An Exceptionally Simple Theory of Everything, by Garrett Lisi (31 p.) -
Deferential Geometry (by Garrett Lisi) -
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Transcript of 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...
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
.%