M-theory from the superpoint

Transcription

1 M-thery frm the superpint Jhn Huerta CAMGSD Institut Superir Técnic Iberian Strings 2017 Lisbn January

2 Prlgue Figure : R 0 1

3 Prlgue Figure : R 0 1 R 0 1 has a single dd crdinate θ, and θ 2 = 0, s a pwer series terminates immediately: f (θ) = f (0) + f (0)θ. In essence, this means we shuld regard θ as infinitesimal. Thus R 0 1 is a single pint with an infinitesimal neighbrhd, as depicted abve.

4 Prlgue We will investigate the superpint with mathematical tls. Inside, we will find all the super-minkwski spacetimes f string thery and M-thery, ging up t dimensin 11. Then we will find the strings, Dp-branes and M-branes themselves, thanks t the brane buquet f Firena, Sati and Schreiber.

5 R 10,1 32 R 9, R 9,1 16 / / R 9, R 5,1 8+8 R 3,1 4+4 R 5,1 8 R 3,1 4 R 2,1 2+2 R 2,1 2 R R 0 1

6 $ / m2brane R 10,1 32 string IIB string het string IIA R 9, R 5,1 8+8 R 3,1 4+4 R 9,1 16 R 5,1 8 R 3,1 4 / R 9, R 2,1 2+2 R 2,1 2 R R 0 1

7 # $ / { m5brane m2brane d5brane d3brane d1brane d0brane d2brane d4brane d7brane R 10,1 32 d6brane d9brane / string IIB string het (pb) string IIA d8brane R 9, R 5,1 8+8 R 3,1 4+4 R 2,1 2+2 R R 9,1 16 R 5,1 8 R 3,1 4 R 2,1 2 R 0 1 / R 9,

8 # $ / { m5brane m2brane d5brane d3brane d1brane d0brane d2brane d4brane d7brane R 10,1 32 d6brane d9brane / string IIB string het (pb) string IIA d8brane R 9, R 5,1 8+8 R 3,1 4+4 R 2,1 2+2 R R 9,1 16 R 5,1 8 R 3,1 4 R 2,1 2 R 0 1 / R 9, The brane buquet.

9 Brane cndensatin M - thery Type IIA SO(32) hetertic Type IIB E x E hetertic 8 8 Type I Figure : Cartn by Plchinski. Type IIA string thery cntains D0-branes.

10 Brane cndensatin M - thery... Type IIA N SO(32) hetertic Type IIB E x E hetertic 8 8 Type I Figure : Cartn by Plchinski. As the number N f D0-branes grws large, type IIA string thery becmes M-thery.

11 Brane cndensatin This means the 10-dimensinal superspacetime where type IIA strings live grws an extra dimensin t becme the 11-dimensinal superspacetime f M-thery. Infinitesimally, R 9, R 10,1 32.

12 Central extensins Given g a Lie superalgebra, ω : Λ 2 g R a 2-ccycle: ω([x, Y ], Z ) ± ω([y, Z ], X) ± ω([z, X], Y ) = 0,

13 Central extensins Given g a Lie superalgebra, ω : Λ 2 g R a 2-ccycle: ω([x, Y ], Z ) ± ω([y, Z ], X) ± ω([z, X], Y ) = 0, we can frm the central extensin: g ω = g Rc,

14 Central extensins Given g a Lie superalgebra, ω : Λ 2 g R a 2-ccycle: ω([x, Y ], Z ) ± ω([y, Z ], X) ± ω([z, X], Y ) = 0, we can frm the central extensin: g ω = g Rc, with ne extra generatr c, even and central, and mdified Lie bracket: [X, Y ] ω = [X, Y ] + ω(x, Y )c.

15 Central extensins Given g a Lie superalgebra, ω : Λ 2 g R a 2-ccycle: ω([x, Y ], Z ) ± ω([y, Z ], X) ± ω([z, X], Y ) = 0, we can frm the central extensin: g ω = g Rc, with ne extra generatr c, even and central, and mdified Lie bracket: [X, Y ] ω = [X, Y ] + ω(x, Y )c. I ll write g ω g fr the map setting c t er, and ften use this arrw t dente a central extensin.

16 The brane buquet, step 1 Nte the parallels: M-thery spacetime has ne mre bsnic dimensin than type IIA string thery spacetime. g ω has ne mre bsnic dimensin than g.

17 The brane buquet, step 1 Nte the parallels: M-thery spacetime has ne mre bsnic dimensin than type IIA string thery spacetime. g ω has ne mre bsnic dimensin than g. In fact, R 10,1 32 is a central extensin f R 9,

18 The brane buquet, step 1 Nte the parallels: M-thery spacetime has ne mre bsnic dimensin than type IIA string thery spacetime. g ω has ne mre bsnic dimensin than g. In fact, R 10,1 32 is a central extensin f R 9, by the 2-ccycle n R 9, dθγ 11 dθ that gives rise t the WZW term f the D0-brane actin.

19 The brane buquet, step 1 Nte the parallels: M-thery spacetime has ne mre bsnic dimensin than type IIA string thery spacetime. g ω has ne mre bsnic dimensin than g. In fact, R 10,1 32 is a central extensin f R 9, by the 2-ccycle n R 9, dθγ 11 dθ that gives rise t the WZW term f the D0-brane actin. The brane buquet prpsal, step 1 Brane cndensatin is central extensin.

20 Fr this t make sense, super-minkwski spacetime R D 1,1 S must be a Lie superalgebra, and it is: [Q α, Q β ] = 2Γ µ αβ P µ Mrever, n R 9, , the 2-frm µ D0 = dθγ 11 dθ must define a 2-ccycle, and it des: µ D0 is left invariant under translatins in superspace. dµ D0 = 0.

21 The superpint This prmpts us t ask Questin Are ther dimensins f spacetime als the result f brane cndensatin/central extensin? At the mst extreme end, let us start with the superpint R 0 1 with a single dd crdinate θ. This has exactly ne 2-ccycle: dθ dθ Extending by this 2-ccycle gives R 1 1, the superline, the wrldline f the superparticle. R 1 1 R 0 1.

22 The superpint Let us a play a game with tw mves: We can extend by 2-ccycles, satisfying a suitable invariance cnditin. We can duble the number f spinrs. This will lead us frm the superpint up t 11 dimensins and beynd.

23 The superpint First, we will duble the number f ferminic dimensins: R 0 2 We will write this peratin as fllws: R 0 2 R 0 1 Nw, R 0 2 has tw dd generatrs, θ 1 and θ 2, and there are three 2-ccycles: dθ 1 dθ 1, dθ 1 dθ 2, dθ 2 dθ 2. Extending by all three we get: R 3 2 R 0 2.

24 Dimensin 3 Nw smething remarkable happens: a metric appears! Aut 0 (R 3 2 ) = R + Spin(2, 1). We didn t put it in, but by lking at the autmrphisms f the algebra, the three even generatrs in R 3 2 transfrm under Spin(2, 1) as vectrs, and the tw dd generatrs as spinrs. Thanks t this metric, we can lk fr Spin(2, 1)-invariant 2-ccycles n R 2,1 2. There are nne, because the nly Spin(2, 1)-invariant map: is antisymmetric

25 Dimensin 4 Duble the number f spinrs again: R 2,1 2+2 R 2,1 2 There is precisely ne Spin(2, 1)-invariant 2-ccycle, and extending by this gives: R 3,1 4 R 2,1 2+2

26 Dimensin 4 Duble the number f spinrs again: R 2,1 2+2 R 2,1 2 There is precisely ne Spin(2, 1)-invariant 2-ccycle, and extending by this gives: R 3,1 4 R 2,1 2+2 Again, the metric is nt a chice: Aut 0 (R 3,1 4 ) = R + Spin(3, 1) U(1). U(1) is the R-symmetry grup. There are n further Spin(3, 1)-invariant 2-ccycles.

27 Dimensin 6 Duble the number f spinrs again: R 3,1 4+4 R 3,1 4 Nw there are tw Spin(3, 1)-invariant 2-ccycles. R 5,1 8 R 3,1 4+4.

28 Dimensin 6 Duble the number f spinrs again: R 3,1 4+4 R 3,1 4 Nw there are tw Spin(3, 1)-invariant 2-ccycles. R 5,1 8 R 3, Again, the metric is nt a chice: Aut 0 (R 5,1 8 ) = R + Spin(5, 1) Sp(1). Sp(1) is the R-symmetry grup. There are n further Spin(5, 1)-invariant 2-ccycles.

29 Dimensin 10 Nw we have a chice f tw different ways t duble the spinrs, a type IIA and type IIB: and R 5,1 8+8 R 5,1 8+8 R 5,1 8 R 5,1 8 There are n Spin(5, 1)-invariant 2-ccycles in type IIB, but n type IIA there are fur: R 9,1 16 R 5,1 8+8.

30 Dimensin 10 Nw we have a chice f tw different ways t duble the spinrs, a type IIA and type IIB: and R 5,1 8+8 R 5,1 8+8 R 5,1 8 R 5,1 8 There are n Spin(5, 1)-invariant 2-ccycles in type IIB, but n type IIA there are fur: R 9,1 16 R 5, Again, the metric is nt a chice: Aut 0 (R 9,1 16 ) = R + Spin(9, 1). There are n further Spin(9, 1)-invariant 2-ccycles.

31 Dimensin 11 Again, we have a chice f tw different ways t duble the spinrs, a type IIA and type IIB: and R 9, R 9, R 9,1 16 R 9,1 16 There are n Spin(9, 1)-invariant 2-ccycles in type IIB, but n type IIA there is ne, the ne we started with: R 10,1 32 R 9,

32 In summary: R 10,1 32 R 9, R 9,1 16 / / R 9, R 5,1 8+8 R 3,1 4+4 R 5,1 8 R 3,1 4 R 2,1 2+2 R 2,1 2 R R 0 1

33 Lie algebra chmlgy What des the 2-ccycle have t d with the D0-brane? µ D0 = dθγ 11 dθ It gives rise t the D0-brane s WZW term: Π0 S D0 = m Π 0 dτ m θγ 11 θdτ.

34 Lie algebra chmlgy In the general, the Lie algebra chmlgy f R D 1,1 S gives rise t the WZW terms fr Green Schwar actins. T cmpute this, write a basis f left-invariant 1-frms n super-minkwski: e µ = dx µ θγ µ θ, dθ α. Find the Lrent-invariant cmbinatins, such as: µ p = e ν 1 e νp dθγ ν1 ν p dθ. This a (p + 2)-ccycle if and nly if it is clsed: dµ p = 0. This happens nly fr special values f D, N and p.

35 The brane scan

36 The brane buquet These ccycles really determine the thery. Schreiber has a mathematical machine that takes ccycles and prduces actin functinals. Lie algebra ccycle n g WZW term n G Centrally extending by these ccycles, we get new algebras.

37 The brane buquet m2brane R 10,1 32 string IIB string het string IIA R 9, R 9,1 16 R 9,

38 Beynd Lie algebras But these are nt 2-ccycles, s: string het, string IIA, string IIB and m2brane are nt Lie algebras! Instead, they are L -algebras.

39 L -algebras An L -algebra g is like a Lie algebra, defined n a chain cmplex: g 0 g 1 g n But the Jacbi identity des nt hld: [[X, Y ], Z ] ± [[Y, Z ], X] ± [[Z, X], Y ] 0. Instead, it hlds up t bundary terms: [[X, Y ], Z ] ± [[Y, Z ], X] ± [[Z, X], Y ] = [X, Y, Z ]. Where this new, trilinear bracket: [,, ]: g 3 g, in turn satisfies an identity like Jacbi up t bundary terms cntrlled by a 4-linear bracket...

40 L -algebras: examples A Lie algebra is an L -algebra cncentrated in degree 0: g 0 0 0

41 L -algebras: examples A Lie algebra is an L -algebra cncentrated in degree 0: g Given any (p + 2)-ccycle ω : Λ p+2 g R, we can cnstruct an L -algebra g ω as fllws: g 0 R

42 L -algebras: examples A Lie algebra is an L -algebra cncentrated in degree 0: g Given any (p + 2)-ccycle ω : Λ p+2 g R, we can cnstruct an L -algebra g ω as fllws: where g 0 R g is in degree 0, R is in degree p. [, ] is the Lie bracket. The (p + 2)-linear bracket, [,, ] = ω, is the ccycle. All ther brackets are 0.

43 L -algebras: examples A Lie algebra is an L -algebra cncentrated in degree 0: g Given any (p + 2)-ccycle ω : Λ p+2 g R, we can cnstruct an L -algebra g ω as fllws: where g 0 R g is in degree 0, R is in degree p. [, ] is the Lie bracket. The (p + 2)-linear bracket, [,, ] = ω, is the ccycle. All ther brackets are 0. All f this generalies t superalgebras in a straightfrward way. This is hw we cnstruct string het, string IIA, string IIB and m2brane frm R D 1,1 S.

44 Dp-branes and the M5-brane Thanks t string het, string IIA, string IIB and m2brane, we can find the branes missing frm the brane scan. Fact The left-invariant frms n g ω are generated by the left-invariant frms n g with ne additinal (p + 1)-frm b such that db = ω. Fr example: On string IIA = R 9, µ IIA, the left-invariant frms are frm R 9, : e ν = dx ν θγ ν dθ, dθ α and a 2-frm F such that df = µ IIA.

45 Dp-branes and the M5-brane Thanks t F, there are new ccycles n string IIA. µ Dp = (p+2)/2 k=0 c p k eν 1 e ν p 2k dθ Γ ν1 ν p 2k dθ F F. c p k are sme cefficients chsen t make dµ Dp = 0. Applying Schreiber s machine t this ccycle gives the Dp-brane actin. Similarly, we can find a ccycle fr the M5-brane n m2brane.

46 The brane buquet m5brane m2brane d5brane d3brane d1brane d0brane d2brane d4brane d7brane R 10,1 32 d6brane d9brane string IIB string het (pb) string IIA d8brane R 9, R 9,1 16 R 9,

47 # $ / { m5brane m2brane d5brane d3brane d1brane d0brane d2brane d4brane d7brane R 10,1 32 d6brane d9brane / string IIB string het (pb) string IIA d8brane R 9, R 5,1 8+8 R 3,1 4+4 R 2,1 2+2 R R 9,1 16 R 5,1 8 R 3,1 4 R 2,1 2 R 0 1 / R 9,

48 Figure : R0 1

49 OBRIGADO

50 References I The use f L -algebras in physics riginates with the wrk f D Auria and Fré, wh call them free differential algebras. L. Castellani, R. D Auria and P. Fré, Supergravity and Superstrings: A Gemetric Perspective, Wrld Scientific, Singapre, R. D Auria and P. Fré, Gemetric supergravity in D = 11 and its hidden supergrup, Nucl. Phys. B201 (1982), pp The cnnectin between Lie algebra chmlgy and Green Schwar p-brane actins is due t de Acárraga and Twnsend: J. A. de Acárraga and P. K. Twnsend, Superspace gemetry and the classificatin f supersymmetric extended bjects, Phys. Rev. Lett. 62 (1989), pp

51 References II The discvery that the WZW terms fr Dp-branes and the M5-branes live n the extended superspacetimes string IIA, string IIB and m2brane appears in tw articles. The case f the type IIA Dp-branes and the M5-brane is in: C. Chryssmalaks, J. de Acárraga, J. Iquierd, and C. Pére Buen, The gemetry f branes and extended superspaces, Nucl. Phys. B 567 (2000), pp , arxiv:hep-th/ while the type IIB Dp-branes are in sectin 2 f: M. Sakaguchi, IIB-branes and new spacetime superalgebras, JHEP 04 (2000), pp. 019, arxiv:hep-th/

52 References III Later, Firena, Sati and Schreiber placed this int the cntext f the hmtpy thery f L -algebras, discvering the brane buquet: D. Firena, H. Sati, U. Schreiber, Super Lie n-algebra extensins, higher WZW mdels, and super p-branes with tensr multiplet fields, Intern. J. Gem. Meth. Md. Phys. 12 (2015), (35 pages). arxiv: Finally, Schreiber and I derive the brane buquet frm the superpint. J. Huerta and U. Schreiber, M-thery frm the superpint. In preparatin.