Standard model and gravity

http://deferentialgeometry.org Garrett Lisi FQXi

Everything as a principal bundle connection

1918;

1954;

1967;

1977;

2002;

2005;

now;

Weyl

Y:M:

F:P:

M:M:

Y:T:

Y:T:

Y:T:

:

:

:

:

:

:

:

A (G) +2 Lie

+

A (G) (1) (2) (3) += B

++W

++G

+2 Lie

+= su

++ su

++ su

+

(G) A!+

= A++ g

!2 Lie

!+

A (G) (1; ) += !

++ e

+2 Lie

+= so

+4

ï != g

!

! A!+

= 21!++ 4

1 e+

+B++W

++G

++ "e

!+ e

!+ u

!+ d

!

(G) (1; ) 2 Lie+

= Cl+

7

! A!+

= 21!++ 4

1 e+

+B++W

++G

++ "e

!+ e

!+ u

!+ d

!

+ "#!+ #

!+ c

!+ s

!+ "$

!+ $

!+ t

!+ b

!

(G) ? 2 Lie+

= e8+

Standard model and gravity in a matrix

Correct interactions and charges from curvature:

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

= H++G

++ ï

!=

! H ++

ï !

"

G "+

"2 so

+7 + so

++ 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!

r

L

d!

r

L

u!

r

R

d!

r

R

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+"G2

+

G 3"iB+"i 3+8

+

G i 6++G7

+

u!

b

L

d!

b

L

u!

b

R

d!

b

R

G i 4+"G5

+

G i 6+"G7

+

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 the 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 part of the 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 part

Modified BF action over 4D base manifold:

F ++

=

=

=

=

H H ! d++

+H++

H+= 2

1!++ 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

1

Z )%+++

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++

Action for everything

Modified BF action for everything, using B :

Fermionic part, using anti-ghost Grassmann 3-form, B ï :

H H G G ï ï G F!+!+

= d+A!+

+ A!+A!+

=%d++

+H++

&+%d++

+G++

&+%d+ !

+H+ !

+ ï! +

&

M "

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

4

3 2

:

++= B

+++B

:

"

S =

=

B (H; ; )

Z ) :

++F!+!+

+ !" +

G+B++

*

B ï ï G F B B % $B

Z ) :

"

%d+ !

+H+ !

+ ï! +

&+B

++++"

4

1s

++

s++

+Bm;h;G++

m;h;G++

*

:

"= e

"

:

e*

S f =

=

=

B ï ï G

Z ) :

"

%d+ !

+H+ !

+ ï! +

&*

ï ï ï !ï ï ï G

Z )e"

:

e*%d+ !

+2

1!+ !

+4

1e+ !

+B+ !

+W+ !

+ ï! +

&*

jej ï% (e ) @ ï ! % ï ï ï G

Zd x4"

) :#

#i%

i!+

4

1

i

#"#"

!+Bi

!+Wi

!" ï

!i

&+ ï

:

!ï!

*

Why this Lie algebra

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

For three generations:

BIG Lie algebra:

jej ï% (e ) @ ï ! % ï ï ï G d x4"

) :#

#i%

i!+

4

1

i

#"#"

!+Bi

!+Wi

!" ï

!i

&+ ï

:

!ï!

*

A!+

= H++G

++ ï

!=

! H ++

ï !

"

G "+

"

=

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!

r

L

d!

r

L

u!

r

R

d!

r

R

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+"G2

+

G 3"iB+"i 3+8

+

G i 6++G7

+

u!

b

L

d!

b

L

u!

b

R

d!

b

R

G i 4+"G5

+

G i 6+"G7

+

G 3"iB+"

2ip3

8+

3

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

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

2 so+

7 + so+

+ 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

...Since has a representation as linear transformations

of , it has two representations on , corresponding to

left and right matrix multiplication; glomming these two

together we get a representation of on .

Similarly we have representations of on

and . Putting all this stuff together we get a

Lie algebra, if we do it right - and it's .

so(8) o(8) nd(V ) nd(S ) nd(S ) + s + e + e + + e "

so(8)

V end(V )

so(8) o(8) + s end(V )

so(8) o(8) + s

end(S ) + end(S ) "

E8

E (E8) = H +G+#I +#II +#III 2 Lie

[H; ] #I

[G; ] #I

=

=

H #I

#I G

[H; ] #II

[G; ] #II

=

=

H+#II

#II G+

[H; ] #III

[G; ] #III

=

=

H"#III

#III G"

E8 T.O.E.

Build a real form of complex E8 by using instead of

. Then E8 T.O.E. connection is:

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

A!+

= H++G

++#

! I+#

! II+#

! III=

something like

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+"G2

+

G 3"iB+"i 3+8

+

G i 6++G7

+

G i 4+"G5

+

G i 6+"G7

+

G 3"iB+"

2ip3

8+

3

7 7 7 5

+

2

6 6 6 4

" !

e

L

e ! L

" !

e

R

e !R

u !

r

L

d !

r

L

u !

r

R

d !

r

R

u !

g

L

d !

g

L

u !

g

R

d !

g

R

u !b

L

d !

b

L

u !

b

R

d !

b

R

3

7 7 7 5 +

2

6 6 6 6 4

" !

#

L

# ! L

" !

#

R

# ! R

c !

r

L

s !

r

L

c !

r

R

s !

r

R

c !

g

L

s !

g

L

c !

g

R

s !

g

R

c !b

L

s !

b

L

c !

b

R

s !

b

R

3

7 7 7 7 5 +

2

6 6 6 4

" !

$

L

$ ! L

" !

$

R

$ !R

t !

r

L

b !

r

L

t !

r

R

b !

r

R

t !

g

L

b !

g

L

t !

g

R

b !

g

R

t !

b

L

b !

b

L

t !

b

R

b !

b

R

3

7 7 7 5

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

L

A

*

d+

= TA g ,A

R*

d+

= g TA

AR*

,BR*

= CAB

CCR*

[ A TB] = CAB

CC

AB = CAC

DBD

C

I+= dyp

+ pR A

A

E (M #G

E

E+

*y = dxi

+ iB ,L

B

*

+ dyp+

p

*

M

A(x) A (x) +

= &0$E+

*I+= dxi

+ iB TB

Cartan subalgebra and charges

Mutually commuting set of Lie algebra generators:

Cartan subalgebra:

Eigenvalues, , and eigenvectors, , using the Lie bracket:

Unique eigenvalue for each of the eigenvectors, corresponding

to roots, ) , in dimensional vector space.

Cartan subalgebra of the standard model and gravity:

C ! % ! % i$ iY i- i-

Eigenvectors are elementary particles, roots are their charges:

)(e ) ) ; ; 1; 1; ; )

r

T ; ; ::; T ; ] f 1 T2 : Trg [ i Tj = 0

T C = ci i 2 Lie(G)

) a V a 2 Lie(G)

[C; ] V ) V Va = )aa =

X

i

ciia

a

(n ) " r

(n ) " ria r

=2

1 0101 + 2

1 1212 +W 3

3 +B +G33 +G8

8

L = (2

1*

2

1" " 0 0

E8 roots

has roots in 8D space — vertices of P 4 :

T.O.E.: Each vertex corresponds to an elementary particle.

Lie(E8) (248 ) 40 " 8 = 2 2;1

E8

Reducing E8 to the standard model

One particularly interesting way can be broken down:

How does this breakdown relate to e8 triality decomposition?

e8

e8 =

=

=

!

e6 u(3) 4 + s + 5 #3

so(1; ) (1) 2 u(3) 4 9 + u + 3 + s + 5 #3

so(1; ) u(2) u(2) (1) (1) 2 u(3) 4 3 + s + s + u + 4#8 + u + 3 + s + 5 #3

! e! ? 2

1 +W +B +4

1 +G+ 3#ï +X

e8

e8 =

=

=

so(1; ) o(8) 7 + s + 3#8#8

so(1; ) o(4) o(6) o(2) 3 + s + 4#4 + s + s + 6#2 + 3#8#8

so(1; ) u(2) u(2) u(4) (1) 3 + s + s + 4#4 + s + u + 6#2 + 3#8#8

Discussion

What is done:

All gauge fields, gravity, and Higgs in one connection, with

fermions as BRST ghosts.

To do:

Will particle assignments work with E8? (Get the mass matrix?)

Why is the action what it is? (How's symmetry breaking happen?)

Is a four dimensional base manifold emergent?

How does this theory get quantized? (LQG methods should apply.)

Natural explanation for QM as a bonus?

What this theory will mean, if it all works:

Gravitational frame and Higgs are intimately related.

Naturally combines standard model with gravity — so it's a T.O.E.

(It's also a U.F.T., but I don't like to call it that.)

Our universe is a very pretty shape!

Gar@Lisi.com

http://deferentialgeometry.org

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&

ï " !=%!+ e

!+ u

!

r;b;g + d!

r;b;g&