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Swansea March 26, 2004. 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. Introductory remarks M-theory ADE (real) U-dualities
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Swansea March 26, 2004 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
Introductory remarks • M-theory • ADE (real) U-dualities • Universal self-duality equation: Nonabelian p-forms (Multigerbes) • Complex algebraic surfaces • Del Pezzo rational surfaces over C (and R) • Blowing up and down, Ak singularities • 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-Theory • « 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=gIIA2 Kepler-Witten formula Compactified on a zero volume two-torus this reduces to IIB superstring theory on 10d Minkowski space again.
M-Theory • 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)/Kone 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 supergroupBE. • 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: Degreetruncations 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 BSA2trc Light O cone g=2 O O O D3=-K/2 Mori cone -K O D7 NS5 D5 (C+K/2)2 =2g -K.C=p+1=degree F1 D1 D1, (0) curve 0=Cartan Kperp D-1, (-2) divisor O g=1 g=0 BA1: iceberg tip g=0
IIB Borcherds algebra Signature of Cartan matrix (1,1) [K]perp.= Z = Roots of SL(2, (R or) C ): degree Root lattice= Z + Z, BPS simple roots: , li.lj=1- , K=-2l1 -2l2, Kperp = C (l1+l2) = l1-l2 , = l2 Weyl-Borcherds (Mori) BPS-cone Cartan-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, 2,0xm)). -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 S = I. c. of invertible sheaves on S = linear equivalence classes of divisors. • Divisors: -effective = collection of points on S = locus of zeroes of section of line bundle, - canonical divisor = W1,0(S) = K. • Degree of a divisor: deg (Sni Pi) = Sni • First Chern class of line bundle E := degree of divisor of meromorphic section • Serre duality: H1(S, O(E)) x G(K-E) C (0,1) [sections] of E / (1,0) sections of –E • c1(E) + cEuler = dim G(E) - dimG(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 + C12 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 –1 Deg(-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’S • Creates a (S.I.=-1) (effective) sphere: e • C2xCP1, 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 • ei2=-1 deg=1, (ei-ei+1)2=-2 deg=0. • {ei=1,k}perp = {ei-ei=1,k-1, -e1-K3d}perpTr. sym.!
Singular Del Pezzo surfaces • Normal complex singularities (pointlike) • Normalisation • Du Val, Watanabe • Normal real singularities, Slodowy Real 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 • 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. • 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 2002CP2 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 Light O cone g=2 O D2 g=1 O O -K D6 NS5 -K.C=p+1=degree D4 (C+K/2)2 =2g Mori cone (0) curve, F1 Kperp D0, (-1) curve 0=Cartan g=0 g=1 g=0
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 matter 10-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 projectbernard.julia@ens.fr