Deferential Geometry (by Garrett Lisi) -

Standard model and gravity

Garrett Lisi FQXi

http://deferentialgeometry.org

Standard model and gravity in one big connection

Correct interactions and charges from curvature:

(1; ) (1; ) (8 ) A!+

= H+

+G++ ï

!=

! H ++ ï

!

"

G "+

"2 so

+7 + so

+7 + C

!# 8

=

2

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

! W 21

L++i 3

+

iW 1+ "W 2

+

" e ! 41

L+0

e ! 41

L+

$

+

W i 1+ +W 2

+

! W 21

L+"i 3

+

e ! 41

L++

e ! 41

L+

$

0

e ! " 41

R+

$

0

e ! 41

R+

$

+

! B 21

R++i+

e ! 41

R++

e ! 41

R+0

! B 21

R+"i+

"! L

e! L

"! R

e!R

B i+

u!

rL

d!

rL

u!

rR

d!

rR

G 3"iB++i 3+8

+

G i 1+ +G2

+

G i 4+ +G5

+

u!

g

L

d!

g

L

u!

g

R

d!

g

R

G i 1+"G

2+

G 3"iB+"i

3+8+

G i 6+ +G7

+

u!

bL

d!

bL

u!

bR

d!

bR

G i 4+"G

5+

G i 6+"G

7+

G 3"iB+"

2ip3

8+

3

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

F!+!+

=

=

d+A!+

+ A!+A!+

H H) G G) ï ï G) (d++

+H++

+ (d++

+G++

+ (d+ !

+H+ !

+ ï! +

Gravitational part of bosonic connection

Using chiral (Weyl) representation of Cl(1,3) Dirac matrices:

Spacetime frame and spin connection:

Note algebraic equivalence: Cl (1; ) l (1; ) o(1; )

C(4 ) # 4

# 0

# 0"

=

=

$1 % 1 =

! 1

1 "

# #0 " =

! "$ "

$ "

" #%

#"%

=

=

$ i 2 % $% =

! "$ %

$ %"

# #" % =

! "i& $ "%' '

"i& $ "%' '

"

!++ e

+=

=

=

dx ! # (e ) # a+ 2

1a("

(" + dxa+ a

((

" ("! $ ! & $ ) 0"

+ " "i2

"%+ "%' '

(e $ ) 0+" e%

+ %

(e $ ) 0+

+ e%+ %

(! $ ! & $ ) 0"+ " "

i2

"%+ "%' '

#

(1; )

2

4

! L+

e L+

e R+

! R+

3

5 2 Cl+

1+2 3

1+2 3 = C 2 4 = s 4

Bosonic connection

Clifford bivector parts:

indices:

H +

=

=

! 21!++ 4

1 e+

+B++W

+=

2

6 6 6 4

! W 21

L++i 3

+

iW 1+ "W2

+

" e ! 41

L+ 0

e ! 41

L+

$

+

iW 1+ +W2

+

! W 21

L+"i 3

+

e ! 41

L++

e ! 41

L+

$

0

" e ! 41

R+

$

0

e ! 41

R+

$

+

! B 21

R++i+

e ! 41

R++

e ! 41

R+ 0

! B 21

R+"i+

3

7 7 7 5

h # (1; ) (1; ) (8 ) dxa+ 2

1a)*

)* 2 so+

7 = Cl+

2 7 & C+

# 8

!+

! e+

=

=

! # dxa+ 2

1a("

(" ï spin connection

# dxa+

e( a)(!!

(!

8 > <

> :

e+

!

=

=

dxa+

e( a)(#( ï frame (vierbein)

# !!!

# ! +! 0

=

=

"! ! ) ( 5+i 6

! ! ) ( 7+i 8

$

ï Higgs

!! M = " 2

B +

W +

=

=

B # "dxa+ 2

1a

%56" #78

&ï # electroweak gauge fields

W # W W # " 21 1+

%67 + #58

&" 2

1 2+

%" #57 + #68

&" 2

1 3+

%56 + #78

&

; ; ; 0 ' a b ' 3 0 ' ( " ' 3 5 ' ! ï ' 8

Curvature of bosonic connection

Modified BF action over 4D base manifold:

F ++

=

=

=

=

H H ! d++

+H++

H+

= 21!++ 4

1 e+

+B++W

+

( ) M

'21 d+!++ 2

1!+!+

+ 116

2 e+e+

(

[ ; ] ! ! B ; ] +'

41%d+e++ 2

1 !+

e+

&" 4

1 e+

%d+

+ [++W

+!&(

B W W +'d++

+ d+ +

+W+ +

(

R M T! D! F 21%+++ 8

1 2 e+e+

&+ 4

1%++

" e++

&+%

B++

+ FW++

&

Fs++

+ Fm++

+ Fh++

ï spacetime #("

ï mixed #(!

ï higher #!ï

2 + 67 58 2 + 57 68 2 + 56 78

S =

=

B (H; ) B B B # $B $B

Z )++F+++ !

+B++

*=

Z )++F++"

4

1s

++

s++

+Bm++

m++

+Bh++

h++

*

F # F $F F $F

Z )s

++

Fs++

" +4

1m++

m++

+4

1h++

h++

*

Gravitational action

S s

S s

S s

=

=

=

B F (B ) B R M B B #

Z )s

++

s++

+ !s s++

*=

Z )s

++ 2

1%+++

8

1 2 e+e+

&"

4

1s

++

s++

*

+B R M # s++

! Bs++

=%+++

8

1 2 e+e+

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

R M R M # F F # 4

1Z )%

+++

8

1 2 e+e+

&%+++

8

1 2 e+e+

&"*=

Z )s

++

s++

"*

RR# # )++++

"*= d

+

)%!+d+!++

3

1!+!+!+

&"*

ï Chern-Simons

# 1

4!

)e+e+e+e+

"*= e

"ï volume element

R R )e+e+++

#"*= e

"ï curvature scalar

M "

12

Ze"R "( + 2 ) cosmological constant: " =

4

3 2

F # F $F F $F s++

Fs++

" +4

1m++

m++

+4

1h++

h++

Why this Lie algebra

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

For three generations:

BIG Lie algebra:

A!+

= H+

+G++ ï

!=

! H ++ ï

!

"

G "+

"

M "

12e"R "( + 2 ) cosmological constant: " =

4

3 2

=

2

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

! W 21

L++i 3

+

iW 1+ "W 2

+

" e ! 41

L+0

e ! 41

L+

$

+

W i 1+ +W 2

+

! W 21

L+"i 3

+

e ! 41

L++

e ! 41

L+

$

0

e ! " 41

R+

$

0

e ! 41

R+

$

+

! B 21

R++i+

e ! 41

R++

e ! 41

R+0

! B 21

R+"i+

"! L

e! L

"! R

e!R

B i+

u!

rL

d!

rL

u!

rR

d!

rR

G 3"iB++i 3+8

+

G i 1+ +G2

+

G i 4+ +G5

+

u!

g

L

d!

g

L

u!

g

R

d!

g

R

G i 1+"G

2+

G 3"iB+"i

3+8+

G i 6+ +G7

+

u!

bL

d!

bL

u!

bR

d!

bR

G i 4+"G

5+

G i 6+"G

7+

G 3"iB+"

2ip3

8+

3

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

(1; ) (1; ) (8 ) A!+2 so

+7 + so

+7 + 3 $ R

!# 8 = ?

n 8 8 4 48 = 2 + 2 + 3 $ 6 = 2

Real simple compact Lie groups

rank group a.k.a. dim name

special unitary group

odd special orthogonal group

symplectic group

even special orthogonal group

G2

F4

E6

E7

E8

"E8 is perhaps the most beautiful structure in all of mathematics, but it's

very complex."

— Hermann Nicolai

r A r SU (r ) + 1 r(r ) + 2

r B r SO(2r ) + 1 r(2r ) + 1

r C r Sp(2r) r(2r ) + 1

r > 2 D r SO(2r) r(2r ) " 1

2 G 2 14

4 F 4 52

6 E 6 78

7 E 7 133

8 E 8 248

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:

Lie brackets between generators (structure constants):

brackets act as multiplication between dimensional Cl(16)

Clifford bivectors, , and positive chiral, dim column spinors, :

E =

2

B b # Q +# =2

1 )*

)*

(16)+ + ïaa+

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

# ; +

)*

(16)+##+

(16)+,

# ; +

)*

(16)+ Qa+,

Q ; +

a+ Q

b

+,

=

=

=

2 # # # # -" ,)#

*+

(16)+ + ,)+ *#

(16)+ + ,*# )+

(16)+" ,*+ )#

(16)+.

# Q Q Q %

)*

(16)+&b

c

%a+&c

b

+ = #)*

(16)+a+

" # # %

(16)+)*&ab )*

(16)+

Lie(E8) 120

B 128 #

B ; [ 1 B2]

B; [ #]

# ; [ 1 #2]

=

=

=

B B B1 2 "B2 1

B+#

$ # "#y1

+2

2 o(16) s

2 S(16)+

2 o(16) s

E8 generator conversion

Build new generators from old ones:

With these basis generators, the elements are:

Lie(E8)

H )*

G )*

# I)*

# IIab

# abIII

=

=

=

=

=

#)*

16 +( )

#16 +( )

)+8 *+8( )( )

#16 +( )

) *+8( )

Q+

16 a"1 +b( )

Q+

16 a"1 +b+8( )

=

=

=

=

=

#)*

(8)+% 1

P+

8( )% #

)*

(8)

#)

(8)+% #

*

(8)

qa+% q

b

+

qa+% q

b

"

2

2

2

2

2

o(8) s + % 1

o(8) 1% s

v(8)+ % v(8)

S(8)+ % S(8)+

S(8)+ % S(8)"

o(8) = s H

o(8) = s G

= SI

= SII

= SIII

Lie(E8)

E =

=

2

H +G+#I +#II +#III

h H g G # # # 2

1 )*)* + 2

1 )*)* + ï

I

)* I)*

+ ïIIab II

ab+ ïab

III abIII

so(8) o(8) H + s G + SI + SII + SIII

E8 triality structure

The brackets between elements in the various parts:

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

Lie(E8)

H ; +

1 H2

,

G ; +

1 G2

,

H; +

#I

,

H; +

#II

,

H; +

#III

,

G; +

#I

,

G; +

#II

,

G; +

#III

,

=

=

=

=

=

=

=

=

H H H1 2 "H2 1

G G G1 2 "G2 1

H #I

H+#II

H+#III

#I G

"#II G+

"#III G"

# ; +

I1 #

I2,

# ; +

1II#2II

,

# ; +

1III

#2III

,

# ; +

I #II

,

# ; +

I #III

,

# ; +

II #III

,

=

=

=

=

=

=

# " 2%

I1#

I2T&H

"2 # # %

I1T

I2&G

# $ # "%

1II

+ 2II

T&H

# $ # "%

1II

T + 2II

&G

# $ # "%

1II

+ 2II

T&H

# $ # "%

1II

T " 2II

&G

# $ # "%

I++

II

&III

# $ # "%

I+"

III

&II

# $ # "%

II++

III

&I

H # G

E8 TOE

Build a real form of complex E8 by using instead of

. Then E8 TOE connection is:

something like

Cl (1; ) o(1; ) 2 7 = s 7Cl (8) o(8) 2 = s

A!+

= H+

+G++#

! I+#

! II+#

! III=

2

6 6 6 6 4

! W 21

L++i 3

+

iW 1+ "W 2

+

" e ! 41

L+0

e ! 41

L+

$

+

W i 1+ +W 2

+

! W 21

L+"i 3

+

e ! 41

L++

e ! 41

L+

$

0

e ! " 41

R+

$

0

e ! 41

R+

$

+

! B 21

R++i+

e ! 41

R++

e ! 41

R+0

! B 21

R+"i+

3

7 7 7 7 5 +

2

6 6 6 4

iB +G 3

"iB++i 3+8+

G i 1+ +G2

+

G i 4+ +G5

+

G i 1+"G

2+

G 3"iB+"i

3+8+

G i 6+ +G7

+

G i 4+"G

5+

G i 6+"G

7+

G 3"iB+"

2ip3

8+

3

7 7 7 5

+

2

6 6 6 4

" !

eL

e ! L

" !

eR

e !R

u !

rL

d !

rL

u !

rR

d !

rR

u !

g

L

d !

g

L

u !

g

R

d !

g

R

u !

bL

d !

bL

u !

bR

d !

bR

3

7 7 7 5 +

2

6 6 6 6 4

" !

(

L

( !L

" !

(

R

( !R

c !

rL

s !

rL

c !

rR

s !

rR

c !

g

L

s !

g

L

c !

g

R

s !

g

R

c !

bL

s !

bL

c !

bR

s !

bR

3

7 7 7 7 5 +

2

6 6 6 4

" !

'

L

' ! L

" !

'

R

' !R

t !

rL

b !

rL

t !

rR

b !

rR

t !

g

L

b !

g

L

t !

g

R

b !

g

R

t !

bL

b !

bL

t !

bR

b !

bR

3

7 7 7 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 symbols...

Gar@Lisi.com

http://deferentialgeometry.org

f10jg

Geometry of Yang-Mills theory

Start with a Lie group manifold (torsor), , coordinatized by .

Two sets of invariant vector fields (symmetries, Killing vector fields):

- (y) g (y) (y) g (y)

Lie derivative: [- ; ] -

Lie bracket: T ; T

Killing form (Minkowski metric): g C

Maurer-Cartan form (frame): (- ) T

Entire space of a principal bundle:

Ehresmann principal bundle connection over patches of :

(x; ) A (x) (y) @

Gauge field connection over :

G y p

LA

*

d+

= TA g -AR*

d+

= g TA

AR*

-BR*

= CAB

CCR*

[ A TB] = CAB

CC

AB = CAC

DBD

C

I+

= dyp+ p

R AA

E (M #GE

E+

*y = dxa

+ aB -L

B

*

+ dyp+

p

*

M

A(x) A (x) +

= $0$E+

*I+

= dxa+ a

B TB

BRST gauge fixing

+L under gauge transformation:

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

, anti-ghosts, B, partners, ., and BRST transformation:

This satisfies +L and ++ .

Choose a BRST potential, , to get new Lagrangian:

BRST partners act as Lagrange multipliers; effective Lagrangian:

"= 0 +A rC C A;

+= "

+= "d

+"++C,

+

C (G) !2 Lie

! g

:

" "

+A !+

+B !++

+. ! "

=

=

=

"rC + !

B; +++C!

,

0

+C ! !

+B !

:

"

=

=

" C; 21+!C!

,

. "

! "= 0

! != 0

# BA :

"=) :

"+

*

L # .A BrC 0"

= L"+ +

!

:

"= L

"+)" g+

*+) :

" + !

*

L [B ; ] Br C eff"

= L"

0++

A0+

+) :

"

0+ !

*

BRST extended connection

Replace pure gauge part of connection with ghosts:

BRST extended curvature:

Effective Lagrangian, with B :

Crazy idea: Fermions are gauge ghosts

L [B ; ] Br C eff"

= L"

0++

A0+

+)"

0+ !

*

A!+

= A0+

+ C!

F!+!+

=

=

; C C; d+A!+

+2

1+A!+

A!+

,= F 0

+++r0

+ !+

2

1+!C!

,

A A C A ; C; %d+

0+

+A0+

0+

&+%d+ !

++ 0+

C!

,&+

2

1+!C!

,

0:

"= B0

"+B

:

"

L B (A ; ) eff"

=) 0

:

"F!+!+

+ !"

0+

B0"

*

A 0+

C !

=

=

H ! +

+G+

=%2

1!++

4

1e+

+B++W

+

&+G

+

ï " !=%!+ e

!+ u

!

r;b;g + d!

r;b;g&

Massive Dirac operator in curved spacetime

Clifford basis vectors for Cl(1,3)

Clifford basis bivectors

Lie algebra basis elements (generators)

orthonormal basis vector components (frame,

vierbein)

spin connection components

; ; Yang-Mills gauge field components (connections)

Higgs scalar field multiplet

Grassmann valued spinor field multiplet

D ( =+ !)ï!= #( e( ()

a@ ! # ; ; T

'a + 4

1a"/

"/ +B W GaA

A

(ï!+ !ï

!

ï " !=%!+ e

!+ u

!

r;b;g + d!

r;b;g&

#(

# #(" = #( "

TA

e ) ( (a

!a"/

BaA Wa

A GaA

!

ï!

Clifford lgebra a

& -! ï

Matrices

Lie lgebraa

.%