11 8 8
(Chronology: Gregory Moore, Jeffrey Harvey, Algebra BPS States, N=2 String theory. Th/9510182 (Conjecture: Dual string theories are manifestations of a single Algebra.)
I.
Brief review of Peter West’s elucidation of the principle that M theory, as the completion of 11d supergravity, a theory with 32 supercharges, is the very extended Kac-Moody group .
(Peter West, 0402140).
I. The 10d anomaly free completion of this theory is , with 16 conserved supercharges, and affine E8xE8. (Chaudhuri, 2004).
II. A matrix realization of this algebra on a Planck sized region of spacetime is conjectured to be generalized EM selfdual Initial State
Universe with the generalized electric magnetic self dual spectrum of 12 pforms .
(Chaudhuri, 2004, 2008-10.)
IV. Conclusions: Conjectures and Future Work.
(Hidden Symmetry Unmasked: E11 & Matrix Theory, hep-th/0404235,

Theory Space or Vacuum Manifold  A Unifying Algebra ensures
Completeness.
Dimensional Reduction of Supergravities on T_n Cremmer -Julia En Series.
(1981).
----begin with T-duality group, observe that E11 contains both O(10-n, 10-n) and GL(11-n,R) as proper subgroups;
----in 6d and below, dualize the NS-NS and R-R fields to extend T-duality to the Cremmer-Julia group: D=9 GL(2,R) ~ SL(n,R)xR D=8 SL(3,R)xSL(2,R)
D=7 SL(5,R) D=6 SO(5,5)
D=5 E_(6,6)
D=4 E_(7,7)
D=3 E_(8,8) D<3 requires string theory.
Nicolai demonstrates E9 in 2d, with SO(2,1) x SO(16) hidden structure 11d sugra.E10 conjectured symmetry algebra for Type II toroidally compactified space. Moore 1993.
[Type IIA-Heterotic 6d effective String duality sheds light on compactifications to
2d, fit neatly within the Monster Group, M_24; is this too large?? Lowe
Chaudhuri, 1995].
Algebra BPS States,N=2 Heterotic String,8 conserved supercharges. Moore
Harvey, 95.
(a) Vacuum structure that of a groupoid, unlike tori; K3 x T2. E10 embedded in gauge and supergravity sectors. (b) Monopoles of the generalized Kac Moody

Peter West elucidated the principle that M theory is the group E11: the elements in the Chevalley basis of E11 can be shown to be isomorphic to the generators of the global algebra of each of the theories with 32 supercharges: 11d sugra, 10d type IIA, type IIB, and massive type IIA.
Hep-th/0402142.
  (1,1)   (1,1)
As for E10, the (1,1) lattices of indefinite signature give real and imaginary roots… the representation theory is subtle. Bosonic E11 has a Minkowskian supersymmetric extension for spacetimes of signature (1,10)/(1,9).
Schnakenburg West 2002-04: (a) enumerate the volume preserving basis for
Lorentz generators, GL(d,R) ~ SL(d,R) x R; d=11, 10; and the full set of pform supergravity potentials, Neveu-Schwarz and Ramond-Ramond, include dilaton-
11 axion 10d. (b) compute explicit structure constants of each 10, 11d supergravity global algebras.(c) identify the isomorphismfrom each Cartan subalgebra to the
Chevalley basis of .

----due to the KMA level structure, tower of fields as an extension of supergravity alone.
No coupling to Yang Mills fields, nor chiral matter fields. Roman’s type IIA “mass”.
-----The 11d sugra and Type IIA T duality group contained in E_11, and Type IIB is a simple modification of the Dynkin diagram of E_11: begin with E_8, extend to affine E_8 (E_9), and on to hyperbolic E_10, E_11. The Dynkin diagram E_11 bifurcates at the 8 th (very extended) node: two inequivalent embeddings of the subgroup A_10, associated to the Lorentz algebra of 10d Type IIA and Type IIB.
Extending the global algebra of non-chiral IIA by a central extension gives Massive II:
[R, P a
] = mb
0
P a
, [P a
, R c
1
· · · c q ] = -mb
0
(
 a c
1 R c
2
...c
q + ... ). q = [0], 2; 6,8 (N S), 1,3; 5,7 (RR).
[R,R c
1
...c
q ] = c q
R c
1
...c
q , [R c
1
...c
p ,R c
1
...c
q ] = c p,q
R c
1
...c
p+q , p,q = 2,6,8,1,3,5,7. c

1

mb
0
.
Lorentz generators 10d : J a b 
SL(10,R) : K a b
, extended by a single generator of translations.
Limit : Take m

0, we recover the global algebra of massless 10d type IIA or 11d supergravity
--to develop a comparison string theory, we still need to introduce spacetime, and mass parameters, namely, the string tension. This bosonic Algebraic theory is topological…
Suppressing the algebra of the Lorentz (volume preserving) generators…..Massive IIA!

Global Algebra of the Type IIB Theory…
Peter West’s elucidation of the principle that M theory is E11 was based in part simple observation: E_11 Dynkin diagram has 2 inequivalent A9 chains, bifurcating at node 8.
Map from global type IIA to the Canonical (Chevalley) Basis:
E
(a)
= K a a +1
,a = 1,...,9. E
(10)
= R
10 10
= C
[1]
, E
(11)
= R
910
= B
910
[2]
----Unlike type IIA, the spinors of IIB have identical chirality, no permissible extension of the supersymmetry algebra by a mass (Roman’s) parameter.
c
1
...c
p [R s
1 c
1
...c
q ,R s
2
] = d p
( s )
R c
1
...c
p
( s )
, [R
(2)
, R c
1
· · · c p
( s
1
)
] = d
( s
2 p
) '
R c
2
...c
p
 s s
1
2 s
2
+ ... ).
[R c
1
( s
1
...c
p , R
) c
1
...c
q
( s
2
)
] = c
( s
1 p,q
, s
2 c
2
...c
p
 q
 s
1 s s
2
2
IIB SL(2;Z) doublets : p,q = (0,0); (2,2);(6,6);(8,8);4 . s = 1,2 (N S,RR).
-----Remarkably, the IIB algebra has a straightforward truncation given by the orientation projection to the global algebra of the type IB supergravity, breaking the SL(2,Z) symmetry, keeping a single term from each doublet.

D
11 8 8
Note! The projection acting on type IIB superstring acts on the global algebra of the type IIB supergravity, breaking the SL(2,Z) symmetry, yields Type IB:
[K b a
,K d c
[R, R c
1
...c
p
] =
 c b
K d a
] = d
-
 d a
K b c
, [K b a
,P c p
R c
1
...c
p , [R c
1
...c
p
] =
,R c
1
...c
q
 c a
P b
, [K b a
,R c
1
...c
p
] = c p,q
R
] =
 b c
1 R ac
2
...c
p c
1
...c
p+q . d = -
+ ...
1  
,q = 1,5. c
2,6
=
1
.
duals. Thehe algebra of generators of gauge symmetries does not mix with the
Lorentz-supergravity-pform potentials in theories of 16 supercharges, the anomaly free extension is affine E8XE8.
(Chaudhuri, April 2004, 0404235).
Note the absence of spacetime in the Algebraic formulation, modifying West’s conjecture, we can state: String/M theory is a realization of the Algebra
D

E
8

E
8
Minkowskian signature (1,9).
Aside: With E_11, (1,10) gives an unambiguous bosonic algebra, Euclidean signature (0,11) gives a bosonic E_11 that does not extend to supersymmetry.
Possibly, fundamnetal starting point for Euclidean formulation of finite temperature string/M theory.
D
11

G
YM

Generalized Electric Magnetic Duality
The heterotic-type IA-type IB theories have the generalized Poincare-Hodge duality of an 11d theory, not necessarily with full 11d Poincare invariance.
Remarkably, we discover the elusive spacetime evidence of the 12 th member in the Dbrane spectrum, magnetic dual to the D8brane, in a worldsheet calculation, the unique extension with modified Dirichlet boundary conditions to
Polchinski’s Dpbrane tension calculation for p = -1, 0, 1, …, 9. In 10d Hodge duality, we find missing dual to the spacefilling D9brane; 11d Hodge duality is complete p = (-2), (-1), 0, 1, 2, 3, and their 11d Hodge duals, =4 —9.
O. Alvarez (‘82). Cohen, Moore, Nelson, Polchinski (86). Chaudhuri, Chen,
 
  r e
9
2
4

7 /2

'
4 

9
2



  e
F ab
= B ab
+ 2š a' F ab
= tanhu
Exact double expansion in external field and distance was obtained, arbitrary field strength. The earliest evidence of sub-string-size monopoles, carry magnetic dual of D8brane charge---End of an F-soliton string on an D8brane, in external B field. Chaudhuri, hep-th/0007056.Pointlike monopoles are the zero size limit of a pair of circular line operators on D8branes. Finite sized loops frame surface operators (Framed BPS States, Gaiotto, Nietzke,Moore, ‘10 )
E
E_11.
(a)
= K a
,a = 1,...,9. E
(10)
= R = A
(KK)
, E
(11)
= R = B
[2]
.

D

E

E
A Planck Scale Generalized Electric Magnetic Dual Initial State
A matrix realization of this algebra on a Planck sized bubble of spacetime is conjectured to be generalized EM selfdual Initial State Universe with the generalized electric magnetic self dual spectrum of 12 pforms.
(Chaudhuri, 2004, 2008-10.)
IV. Conclusions: Conjectures and Future Work. In 9d and below, and with 8 susy,
4 susy on K3 x T2, likely the algebra will extend to E_10, E_11. Needs an update, a systematic analysis, fascinating prospects, exciting new stage for String/M
Theory!!
(Open Questions in hep-th/0404235, Chaudhuri; w/Lowe, 9512226; and w/ H.
Verlinde.)
In 0201129, 0408057, I showed what modifications of M(atrix Theory) and the
IKKT
IIB Matrix Model were needed for a matrix realization of the algebra ; large N is a flavor symmetry, G_YM is finite rank, and (volume preserving)
Lorentx x Supersymmetry extends to the full Electric-Magnetic Duality group
D_11.
Spacetime & Matter emergent.