Periodic table of the standard model
An Exceptionally Simple Theory of Everything
http://arxiv.org/abs/0711.0770
Garrett Lisi
FQXi
A connection with everything
!
=
!
W
=
!
2 Cl
(3; 1)
!
kW % i $
dx
%
!
k 2
2 su
(2)
!
kB i
B
=
dx
k
!
!
kg
g = dx
!
!
2
k 1 ! "# !
dx
"#
! 2 k
k
Ai&
2
e! =
2 u
(1)
!
A
k (e )" !
dx
"
k
!
!
'+
'0
"
Y
1
2 Cl
(3; 1)
!
!
"
#eL
eL
[ur ; ug ; ub ]
2 su
(3)
!
l
1
1
A
=
!
+
'+W
+B
+ g + (#" e + e" + u" + d" )
2!
4e
!"
!
!
!
!
+ (#" " + " + c" + s" ) + (#" ( + (" + t" + b" )
"
F
= d!!"
A+
="
#
1
A; A
2 !" !"
$
2
e^L
3
6 7
6 e_ 7
6 L7
6 ^7
6e 7
4 R5
e_R
Review of some representation theory
Cartan subalgebra:
C = C a Ta # Lie(G)
Built from a maximal commuting set of R generators,
#
$
Ta ; Tb = Ta Tb ! Tb Ta = 0
8
1 $ a; b $ R
Root vectors, V* , are eigenvectors of C in the Lie bracket,
[C; V* ] = )* V* =
X
iC a )a* V*
a
Roots, )a* , are the eigenvalue coefficients. The pattern of roots in R
dimensions corresponds to the Lie algebra,
[V* ; V! ] = V+
,
)* + )! = )+
Weight vectors and weights are eigenvectors and eigenvalue
coefficients of C acting on some representation space,
C V* = ) * V*
Weight vectors are particles, weights are their quantum numbers.
Gluon and quark weights
2
3
g 3 + p1 3 g 8 g 1 !ig 2 g 4 !ig 5
6
7
6
7
i
i
g = g A TA = g A &A = 6 g 1 +ig 2 !g 3 + p1 3 g 8 g 6 !ig 77
2
24
5
g 4 +ig 5 g 6 +ig 7
! p2 g 8
3
Cartan subalgebra:
C = g 3 T3 + g 8 T8
Roots and root vectors:
#
$
C; Vg gb! = i
%&
(the diagonal)
& p3 ' (
1
! g3 +
g 8 Vg gb!
2
'
2
!
for the g gb gluon. Weights and weight vectors:
C Vqr = i
ï
& '
1 3
g +
2
&
1
p
2 3
'
g8
!
Vqr
Vg gb!
2
0
6
= 40
0
0
0
0
3
0
7
15
0
Vqr = [1; 0; 0]
for a red quark, q r, and for their duals acted on by !C T, the anti-quarks.
Strong G2
G2
&% grg!
%& grg!
%& grb!
%& grb!
%& gg!b
%& ggb!
N
M
M
N
M
N
O
H
O
H
O
H
qr
g3
g8
(T 2 ! i T 1 ) 1
0
V*
(!T 2 ! i T 1 ) !1
(T 5 ! i T 4 )
1
2
(!T 7 ! i T 6 )
1
2
0
p
3
2
1
2
(!T 5 ! i T 4 ) ! !
(T 7 ! i T 6 ) !
[1; 0; 0]
qg
[0; 1; 0]
qb
[0; 0; 1]
q!r
[1; 0; 0]
q!g
[0; 1; 0]
q!b
[0; 0; 1]
1
2
1
2
!
1
2
!
3
2
3
2
2
p1
2
p1
3
3
p1
3
1
p
2 3
p1
2 3
p1
3
! 12 !
0
3
2
p
p
0 !
1
2
p
!
Exceptional Lie brackets
The 14 Lie algebra elements of the smallest exceptional Lie group, G2:
g2 = s u(3) + 3 + !
3
g + q + q! 2 g2
!
"
"
!"
Structure of G2 implies Lie bracket equivalent to fundamental action,
#
[g; q ] = g A TA ; q B TB
$
2
6
6
=gq =6
4
i
2
g 3 + pi
2
g r!g
3
i
! i2 g 3 + p
g r!b
corresponding to the strong interactions, such as
#
rg!
g ;q
g
$
= qr
!
g rb
g rg!
g8
2
!
g gb
3
g8
3
2
qr
3
7
74 g5
7 q
5 b
q
g8
!
g gb
! pi
3
G2 in SO(7) .
G2+
%& grg!
%& grg!
%& grb!
%& grb!
%& gg!b
%& ggb!
N
M
N
M
N
M
H
O
H
O
H
O
x y
z
g3
!1 1
0
1
1 !1 0 !1
!1 0
1
0
N
M q II
H
O
M
N
O q III
H
1
2
1
2
1
2
! 21
p
8
p B2
0
0
0
0
p
0 !1 ! 21 !
1 !1
1
2
!
1
2
1
2
1
2
1
2
1
2
1
2
!
1
2
!
3
1
2
!
p
p
p
! 21 ! 21
'1
1
2
1
2
0
0
3
0
3
2
0
2
1
p
2
1
p
! 31
3
3
1
p
3
1
p
2 3
1
p
2 3
p1
3
!
"
"
"
"
N
M l ' 21 ' 21 ' 21 0
H
O
0
(1 (1
3
2
0 !
! 21
0
2
! 21 ! 21 ! 21 !
1
2
3
2
0 !1 1 ! 21
q Ir ! 21
q Ig
q bI
q!rI
q!gI
q!bI
1
2
1
g8
! 31
! 31
1
3
1
3
1
3
( 32
' 34
(1
G2 in SO(7) ..
G2+
%& grg!
%& grg!
%& grb!
%& grb!
%& gg!b
%& ggb!
N
M
N
M
N
M
H
O
H
O
H
O
x y
z
g3
!1 1
0
1
1 !1 0 !1
!1 0
1
0
N
M q II
H
O
M
N
O q III
H
1
2
1
2
1
2
! 21
p
8
p B2
0
0
0
0
p
0 !1 ! 21 !
1 !1
1
2
!
1
2
1
2
1
2
1
2
1
2
1
2
!
1
2
!
3
1
2
!
p
p
p
! 21 ! 21
'1
1
2
1
2
0
0
3
0
3
2
0
2
1
p
2
1
p
! 31
3
3
1
p
3
1
p
2 3
1
p
2 3
p1
3
!
"
"
"
"
N
M l ' 21 ' 21 ' 21 0
H
O
0
(1 (1
3
2
0 !
! 21
0
2
! 21 ! 21 ! 21 !
1
2
3
2
0 !1 1 ! 21
q Ir ! 21
q Ig
q bI
q!rI
q!gI
q!bI
1
2
1
g8
! 31
! 31
1
3
1
3
1
3
( 32
' 34
(1
G2 in SO(7) ...
G2+
%& grg!
%& grg!
%& grb!
%& grb!
%& gg!b
%& ggb!
N
M
N
M
N
M
H
O
H
O
H
O
x y
z
g3
!1 1
0
1
1 !1 0 !1
!1 0
1
0
N
M q II
H
O
M
N
O q III
H
1
2
1
2
1
2
! 21
p
8
p B2
0
0
0
0
p
0 !1 ! 21 !
1 !1
1
2
!
1
2
1
2
1
2
1
2
1
2
1
2
!
1
2
!
3
1
2
!
p
p
p
! 21 ! 21
'1
1
2
1
2
0
0
3
0
3
2
0
2
1
p
2
1
p
! 31
3
3
1
p
3
1
p
2 3
1
p
2 3
p1
3
!
"
"
"
"
N
M l ' 21 ' 21 ' 21 0
H
O
0
(1 (1
3
2
0 !
! 21
0
2
! 21 ! 21 ! 21 !
1
2
3
2
0 !1 1 ! 21
q Ir ! 21
q Ig
q bI
q!rI
q!gI
q!bI
1
2
1
g8
! 31
! 31
1
3
1
3
1
3
( 32
' 34
(1
G2 in SO(7) ....
G2+
%& grg!
%& grg!
%& grb!
%& grb!
%& gg!b
%& ggb!
N
M
N
M
N
M
H
O
H
O
H
O
x y
z
g3
!1 1
0
1
1 !1 0 !1
!1 0
1
0
N
M q II
H
O
M
N
O q III
H
1
2
1
2
1
2
! 21
p
8
p B2
0
0
0
0
p
0 !1 ! 21 !
1 !1
1
2
!
1
2
1
2
1
2
1
2
1
2
1
2
!
1
2
!
3
1
2
!
p
p
p
! 21 ! 21
'1
1
2
1
2
0
0
3
0
3
2
0
2
1
p
2
1
p
! 31
3
3
1
p
3
1
p
2 3
1
p
2 3
p1
3
!
"
"
"
"
N
M l ' 21 ' 21 ' 21 0
H
O
0
(1 (1
3
2
0 !
! 21
0
2
! 21 ! 21 ! 21 !
1
2
3
2
0 !1 1 ! 21
q Ir ! 21
q Ig
q bI
q!rI
q!gI
q!bI
1
2
1
g8
! 31
! 31
1
3
1
3
1
3
( 32
' 34
(1
G2 in SO(7) ....x
G2+
%& grg!
%& grg!
%& grb!
%& grb!
%& gg!b
%& ggb!
N
M
N
M
N
M
H
O
H
O
H
O
x y
z
g3
!1 1
0
1
1 !1 0 !1
!1 0
1
0
N
M q II
H
O
M
N
O q III
H
1
2
1
2
1
2
! 21
p
8
p B2
0
0
0
0
p
0 !1 ! 21 !
1 !1
1
2
!
1
2
1
2
1
2
1
2
1
2
1
2
!
1
2
!
3
1
2
!
p
p
p
! 21 ! 21
'1
1
2
1
2
0
0
3
0
3
2
0
2
1
p
2
1
p
! 31
3
3
1
p
3
1
p
2 3
1
p
2 3
p1
3
!
"
"
"
"
N
M l ' 21 ' 21 ' 21 0
H
O
0
(1 (1
3
2
0 !
! 21
0
2
! 21 ! 21 ! 21 !
1
2
3
2
0 !1 1 ! 21
q Ir ! 21
q Ig
q bI
q!rI
q!gI
q!bI
1
2
1
g8
! 31
! 31
1
3
1
3
1
3
( 32
' 34
(1
Pati-Salam model plus gravity
)
*
)
SO(3; 1) + 4 ) 4 + SU (2)L + SU (2)R + U (1) + SU (3)
*
Gravitational SO(3,1)
! = 21 ! "# !"# =
!
!L
!R
"
! L=R =
e = e" ! " =
!
eR
eL
"
eL=R =
!
fL
f=
fR
"
f L=R =
SO(3; 1)
"
"
"
i! 3L=R
! _L=R
eT^=_
'eS_
^
f L=R
f _L=R
#
^
! L=R
!i! 3L=R
^#
'eS
eT_=^
#
%&
%&
%&
%&
1
2
! 3L 21 ! 3R
! L^
1
0
! L_
!1
0
0
1
! _R
0
!
"
eS^
1
2
!1
!
"
eS_
!
"
e^T
!
"
e_T
N
M
N
M
N
M
N
M
! ^R
1
2
! 21 ! 21
! 21
1
2
1
2
! 21
f L^
1
2
f L_
! 21
0
0
1
2
0
! 21
f R^
f R_
0
Electroweak SU(2) and U(1)
!
"
"
#
+
i
3
B
B
W3 W+
2 1
1
W =
B
=
1
W ! ! i2 W 3
B1! ! i2 B13
!
" !
"
# W
'B $
;
B1
'W
"
#
!'0=1 '+
'W=B =
'!
'1=0
2
3 2 3
#eL
uL
!
" 6
7 6 7
W
6 eL 7 6 d L 7
6
7 6 7
4
B1
#eR 5 4 uR 5
eR
dR
p
p
) p3 3 p2 *
3 1
3
p B ! p B2 = ( p ) 2 Y ! g 1 = p
1
5
i
2
5
5
5
SO(4)
%& W +
%& W !
%& B1+
%& B1!
!
" '+
W3
1
p
2
3
p
B1 3 B2 21 Y
0
0
0
!1 0
0
0 !1
0
1
0 !1
1
2
1
2
#
$ '! ! 21 ! 21
!
" '0 ! 21 21
#
$ '1
1
2
#eL
1
2
N
M
N
M
N
M
N
M
Q
! 21
0
0
0
0
0
0
0
1
2
eL ! 21 0
1
2
1
2
#eR
0
eR
0 ! 21
N
M
N
MN
M u L 21 0
1
N
M
N
MN
M dL ! 2 0
N
M
N
MN
M u R 0 21
1
N
M
N
MN
M dR 0 ! 2
1
2
1
2
! 61
! 61
! 61
1
1
1
!1 !1
1
2
1
! 21 !1
1
2
0
! 21 0
! 21 0
! 21 !1
0
0
!1 !1
1
6
2
3
1
6
! 31
2
3
2
3
! 61 ! 31 ! 31
Graviweak SO(7,1)
D4
%&
%&
%&
%&
H1 = ( 21 ! + 41 e' + W + B1 )
)
*
2 s o(3; 1) + 4 ) 4 + su(2) + s u(2)
=
8S+
Cl2 (7; 1)
!
= s o(7; 1) = d4
=
21
i
1
3
+
!
+
W
W
!
L
2
2
4 e R '1
6 W ! 21 !L ! i2 W 3 41 eR '!
6 1
4 ! 4 eL '0 41 eL '+ 21 !R + i2 B 3
1
1
4 e L '!
1
4 e L '1
B1!
32
1
4 e R '+
1
4 e R '0
B1+
1
i
3
!
!
2 R
2 B1
3
#eL
76 e 7
76 L 7
5 4 #eR 5
eR
N
M
N
M
N
M
N
M
! 3L 21 ! 3R W 3 B13
!L
(1
!R
0
W(
0
B1(
0
^=_
^=_
'+
T
^=_
!
" e
'+
S
^=_
#
$ eT ' !
^=_
#
$ eS ' !
^=_
!
" e
'0
T
^=_
!
" e
'0
S
^=_
#
$ eT ' 1
^=_
#
$ eS ' 1
^=_
#eL
^=_
eL
^=_
#eR
^=_
eR
!
" e
H 1 ( # e + e)
^=_
1
2
0
0
0
(1 0
0
0
0
' 21 ( 21
( 21 ( 21
(1 0
0 (1
1
2
1
2
1
2
1
2
' 21 ( 21 ! 21 ! 21
( 21 ( 21 ! 21 ! 21
' 21 ( 21 ! 21
( 21 ( 21 ! 21
' 21 ( 21
1
2
( 21
1
2
( 21 ( 21
( 21
0
0
0
0
( 21
( 21
1
2
1
2
1
2
! 21
! 21
0
! 21 0
0
1
2
0 ! 21
Graviweak F4
A triality rotation, T , of D4:
21
6
6
4
03
!
2
L
3
1 0
2! R
W 03
3
B0 1
3
3 2 1 !3 3 2 1 !3 3
2 L
2 R
0 1 0 0
7 6 0 0 0 1 7 6 1 !3 7 6 B 3 7
2 R7
7=4
6 1 7
56
=
5
4
5
4 W3 5
0 0 1 0
W3
TTT
2
1
! ^R
0 0 0
=TT
!L^
1 3
2 !L
B13
=T
B1+
=
! ^R
Roots invariant under this T :
fW + ; W ! ; eS^ '+ ; eS^ '0 ; eS_ '! ; eS_ '1 g
Rotations to triality-equivalent vector and
negative chiral spinor representation spaces:
T 8S+ = 8V
T 8V = 8S!
T 8S! = 8S+
Three generations, related by triality:
T e^L = " ^L
T " ^L = (L^
T (L^ = e^L
8V
1
2
! 3L 21 ! 3R W 3 B13
tri
1
2
! 3R B13 W 3 21 ! 3L
^=_
0
0
N
M "L
^=_
0
0
N
M #"R
^=_
( 21
1
2
N
M #"L
^=_
N
M "R
8S!
( 21 ! 21
1
2
( 21
! 21 ( 21
0
0
0
0
! 3L 21 ! 3R W 3 B13
tri
B13
^=_
(L
^=_
M(
N
L
^=_
M#
N
(R
^=_
M(
N
R
0
M#
N
1
2
0
1
2
! 21
1
2
! 3L W 3 21 ! 3R
( 21
1
2
( 21 ! 21
0
0
0
0
0
0
( 21
( 21
F4 root system
F4 and G2 together
F4 : (
G2
1
2
!L3 ; 21 !R3 ; W 3 ; B13 )
:
+
graviweak interactions
three generations
+
strong interactions
(B2 ; g 3 ; g 8 )
anti-particles
E8 : ( 21 !L3 ; 21 !R3 ; W 3 ; B13 ; w; B2 ; g 3 ; g 8 ) f everything
Breakdown of E8 to the standard model and gravity:
e8 = f 4 + g 2 + 26 )7
= so(7; 1) + s u(3) + (8S+ +8V +8S! ) )(1 +1+3+3!) + 3)(3 +3!) + 2
)1
*
1
A = 2 ! + 4 e' + W + B1 + g + 3)# + x" + B2 + w
Two new quantum numbers and some non-standard particles:
fw
(B13 +B2 )
B1(
x1=2=3 "r=g=b
!
!
x1=2=3 "r!=g=b
g
E8 root system .
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ..
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ...
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x.
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x..
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x...
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x....
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x....x
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x....x.
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x....x..
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x....x..x
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x....x..x.
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x....x..x..
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x....x..x...
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 root system ....x.x....x..x...x
E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.
E8 periodic table
"E8 is perhaps the most beautiful structure in all of mathematics, but it's
very complex."
— Hermann Nicolai
E8 connection
A
=H
+H
+ #" I + #" II + #" III
!"
!1
!2
1
1
H
=
!
+
e!' + W
+B
1
2
4
!
!
!
!1
!
!
2 e8
!"
2 so(7;
1)
!
2 so(3;
1)
!
e!' = (e!1 + e!2 + e!3 + e!4 ) ) ('+=0 + '!=1 ) 2 4! ) (2 + !2)
W
+B
!
!1
H
=w
+B
+x
"+g
!2
!
!2
!
!
w
+B
!
!2
! !
r!=g=b
r=g=b
x"
= (x
+x
+x
) ) ("
+"
)
!
!1
!2
!3
g
!
#" I = #" e + e" + u" + d"
#" II = #" " + " + c" + s"
"
#" III = #" ( + (" + t" + b"
2 su(2)
+ su
(2)
!
!
2 so(8)
!
2 u(1)
+u
(1)
!
!
2 3! ) (3 + !3)
2 su(3)
!
2 8S+ ) 8S+
2 8V ) 8V
2 8S! ) 8S!
E8 curvature
)
F
= d!!"
A+A
A = F= 1 + F= 2 + D
# + #" II + #" III
!"!"
! "I
="
F= 1 =
1
2
)
R
!
=
e e '2
1
8 !!
*
+
1
4
)
*
)
T= ' ! e!D'
+ F= B1 + F= W
!
*
*
2 e8
="
2 so(7;
1)
!
1
R
=
d
!
+
!!
2 so(3;
1)
2
!!
!!
!
=
)
*
)
*
1
!2)
T= ' !e!D'
=
d
e
+
[!;
e
]
'
!
e
d'
+[
B
+W
;
'
]
2
4
)
(
2
+
1
2
!
!!
! !
! !
!
!
!
F= B1 + F= W = (dB
+B
B ) + (dW
+W
W)
2 su(2)
+su
(2)
! !1
!1 !1
! !
! !
!
!
)
* )
*
F= 2 = F= w + F= B2 + x
"x" + (Dx)"
!x
D" + F= g 2 so(8)
! !
!!
!!
!
F= w + F= B2 = d!w
+ d!B
2 u(1)
+u
(1)
!
!2
!
!
)
*
)
*
!3)
(Dx)"
!x
D"
=
dx
+[
w
+B
;x]
"
!x
d"
+[g
;"]
2
3
)
(
3
+
2
!!
!!
!!
!
! !
! !
!
!
F= g = d!g + g g
!
)
D#
= d! +
! "
!!
1
!
2!
+
1
e'
4!
*
)
2 su(3)
!
*
#" + W
# +B
# ! #" w
+B
+x
" ! #" q g
! "L
!1 " R
!
!2
!
!
Action for everything
:
:
Modified BF action, using B
=B
+B
:
=
=
*
S=
=
=
Z
Z
Z
,
BF
+"
(H ; H ; B )
+ !1 !2 =
= ="
,
)
1
3 2*
1 0
2
0
BD#
+ e+
' R ! ' + F= ,F
* ! "
=
2
4
16%G
,
:
:
BD#
+B
F+
* ! "
= =
-
%G
B B !
4 =G =G
+
0
B=0 ,B
=
:
Cosmological constant from the Higgs VEV:
Implies frame VEV is de Sitter:
$
R
=
e
6e
!!
=
-
$ = 43 '2
R = 4$
Vacuum expectation value of the curvature vanishes:
F
=0
="
Gravitational part of the action
SG =
Z
,
B
F +
=G =G
%G
B B !
4 =G =G
-
F= G =
*
1 )
1
2
+B
!B
=
R ! 8 e!e!' !
=G
=G
%G =
SG =
1
%G
,
Z
,
-
F= G F= G ! =
-
,)
1
4%G
RR!
= d! !d!
+
!! !
==
,
1
e e e e ! = !e+
4! ! ! ! !
,
e!e!R
! = !e+R
=
SG =
1
16%G
Z
&
e+ '2 R !
3 2
'
2
Z
,)
1
!!!
3 !!!
R
!
=
* !
1
2
)
R
!
=
1
2
e
e
'
8 !!
*
2 so
(3; 1)
=
! = !1 !2 !3 !4
1
2
e
e
'
8 !!
*)
R
!
=
1
2
e
e
'
8 !!
* !
ï Chern-Simons
ï volume element
ï curvature scalar
'
cosmological constant:
$ = 43 '2
Fermionic part of the action
:
:
*
Choosing the anti-Grassmann 3-form to be B
= e+# e gives the
*
massive Dirac action in curved spacetime:
Sf =
=
=
=
:
Z
Z
Z
Z
,
,
,
:
Z
-
BF
=
* ="
:
*)
,
:
BD#
* ! "
-
e+# e d#
+H
# ! #" H
! "
!1 "
!2
:
*)
e+# e (d! +
d4+x
,
:
1
!
2!
*-
1
e' + W
!
4!
+
+B
)# ! #" (w
+B
+x
" + g)
!1 "
!
!2
!
)
jej #! (e" ) @i #" + 41 !i "# !"# #" + Wi #" + B1i #"
"
i
*
:
+ #" wi + #" B2i + #" xi " + #" gi + # ' #"
The # ' #" is the standard Higgs mass term.
:
!
The #! " #x
" term... I don't understand yet — promising for CKM.
" "
-
*-
E8 Theory summary
Everything in an E8 principal bundle connection,
A
2 e8
!"
!"
Periodic table of interactions (Feynman vertices) from curvature,
F
= d!!"
A+
="
#
1
A; A
2 !" !"
$
described by the E8 root polytope. Three generations through triality,
Te="
T"=(
T( =e
Pati-Salam SU (2)L ) SU (2)R ) SU (4)GUT and MM gravity together,
S=
Z
,
:
BF
+
= ="
%
B B !
4 =G =G
+
0
B=0 ,B
=
No free parameters — masses from Higgs VEV's,
g1 =
r
3
5
g2 = 1
g3 = 1
3
4
$ = '2
Everything is pure geometry, and it's very beautiful.
-
'0 ; '1 ; " : : :
E8 Theory discussion
Quantization
Coupling constants run.
Large $ compatible with UV fixed point.
Just a connection — amenable to LQG, spin foams, etc.
Understand triality-generation relationship better
Possible collapse or mixing to graviweak SL(2; C).
The role of w
+x
" and symmetry breaking.
!
!
Getting the CKMPMNS matrix would be nice.
Why is the action what it is?
:
Pulling e! out and putting it into F
,F and B
seems weird.
= =
*
Why e!' simple?
Four dimensional base manifold emergent?
What this theory will mean, if it all works:
Combines standard model with gravity — with LQG, it's a T.o.E.
Our universe is very pretty.
http://deferentialgeometry.org
Garrett Lisi
BRST extended connection
Start with E8 principal bundle connection and its curvature,
A
=H
+#
!
!
!
F= = (dH
+H
H +#
#) + (d#
+H
#+#
H)
!!
! !
! !
!!
! !
! !
Action such that #
part is pure gauge,
!
S=
Z
,
BF
+
= =
%G
B B !
4 =G =G
+
0
B=0 ,B
=
-
BRST: Replace #
part with ghosts, #" , in extended connection,
!
)
A
=H
+ #"
!"
!
*
)
F
= dH
+H
H + d#
+ [H
; #] = F= H + D
#
!!
! !
! "
! "
! "
="
Effective action for gauge fields, ghosts, and anti-ghosts:
S=
=
*
Z
Z
,
,
:
BF
+
= ="
%G
B B !
4 =G =G
+
0
B=0 ,B
=
-
)
1
3 2*
1 0
2
0
BD#
+ e+
' R ! ' + F= ,F
* ! "
=
2
4
16%G
:
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) d!g
= TA g (y)
Lie derivative:
Lie bracket:
* *
[,AR ; ,BR ]
=
[ TA ; TB ] =
*
,AR (y) d!g
*
C
CAB ,CR
CAB C TC
= g (y) TA
Killing form (Minkowski metric): gAB = CAC D CBD C
Maurer-Cartan form (frame):
I! = dy p (,pR )A TA
!
Entire space of a principal bundle: E + M ) G
Ehresmann principal bundle connection over patches of E:
*
i
E! (x; y ) = dx
Ai
!
B
*
(x) , LB (y)
Gauge field connection over M :
A(x)
=
!
*
,
$0 E! I!
+ dy
p
!
i
B
= dx
A
(x) TB
!
i
*
@p
The Coleman-Mandula theorem
Let G be a connected symmetry group of the S matrix, and
let the following five conditions hold: (1) G contains a
subgroup locally isomorphic to the Poincaré group. (2)...
Then, we show that G is necessarily locally isomorphic to the
direct product of an internal symmetry group and the
Poincaré group.
E8 Theory does not allow a subgroup locally isomorphic to the Poincaré
group. The S matrix only exists as an approximation, in which the
theorem is satisfied.
K. Cahill, "On the unification of the gravitational and electronuclear
forces," Phys. Rev. D 26, 1916 (1982).
T. Love, "Geometry of Unification," Int. J. Th. Phys., 801 (1984).
...
F. Nesti and R. Percacci, "GraviWeak Unification" arxiv{0706.3307}
S. Alexander, "Isogravity: Toward an Electroweak and Gravitational
Unification," arxiv{0706.4481}.