Deferential Geometry (by Garrett Lisi) - Www.oloscience.com

Standard model and gravity

Garrett Lisi FQXi http://deferentialgeometry.org

Standard model and gravity in one big connection A=H +G +ï = !+ + + !

21

3 L +iW 2 !+ +

6 1 6 iW+ "W+2 6 6 1 6 " 4 e L !0 6 6 1 + $ 6 e ! =6 4 +L + 6 6 6 6 6 6 4

!

H++

1 2 iW + +W + 1 3 2 !+L "iW + 1 L !+ 4 e+ 1 $ L!0 4 e+

ï !

"

G+"

"

2 so (1; 7) + so (1; 7) + C! (8 # 8) + +

" 41 e R !$0 +

1 R !+ 4 e+

1 $ R!+ 4 e+

1 R !0 4 e+

u! gL d! gL u! gR d! gR

u! bL

4 5 iG + +G +

6 7 iG + +G +

"i 2i 8 3B + " p3 G +

"! L

7 7 e! L 7 7 r b 1 7 "! R u! R u! R R +iB 2 !+ + 7 7 r b 1 7 e! R d! R d! R R "iB 2 !+ + 7 7 iB 7 + 7 "i 3+8 1 2 4 5 7 iG 3B + +iG+ iG + "G + + "G + 7 1 +G 2 "iB "iG 3+8 iG 6 "G 7 7 iG + + 3 + + + + 5 d! rL

Correct interactions and charges from curvature:

F =+ d !+ A + !+ A !+ A

!+ !+

3

u! rL

= (d H+H H) + (d G+G G) + (d ï+H ï + ï G) ++ ++ ++ ++ + + + !

!

!

d! bL

Gravitational part of bosonic connection Using chiral (Weyl) C(4 # 4) representation of Cl(1,3) Dirac matrices:

#0 = $ 1 % 1 = #0" = #0 #" =

!

!

1

1

"$"

"

$"

"

#% = i $ 2 % $ % = #"% = #" #% =

!

!



%$"

%$"i&"%' $'

"

"i&"%' $'

Spacetime frame and spin connection: (" a1 a ( ! + e = dx ! # + dx (e ) #( (" a + + + 2 a + " # i "% 0" 0 % ("!+ $" " 2 !+ &"%' $' ) (e+ + e+ $% ) = (e+0 " e+%$% ) (!+0" $" " i2 !+"%&"%' $' ) 2 3 ! L eR + + 5 1+2 =4 2 Cl (1; 3) + eL ! R +

+

Note algebraic equivalence:

Cl1+2 (1; 3) = Cl2 (1; 4) = s o(1; 4)

"

Bosonic connection 21

3 + 2 !+L +i W

6 iW 1 "W 2 6 + + 1 1 H = 2! + 4+ e! + B +W =6 1 + + + + 4 " 4 e+L !0 1 $ L!+ 4 e+

1 2 iW + +W +

1 3 + 2 !+L "i W 1 L !+ 4 e+ 1 $ L!0 4 e+

1 " 4 eR !$0 + 1 $ R!+ 4 e+ 1 R +i B + 2 !+

1 R !+ 4 e+ 1 R !0 4 e+

3 7 7 7 5

1 R "i B + 2 !+

2 a 1 h )* # = dx 2 so (1; 7 ) = Cl (1; 7) & C (8 # 8) )* + 2 a + + +

Clifford bivector parts: a 1 ! (" # ! = dx (" + + 2 a

ï spin connection

8 a (e )( # > e = dx ï frame (vierbein) a ( <+ + # $ ( ! a 5 6 e ! = dx (ea ) ! #(! !+ = ("! +i! ) ï Higgs + + ! > : ! = ! #! !0 = (!7 +i!8 ) !! = "M 2 % & 1 a B = "dx B #56 " #78 ï # electroweak gauge fields + + 2% a & 1 2% & 1 3% & 1 1 W = "2W #67 + #58 " 2 W " #57 + #68 " 2 W #56 + #78 + + + + indices:

0 ' a; b ' 3

0 ' (; " ' 3

5 ' !; ï ' 8

Curvature of bosonic connection 1 1 F = d H + H H H = ! + e! + B +W 2+ 4+ + + + + + + + + + ' ( 1 1 2e e = 21 (+ d! + ! ! ) + M ï spacetime #(" 2 16 + ++ ++ ' % ( & % & 1 + 41 + d+ e + 21 [! ; e ] ! " e + d ! + [B +W ; !] ï mixed #(! 4+ + + + + ' ( + ++ dB + + dW +W W ï higher #!ï + + + % & 1% & % & 1 1 2 = 2 R + 8M + e+ e +4 T !"+ e D! + FB + FW + + +

+

+ +

+

+ +

= Fs + Fm + Fh + +

+ +

+ +

Modified BF action over 4D base manifold:

S= =

Z Z

) )

*

B F + !(H; B) = + + + + + +

+

Z

)

* 1 B F " Bs Bs # + Bm $Bm + Bh $Bh + + + + + 4 + + + + + + + + + + +

* 1 1 Fs Fs # + Fm $Fm + Fh $Fh + + + 4 + 4 + + + + + + + + "

Gravitational action Ss =

Z

)

*

Bs Fs + !s (Bs ) = + + + +

+ +

)

& 1 * 1% 1 2 Bs R + M + e+ e " Bs Bs # + 2 + + 8 4 + + + + +

& " 1 2 +Bs ! Bs = R + M + e+ e # + + + 8 + + +

1 Ss = 4

)

Z

)%

RR# ++ ++

%

Z

1 2 R + M + e+ e + 8 +

"

*

)%

=+ d

Ss =

e eR #

+++ +

" 12

Z

"

*

= e"R

e" (R + 2")

&

1 2 R + M + e+ e # + 8 +

& "* 1 ! d! + ! !! # +++ 3 +++

* 1) " e+ e+ e+ e # = e" + 4!

)

&%

pseudoscalar: # = #0 #1 #2 #3 * "

=

Z

)

Fs Fs # + + + +

"

*

ï Chern-Simons ï volume element ï curvature scalar

cosmological constant:

3 4

" = M2

Why this Lie algebra A=H +G +ï = !+ + + 21

3 L +iW 2 !+ +

6 1 6 iW+ "W+2 6 6 1 6 " 4 e L !0 6 6 1 + $ 6 e ! =6 4 +L + 6 6 6 6 6 6 4

!

!

H++

1 2 iW + +W + 1 3 2 !+L "iW + 1 L !+ 4 e+ 1 $ L!0 4 e+

ï !

"

G+"

"

" 41 e R !$0 +

1 R !+ 4 e+

1 $ R!+ 4 e+

1 R !0 4 e+

"! L

u! gL d! gL u! gR d! gR

u! bL

4 5 iG + +G +

6 7 iG + +G +

"i 2i 8 3B + " p3 G +

7 7 e! L 7 7 r b 1 7 "! R u! R u! R R +iB 2 !+ + 7 7 r b 1 7 e! R d! R d! R R "iB 2 !+ + 7 7 iB 7 + 7 "i 3+8 1 2 4 5 7 iG 3B + +iG+ iG + "G + + "G + 7 1 +G 2 "iB "iG 3+8 iG 6 "G 7 7 iG + + 3 + + + + 5 d! rL

d! bL

Note: Only one generation, and fermion masses not quite right.

A 2 so (1; 7) + so (1; 7) + 3 $ R! (8 # 8) = For three generations: !+ + + BIG Lie algebra:

3

u! rL

n =

28 +

28

+ 3 $ 64

?

= 248

Real simple compact Lie groups rank group

r r r r>2 2 4 6 7 8

Ar Br Cr Dr G2 F4 E6 E7 E8

a.k.a.

dim

name special unitary group

SU (r + 1) r(r + 2) SO(2r + 1) r(2r + 1) odd special orthogonal group Sp(2r) r(2r + 1) symplectic group SO(2r) r(2r " 1) even special orthogonal group 14 G2 52 F4 78 E6 133 E7 248 E8

"E8 is perhaps the most beautiful structure in all of mathematics, but it's very complex." — Hermann Nicolai

Triality decomposition of E8 John Baez in TWF90: ... we now look at the vector space so(8) + so(8) + end(S+) + end(S-) + end(V) ...Since so(8) has a representation as linear transformations of V, it has two representations on end(V), corresponding to left and right matrix multiplication; glomming these two together we get a representation of so(8) + so(8) on end(V). Similarly we have representations of so(8) + so(8) on end(S+) and end(S-). Putting all this stuff together we get a Lie algebra, if we do it right - and it's E8.

Pieces of E8 Pirated from GS&W, Superstring Theory: 1 2

E = B + # = b)* # (16)+ + ï a Qa+ )*

2 so(16) + S (16)+ = Lie(E8)

Lie brackets between generators (structure constants):

+

(16)+ # ; )*

+

(16)+ # #+

(16)+ #)* ;

Qa+

Qa+ ;

Q+ b

+

,

,

,

= = =

-

(16)+ (16)+ (16)+ 2 " ,)# # + ,)+ # + ,*# # *+ *# )+ c (16)+ b (16)+ + + + #)* Q Q = # Qa c a b )* )* (16)+ " # (16)+ # ab )*

%

%

& %

&

&

"

(16)+ ,*+ #)#

.

Lie(E8) brackets act as multiplication between 120 dimensional Cl(16) Clifford bivectors, B , and positive chiral, 128 dim column spinors, #: [ B1 ; B2 ] = B1 B2 " B2 B1 [B; #] = B + # [#1 ; #2 ] = " #y1 $+ #2

2 s o(16) 2 S (16)+ 2 s o(16)

E8 generator conversion Build new Lie(E8) generators from old ones: ( 16) +

H)* = #)*

G)* = # (16)+ #I)* = # #II = ab # III = ab

( )+8)( *+8) ( 16) +

(8)+

2 s o(8)+ % 1

= #)* % 1

= P+(8) % # (8) 2 1 % s o(8) (8)+

= #)

)( *+8) Q+ = 16( a"1) +b Q+ = 16( a"1) +b+8

%

)* (8) #*

qa+ % qb+

2 v (8)+ % v (8)

= s o(8)H = s o(8)G = SI

2 S (8)+ % S (8)+ = S II

qa+ % q "

2 S (8)+ % S (8)" = S III

b

With these basis generators, the Lie(E8) elements are:

E = H + G + #I + #II + #III 1 2

1 2

ab II III = h)* H)* + g )* G)* + ï )* #I)* + ïII # ab + ï ab # III ab I

2 so(8)H + s o(8)G + S I + S II + S III

E8 triality structure The Lie(E8) brackets between elements in the various parts:

+

+ , H1 ; H2 = H1 H2 " H2 H1 + , G1 ; G2 = G1 G2 " G2 G1

+ , H; #I = H #I + , H; #II = H + #II + , H; #III = H + #III

+ , G; #I = #I G + , G; #II = " #II G+ + , G; #III = " #III G"

#I1 ; #I2

,

#1II ; #2II

,

#1III ; #2III

,

+ +

=

=

=

%

T& 1 2 " 2 #I #I H % 1T 2& "2 #I #I G % T& 1 + 2 " # II $ # II H % T + 2 & 1 " # II $ # II G % 1 + 2 T& " # II $ # II H % 1 T " 2 & " # II $ # II G

+ , % & ++ #I ; #II = " #I $ #II III + , % & +" #I ; #III = " #I $ #III II + , % & ++ #II ; #III = " #II $ #III I

Note: H acts on #'s from the left and G acts from the right.

E8 TOE Build a real form of complex E8 by using Cl2 (1; 7) = s o(1; 7) instead of Cl2 (8) = s o(8). Then E8 TOE connection is:

A=H +G + #! I + #! II + #! III = + +

!+

something like 21

3 ! +i W L 2 + +

6 1 6 iW+ "W+2 6 6 " 1e ! 4 4 +L 0 1 $ L!+ 4 e+

2

"! eL

6e 6 !L +6 e 4 "! R e! R

1 +W 2 " 1 e !$ 1 e ! iW R 0 4 + R + 4 + + + 1 1 1 3 $ ! "i W e ! L R + 4 e+ R !0 2 + 4 + + 1 1 L !+ R +iB 4 e+ 2 !+ + 1 1 $ e ! L 0 R "iB 4 + 2 !+ +

u! rL d! rL u! rR d! rR

u! gL d! gL u! gR d! gR

u! bL d! bL u! bR d! bR

3

2

"! (L

6( 7 6 !L 7 7 + 6 6 "( 5 4 !R

(R !

3 2 iB + 7 6 "i 3+8 iG 1 "G 2 4 "G 5 +iG+ iG 7 6 3B + + + + + 7+6 1 +G 2 "iB "iG 3+8 iG 6 "G 7 7 4 i G + + 3 + + + + 5

c! rL s! rL c! rR s! rR

4 5 iG + +G +

c! gL s! gL c! gR s! gR

c! bL s! bL c! bR s! bR

3

2

6 7 iG + +G +

3 7 7 7 5

"i 2i 8 3B + " p3 G +

"! 'L

t! rL

t! gL

'! R

b! rR b! gR b! bR

t! bL

3

7 6 ' br bg bb 7 7 !L !L !L !L7 7 + 6 6 7 ' r g b 7 4 "! R t! R t! R t! R 5 5

Discussion What I just did: All gauge fields, gravity, and Higgs in an E8 connection, with fermions as BRST ghosts. To do: Particle assignments not perfect yet. Get the CKM matrix. Might just not work. Where does the action come from? Symmetry breaking. Natural explanation for QM. What this E8 theory means for LQG: Modified BF gravity (MM) is favored — intimate frame and Higgs. Flexible as to what (or how) spacetime base manifold happens. Keep up the good work! Extending LQG methods to E8 gives a TOE. E8 f10jg symbols... [email protected] http://deferentialgeometry.org

Geometry of Yang-Mills theory Start with a Lie group manifold (torsor), G, coordinatized by y p . Two sets of invariant vector fields (symmetries, Killing vector fields): * - LA (y) + dg

= TA g (y)

Lie derivative:

* * [-AR ; -BR ]

Lie bracket:

= [ TA ; TB ] =

* -AR (y) + dg * C CAB -CR CAB C TC

= g (y) TA

Killing form (Minkowski metric): gAB = C Maurer-Cartan form (frame):

I = +

DC

C

AC BD dy p (-pR )A TA +

Entire space of a principal bundle: E ( M # G Ehresmann principal bundle connection over patches of E: *

E (x; y ) =

+

* a B L dx A (x) (y) a + B

+

* p dy @p +

Gauge field connection over M :

A(x) = +

* $ $0 + EI +

a B = dx A (x) TB a +

BRST gauge fixing

+

+L = 0 under gauge transformation: "

+A = "rC = "d C " A; C + + + +

,

Account for gauge part of A by introducing Grassmann valued ghosts, + :

C! 2 Lie (G)g, anti-ghosts, B , partners, . , and BRST transformation: ! " " + , 1 +A = "rC +C = " 2 C; C! !+ + ! ! ! ! : + , +B = B; C +B = . !+ + ! !" " +

+

+. = 0 !"

This satisfies +L = 0 and ++ = 0. !" ! ! :

)

:

*

= BA Choose a BRST potential, # , to get new Lagrangian: " "+ L"0

:

)

*

)

:

=L + +! # =L + .A + BrC " " " " g "+ ! +

*

BRST partners act as Lagrange multipliers; effective Lagrangian:

L"eff

=

0 0 L [B ; A ] " + + +

+

)

:

0 Br C " + !

*

BRST extended connection Replace pure gauge part of connection with ghosts:

BRST extended curvature:

0 A = A + C! !+ +

, , 1+ 1+ 0 0 F =+ d !+ A + !+ A ; !+ A = F+ + r C! + C; C! !+ + ! 2 2 + !+ =

%

&

0 d A ++

Effective Lagrangian,

Crazy idea:

%

+

,&

0 0 0 +A A + d C + A ;C + + + + ! + ! : : 0 0 with B = B" + B : " " ) :0 * eff 0 0 L" = B" !+ F +! (A ; B ) " + " !+

, 1+ C; C 2 ! !

Fermions are gauge ghosts

0 A +

& 1 =H +G = ! + + e! + B +W +G + + + + + 2+ 4 %

%1

C! = ï = "! + e! + u! !

r;b;g

+ d!

r;b;g

&

Massive Dirac operator in curved spacetime (D = + !) ï = # ( (e( ) !

#("

#( = #( #" TA

( e( ) a !a "/

a

'

(

1 @a + !a "/ #"/ + B; W; GaA TA ï + ! ï 4 ! !

Clifford basis vectors for Cl(1,3) Clifford basis bivectors

Lie algebra basis elements (generators) orthonormal basis vector components (frame, vierbein) spin connection components

Ba A ; Wa A ; GaA Yang-Mills gauge field components (connections) ! Higgs scalar field multiplet ï Grassmann valued spinor field multiplet !

Clifford algebra &

ï! Matrices

Lie algebra % .