On a Final Theory of Mathematics and Physics by Michael Rios

The connection between SL(4,C) and CP^3 is seen with

h_5(C) ~= h_4(C)вЉ•C^4вЉ•R

That is a 2x2 matrix of 4-component objects, spinors П€ and vectors П† in C^4 and h_4(C). The Jordan algebra h_4(C) ~ CP^3 and in conjuction with the C^4 tells us that we have a CP^{3|4} superspace for twistor theory.

It would appear best at first to consider

h_3(H) ~= h_2(H)вЉ•H^2вЉ•R.

This is just a way of twistor theory based on H^2 is that H^2 ~ C^4, and of course we have a different meaning to projective spaces. The standard theory has T = C^4 with projective twistor space CP^3 = P(C^4) < --- F_12(T) --- > G_{2,4}(C). The G_{2,4}(C) = U(4)/U(2)^2, or in signatured spacetime SU(2,2)/SU(2)xSU(1,1). The quotient between SU(2,2) = SO(4,2) and SU(2)xSU(1,1) ~ SO(3,1) is the Grassmannian space of this space of 2-planes in 4-space. The quotient SO(4,2)/SO(4,1) is the AdS_5 or moduli space for the quaternion bundle. This has a relationship to G_{2,4}(C) as its quotient is with SO(3,1) вЉ‚ SO(4,1). The CP^3 is projective twistor space PT and the manifold F_{12}(T) is a five dimensional space that through this double fibration maps information in this space to CP^3.

This change is a bit similar to considering the Dirac equation according to the Dirac equation that is a 4-column made of complex objects. We can of course consider this equivalently as a a pair of complex fields that define a quaterionion. This is a bit like working with the Dirac equation directly or with a pair of Weyl equations that define the same thing. Twistor theory works in exactly this construction.

What I propose is that we consider

h_5(H) ~= h_4(H)вЉ•H^4вЉ•R.

The group SL(4,H) constains SL(4,C)xSL(4,C) and we may then embed two copies of twistor space in here. We have instead of conjugation *i --- > -i there is in addition so(n) algebras for the elements with z = x + iu + jv + ky, from which we can build up Clifford algebras. The Clifford algebras correspond to a gauge theory. I illustrated in an attachment how quaternions give rise to gauge theory. In this way the theory obeys the symmetry of a Clifford algebra corresponding to the symmetry of the gauge fields.

LC

I can now see the connection to D-branes. The triplet system most fundamentally involves the cubic equation with eigenvalues. The d = 5 cosets of states G_5/K_5 involves G_5 = SL(3,K), K = R, C, H or O with quotients with SO(3), SU(3), USp(6), and F_4 for E6(-26) and USp(8) for E6(6). In four dimensions, with an additional vector, a similar quotient of states exists. These are G_4/K_4 with G_4 = Sp(6,R), SU(3,3), SO(12,C) and E7(-25) for R, C, H and O with K_4 respectively SU(3)xU(1), SU(3)xSU(3)xU(1), SU(6)xU(1) and E6xU(1). The quotients of these states have dimension 5, 8, 14, and 26. The vector fields in these supergravities is one added to this.

If we work in E6(6) or E6(-26) we can work with their mcs, and in the case of E6(6) there is a USp(8) symplectic system of 4x4 matrices, and for E6(-26) USp(6,2). The triplet system is then a cubic invariant of these groups with an entropy/action given by C_{ijk}z^i z^j z^k which is invariant under z^i = Ω^i_jz^j for Ω^i_j in USp(8) or USp(6,2). In this case we think of diagonalize the E6(-26) with F4 mcs, where SO(8) ⊂SO(9) a subset of F4. This entropy S = sqrt{z^3} is then a product of charges on a 5-dimensional surface or brane. This is in fact the same as the QCD-ish brane physics with S = sqrt{N_5N_s}, for N_s string charges (D1-brane) and N_5 is the charge on the 5-brane. The real valued system with n = 3(dimK + 1) for K = R gives the real dimension on the brane. The NS charge on the 5-brane (NS5-brane) is dual to the D1-brane charge on the 5-brane that is R-R charge for a U(1) symmetry. This is the theory of SO(9) in the BSFF theory.

The corresponding theory with d = 4 corresponds to the NS5-brane compactified in one dimension. In doing this we can build a black hole. The K = O theory with E7(-25)/E6xU(1) is 56 with 27 gauge potentials and 27 of their magnetic monopole duals, with a graviton and the NUT dual graviton. This is some form of holographic duality. The 27 charges are equivalent to the 26 of the E6(-26)/F4 plus one unit and there are the gravitons. The additional vector is then the 3(dimK + 1) + 1 and the graviton has some sort of duality as well.

I am tempted to think of the 26 charges as being themselves dimensions, and in fact define the Lorentz geometry of the bosonic string. Each dimension defines a particular mode of vibration of the string, and we may think of these as "charges."

Cheers LC

The H_1 and H_2 I presume as elements of the J3(O) then I presume that X is then some nonassociating element The equation

dX = [H_1, X, H_2]

corresponds to the associator. The odd think is that the Jacobi identity for

6[x, y, z] = [x, [y, z]] + [y, [z, x]] + [z, [x, y]],

which in standard group theory is zero. This is probably one reason that most people reject the idea of nonassociative mathematics. I myself would be tempted to say that for the real eigenvalues of J3(O) diagonalized with F4 that the associator has zero contribution.

The extension to E6 = SL(3, O) involves the subgroups sl(2,O) and SU(2, O). We the include the real forms of D5 and B4 in addition to the eigenvectors of F4 that are the Hurwitz quaternions. This means cranking out the roots and weights of PSO(9,1) and SO(9). The SO(9) is contained in the F4, and so we now have to contend with SO(10) or SO(9,1). For the SO(9,1) we have Lorentz boosts now to contend with. The elements H_3в--¦X, the Jordan product then evaluates according to the work I did last month with

Tr(Aв--¦Q_О») = 2П‰([П‡, П‡'])

For Q = v_λ(v_λ)^†/\langle (v_λ)^†|v_λ\rangle

Which is a projector in J3(O). Now since the eigenvalues can be imaginary it poses I think no particular problem with the closure of the Jacobi identity.

That H3 is 26 dimensional does give a sort of triality situation. The 26 dimensions are "half" of the F4, where there are two dual root systems. The 24 roots in the 24 cell are (+/-ВЅ, +/-ВЅ, +/-ВЅ, +/-ВЅ) = 16 roots and (+/-1, 0, 0, 0) = 8 roots with units permuted. These Hurwitz quaternions are z = a + bi + cj + dk. There is a dual set of 24 roots (+/-1, +/-1, 0, 0). These fill out the 48 roots of the F4 with the additional 4 weights.

We have then the E6(-26) is a set of Hurwitz quaternions plus another "half" or fundamental representation of F4. The invariant S = ПЂ sqrt{I_1}is then cubic in the elements from the three fundamental representations. The quaternion roots when summed give three fundamental quantities associated with the action or invariant. It is then interesting to note that E7(-25) is then 133 - 25 which contain 108 dimensions, suggesting a span equivalent to two F4s.

Just to further the argument with respect to 5-branes, the generator of the partition function for the 5-brane (NS5-brane) is ПЂ sqrt(Q_1Q_5n_5) with n_5 = EL/2ПЂ for E the energy of the brane and L its scale. The charge Q_5 is the charge of the NS5-brane and Q_1 that of a D1 on the brane. The determinant system is of the same form. Given that the cubic system is for the d = 5 system the space in five dimensions is identified as the NS5-brane.

The invariant I_1(p, q) defines the metric with z = (p, q)

ds_4^2 = (1/I_1(z))(dt + ПЃ_idz^i) + I_1(z)dx_idx^i

where the entropy function I(z + δz) = I(z) + J(I)(z)δz + δz^†H(I)(z)δz. For J(I)(z) = Jacobian matrix and H(z) the Hessian matrix with second derivatives. Clearly H(I)(z) = J(∇I)(z). The near horizon moduli can then be

q_ОЈ + (∂I(z)/∂p)_ОЈ + X(∂^I(z)/∂q∂p)_ОЈ = 2iZ(О")M_ОЈ

p^ОЈ + (∂I(z)/∂q)^ОЈ + X(∂^I(z)/∂q∂p)^ОЈ = 2iZ(О")L^ОЈ

where the X is meant to indicate twistor geometry with

П‰'^A = П‰^A + X^{AA'}ПЂ_A'

In effect the inclusion of the second derivative "twistorizes" this. We then have the twistor connection ПЃ^i = Пѓ^i_{AA'}X^{AA'}. The holomorphic coordinate variable of interest is

t^ОЈ(p,q) = (M_ОЈ/M^0) =

(p^Σ + (∂I(z)/∂q)^Σ + X(∂^I(z)/∂q^2)^Σ)

(p_0 + (∂I(z)/∂q)_0 + X(∂^I(z)/∂q^2)_0).

For holomorphy we have ∂^I(z)/∂z^2 = 0 so that ∂^I(z)/∂q^2 = ∂^I(z)/∂p^2. The invariant then obeys

∂^I(z)/∂z∂z-bar = ∂^I(z)/∂q^2 + ∂^I(z)/∂p^2,

and the invarant is a Kahler potential.

The harmonic symplectic

H(z) = (H^a, H_a) = h + sum_s=1^nО"_s/|z - z_s|^p = h + О"/|z - z_s|^p

for the far field situation z >> z_s the invariant is I_1 ~= h + О"/z^p. The metric element is (∇xПЃ)_i = _i and ∇^2ПЃ_i = _i. The invariant is written as a product so that

(1/3!)c_{ijk}t^i t^j t^k = h + О"/z^p = e^{-2U(z)}.

The holomorphic coordinates corresponds to the NS5-brane a string on the brane and momentum associated with the energy scale of the brane.

In ten dimensions the line element determined by the 5 dimensions tangent to the branes, Greek indices, the directions tangent to the branes and transverse to the string (D1-brane) on the brane, Latin index n, and the spatial directions 1, 2, 3, 4 transverse to both the NS5-brane and the D1-brane, Lattice index j, and finally the index 5 in the direction the NS5-brane is moving,

ds^2 = A^{-1}[g_{ОјОЅ}dx^Ојdx^ОЅ + B(-dt^2 + dx_5^2)] + Adx^jdx^j + Cdx^ndx^n.

The event horizon in 10 dimensions is 8 dimensional. This has 5 dimensions tangent to the brane and 3 dimensions transverse to it. This then occurs in the metric term

A^{-1}r^2dО©^2.

The metric coefficient is ~ 1/r^p, for p = 5 - horizon dimension on the brane = 2. It is also dependent on the brane and string scales r_1 and r_5, from which we compute A^{-1}B. This is compared to the metric for I_1 above. These occur separately so that

A^{-1} ~ r_1^2r_5^2/r^2.

The area is then Area = r_1^2r_5^2. For the direction along the motion of the brane we have in Euclideanized metric the area is then

Area = r_1^2r_5^2n_5^2LV^4.

For L the scale of the brane and V^4 the volume in the 1, ..., 4 transverse directions. The distances are the square of the respective charges so that

Area = K sqrt{Q_1Q_5n_5}

This then connects S = ПЂ sqrt(det A^3) to the area of a black-brane or black hole.

LC

The Jordan product xâ--¦y = ½ (xy + yx) involves obvious Fermi-Dirac statistics as an anticommutator. The associator [x, y, z] = (xâ--¦y) â--¦z - xâ--¦(yâ--¦z) is computed as

[x, y, z] = ¼ ({x, y}z + z{x, y})

where if we employ {x, y} = g(x, y), the associator is then

[x, y, z] = g(x, y)z

Usually the anticommutator evaluates as a delta function, and so the associator with the Jordan product appears to be an operation with three elements that returns one element multiplied by this metric.

We might consider this in fermion space with V = C^N as the vector space for N particles, and V* the dual vector space. The vector space V contains elements of v = a^iv_i, for the index i = 1, ..., N and the dual space contains elements dual to these vectors, which can be thought of as 1-forms ω^i, so that v* = b_iω^i. In this way an elements of W = V⊕V* = (v, v*). The anticommutator is then {e_i, ω^j} = δ_i^j. We think of e_i as the same as a^† for the ith particle and ω^i as the annihilation operator a for the ith particle. From this of course any state my be built up as a sum of products of these operators e_i acting on the vacuum.

We are then able to look at the metric/anticommutator g(x, y) with x = a^ie_i + b_iω^i and y = c^ie_i + d_iω^i

g(x, y) = {x, y} = {a^ie_i + b_iω^i, c^je_j + d_jω^j} = a^id_jδ_j^i. + c^jb_iδ_i^j.

This then gives the associator

[x, y, z] = (a^id_i + c^ib_i)z.

The term a^id_i + c^ib_i = as a form of inner product. We may then of course extend this to V --- > V⊕R, where the states have a scalar and vector/form component to them as seen in Clifford algebras. This J(W) = V⊕R, J for Jordan product is such that for (x, a) and (y, b) then

(x, a)â--¦(y, b) = (ax + by, ab + )

For a change of signature on the second component, or equivalently (x, a) --- > (x, Ja) for J the pseudocomplex matrix, - ab is then the Lorentzian metric interval and vanishes on the light cone.

We may then consider what this means for the e6(-26). This has mcs f4 ⊕ R^26. The R^{26} is treated as the Lorentz space for the bosonic string. These 26 dimensions have one dimension for time, the direction of the string motion and 24 dimensions for the transverse polarizations of the string. The transverse directions correspond to the rank of the gauge group, which is then SO(24). The Regge trajectory α'M = k(N - 1), k = 1 or 4 for open and closed strings. We ignore the negative states as an unstable vacuum or tachyon and we then have the first massless states |Ω^{ij}> = a^ia^j|0>. The operator a^i is for -1. The antisymmetric part of |Ω^{ij}> corresponds to a gauge field, the antisymmetric part due to gravity, and the trace forms the dilaton and axion fields. The gauge field is SO(24), and the symmetric part contains independent 300 elements, which can form an SO(25), such that the traceless part is again SO(24), and the remaining 24 elements of the trace can form an SU(5) group. In fact the α'M = 2 states, a^i_{-3}|0>, a^i_{-1}a^j_{-2}|0> and a^ia^ja^k|0> give 24, 576 and 2600 and combine to give the 300 of SO(25).

This is not a physically reasonable theory, largely from experimental grounds. We then consider the replacement of the Virasoro algebra for the bosonic string with D = 26 anomaly cancellation with the superVirasoro algebra with the D = 10 anomaly cancellation term. In this case we now have a theory with 10 Dirac matrices. I of course am not going into the complexity of this, which is fairly extensive. With 10 Dirac operators we are now in the domain of SL(2, O) ~ SO(9, 1). Weyl-Majorana spinors or Î" in D = 10 have 16 components. With this we can now extract the SUGRA out of this theory of entanglement of states. It is recalled that this construction with E6(-26) and F4 does not pertain to field theory, but instead points to the coset construction of quantum states. The coset e6/f4 is ½ SUSY BPS BH, but the gauge theory and gravity emerges from this.

This connects back to the associator. In the context of the Jordan matrix this contains fermion theory. The above space W = V⊕V* = (v, v*) transforms by SL(2N, C). For N = 2 this is SL(4,C), for N = 4 this transforms by SL(8, C) ~ SL(4, N) and for N = 8 we have SL(4, O). SL(4, O) ~ SL(16, C) is the 8qubit SLOCC group. It appears that we can connect up SUGRA this way.

LC

I have another question on all of this, which might in some ways be an insight. The invariant of E6(6) is this cubic invariant. The matrix

J3 = q_{ab}П‰^{bc}q_{cd}П‰^{de}q_{ef}П‰^{fa}

is written with the 54 = 27⊕27, for 27 = 1⊕26, where these are sp(8). The action is then S = π sqrt{|J3|}. The determinant of this matrix is a 3-volume that has zero volume and if its bounding area ∂V = ∂J3/∂q = 0 ½ of supersymmetries are preserved. If the volume is zero and the area nonzero then ¼ SUSYs preserved and if both are nonzero then 1/8.

I have been thinking that this has a connection to the Weyl curvature and its action. Since this has gotten into conformal relativity this is a time where I can use this forum as a sound board to present some physics involving the Weyl curvature. Consider the Lagrangian

L = (1/2)sqrt{-g}C_{abcd}C^{abcd}

and the variation with respect to the metric g_{ij} is then

sqrt{-g}^{-1}ОґL/Оґg_{ij} = B^{ij} = 2C_{aijb}^{;ab} + C_{aijb}R^{ab} = 0.

This is comparatively simple for a conformally flat spacetime where R^{ab} = 0. An Einstein spacetime with R^{ab} = О»g^{ab} is also a solution to the equation. This is the Bach equation.

Suppose these two actions are the same. This then equates a quadratic function with a cubic function. The above functional differential with the metric is essentially the same as a derivative with respect to the 26 charges of the sp(8). The metric is then given by the 26 electromagnetic charges. This then gives action as an elliptic curve. This seems to be a physical way of looking at ideas of using Shimura and symmetric varieties in physics.

LC

To introduce more physical ideas into this subject, in particular magic SUGRA I have been musing over this for a while. I think it is important to keep in contact with physical ideas. I do at the end here consider the prospect that this connects with J3 and E6. This has to do with the general uncertainty principle and the tension on the brane of our universe. If the tension is zero then the fluctuations in the spacetime are tiny. However, if we are observing spacetime under a high magnification of a Heisenberg microscope then we are imposing high transverse momenta and the tension is huge. Under those conditions the fluctuations are quite large.

We might think of the situation like this. Suppose the uncertainty principle is

О"pО"x = Д§ + F(x)О"xО"t.

This is a version of the uncertainty principle modified for string theory

О"pО"x = Д§ + ОјО"x^2,

where ОјО"x = F(x)О"x/c = F(x)О"t. The constant Ој has been replaced with a force that depends upon distance or a region in space/spacetime. I have though a term F(x) that is written to indicate force. This leads to the spread in the width of the uncertainty

О"x = О"pc/2F(x) +/- c sqrt{О"p^2 - 4Д§F(x)/c}/2F(x)

From О"pО"x = Д§ + F(x)О"xО"t we can divide by О"t with О"x/О"t ~ c to get

F(x)О"x = О"pc - Д§/О"x

where the LHS is the work equivalent due to this force. We may write this according to О"W = О"pc - Д§/О"x, so that the work is seen to increase with О"p and decrease with О"x. This may also be expressed to connect the momentum-position uncertainty with a "quantum work-time uncertainty" so that

О"WО"t = О"pО"x - Д§.

For F = 0 then О"pО"x = Д§ and the LHS is zero. This illustrates how О"WО"t is associated with an increase in uncertainty.

If you think this looks similar to the entropy force of gravity results you are not too far off! The entropy force of gravity concerns the transverse displacement of a holographic screen. Here we are concerned with the stretching or longitudinal motion along the screen. The entropy force of gravity F = T∇S determines work as

О"W = FО"x = T∇SО"x = TО"S.

We employ E = (1/2)NkT which equals Mc^2, We then see that temperature is T = Mc^2/Nk. This is inserted into the equation for the entropy force

FО"x = (2Mc^2/Nk)О"S

The motion of the holographic screen by a distance О"x = Д§/mc results in

F = (Mmc^3/Nk Д§)О"S

and we use N = A/4L_p^2 = πR^2/(Għ/c^3). We then have that

F = 2GMm/(kR^2)О"S

Where we then assign the unit of entropy О"S = k/2 to get the result.

The NS5-brane has spacetime of four dimensions on a D-brane sandwiched between two other branes embedded in 5 dimensions. This sandwich is contained in an anti-de Sitter spacetime (AdS_5) of five dimensions. In four dimensions of spacetime a static gravity field in 3-dimensional space has the property that a two-dimensional surface bounding a mass will have the same number of line of flux through it independent of the radial size of the surface. Thus the force F multiplied by the surface area 4ПЂR^2 is a constant and equal to the gravity constant in 4-dimensions multiplied by the masses

GMm = 4ПЂFR^2,

Which leads to Newton's law of gravity

F = GMm/R^2

The 4ПЂ has been "absorbed into the definition of G. In five dimensions the surface bounding the mass is not two dimensional but three dimensional, which gives us the law

G_5Mm = 4ПЂFR^3

And so the force law in five dimensions is

F = G_5Mm/R^3.

The G_5 is the gravitational constant in five dimensions, and it reflects how the branes are "warped" in the embedding space by interacting with each other. G_5 in these units is Nm^3/kg^2 (N = Newton, m = meter, kg = kilogram). The relationship with the standard gravitational constant G with units Nm^2/kg^2 is that G_5 = GL, where L is a unit of length.

The brane tension, or equivalently the tension of strings attaching them together, is T = kL, for k = constant. The string length is L = 1/sqrt{R}, for R the curvature of the brane which we write as L = L_0^2/s, for L_0 the zero tension string with F(x) = 0 and s the "warp length," which is something that Randall worked out.

G_4 = (4ПЂ/3)О»G_5^2 = G_5/s

It is kind of neat. The big question I have is whether or not our four dimensional spacetime is not in fact on the conformal boundary of the AdS_5. This does go somewhat into the idea about changing gravity, which appeals to the physics of our brane in or on the AdS_5 and how gravity is related to the brane tension.

We also see that the tension depends upon the length L and for the tension at a minimum the Newton gravitational constant G_4 is small. For T or О» large the gravitational constant is huge. We then have a renormalization group flow for the gravitational constant. Also in addition G_4 = (Planck area)c^3/Д§, and for the tension large this is larger and the area of a Planck unit is larger. This is due to an impulse F(x)О"x = О"pc - Д§/О"x that "stretches" the horizon or Planck unit of area. This force is what adjusts the warp length.

Quantum gravity involves noncommutative geometry. In particular there are coordinates of spacetime that are not commensurate or which have an uncertainty relationship. This can be seen with the argument Bohr gave at the Solovay conference in 1930. Einstein argued that a box containing a photon could have a door opened and the mass of the box plus photon measured. This would get around the uncertainty principle. However, the uncertainty in momentum is О"p = TgО"m, for T an interval of time, g the acceleration of gravity and О"m the uncertainty in mass. The spread in distance is О"x = cО"t, and the product in О"pО"x = TgcО"mО"t = (Tg/c)О"mc^2О"t. With О"pО"x = Д§ then О"mc^2О"t = Д§ and the uncertainty in time is О"t = (Tg/c^2)О"x.

We may now consider this with the uncertainty in time and radial distance with the metric coefficients. We have then that t' = t sqrt{1 - 2GM/r'c^2}, and for О"t = t" - t' measured between two small radial distances in a Schwarzschild metric small enough then

О"t = t(sqrt{1 - 2GM/r"c^2} - sqrt{1 - 2GM/r'c^2}) ~ t(GM/c^2)(1/r" - 1/r').

For r" - r' = О"r then

О"t ~ -tGMО"r/(rc)^2,

We now consider the spread in the potential V = -GM/r, О"V = GMО"r/r^2 - GО"M/r. This leads to

О"t ~ (t/c^2)(GО"M/r + О"V)

We now let cО"t = О"x, for x a distance related to the radius. The time t is equated to GM/rc, as above. In addition the fluctuation in the radius and mass are for the Schwarzshild radius equal, so the uncertainty in the potential GО"M/r =~ О"V. This gives us

О"x ~ 2GО"M(t/rc) = 2GО"EV(t/rc^3).

A multiplication by О"t is

О"xО"t ~ 2GО"EО"t(t/rc^3) = 2GД§(t/rc^3).

The t and r are equated and this leads to

О"rО"t = 2GД§/c^4 = 8.5x10^{-75}ms.

This is a Planck unit of area divided by c. From О"pО"x = Д§ + F(x)О"xО"t it is easy to see that

О"pО"x = Д§ + (GД§/c^4)F(x)

= Д§(1 + GF(x)/c^4)

We then see that the uncertainty in the spacetime variables is due to this force. This force is what is responsible for the brane tension. If this tension is zero or minimal then the fluctuations on our brane are minimal. Measurements across huge distances are using a telescope to measure across vast distances. An extreme high energy experiment imparts huge transverse momenta/energy and increases the brane tension.

The entropy force of gravity involves motion of a holographic screen along a direction perpendicular to the screen. Here the concern is with the dynamics transverse to the motion derived by the entropy force of gravity. This means that we have a general problem with fluid dynamics. This seems to suggest that maybe the general form of fluid dynamics is given by the 3x3 matrix, maybe diagonalized by the E6.

LC

The paper http://arxiv.org/pdf/1407.5977v2.pdf by Bonezzi, Corradini and Waldron is pretty interesting. Much of the start seems to hinge around

1/sqrt{det(H)} = exp(-ВЅ Tr log(H)).

We have then that

I* = ½ Tr(H) = ½ ∫_0^∞ ds/s exp(sH).

This is an elliptic integral that diverges at s = 0. We then set the lower integration s = 1, which corresponds to the area of the string world sheet with s = t/x, for t the time and x the spatial coordinate on the sheet. The s = t/x is a form of the S-duality of the string, or a form of t and s channels in Mandelstam variables are equivalent.

The same seems plausible with the Grassmannian coordinates c. The equation 13

∫Dχexp(-Tr∫d^3cχF_A) = sum_A'δ(A - A')det[δF_A/δA]

is integrated over the space of Grassmannian variables. This could be represented with a string target map. The Grassmannian coordinates c^a = ОґОё^a are anticommuting variables and the target map is given by a determinant of the c^a's, similar to a Slater determinant of fermion states, that defines the string world sheet.

LC

I am looking into Motives. I have found the following http://www.ams.org/notices/200410/what-is.pdf to be of some help. The Wikipedia article assumes too much understanding of the preliminaries.

I do see that this has some connection to groupoids and magma constructions. I also think this has something to do with nature of the Leech lattice in J3(O). The 24-cell, indeed the very number 24, keeps cropping up. The F4 group has as its roots space the 24-cell, and I have been working on something involving F4 and its B4 reduction with respect to projective varieties of rays. These as putative hidden variables fail to define a Ω [φ, ψ] function as unit for a ψ-epistemic or 0 for a ψ-ontic theory. The F4 also diagonalizes J3(O).

There seems to be some sort of relationship of the sort

Leech lattice ---------------------------- modular forms

Golay Code ---------------------------- Elliptic curve (exotic varieties, Goppa codes)

Sporadic groups ------------------------ Special functions ( Jacobi θ-functions, Ramanujan mock Θ-function)

The space of quaternions H = {pm 1. pm i, pm j, pm k} ---> SL(2, 3) is 24 dimensional. In the sequence

F4: B4 --- > F_{52/16} ---> OP^2

the F4 when modded by the quaternion space, is then the space of J3(O) on the RHS minus the pairs of roots which is an elliptic curve.

My main interest is to connect the M_{24} in this system, but this is just one rung on the whole sporadic group ladder leading up to the FG or monster group.

LC

This mathematics of motives is really a set of varieties that connect cohomological systems. I had a course in algebraic topology, I have a masters in math, and have used it some, though again I am more of a physicist.

What is interesting is that this approach with algebraic varieties is something I worked with looking at nonlocal hidden variables in QM. These have in four dimensions 6 collineations of 24 vectors, 48 in total with their duals, and this is the root space of F4. In three dimensions this is reduced to 3 collineations that share a normalizer, and there is an automorphism that connects these. As a result there is no "frame independent" manner in which the correlation function can be evaluated. There is then no manner by which we can establish the epistemic or ontic nature of the quantum. This puts a lot of, and I hope all, quantum interpretations into the junk heap. These automorphisms are on projective varieties. This system of 3 collineations is a form of Mermin's trick with 9 observables.

The full spacetime form involves the F4, and this is the automorphism of J3. So this seems to be meshing together in some physical manner. The F4 is the real eigenvalued diagonalizer of the "3x3." E6, and its variants, are involved with "real plus imaginary" eigenvalues, which seems to connect to quantum field theory in curved spacetime. The imaginary evs appear to be Bogoliubov coefficients for quantum fields in curved spacetime. E7 and E8 then connects with quantum gravity where

e8 --- > 1 ⊕ 56 ⊕ (133 ⊕ 1) ⊕ 56 ⊕ 1

for the 133 ---- > e7, and the 1 a graviton and the other 1 the Taub-NUT mag-graviton dual, and the E7 corresponds to states that connect with gravity.

There is the general 3x3 of O's or the vector E8 plus the two spinor E8s in a SUSY setting. The systems for evaluating this matrix then looks like the hierarchy F4, E6, E7 and E8, which takes us from standard quantum field theory, to QFT in curved spacetime and up to I think quantum gravity.

There seems to be a lot here I yet do not quite understand. I am trying to rely on physics to fill in the gaps. The modular functions, such as the Jacobi θ-functions obey diff-eqs that are a form of the heat equation, or Schrodinger equation in a complex form, which can evaluate a heat kernel. I am not sure but I have thought these are some sort of underlying unital structure. The modular functions (forms) the conserve quantum information, which is another way of looking at SLOCC algebras.

Cheers LC

The paper by Yang-Hui He & John McKay centers initially upon the linear fractional group. This is of course an aspect of T-duality in string theory. It is also a part of what I have worked with respect to quantum phases of the event horizon. A highly accelerated observer that hovers closer to the horizon must do so under increasing acceleration. The distance to the horizon is d = c^2/a, where near the Planck acceleration a =~ 10^{53}m/s^2 the observer is on the holographic screen of the stretched horizon. In this limit there is a quantum critical phase transition. The phase transition changes the horizon from a conducting phase to a symmetry protected topological phase similar to a Mott insulator.

The SPT is an energy gapped phase at zero temperature, which in our case is near the Hagedorn temperature, where the heat capacity of event horizons is negative. The SPT states with a symmetry given by group G have topological orders given by the cohomology H^2(G, U(1)). For the case d = 2 the 2-space plus ime model with the Euclideanized G = SO(3) gives the 2 + 1 spin Hall effect and he time reverse symmetry group Z. The projective representation of this group is found by the quotient with the normalizer of that group PG = G/N with the map π : G = PG that defines a bundle of lift elements. These elements λ obey for g \in G the rule λ (g; g') = σ(g; g') λ (g) λ (g'), where σ(g; g') is a Schur multiplier. The projective representation of the group \mods out rq' the action of this normalizer. This is a cocycle in the cohomology H^2(G; N) with the normalizer N = U(1).

The Schur multiplier is a way in which the projective realization of a group can be represented according to the second cohomology that is an abelian group that defines a covering or line bundle. The projective representation is a homomorphism into the projective linear group PGL(n, K),specifically for K = R or C, and for n = 2. A field F is mapped into K, the group C and B have the sequence 1 --- > F --- > C --- > G --- > 1 for C the centeral of G and this sequence is the central extension of the group G. The following diagram ensues

F ---- > C ------------- > G

| | |

K --- > GL(n, K) --- > PGL(n, K)

The group SO(2, 1) ~ SU(1; 1) is homomorphic to SL(2, R) x SL(2, R)/Z_2. The linear fractional group SL(2, Z) the gives the braid group as

B_3 ------------ > PSL(2, Z)

| |

SL(2, R) ------- > PSL(2, R),

where vertical lines are down arrows. The braid group is then the central extension of the linear fractional or modular group PSL(2, Z). The braid group B_3 then contains a center which under this map gives PSL(2, Z) ~ B_3/C. This center is under the correspondence between C and G = B_3 with the normalizer N = U(1) projective realization PSL(2, R) = SL(2, R)/U(1) a set of elements that definnes the modular group as a projective realization of the braid group.

I like trying to make contact with physics, and I have this idea that physics found in things like condensed matter or quantum optics can be general principles that occur in other forms. I think in addition that this physics with accelerated frames means that physics in inertial and accelerated frames are ultimately equivalent.

LC