Nonabelian p-forms, algebraic surfaces and M-theory

P. Henry, L. Paulot and B. Julia.

ENS Paris. hep-th/(20)02-03070,-12346, 0403xxx +Y Dolivet

Swansea March 26, 2004

Introductory remarks1. M-theory

2. ADE (real) U-dualities

3. Universal self-duality equation: Nonabelian p-forms (Multigerbes)

4. Complex algebraic surfaces

5. Del Pezzo rational surfaces over C (and R)

6. Blowing up and down, Ak singularities

7. Borcherds’ root systems and solvable subalgebras i

Prospects

Outline

Introductory remarks

It is a privilege to have been invited on this occasion to honor David Olive

We shared over the years passions for

duality in particular the interconnections betweenn fermionic and bosonic implementations

infinite dimensional Lie algebras

and some guarded attitude towards supersymmetry using its helpful constraints and richness to construct unique models

I. M-Theorya) « Quantized 11d Supergravity » with

extended objects M2, M5, m=0 fields: G metric+A(3) gauge three-form+ spinor valued one-form.b) Compactified on a large spacelike circle

this is a description of IIA superstring theory at large string coupling (on 10d Minkowski space), (R11/l p)3=gIIA

2 Kepler-Witten

formula

Compactified on a zero volume two-torus this reduces to IIB superstring

theory on 10d Minkowski space again.

M-Theoryc) BPS branes for IIB: Fields= (metric), A+

(4), B(2), C(2), SL(2,R)/U(1)=BSL(2,R)=Poincaré half plane of scalars whose sources are:

D-1-branes, F1 coupled to B(2) , D1 to C(2) , D3

to A+(4), and duals:

NS5 and D5 coupled to the 6-forms (Hodge Dirac dual to the 2-forms) , D7 and D9.

(pp-wave, KK6 of M fit in 10d as 0- resp 5- and 6-branes, they are partly gravitational)

U duality preserves the degree of forms Scalars take values in coset, vectors form a representation…

II. U-dualities ADE (real) symmetries appear• In D 3d reduced gravity one gets

duality symmetry, for instance stationary pure gravity leads to Ehlers A1 S.G.Alg.

• In open string model one finds symmetries, in M-theory reduced to (11-n) dimensions one gets split symmetry .

• Starting from 3d symmetric space models (n)/K one gets upon decompactification

the Symmetric magic triangle. CJLP hep-th/9909099

D |11|10| 9| 8| 7| 6| 5| 4| 3|

C J L P 1999 H J P 2002

True magic triangle of pure SUGRAS CJ, J 1980

III. Universal self-duality equation

• CJLP noted enlargment of U-duality group E to spin mixing superalgebra that mixes scalars coordinatizing the Borel subgroup BE of E with p-forms coefficients for supergenerators of degree: –p (total degree is 0) of a supergroup BE.

• E-1dE=S * (E-1dE) • These algebras are highly nonsimple

(solvable). They were Mysterious, although up to a gravitational deformation they have names:

Superalgebras mixing bosonic p-forms They are deformations of:[SL(1|11-D)+ x sSym.3-tensor] x[its dual

rep.] (coadjoint) • But these algebras do not belong to a nice classification! No names after deformation… Except in 11d: {Q3,Q3}= Q~6 Note: this is the nilpotent part Osp(1|2)++ of Osp(1|2) : E=eAQ3 e A~ Q~6 • Miracle? they are: Degree truncations of Borels of Borcherds superalgebras.

Nonabelian p-forms• A a 3-form, B (dual) 6-form, D=11 F=dA, dF=0, d*F+F.F=0• G=dB+A.F dG +F.F=0• E=eAQ3 e BQ~6

• {Q3,Q3} = Q~6

• F=E-1dE=S * (E-1dE)=S * F• Toy example 4D

Electromagnetism {Q1,Q1} = 0 AND Q~1 idem, F=S*F

IIB Iceberg: from BA1 to BSA2

trc

D1D-1,

D7D5 NS5

F1

-K

0=Cartan Kperp

O

OO O

g=1O g=0

Light O cone

g=0

g=2

BA1: iceberg tip

(C+K/2)2 =2g -K.C=p+1=degree

D1, (0) curve (-2) divisor

Mori cone

D3=-K/2

IIB Borcherds algebraSignature of Cartan matrix (1,1)[K]perp. = Z = Roots of SL(2, (R or) C ):

degreeRoot lattice= Z + Z, BPS simple

roots: ,li.lj=1- , K=-2l1 -2l2, Kperp = C (l1+l2) = l1-l2 , = l2

Weyl-Borcherds (Mori) BPS-coneCartan-Borcherds matrix :

IV. Times and names_C. Jordan 1869 , 27 lines on cubic in CP3

_Del Pezzo 1887, Comessatti 1912. Kodaira 1960-68, Silhol 1989 (1860’s real forms)_Manin 1974._Du Val 1934, Demazure et al. 1980, Slodowy 1980, (Arnold…).

_Ramond-Neveu-Schwarz 1969 Gliozzi-Scherk-Olive 1977, Cremmer-Julia-Scherk 1978, Cremmer-Julia 1979, Julia 1980-82, Townsend Hull, Witten 1995._Cremmer, Julia, Lu and Pope 1998-99._Iqbal, Neitzke and Vafa 2001.

Algebraic complex surfaces -Birational discrete invariants: (q;pg)=I, Pm.2q=dimH1(S,R)=b1=2h0,1 (q is called irregularity) Geometric genus: pg=(b2+ - 1)/2=h0,2 = P1= dim (H°(S, 2,0)).Plurigenera: Pm=dim (H°(S, 2,0

xm)).-Classification (Smooth compact (complex) case)Four disjoint classes: all Pm=0; =0’s + 1’s;all less than C.m; Resp. asymptotically quadratic in m. They are: A.Ruled (S->Cq, CP1 fibers);B.Enriques, K3, EllxEll’/G, CT2; C. Elliptic (S->Cg, Ell. generic fiber); D. « General type ».

Minimal models/ invariants with (q,pg)Two ruled cases I.1: (0,0) rational surfaces: CP2, and

minimal CP1xCP1=F1, Fn: Hirzebruch surfaces i.e. CP1 bundles on CP1.

I.2: (q,0) CP1xCq or CP1 bundles on Cq. q= genus of Cq . Kodaira dimension = - . Other Kodaira dimension = 0,1 or 2 M-PHYSICS likes I.1 !Higher dim. Geometry too= Building

block

Riemann surface (C alg. curve)• Picard group: Isomorphism classes of line

bundles on = I. c. of invertible sheaves on = linear equivalence classes of divisors.

• Divisors: -effective = collection of points on = locus of zeroes of section of line bundle,

- canonical divisor = 1,0() = K. • Degree of a divisor: deg (ni Pi) = ni • First Chern class of line bundle E := degree of divisor of meromorphic section• Serre duality: H1(O(E)) x (K-E) C(0,1) [sections] of E / (1,0) sections of –E • c1(E) + Euler = dim (E) - dim(K-E)

V. Del Pezzo surfaces (I.1)• Picard group: Isomorphism classes of line

bundles on S = I.c. of invertible sheaves on S = linear equivalence classes of (Cartier) divisors (for Gorenstein surfaces).

• Divisors: effective = analytic curve on S = locus of zeroes of global section of bundle,

:canonical = (S) = K, f(z)dz1.dz2

:very ample = projective via global sections.• Complex Del Pezzo surface:[-nK] is very ample (n positive integer). –K.D:=degree of D (Del Pezzo’s are Kaehler + C1

2 positive: C1 timelike.)

Cohomologies: Picard H2

(S,Z). Intersection form: Z-valued on Cartier

divisors, signature (1,d) on H 1,1 (S,R). On H2(K3) metric is –(3H2+2E8) (even) dalg 19

• Regular Del Pezzos: dim([K]perp)= b2 –1Deg(-K):=K2=9 CP2 Roots of E algebras: 10-K2=b2=2 CP2,1 (-) or CP1xCP1 (A1) b2=3 CP2,2 (A1x-) b2=4 CP2,3 (A2xA1) b2=5 CP2,4 (A4) b2=6 CP2,5 (D5) b2=7 ‘‘the cubic’’: CP2,6 (E6) b2=8 CP2,7 (E7) b2=9 CP2,8 (E8) =[K]perp

VI. Blowing up and down• Blowing up one regular point S’SCreates a (S.I.=-1) (effective) sphere: eC2xCP1, zi lj=zj li , [K’]=[K]+(S.I.+2)[e]Crepant resolution S.I.=-2 ([K’]=[K])

• Blowing down (-n) sphere projects to [e]perp : n=1 spheresMinimal model (particles,1-forms) n=2: e an (S.I=-2) sphere (instantons, scalars)

• Adjunction or genus formula: K.e+e2=2(g-1)=-2• ei

2=-1 deg=1, (ei-ei+1)2=-2 deg=0.• {ei=1,k}perp = {ei-ei=1,k-1, -e1-K3d}perp Tr. sym.!

Singular Del Pezzo surfaces

• Normal complex singularities (pointlike)• Normalisation• Du Val, Watanabe• Normal real singularities, SlodowyReal point, Virtual (compact) point, Conj.

Pair O , X , <->• Non-normal complex Del Pezzo surfaces• Reid: Type Ia projected normal scroll..

• Non-normal real Del Pezzo surfaces??

VII. Borcherds (super-)algebras

a) Among infinite dimensional algebras one meets first affine Kac-Moody algebras and next hyperbolic ones. But better behaved are Borcherds algebras generalizing symmetrizable Kac-Moody algebras.

b) Generalised Kac-Moody: Cartan+quadratic form, Positive part from « simple » or degree one part, Negative part.

c) Borel=Cartan torus Cl+ Positive root spaces

Cartan-Kac-Moody-Borcherds matrices

Chevalley-Serre-Borcherds relations

D |11|10| 9| 8| 7| 6| 5| 4| 3|

C J L P 1999 H J P 2002

CP2 blown up in (11-D) points in general position

for n=8 or more generally with A(8-n) singularity

IIA Iceberg: from R to BSB2trc

D2

D6D4

NS5

(0) curve, F1

-K

0=CartanKperp

OOO

g=1 g=0g=0

g=1

(C+K/2)2 =2g -K.C=p+1=degreeD0, (-1) curve

Light O cone

Mori cone

g=2

Slansky’s example of Borcherds algebra (2) O--------O (0) = First Lie algebra case = IIB

IIA Superalgebra (0F) O--------O (0)

Strings (of type) I with G=0 := Borcherds SI2

D=10 (o) O======O (-4) = Lie algebra case

Dictionary• Classification: dim([K]perp)= b2 –1-->

Roots Deg(-K)=K2 =D-2=10-b2 The crux of the matter10-K2=1 CP2 : M-theory 2: IIA CP2,1 (-) or CP1xCP1 (A1): IIB 3: M on T2 CP2,2 (A1x-) 4: CP2,3 (A2xA1) 5: CP2,4 (A4) 6: CP2,5 (D5) 7: CP2,6 (E6) 8: CP2,7 (E7) 9 : M on T8 CP2,8 (E8)

Prospects

1f-0b=? (not 0) From Cartan algebra to Group

(better real ADE) Incorporate Einstein, SUSY and

then discard the latter maybe. Merci

Thank you You are welcome to join

this project

bernard.julia@ens.fr