Gravity =(Yang-Mills) 2

Transcription

1 Gravity =(Yang-Mills) 2 Michael Duff Imperial College Hawking Radiation, Stockholm August 2015

2 1.0 Basic idea Strong nuclear, Weak nuclear and Electromagnetic forces described by Yang-Mills gauge theory (non-abelian generalisation of Maxwell). Gluons, W, Z and photons have spin 1. Gravitational force described by Einstein s general relativity. Gravitons have spin 2. But maybe (spin 2) =(spin 1) 2.Ifso (1) Do gravitational symmetries follow from those of Yang-Mills? (2) What is the square root of a black hole?

3 1.2 Local and global symmetries from Yang-Mills LOCAL SYMMETRIES: general covariance, local lorentz invariance, local supersymmtry, local p-form gauge invariance [ arxiv: A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes and S. Nagy] GLOBAL SYMMETRIES eg G = E 7 in D = 4, N = 8 supergravity [arxiv: arxiv: arxiv: A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes and S. Nagy] YANG-MILLS SPIN-OFF (interesting in its own right): Unified description of (D = 3; N = 1, 2, 4, 8), (D = 4; N = 1, 2, 4), (D = 6; N = 1, 2), (D = 10; N = 1) Yang-Mills in terms of a pair of division algebras (A n, A nn ), n = D 2 [arxiv: A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes and S. Nagy] GENERALIZED SELF-MIRROR CONDITION, VANISHING CONFORMAL ANOMALIES AND LOG CORRECTIONS TO BLACK HOLE ENTROPY

4 1.3 Product? Although much of the squaring literature invokes taking a product of left and right Yang-MiIls fields A µ (x)(l) A (x)(r) it is hard to find a conventional field theory definition of the product. Where do the gauge indices go? Does it obey the Leibnitz rule If not, why µ (f g) =(@ µ f ) g + f (@ µ g)

5 1.4 Convolution Here we present a G L G R product rule : [A µ i (L)? ii 0? A i 0 (R)](x) where ii 0 is the spectator bi-adjoint scalar field introduced by Hodges [Hodges:2011] and Cachazo et al [Cachazo:2013] and where? denotes a convolution Z [f? g](x) = d 4 yf (y)g(x y). Note f? g = g? f, (f? g)? h = f? (g? h), and, importantly obeys and not µ (f? g) =(@ µ f )? g = f? (@ µ µ (f g) =(@ µ f ) g + f (@ µ g)

6 1.5 Gravity/Yang-Mills dictionary For concreteness we focus on N = 1 supergravity in D = 4, obtained by tensoring the (4 + 4) off-shell N L = 1 Yang-Mills multiplet (A µ (L), (L), D(L)) with the (3 + 0) off-shell N R = 0 multiplet A µ (R). Interestingly enough, this yields the new-minimal formulation of N = 1 supergravity [Sohnius:1981] with its multiplet (h µ, µ, V µ, B µ ) The dictionary is, Z µ h µ + B µ = A µ i (L)? ii 0? A i 0 (R) = i (L)? ii 0? A i 0 (R) V = D i (L)? ii 0? A i 0 (R),

7 1.6 Yang-Mills symmetries The left supermultiplet is described by a vector superfield V i (L) transforming as V i (L) = i (L)+ i (L)+f i jkv j (L) k (L) + (a,, ) V i (L). Similarly the right Yang-Mills field A i 0 (R) transforms as A i 0 (R) =@ and the spectator as i 0 (R)+f i 0 j 0 k 0A j 0 (R) k0 (R) + (a, ) A i 0 (R). ii 0 = f j ik ji 0 k (L) f j 0 i 0 k 0 ij 0 k0 (R)+ a ii 0. Plugging these into the dictionary gives the gravity transformation rules.

8 1.7 Gravitational symmetries where Z µ µ (L)+@ µ (R), µ µ, V µ µ, µ (L) = A µ i (L)? ii 0? i 0 (R), (R) = i (L)? ii 0? A i 0 (R), = i (L)? ii 0? i 0 (R), = D i (L)? ii 0? i 0 (R), illustrating how the local gravitational symmetries of general covariance, 2-form gauge invariance, local supersymmetry and local chiral symmetry follow from those of Yang-Mills.

9 1.8 Supersymmetry We also recover from the dictionary the component supersymmetry variation of [Sohnius:1981], Z µ = 4i µ, µ = i 4 k g µ + 5 V µ 5 H µ V µ = µ apple i 2 µ 5 H, 5@ apple.

10 1.9 Lorentz multiplet New minimal supergravity also admits an off-shell Lorentz multiplet ( µab, ab, 2V ab + ) transforming as V ab = ab + ab + (a,, ) V ab. (1) This may also be derived by tensoring the left Yang-Mills superfield V i (L) with the right Yang-Mills field strength F abi 0 (R) using the dictionary V ab = V i (L)? ab = i (L)? ii 0? F abi 0 (R), ii 0? F abi 0 (R).

11 1.10 Bianchi identities The corresponding Riemann and Torsion tensors are given by R + µ = F µ i (L)? ii 0? F i 0 (R) =R µ. T + µ = F [µ i (L)? ii 0?A ] i 0 (R) = A [ i (L)? ii 0?F µ ] i 0 (R) = T µ One can show that (to linearised order) the gravitational Bianchi identities follow from those of Yang-Mills D [µ (L)F ] I (L) =0 = D [µ (R)F ] I 0 (R)

12 1.11 To do Convoluting the off-shell Yang-Mills multiplets (4 + 4, N L = 1) and (3 + 0, N R = 0) yields the new-minimal off-shell N = 1 supergravity. Clearly two important improvements would be to generalise our results to the full non-linear transformation rules and to address the issue of dynamics as well as symmetries.

13 2.1 Division algebras Mathematicians deal with four kinds of numbers, called Divison Algebras. The Octonions occupy a privileged position : Name Symbol Imaginary parts Reals R 0 Complexes C 1 Quaternions H 3 Octonions O 7 Table : Division Algebras

14 2.2 Lie algebras They provide an intuitive basis for the exceptional Lie algebras: Classical algebras Rank Dimension A n SU(n + 1) n (n + 1) 2 1 B n SO(2n + 1) n n(2n + 1) C n Sp(2n) n n(2n + 1) D n SO(2n) n n(2n 1) Exceptional algebras E E E F G Table : Classical and exceptional Lie alebras

15 2.3 Magic square Freudenthal-Rozenfeld-Tits magic square A L /A R R C H O R A 1 A 2 C 3 F 4 C A 2 A 2 + A 2 A 5 E 6 H C 3 A 5 D 6 E 7 O F 4 E 6 E 7 E 8 Table : Magic square Despite much effort, however, it is fair to say that the ultimate physical significance of octonions and the magic square remains an enigma.

16 2.4 Octonions Octonion x given by x = x 0 e 0 + x10e 1 + x 2 e 2 + x 3 e 3 + x 4 e 4 + x 5 e 5 + x 6 e 6 + x 7 e 7, One real e 0 = 1 and seven e i, i = 1,...,7 imaginary elements, where e 0 = e 0 and e i = e i. The imaginary octonionic multiplication rules are, e i e j = ij + C ijk e k [e i, e j, e k ]=2Q ijkl e l C mnp are the octonionic structure constants, the set of oriented lines of the Fano plane. {ijk} = {124, 235, 346, 457, 561, 672, 713}. Q ijkl are the associators the set of oriented quadrangles in the Fano plane: ijkl = {3567, 4671, 5712, 6123, 7234, 1345, 2456}, Q ijkl = 1 3! C mnp" mnpijkl.

17 2.5 Fano plane The Fano plane has seven points and seven lines (the circle counts as a line) with three points on every line and three lines through every point. Fano plane

18 Big Bang PhysRevLett ½ BPS and non-bps S0(4,4) SO(2;3;R) x [(5+1)^2] Page 52 Imperial College London

19 2.6 Gino Fano Gino Fano (5 January 1871 to 8 November 1952) was an Italian mathematician. He was born in Mantua and died in Verona. Fano worked on projective and algebraic geometry; the Fano plane, Fano fibration, Fano surface, and Fano varieties are named for him. Ugo Fano and Robert Fano were his sons.

20 2.8 Cayley-Dickson Octonion O = O 0 e 0 + O 1 e 1 + O 2 e 2 + O 3 e 3 + O 4 e 4 + O 5 e 5 + O 6 e 6 + O 7 e 7 Quaternion = H(0)+e 3 H(1) H(0) =O 0 e 0 +O 1 e 1 +O 2 e 2 +O 4 e 4 H(1) =O 3 e 0 O 7 e 1 O 5 e 2 +O 6 e 4 Complex H(0) =C(00)+e 2 C(10) H(1) =C(01)+e 2 C(11) C(00) =O 0 e 0 + O 1 e 1 C(01) =O 3 e 0 O 7 e 1 C(10) =O 2 e 0 O 4 e 1 C(11) = O 5 e 0 O 6 e 1 C(00) =R(000)+e 1 R(100) C(01) =R(001)+e 1 R(101) C(10) =R(010)+e 1 (110) C(11) =R(011)+e 1 R(111) Real R(000) =O 0 R(100) =O 1 R(001) =O 3 R(101) = O 7 R(010) =O 2 R(110) = O 4 R(011) = O 5 R(111) = O 6

21 2.7 Division algebras Division: ax+b=0 has a unique solution Associative: a(bc)=(ab)c Commutative: ab=ba A construction division? associative? commutative? ordered? R R yes yes yes yes C R + e 1 R yes yes yes no H C + e 2 C yes yes no no O H + e 3 H yes no no no S O + e 4 O no no no no As we shall see, the mathematical cut-off of division algebras at octonions corresponds to a physical cutoff at 16 component spinors in super Yang-Mills.

22 2.8 Norm-preserving algebras To understand the symmetries of the magic square and its relation to YM we shall need in particular two algebras defined on A. First, the algebra norm(a) that preserves the norm hx yi := 1 2 (xy + yx) =x a y b ab norm(r) =0 norm(c) =so(2) norm(h) =so(4) norm(o) =so(8)

23 2.9 Triality Algebra Second, the triality algebra tri(a) tri(a) {(A, B, C) A(xy) =B(x)y+xC(y)}, A, B, C 2 so(n), x, y 2 A. tri(r) =0 tri(c) =so(2)+so(2) tri(h) =so(3)+so(3)+so(3) tri(o) =so(8) [Barton and Sudbery:2003]:

24 3.1 Yang-Mills spin-off: interesting in its own right We give a unified description of D = 3 Yang-Mills with N = 1, 2, 4, 8 D = 4 Yang-Mills with N = 1, 2, 4 D = 6 Yang-Mills with N = 1, 2 D = 10 Yang-Mills with N = 1 in terms of a pair of division algebras (A n, A nn ), n = D 2 We present a master Lagrangian, defined over A nn -valued fields, which encapsulates all cases. The overall (spacetime plus internal) on-shell symmetries are given by the corresponding triality algebras. We use imaginary A nn -valued auxiliary fields to close the non-maximal supersymmetry algebra off-shell. The failure to close off-shell for maximally supersymmetric theories is attributed directly to the non-associativity of the octonions. [arxiv: A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes and S. Nagy]

25 3.2 D = 3, N = 8Yang-Mills The D = 3, N = 8 super YM action is given by Z 1 S = d 3 x 4 F µ F A Aµ 1 2 D µ A i D µ A i + i A µ A a D µ a 1 4 g 2 f A A B D C E A B BC f DE i i j j gf Aa i Cb BC i ab, i where the Dirac matrices ab, i = 1,...,7, a, b = 0,...,7, belong to the SO(7) Clifford algebra. The key observation is that structure constants, i ab can be represented by the octonionic i ab = i( bi a0 b0 ai + C iab ), which allows us to rewrite the action over octonionic fields

26 3.3 Transformation rules The supersymmetry transformations in this language are given by A = 1 2 (F Aµ + " µ D A ) µ gf BC A B i C j ij, A A µ = i 2 ( µ A A µ ), (2) A = i 2 e i[( e i ) A A (e i )], where µ are the generators of SL(2, R) = SO(1, 2).

27 4.1 Global symmetries of supergravity in D=3 MATHEMATICS: Division algebras: R, C, H, O (DIVISION ALGEBRAS) 2 = MAGIC SQUARE OF LIE ALGEBRAS PHYSICS: N = 1, 2, 4, 8 D = 3 Yang Mills (YANG MILLS) 2 = MAGIC SQUARE OF SUPERGRAVITIES CONNECTION: N = 1, 2, 4, 8 R, C, H, O MATHEMATICS MAGIC SQUARE = PHYSICS MAGIC SQUARE The D = 3 G/H gravsymmetries are given by ym symmetries G(grav) =tri ym(l)+tri ym(r)+3[ym(l) ym(r)]. eg E 8(8) = SO(8)+SO(8)+3(O O) 248 = (8 v, 8 v )+(8 s, 8 s )+(8 c, 8 c )

28 4.2 Squaring R, C, H, O Yang-Mills in D = 3 Taking a left SYM multiplet {A µ (L) 2 ReA L, (L) 2 ImA L, (L) 2 A L } and tensoring with a right multiplet {A µ (R) 2 ReA R, (R) 2 ImA R, (R) 2 A R } we obtain the field content of a supergravity theory valued in both A L and A R : Grouping spacetime fields of the same type we find, AL g µ 2 R, µ 2, ' 2 A R AL A R, 2 A L A R AL A R A L A R

29 4.3 Grouping together Grouping spacetime fields of the same type we find, AL AL A g µ 2 R, µ 2, ', 2 R. (3) A R A L A R Note we have dualised all resulting p-forms, in particular vectors to scalars. The R-valued graviton and A L A R -valued gravitino carry no degrees of freedom. The (A L A R ) 2 -valued scalar and Majorana spinor each have 2(dim A L dim A R ) degrees of freedom. Fortunately, A L A R and (A L A R ) 2 are precisely the representation spaces of the vector and (conjugate) spinor. For example, in the maximal case of A L, A R = O, we have the 16, 128 and of SO(16).

30 4.4 Final result The N > 8 supergravities in D = 3 are unique, all fields belonging to the gravity multiplet, while those with Napple8 may be coupled to k additional matter multiplets [Marcus and Schwarz:1983; dewit, Tollsten and Nicolai:1992]. The real miracle is that tensoring left and right YM multiplets yields the field content of N = 2, 3, 4, 5, 6, 8 supergravity with k = 1, 1, 2, 1, 2, 4: just the right matter content to produce the U-duality groups appearing in the magic square.

31 4.5. Conclusion In both cases the field content is such that the U-dualities exactly match the groups of of the magic square: A L /A R R C H O R SL(2, R) SU(2, 1) USp(4, 2) F 4( 20) C SU(2, 1) SU(2, 1) SU(2, 1) SU(4, 2) E 6( 14) H USp(4, 2) SU(4, 2) SO(8, 4) E 7( 5) O F 4( 20) E 6( 14) E 7( 5) E 8(8) Table : Magic square The G/H U-duality groups are precisely those of the Freudenthal Magic Square! G : g 3 (A L, A R ):=tri(a L )+tri(a R )+3(A L A R ). H : g 1 (A L, A R ):=tri(a L )+tri(a R )+(A L A R ).

32 4.6 Magic Pyramid: G symmetries

33 4.7 Summary Gravity: Conformal Magic Pyramid We also construct a conformal magic pyramid by tensoring conformal supermultiplets in D = 3, 4, 6. The missing entry in D = 10 is suggestive of an exotic theory with G/H duality structure F 4(4) /Sp(3) Sp(1).

34 4.8 Conformal Magic Pyramid: G symmetries

35 5.1 Generalized mirror symmetry: M-theory on X 7 We consider compactification of (N = 1, D = 11) supergravity on a 7-manifold X 7 with betti numbers (b 0, b 1, b 2, b 3, b 3, b 2, b 1, b 0 )and define a generalized mirror symmetry under which changes sign [Duff and Ferrara:2010] (b 0, b 1, b 2, b 3 )! (b 0, b 1, b 2 /2, b 3 + /2) (X 7 ) 7b 0 5b 1 + 3b 2 b 3! Generalized self-mirror theories are defined to be those for which = 0

36 5.2 Anomalies Field f A 360A 360A 0 X 7 g MN g µ b 0 A µ b 1 A b 1 + b 3 M µ b 0 + b b 2 + b 3 A MNP A µ b 0 A µ b 1 A µ b 2 A b 3 total A 0 total A /24 total A 0 /24 Table : X 7 compactification of D=11 supergravity

37 5.3 Vanish without a trace! Remarkably, we find that the anomalous trace depends on A = 1 24 (X 7 ) So the anomaly flips sign under generalized mirror symmetry and vanishes for generalized self-mirror theories. Equally remarkable is that we get the same answer for the total A and total A ( the numbers of [Grisaru et al 1980]). See, however, [Bern et al 2015]: 2-loop counterterms different but finite parts the same. Suggests (to me) that the Euler density coefficienct c is defined only up to an integer.

38 5.4 Squaring Yang-Mills in D = 4andtheself-mirror condition Tensoring left and right supersymmetric Yang-Mills theories with field content (A µ, N L, 2(N L 1) ) and (A µ, N R, 2(N R 1) ) yields an N = N L + N R supergravity theory. L \ R A µ N R 2(N R 1) A µ g µ + 2 N R ( µ + ) 2(N R 1)A µ N L N L ( µ + ) N L N R (A µ + 2 ) 2N L (N R 1) 2(N L 1) 2(N L 1)A µ 2(N L 1)N R 4(N L 1)(N R 1) Table : Tensoring N L and N R super Yang-Mills theories in D = 4. Note that we have dualized the 2-form coming from the vector-vector slot

39 5.5 Betti numbers from squaring Yang-Mills The betti numbers may then be read off from the Table and we find (b 0, b 1, b 2, b 3 )= (1, N L + N R 1, N L N R + N L + N R 3, 3N L N R 2N L 2N R + 3) Consequently X 7 = 7b 0 5b 1 + 3b 2 b 3 = 0 (4) Similar results hold in D = 5where (c 0, c 1, c 2, c 3 )=(1, N L + N R 2, N L N R 1, 2N L N R 2N L 2N R + 4) Consequently X 6 = 2b 0 2c 1 + 2c 2 c 3 = 0 (5)

40 6.1 Black hole entropy Log corrections=anomaly +zero mode terms [Sen 2014]: S = 1 12 (23 11(N 2) n V + n H )loga H (6) N = 4 n H n V = 1 N = 6 n H n V = 3 N = 8 n H n V = 5 S vanishes for N=4 but not N=8. Counter-intuitive.