed string is red.\     |          It interacts with a partially shown gray string.) \   / \...  \ /    *  
perhaps with lots of interactions with lots of other particles/antiparticles of other world-lines in the meantime, so that part of each string might represent a photon or other particle of any type formed by interaction of one of the particle/antiparticle pair.

Note that since our Universe began with a Big Bang, all its particles originate from pair creation since then. For pairs that do not appear to reconnect for mutual annihilation within the volume of 26-dimensional spacetime being considered in working with the String Theory,

****************    \    ...   /   \      \ /    \      |         (The illustrated string is red.     \     |          It interacts with a partially shown gray string.      \   / \...      A perfect absorber in the future         \ /            is indicated by ******* ).        *  
the string is closed by considering the 26-dimensional spacetime to be a compactified 25+1 dimensional Minkowski spacetime due to considering the Universe to "... be a perfect absorber in the future ...[as in]... the Wheeler-Feynman ... absorber theory of radiation ..." described by Narlikar in his book Introduction to Cosmology (Cambridge 1997) (Section 8.8.1) and related to the Collective Electrodynamics of Carver Mead. For most of the matter in our Galactic Cluster, such an absorber could be a Black Hole of the Black Hole Era. Such a compactification is also similar to the conformally compactified 3+1 dimensional Minkowski spacetime M# used by Penrose and Rindler in their book Spinors and Space-Time, Volume 2 (Cambridge 1986) (particularly Chapter 9). ]]


Bosonic Unoriented Closed String Theory describes the structure ofBohm's SuperImplicate Order MacroSpace andis related through (1+1) conformal structures to the LargeN limit of the AN Lie Algebras. For a nice introductorydiscussion of the mathematics of Bosonic Closed Strings, seeWeek 126 andWeek 127 andother relevant works of JohnBaez.


Branching among the Worlds of the Many-Worlds may bedescribable in terms of Singularities,such as:


An M-theoryof the full 27-dimensionalJordan algebra J3(O)

that could be S-dual to BosonicString theory representing MacroSpace on 26-dim J3(O)o has beendiscussed in some recent (1997 and later) papers. In this model,27-dimensional M-theory of bosonic strings has the geometry ofE7 /E6xU(1).

A physical interpretation of27-dimensional J3(O) ( corresponding to J4(Q)o ) M-theory could be asa theory of Timelike Brane-Universes.

Timelike Brane-Universes might be considered as World-Lines(1-dimensional World-Lines with respect tothe Shilov BoundaryPointlike States, but 1+1-dimensionalComplex World-Lines (Complex Lines being like 2-dimensional Sheets orMembranes) with respect to theComplex Bounded Domain Stringlike States) in a Many-WorldsQuantum Theory.

In this view, each World of the MacroSpace ofMany-Worlds can be seen as a 1-Timelike-dimensional String ofSpacelike States, like a World Line or World String, and theMacroSpace of Many-Worlds can berepresented geometrically by E7/ E6xU(1) with 54 real dimensions and 27 complexdimensions corresponding to the complexification of the27-dimensional Jordan algebra J3(O) andalgebraically by structurerelated to the same 27-dimensional Jordanalgebra J3(O).


Discussing both open and closed bosonic strings, Soo-Jong Rey, inhis paper hep-th/9704158,Heterotic M(atrix) Strings and Their Interactions, says:

"... We would like to conclude with a highly speculative remark on a possible M(atrix) theory description of bosonic strings. It is well-known that bosonic Yang-Mills theory in twenty-six dimensions is rather special ... The regularized one-loop effective action of d-dimensional Yang-Mills theory ... For d=26, the gauge kinetic term does not receive radiative correction at all ... We expect that this non-renormalization remains the same even after dimensional reductions. ... one may wonder if it is possible to construct M(atrix) string theory ... for bosonic string as well despite the absence of supersymmetry and BPS states.

The bosonic strings also have D-brane extended solitons ... whose tension scales as 1 / gB for weak string coupling gB << 1. Given the observation that the leading order string effective action of graviton, dilaton and antisymmetric tensor field may be derived from an Einstein gravity in d = 27, let us make an assumption that the 27-th `quantum' dimension decompactifies as the string coupling gB becomes large. For D0-brane, the dilaton exchange force may be interpreted as the 27-th diagonal component of d = 27 metric. Gravi-photon is suppressed by compactifying 27-th direction on an orbifold [ such as S1 / Z2 ] rather than on a circle. Likewise, its mass may be interpreted as 27-th Kaluza-Klein momentum of a massless excitation in d = 27. In the infinite boost limit, the light-front view of a bosonic string is that infinitely many D0-branes are threaded densely on the bosonic string. ...".


Gary T. Horowitz and Leonard Susskind, in their paper hep-th/0012037,Bosonic M Theory, say:

"... The possibility that the bosonic string has a 27 dimensional origin was ... discussed ...[ by Soo-Jong Rey in his paper hep-th/9704158 ]... in the context of a proposed matrix string formulation. ... We conjecture that there exists a strong coupling limit of bosonic string theory which is related to the 26 dimensional theory in the same way that 11 dimensional M theory is related to superstring theory. More precisely, we believe that bosonic string theory is the compactification on a line interval of a 27 dimensional theory whose low energy limit contains gravity and a three-form potential. The line interval becomes infinite in the strong coupling limit, and this may provide a stable ground state of the theory. ...

we ... argue that the tachyon instability may be removed in this limit. ... The main clue motivating our guess comes from the existence of the dilaton and its connection to the coupling constant. ... Evidently, as in IIA string theory, the dilaton enters the action just as it would if it represented the compactification scale of a Kaluza Klein theory. We propose to take this seriously and try to interpret bosonic string theory as a compactification of a 27 dimensional theory. We will refer to this theory as bosonic M theory. ...

Closed bosonic string theory does not have a massless vector. This means it cannot be a compactification on an S1 . ... Accordingly, we propose that

closed bosonic string theory is a compactification of 27 dimensional bosonic M theory on [ an orbifold ] S1 / Z2. ...

In the bosonic case, since there are no fermions or chiral bosons, there are no anomalies to cancel. So there are no extra degrees of freedom living at the fixed points. ... the weakly coupled string theory is the limit in which the compactification length scale becomes much smaller than the 27 dimensional Planck length and the strong coupling limit is the decompactification limit. The 27 dimensional theory should contain membranes but no strings, and would not have a dilaton or variable coupling strength. The usual bosonic string corresponds to a membrane stretched across the compactification interval. ... the low energy limit of bosonic M theory ... is a gravity theory in 27 dimensions ... In order to reproduce the known spectrum of weakly coupled bosonic string theory, bosonic M theory will have to contain an additional field besides the 27 dimensional gravitational field, namely a three-form potential CFT. Let us consider the various massless fields that would survive in the weak coupling limit.

  • First of all, there would be the 26 dimensional graviton. As usual, general covariance in 26 dimensions would insure that it remains massless.
  • The component of the 27 dimensional gravitational field g27;27 is a scalar in the 26 dimensional theory. It is of course the dilaton. No symmetry protects the mass of the dilaton. In fact we know that at the one loop level a dilaton potential is generated that lifts the dilatonic at direction. Why the mass vanishes in the weak coupling limit is not clear.
  • Massless vectors have no reason to exist since there is no translation symmetry of the compactification space. This is obvious if we think of this space [ the orbifold S1 / Z2 ] as a line interval.
  • ...[ with respect to tachyons ]... Even if 27 dimensional flat space, M27, is a stable vacuum, one might ask what is the "ground state" of the theory at finite string coupling, or finite compactification size? Tachyon condensation is not likely to lead back to M27, and there is probably no stable minimum of the tachyon potential in 26 dimensions ... Instead, we believe tachyon condensation may lead to an exotic state with zero metric guv = 0. It is an old idea that quantum gravity may have an essentially topological phase with no metric. We have argued that the tachyon instability is related to nucleation of "bubbles of nothing" which is certainly reminiscent of zero metric.

... As an aside, we note that there is also a brane solution of 26 dimensional bosonic string theory which has both electric and magnetic charge associated with the three-form H. It is a 21-brane with fundamental strings lying in it and smeared over the remaining 20 directions. Dimensionally reducing to six dimensions by compactifying on a small T 20 , one recovers the usual self dual black string in six dimensions. ...

... We have proposed that a bosonic version of M theory exists, which is a 27 dimensional theory with 2-branes and 21-branes. One recovers the usual bosonic string by compactifying on S1 / Z2 and shrinking its size to zero. In particular, a Planck tension 2-brane stretched along the compact direction has the right tension to be a fundamental string. This picture offers a plausible explanation of the tachyon instability and suggests that uncompactified 27 dimensional flat space may be stable. A definite prediction of this theory is the existence of a 2+1 CFT with SO(24) global symmetry, which should be its holographic dual for AdS4 x S23 boundary conditions. ... if there does not exist a 2+1 CFT with SO(24) global symmetry, bosonic M theory would be disproven.

... What kind of theory do we get if we compactify bosonic M theory on a circle instead of [ the orbifold S1 / Z2 ] a line interval? ... we believe the limit of bosonic M theory compactified on a circle as the radius R --> 0 is the same as the limit R --> infinity, i.e., the uncompactified 27 dimensional theory. If we compactify bosonic M theory on S1 x ( S1 / Z2 ), and take the second factor very small, this is a consequence of the usual T-duality of the bosonic string. More generally, it appears to be the only possibility with the right massless spectrum. ...".


Lee Smolin, in hispaper The exceptional Jordan algebra and the matrix string,hep-th/0104050, says: "A new matrix model is described, based onthe exceptional Jordan algebra, J3(O). The action is cubic, as inmatrix Chern-Simons theory. We describe a compactification that, weargue, reproduces, at the one loop level, an octonioniccompactification of the matrix string theory in which SO(8) is brokento G2. There are 27 matrix degrees of freedom, which under Spin(8)transform as the vector, spinor and conjugate spinor, plus threesinglets, which represent the two longitudinal coordinates plus aneleventh coordinate. Supersymmetry appears to be related to trialityof the representations of Spin(8).".

 YuhiOhwashi, in his paper E6 Matrix Model, hep-th/0110106, says: "...Lee Smolin's talk presented at The 10thTohwa University International Symposium (July 3-7, 2001, Fukuoka,Japan) was my motive for starting this work. ...

... Smolin's matrix model [is] based on the groups of typeF4. ... The action of Smolin's model is given ...[in termsof]... elements of exceptional Jordanalgebra J..... The exceptional Jordan algebra J is a27-dimensional R-vector space. This space can be classified intothree main parts.

 ... If the standard model were described by using Majoranaspinors only, F4 might be the underlying symmetry of the universe.... However, the actual world requires complex fermions withoutdoubt. This is the reason why we have to abandon the simply connectedcompact exceptional Lie group F4. ... In accordance with thiscomplexification, the groups of type F4 are upgraded to the groups oftype E6. ...

... we consider a new matrix model based on the simply connectedcompact exceptional Lie group E6 ... The action of the model isconstructed from cubic form which is the invariant on E6 mapping.

This action is an essentially complex action. Of course if onewants, one may take up only real part of that ...

Our model has twice as many degrees of freedom as Smolin's modelhas because we are considering E6 instead of F4. ... Thisis a future problem which needs to be asked. ...".


The ideas of Smolin and Ohwashi are related to my D4-D5-E6-E7-E8VoDou Physics model in interesting ways:


However, there are differences between the approaches of Smolinand Ohwashi, and the approach of the D4-D5-E6-E7-E8VoDou Physics model. For example:


Metod Saniga, inphysics/0012033, discusses in the context of string theory (although in a different context ( heterotic superstrings ) from thatof the D4-D5-E6-E7-E8 VoDou Physicsmodel ) another 27-dimensional structure, saying:

"...  It is a well-known fact that on a generic cubic surface, K3, there is a configuration of twenty-seven lines ... the lines are seen to form three separate groups. The first two groups, each comprising six lines, are known as Schlafli's double-six. The third group consists of fifteen lines ... The basics of the algebra can simply be expressed as 27 = 12 + 15 ...".

It is interesting to contemplate the relationship between the 3x3matrix structure

1 8 8- 1 8- - 1

of the 27-dimensional Jordan algebraJ3(O) and the 27-line geometrystructure

6+6 + 6/\6 = 6+6 + 15 = 27

Let the 8 be represented by 8-dimensional octonions,with basis {1,i,j,k,E,I,J,K}, and let the 6 be represented by a6-dimensional subspace, with basis {i,j,k,I,J,K}. Let the two 6s of6+6 be represented as subspaces of the two next-to-diagonal 8s of theJ3(O) matrix:

- 6 -- - 6- - -

then the 6/\6 = 15 lines of the 27-line might correspond tothe

1 2 8- 1 2- - 1

in terms of the J3(O) matrix. Here are some more relevantrelationships:


Metod Saniga's ideas have been referencedby Carlos Castro in physics/0104016,in which Carlos Castro says:

"... Motivated by the fact that the bosonic membrane is devoid of anomalies in d = 27, and the supermembrane is anomaly free in d = 11, and that the anomaly free ( super) string actions ( d = 26, 10 ) are directly obtained by a double-dimensional reduction process of both the world-volume of the ( super) membrane and the target spacetime dimension, where the (super) membrane is embedded, we shall derive rigorously the transfinite M theoretical corrections ... to El Naschie's inverse fine structure constant ... which were based on a transfinite perturbative Heterotic string theory formalism ...".

Although Carlos Castro uses some similar mathematical structures,such as Clifford algebra, his physics model is different from theD4-D5-E6-E7-E8 VoDou Physics model in anumber of respects, such as, particularly, his use of conventionalsuperstring theory instead of using the D4-D5-E6-E7-E8VoDou Physics model viewpoint of seeing bosonic strings as WorldLine Worlds in the MacroSpace of the Worlds of the Many-Worlds.

However, some of the interesting similarities that I perceiveinclude:



Branchingamong the Worlds of the TimelikeBranes of 27-dim M-Theory may be describable in terms ofSingularities, suchas:


In bosonic string theory, 27-dimensionalM-theory is a subspace of

28-dimensional F-theorywith Jordan algebraJ4(Q).

A physical interpretation of28-dimensional J4(Q) F-theory could be as a theory of SpacelikeBrane-Universes.

Spacelike Brane-Universes might be considered as Spatial Worlds(3-dimensional Spatial Worlds with respect tothe Shilov BoundaryPointlike States, but 3+3-dimensionalComplex Spatial Worlds with respect to theComplex Bounded Domain Stringlike States) in a Many-WorldsQuantum Theory.

 28-dimensional F-theory of bosonic strings has thegeometry of E8 /E7xSU(2).

Bosonic string F-theory is described by Jose M Figueroa-O'Farrill,in his paperF-theory and the universal string theory, hep-th/9704009:

"... Let us first consider a bosonic string background. ... The graviton couples to the energy-momentum tensor T. If we now add a U(1) gauge field, it will couple to a vector J. We therefore would like to investigate under which conditions the algebra generated by T and J can be used consistently to define a (generalised) string theory. ...

... A particular realisation of this algebra is provided by a bosonic string propagating in a 28-dimensional pseudo-euclidean space with signature ( 26, 2 ). The signature can be understood from unitarity ...

... the BRST cohomology of this system agrees with that of an underlying bosonic string propagating in a 26-dimensional Minkowski subspace perpendicular to v and not containing v, provided that we identify states whose momenta differ by a multiple of v ...

... Suppose that T is the energy-momentum tensor of a critical bosonic string propagating in 26-dimensional Minkowski spacetime.

Then T corresponds to the string propagating on a ( 26 +2 )-dimensional pseudo-euclidean space.

The BRST operator is invariant under the subgroupof the ( 26 + 2 ) pseudo-euclidean group of motions which preserves the null vector v. This is nothing but the ( 25 + 1 ) conformal group, which does not act linearly in Minkowski spacetime but does on the larger space.

Symmetries of the BRST operator induce symmetries in the cohomology, hence we would expect that the spectrum should assemble itself into representations on the conformal group. We know that the physical spectrum of the bosonic string only possesses ( 25 + 1 ) Poincare covariance, so what happens to the special conformal transformations?

... bosonic ghosts ... have a (countably) infinite number of inequivalent vacua which can be understood as the momenta in one of two auxiliary compactified dimensions introduced by the bosonisation procedure. The picture changing operator interpolates between these different vacua, commuting with the BRST operator and thus introducing an infinite degeneracy in the cohomology. ...

.. the special conformal transformations ... change the picture. By definition a picture-changing operator is a BRST invariant operator which changes the picture, whence the special conformal transformations are picture-changing operators.

A remarkable fact of this treatment is that the appearance of the lorentzian torus is very natural. In other words, by enhancing the gauge principle on the worldsheet to incorporate the extra U(1) gauge invariance we are forced to reinterpret bosonic string vacua corresponding to propagation on a given manifold M, as propagation in a manifold which at least locally is of the form M x T 2 where T 2 is the lorentzian torus corresponding to the bosons ... This theory is precisely the F-theory introduced ...[ by Vafa in hep-th/9602022 ]... except that there the compactness of the extra two coordinates was an ad hoc assumption. ...".


Branchingamong the Worlds of the SpacelikeBranes of 28-dim F-Theory may be describable in terms ofSingularities, suchas:


Here are some descriptions of a few relevant terms:


Michio Kaku, in his book Introduction to Superstringsand M-Theory (second edition, Springer 1999), says:

"... Superstrings ...
  • Type I = open and closed string, same chiralities
  • Type IIA = closed string, opposite chiralities
  • Type IIB = closed string, same chiralities ...

... the closed [super] string ( Type II ) ... the fields can either be chiral or not. Closed strings are, by definition, periodic in sigma, which yields the following normal mode expansion:

  • S1a(s,t) = Sum( n = -infinity; n = + infinity ) San exp( - 2 i n( t - s ) ) ,
  • S2a(s,t) = Sum( n = -infinity; n = + infinity ) S'an exp( - 2 i n( t + s ) ) .

If these two fields have different chiralities, then they are called Type IIA. ... this represents the N = 2, D = 10-dimensional reduction of ordinary N = 1, D = 11 supergravity. ...

... there exists a new 11-dimensional theory, called M-theory, containing 11-dimensional supergravity as its low-energy limit, which reduces to Type IIA [super] string theory (with Kaluza-Klein modes) when compactified on a circle. ... the strong coupling limit of 10-dimensional Type IIA superstring theory is equivalent to the weak coupling limit of a new 11-dimensional theory [ M-theory ], whose low-energy limit is given by 11-dimensional supergravity. ... Using perturbation theory around weak coupling in 10-dimensional Type IIA superstring theory, we would never see 11-dimensional physics, which belongs to the strong coupling region of the theory. ... M-theory is much richer in its structure than string theory. In M-theory, there is a three-form field Amnp, which can couple to an extended object. We recall that in electrodynamics, a point particle acts as the source of a vector field Au. In [open] string theory, the [open] string acts as the source for a tensor field Buv. Likewise, in M-theory, a membrane is the source for Amnp. ...

... Ironically, 11-dimensional supergravity was previously rejected as a physical theory because:

  • (a) it was probably nonrenormalizable (i.e., there exists a counterterm at the seventh loop level);
  • (b) it does not possess chiral fields when compactified on manifolds; and
  • (c) it could not reproduce the Standard Model, because it could only yield SO(8) when compactified down to four dimensions.

Now we can veiw 11-dimensional supergravity in an entirely new light, as the low-energy sector of a new 11-dimensional theory, called M-theory, which suffers from none of these problems. The question of renormalizability is answered because the full M-theory apparently has higher terms in the curvature tensor which render the theory finite. The question of chirality is solved because ... M-theory gives us chirality when we compactify on a space which is not a manifold (such as [ orbifolds such as S1 / Z2 ] line segments). And the problem that SO(8) is too small to accommodate the Standard Model is solved when we analyze the theory nonperturbatively, where we find E8 x E8 symmetry emerging when we compactify on [ orbifolds such as S1 / Z2 ] line segments. ...".

Note that the D4-D5-E6-E7-E8 VoDou Physics Model solves the problems of 11-dimensional supergravity in different ways, but uses many similar mathematical structures and techniques.

Michio Kaku, in his book Strings, Conformal Fields and M-Theory(second edition, Springer 2000), says:

"... S: M-theory on S1 <---> IIA ... Type IIA [super] string theory is S dual to a new, D = 11 theory called M-theory, whose lowest-order term is given by D = 11 supergravity. ...

... S: M-theory on S1 / Z2 <---> E8 x E8 ...[ 11-dimensional ]... M-theory, when compactified on a line segment [S1 / Z2 ], is dual to the ... [ E8 x E8 heterotic ]... string ...".


Lisa Randall and Raman Sundrum, in their paper hep-ph/9905221,say:

"... we work on the space S1 / Z2. We take the range of PHI to be from -pi to pi; however the metic is completely specified by the values in the range 0 < PHI < pi. The orbifold fixed points at PHI = 0, pi ...[may]... be taken as the locations of ... branes ...".
Note that S1 / Z2 can have two different interpretations.       John Baez says: "... Z_2 acts in various ways on the circle.  Let's think of the circle as the subset{(x,y): x^2 + y^2 = 1}  of R^2.  Z_2 can act on it like this:(x,y) |-> (-x,-y) and then S^1/Z_2 = RP1 [Real Projective 1-space] which is a manifold, in fact a circle.... Z_2 also can act on the circle like this:(x,y) |-> (-x,y) and then S^1/Z_2 is an orbifold, in fact a closed interval. ...".       The physical interpretations of RP1 in the D4-D5-E6-E7-E8 VoDou Physics model as Time of SpaceTime andas representation space for Neutrino-type (only one helicity state) Fermions might be viewed as having some of the characteristics of a orbifold line interval. 

Joseph Polchinski, in his book String Theory (volume 1, Cambridge1998), says:

"... orbifold
  • 1. ... a coset space M / H , where H is a group of discrete symmetries of a manifold M. The coset is singular at the fixed points of H ;
  • 2. ... the CFT or string theory produced by the gauging of a discrete world-sheet symmetry goup H. If the elements of H are spacetime symmetries, the result is a theory of strings propagating on the coset space M / H . A non-Abelian orbifold is one whose point group is non-Abelian. An asymmetric orbifold is one where H does not have a spacetime interpretation and which in general acts differently on the right-movers and left-movers of the string;
  • 3. ... to produce such a CFT or string theory by gauging H ; this is synonymous with the second definitioin of twist.

... S-duality ... a duality under which the coupling constant of a quantum theory changes nontrivially, including the case of weak-strong duality. ... In compactified theories, the term S-duality is limited to those dualities that leave the radii invariant, up to an overall coupling-dependent rescaling ...

... T-duality ... a duality in string theory, usually in a toroidally compactified theory, that leaves the coupling constant invariant up to a radius-dependent rescaling and therefore holds at each order of string perturbation theory. Most notable is R --> a' / R duality, which relates string theories compactified on large and small tori by interchanging winding and Kaluza-Klein states. ...

... U-duality ... any of the dualities of a string theory ... This includes the S-dualities and T-dualities, but in contrast to these includes also transformations that mix the radii and couplings. ...".


26-dimensional Bosonic Strings and the FakeMonster


R. E. Borcherds,in hispaper Problems in Moonshine, says: "... The classification offinite simple groups shows that every finite simple group eitherfits into one of about 20 infinite families, or is one of 26exceptions, called sporadic simple groups. The monstersimple group is the largest of the sporadic finite simple groups,and was discovered by Fischer and Griess ... Its order is

8080, 17424, 79451, 28758, 86459, 90496, 17107, 57005, 75436,80000, 00000


2^46 .3^20 .5^9 .7^6 .11^2 .13^

(which is roughly [ 8 x 10^53 ] the number of elementaryparticles in the earth [ actually, the earth's mass is about 6 x10^51 GeV, and it is Saturn that has mass about 6 x 10^53 GeV, orabout 6 x 10^53 hydrogen masses ]). The smallest irreduciblerepresentations have dimensions 1, 196883, 21296876, . . ..

On the other hand the elliptic modular function j(t) ... has thepower series expansion

j(t)= q^(-1) + 744 + 196,884 q + + 21,493,760q 2+...

where q = exp( 2 pi i t ). John McKay noticed some rather weirdrelations between coefficients of the elliptic modular function andthe representations of the monster as follows:

where the numbers on the left are coefficients of j(t) and thenumbers on the right aredimensions of irreducible representations ofthe monster. The term "monstrous moonshine" (coined by Conway) refersto various extensions of McKay's observation, and in particular torelations between sporadic simple groups and modular functions....

Allcock ... recently constructed some striking examples of complexhyperbolic reflection groups from the Leech lattice, or moreprecisely from the complex Leech lattice, a 12 dimensional latticeover the Eisenstein integers. This complex reflection group lookssimilar in several ways to Conway's real hyperbolic reflection groupof the lattice /\25,1 ...".


Gregory Moore, in hispaper Finite in All Directions, hep-th/9305139, says:

"... At a generic point [ g of a string theory toroidal compactification lattice, the Lie algebra ] Lg = IR^26 + IR^26 ...

... The distinguished point g* may be regarded as a point of maximal symmetry in the moduli space of toroidal compactifications ...

... Given a point of maximal symmetry it is natural to ask if L* = Lg* is a universal symmetry of string theory in the sense that all other unbroken symmetry algebras which arise in toroidal compactification are subalgebras of L* . Unfortunately, maximal symmetry does not imply that L* is universal. ...

... For the bosonic string the Lie algebra L* is related to the Monster group. L* = A x A where A is the "Fake Monster Lie algebra" studied by Borcherds ...".


R. E. Borcherds,J. H. Conway, L. Queen and N. J. A. Sloane, in theirpaper A Monster Lie Algebra?, say: "... [A version of thispaper was originally published in Advances in Mathematics, vol.53(1984), no. 1, pp. 75{79. A revised version appeared as Chapter 30of "Sphere packing, lattices and groups" by J. H. Conway and N. J. A.Sloane, Springer-Verlag, 1988.] ... Remark added 1998:

The Lie algebra of this paper is indeed closely related to the monster simple group. In order to get a well behaved Lie algebra it turns out to be necessary to add some imaginary simple roots to the "Leech roots". This gives the fake monster Lie algebra,which contains the Lie algebra of this paper as a large subalgebra.

See R. E. Borcherds, "The monster Lie algebra", Adv. Math. Vol. 83, No. 1, Sept. 1990, for details (but note that the fake monster Lie algebra is called the monster Lie algebra in this paper). The term "monster Lie algebra" is now used to refer to a certain \"=2Z-twisted" version of the fake monster Lie algebra. The monster Lie algebra is acted on by the monster simple group, and can be used to show that the monster module constructed by Frenkel, Lepowsky, and Meurman satisfies the moonshine conjectures; see R. E. Borcherds, "Monstrous moonshine and monstrous Lie superalgebras", Invent. Math. 109, 405-444 (1992). ...". Also, see the web seminar What is Moonshine?, Richard Borcherds, 25 November 1998. and the paper What is Moonshine?, math.QA/9809110.


Reinhold W. Gebert, in hispaper Introduction to Vertex Algebras,Borcherds Algebras, and theMonster Lie Algebra, hep-th/9308151, says:

"... Borcherds algebras arise as certain "physical" subspaces of vertex algebras ... As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake Monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analysed from the point of view of symmetry in quantum theory and the construction of the Monster Lie algebra is sketched. ...

the fake Monster Lie algebra seemingly plays an important role in bosonic string theory ... Vertex algebras associated with even lattices have their origin in toroidal compactifications of bosonic strings. ... As an easy application we demonstrate how affine Lie algebras arise in this context. Furthermore, the fake Monster Lie algebra which is the first generic example of a Borcherds algebra, is worked out in detail. ...

the so called Moonshine Module ... constructed by Frenkel, Lepowsky, and Meurman is a vertex operator algebra ... and it turns out that ...[its]... weight two piece ... is a non-associative algebra with symmetric product ... and associative bilinear form ...[that]... is precisely the 196,884-dimensional Griess algebra which possesses the Monster group ... as its full automorphism group...

Things become more complicated when we move away from the lattice /\ being Euclidian. Let us consider the unique 26-dimensional even unimodular Lorentzian lattice /\25,1 ... In physics this corresponds to an open bosonic string moving in 26-dimensional spacetime compactified on a torus so that the momenta lie on a lattice. Calculations in connection with the automorphism group of /\25,1 show that ... [the] simple roots generate the reflection group of /\25,1 ... We shall also call the positive norm simple roots of /\25,1 Leech roots since Conway has shown that this subset is indeed isometric to the Leech lattice, the unique 24-dimensional even unimodular Euclidian lattice with no vectors of square length two. ... We now define a Kac-Moody algebra Linfinity, of infinite dimension and rank ... Linfinity has three generators ... for each Leech root ...

... Let us summarize: We define the fake Monster Lie algebra g/\25,1 to be the Lie algebra with root lattice /\25.1, whose simple roots are the simple roots of the Kac-Moody algebra Linfinity, together with the positive integer multiples of the Weyl vector ... each with multiplicity 24. ... the fake Monster Lie algebra is not a Kac-Moody algebra due to the presence of the lightlike simple Weyl roots which violate an axiom for these algebras ... Nevertheless, the structure of g/\25,1 resembles a Kac-Moody algebra very well. ...

Frenkel, Lepowsky and Meurman constructed the Monster vertex algebra which is acted on by the Monster simple sporadic group. The underlying vector space which is called Moonshine Module ...[It]... provides a natural infinite-dimensional representation of the Monster [and it] is characterized by the following properties:

  • (i) ...[It]... is a vertex operator algebra with a conformal vector ... of dimension 24 and a positive definite bilinear form
  • (ii) ...[It is the sum of eigenspaces of L(0) with eigenvalues n+1 and with dimension ]... given via the generating function ...
[ SUM(n>-1) dim(Fn) q^n = J(q) = j(q) - 744 = q^(-1) + 196,884 q + ... ]...
  • (iii) The Monster group acts on ...[it]... preserving the vertex operator algebra structure, the conformal vector ... and the bilinear form. ...

The Monster vertex algebra is realized explicitly as

F = F+/\Leech+ + F+/\Leech

where F/\Leech denotes the vertex operator algebra associated with the Leech lattice, the unique 24-dimensional even unimodular Euclidian lattice with no elements of square length two. ... the Monster module [can be seen] as Z2-orbifold of a bosonic string theory compactified to the Leech lattice ...[and as]... a Zp-orbifold ... It is interesting that there is also an approach to the Monster module based on twisting the heterotic string. ...

The starting point for the definition of a Monster Lie algebra should be the fake Monster Lie algebra. We use the fact that the Lorentzian lattice /\25.1 can be written as the direct sum of the Leech lattice and the unique two-dimensional even unimodular Lorentzian lattice /\1,1. ... the vertex algebra associated with the Lorentzian lattice /\25,1 is the tensor product of thevertex algebras corresponding to F/\Leech and F/\1,1 . One finds that the Leech lattice gives rise to a vertex operator algebra with conformal vector of dimension 24 and a positive definite bilinear form. Furthermore, F/\Leech = SUM(n>-1) Fn/\Leech where Fn/\Leech ... is the eigenspace of L(0) with eigenvalue n+1 and the dimension of Fn/\Leech is given via the generating function ...

SUM(n>1) dim(Fn/\Leech) q^n = J(q) = j(q) - 720 = q^(-1) + 24 + 196,884 q + ...

... the Monster Lie algebra is seen to be a generalized Kac-Moody algebra. ...".  


Tony's Home