Katrin Becker, Texas A&M University. Strings 2016, YMSC,Tsinghua University

Transcription

1 Katrin Becker, Texas A&M University Strings 2016, YMSC,Tsinghua University

2 ± Overview

3 Overview ± II. What is the manifestly supersymmetric complete space-time action for an arbitrary string theory or M-theory compactification?

4 Overview ± II. What is the manifestly supersymmetric complete space-time action for an arbitrary string theory or M-theory compactification?

5 Overview ± II. What is the manifestly supersymmetric complete space-time action for an arbitrary string theory or M-theory compactification?

6

7 General Remarks A vacuum of type II supergravity is 1,2 R M Minkowski space parallel spinor Supersymmetry in space-time η = a 0 Calabi-Yau, Spin(7) manifold

8 Tools Given a 7d spin manifold M there is a unit spinor η abc T ϕ = η Γ η abc unit spinor T η η = 1 and a 4-form... T ψ abcd = η Γabcdη

9 Tools Given a 7d spin manifold M there is a unit spinor η abc T ϕ = η Γ η abc unit spinor T η η = 1 and a 4-form... T ψ abcd = η Γabcdη related by ψ = ϕ

10 Tools Given a 7d spin manifold M there is a unit spinor η abc T ϕ = η Γ η abc unit spinor T η η = 1 and a 4-form... T ψ abcd = η Γabcdη related by ψ = ϕ [ ] ( ) ϕ det 1/9 g = g = s s ab ab ab Metric 1 sab = ϕamnϕbpqϕrstε 144 mnpqrst

11 dϕ dψ = 0 = 0 dϕ = τ ψ + 3τ ϕ + τ dψ = 4τ ψ + τ ϕ 1 2

12 dϕ dψ = 0 = 0 dϕ = τ ψ + 3τ ϕ + τ dψ = 4τ ψ + τ ϕ 1 2

13 dϕ dψ = 0 = 0 dϕ = τ ψ + 3τ ϕ + τ dψ = 4τ ψ + τ ϕ 1 2 τ, τ, τ, τ are torsion classes

14 Leading Order Correction

15 Gravitino supersymmetry transformation δψ a = a η A a = 0 1 = 32g b 3 c dc1 c6 bd1 d6 Ba α ' ϕacd ε ε Rc 1c2d1d R 2 c3c4d3d R 4 c5c6d5d6

16 Gravitino supersymmetry transformation δψ a = a η A η ib b a a b + + Γ η α corrections A a = 0 1 = 32g b 3 c dc1 c6 bd1 d6 Ba α ' ϕacd ε ε Rc 1c2d1d R 2 c3c4d3d R 4 c5c6d5d6

17 Gravitino supersymmetry transformation δψ a = a η A η ib b a a b + + Γ η In 7d dimensions spinors have 8 real components. A basis is η Γ η, a=1,...,7. { }, a α corrections A a = 0 1 = 32g b 3 c dc1 c6 bd1 d6 Ba α ' ϕacd ε ε Rc 1c2d1d R 2 c3c4d3d R 4 c5c6d5d6

18 Gravitino supersymmetry transformation δψ a = a η A η ib b a a b + + Γ η In 7d dimensions spinors have 8 real components. A basis is η Γ η, a=1,...,7. { }, a The coefficients are tensors A A [ ϕ] a α corrections b b = B = B [ ϕ] a a a A a = 0 1 = 32g b 3 c dc1 c6 bd1 d6 Ba α ' ϕacd ε ε Rc 1c2d1d R 2 c3c4d3d R 4 c5c6d5d6

19 Gravitino supersymmetry transformation δψ a = a η A η ib b a a b + + Γ η In 7d dimensions spinors have 8 real components. A basis is η Γ η, a=1,...,7. { }, a The coefficients are tensors A A [ ϕ] a α corrections b b = B = B [ ϕ] a a a A a = 0 1 = 32g b 3 c dc1 c6 bd1 d6 Ba α ' ϕacd ε ε Rc 1c2d1d R 2 c3c4d3d R 4 c5c6d5d6

20 Supersymmetric Vacuum [ ] ib b [ ] ' A a a a a b δψ = η ' + ϕ η + ϕ Γ η = 0

21 Supersymmetric Vacuum [ ] ib b [ ] ' A a a a a b δψ = η ' + ϕ η + ϕ Γ η = 0 Primed quantities include corrections: 3 η ' = η + O( α ' )

22 Supersymmetric Vacuum [ ] ib b [ ] ' A a a a a b δψ = η ' + ϕ η + ϕ Γ η = 0 Primed quantities include corrections: 3 η ' = η + O( α ' )

23 Supersymmetric Vacuum [ ] ib b [ ] ' A a a a a b δψ = η ' + ϕ η + ϕ Γ η = 0 Primed quantities include corrections: 3 η ' = η + O( α ' ) Set up PDE dϕ dψ ψ [ ] [ ] ' = α ϕ ' = β ϕ ' = ' ϕ '

24 Supersymmetric Vacuum [ ] ib b [ ] ' A a a a a b δψ = η ' + ϕ η + ϕ Γ η = 0 Primed quantities include corrections: 3 η ' = η + O( α ' ) Set up PDE dϕ dψ ψ [ ] [ ] ' = α ϕ ' = β ϕ ' = ' ϕ ' α = 8A ϕ 8B ψ e abcd [ a bcd ] [ a bcd ] e β = 10A ψ 40B ϕ abcde [ a bcde] [ ab cde]

25

26 All Orders in α

27 Using induction over the order in α ' it is possible to show that a solution of the supersymmetry conditions exists to any order in α ' provided the corresponding dϕ ' dψ ' n n = = ψ ' = ' ϕ ' [ '] α ϕ n [ '] β ϕ n

28 Using induction over the order in α ' it is possible to show that a solution of the supersymmetry conditions exists to any order in α ' provided the corresponding contains the unknown which is the order n of φ dϕ dψ ψ ' ' n n = = ' = ' ϕ ' [ '] α ϕ n [ '] β ϕ n

29 Using induction over the order in α ' it is possible to show that a solution of the supersymmetry conditions exists to any order in α ' provided the corresponding contains the unknown which is the order n of φ dϕ dψ ψ ' ' n n = = ' = ' ϕ ' [ '] α ϕ n [ '] β ϕ n involves φ to order less than n

30 Using induction over the order in α ' it is possible to show that a solution of the supersymmetry conditions exists to any order in α ' provided the corresponding contains the unknown which is the order n of φ dϕ dψ ψ ' ' n n = = ' = ' ϕ ' [ '] α ϕ n [ '] β ϕ n = = dχ dξ n n involves φ to order less than n

31 Using induction over the order in α ' it is possible to show that a solution of the supersymmetry conditions exists to any order in α ' provided the corresponding contains the unknown which is the order n of φ dϕ dψ ψ ' ' n n = = ' = ' ϕ ' [ '] α ϕ n [ '] β ϕ n = = dχ dξ Exactness of α and β is not only necessary but also sufficient... n n involves φ to order less than n

32 Using induction over the order in α ' it is possible to show that a solution of the supersymmetry conditions exists to any order in α ' provided the corresponding contains the unknown which is the order n of φ dϕ dψ ψ ' ' n n = = ' = ' ϕ ' [ '] α ϕ n [ '] β ϕ n = = dχ dξ Exactness of α and β is not only necessary but also sufficient... There exists a solution of δψ=0 to all orders in α! n n involves φ to order less than n

33 II. The Space-Time Action of M-theory Compactified to 4d in N=1 Superspace

34 We wish to describe the fluctuations around the background...! 1,3 M

35 We wish to describe the fluctuations around the background...! 1,3 M... these include massless states as well as massive KK modes.

36 Guiding Principles 4d supersymmetry Assemble fields into 4d superfields

37 Guiding Principles 4d supersymmetry Locality Assemble fields into 4d superfields Keep locality along space-time and M. φ = φ ( x, y) 1 C = Cabc ( x, y) dy dy dy 3! a b c

38 Guiding Principles 4d supersymmetry Locality Assemble fields into 4d superfields Keep locality along space-time and M. φ = φ ( x, y) 1 C = Cabc ( x, y) dy dy dy 3! a b c 11d fields decompose into many 4d fields { CMNP, GMN C, C, C, C abc abµ aµν µνρ g, g, g ab aµ µν

39 Manifest Global 4d Supersymmetry ( x, θ, θ µ! ) The coordinates of flat 4d superspace are m µ. Superfields are functions of these coordinates...

40 Manifest Global 4d Supersymmetry ( x, θ, θ µ! ) The coordinates of flat 4d superspace are m µ. Superfields are functions of these coordinates... D = iθ σ θ α m α! α! αα! m Chiral superfields D α Φ = 0! Φ x, θ = C x + 2 θψ x + θθ F( x ) ( ) ( ) ( ) C ( = x, y) = ϕˆ ( x, y) + ic ( x, y) abc abc abc m m m x = x ± ± iθσ θ To leading order in α ': ˆϕ = ϕ

41 1 2 Action For Chiral Superfields ( + ) [ ] D 4 4 I = d x K Φ Φ + d x f Φ F + c c, ( )..

42 1 2 Action For Chiral Superfields ( + ) [ ] D 4 4 I = d x K Φ Φ + d x f Φ F + c c, ( ).. Lagrangian density for bosonic fields δ K ' C( ) ( C') ( ) ( ') δ C L = d yd y y y F y F y M M ( ) ( ) ( µ y y ' ) µ δ C + ( C) 7 2Re d y F( y) M δ f δ C( y)

43 Superpotential A good candidate is f ( ) In a supersymmetric ground state Φ = β Φ dφ constant M δ f δφ = 0 dφ = 0 dϕˆ = 0, G = 0 4 ϕˆ = ϕ ' χ There is a closed 3-form!

44 C = ϕ + ic Kähler Form abc abc abc are coordinates of an infinite dimensional Kaehler manifold. Eleven-dimensional gauge transformations gives rise to isometries of the metric. δ C = dλ the space Λ V of 2-forms mod closed 2-form The Kähler form is invariant and as a result there is a moment map (a concept we borrow from symplectic geometry). As we show in more detail in our paper the vanishing of the moment map implies µ δ K = 0 a = 0 δ abc ( y) C Closed 4-form!

45 Needles to say it would be interesting to derive these conditions from a Kaluza-Klein reduction of M-theory. We envision this as a two step process: 1) we rewrite the action of 11d supergravity in a form that displays manifest N=1 supersymmetry in 4d. 2) non-renormalization theorems should then give us information about which results hold to all orders in perturbation theory.

46 Kaluza-Klein Reduction of M-Theory

47 MNP Bosonic Fields Fields are decomposed into a 4+7 split: C C, C, C, C abc abµ aµν µνρ G MN c d c hµν + gcd A A gbc A µ ν µ = c gac Aν gab µ,ν=0,..,3 Symmetries: C C + dλ x x ξ M M M 4d system is very complicated but known in detail...

48 α, β, are space-time indices a,b,c, are 7d indices K. B, M. Becker, D. Robbins,

49 α, β, are space-time indices a,b,c, are 7d indices K. B, M. Becker, D. Robbins,

50 α, β, are space-time indices a,b,c, are 7d indices K. B, M. Becker, D. Robbins,

51 α, β, are space-time indices a,b,c, are 7d indices K. B, M. Becker, D. Robbins,

52 α, β, are space-time indices a,b,c, are 7d indices K. B, M. Becker, D. Robbins,

53 α, β, are space-time indices a,b,c, are 7d indices K. B, M. Becker, D. Robbins,

54 α, β, are space-time indices a,b,c, are 7d indices K. B, M. Becker, D. Robbins,

55 α, β, are space-time indices a,b,c, are 7d indices K. B, M. Becker, D. Robbins,

56 Goal: Write this Action in Superspace

57 Use the Kaehler potential Kinetic Terms 3 κ 7 K = d y g F 2 M ( ) [ ] ( ) ϕ det 1/9 g = g = s s ab ab ab Metric 1 sab = ϕamnϕbpqϕrstε 144 mnpqrst F 1 = Φ Φ 2i ( ) 3 [ V ] abc abc abc a bc Real superfield for C ab µ

58 The kinetic terms obtained from M-theory compactification are Skin = g 2 ( π1 µ ϕ) ( π 27 µ ϕ) 24κ g C C C C C C 2 24κ { } π1( µ 3 µ ) π 7( µ 3 µ ) π 27( µ 3 µ ) Expanding in components the kinetic terms obtained from superspace are Skin = g ( π 2 1 µ ϕ) ( π 7 µ ϕ) ( π 27 µ ϕ) 24κ g C C C C C C 2 24κ { } π1( µ 3 µ ) + π 7( µ 3 µ ) π 27( µ 3 µ )

59 The kinetic terms obtained from M-theory compactification are Skin = g 2 ( π1 µ ϕ) ( π 27 µ ϕ) 24κ g C C C C C C 2 24κ { } π1( µ 3 µ ) π 7( µ 3 µ ) π 27( µ 3 µ ) Expanding in components the kinetic terms obtained from superspace are Skin = g ( π 2 1 µ ϕ) ( π 7 µ ϕ) ( π 27 µ ϕ) 24κ g C C C C C C 2 24κ { } π1( µ 3 µ ) + π 7( µ 3 µ ) π 27( µ 3 µ ) This coefficient only agrees after integrating out auxiliary fields in the gravity multiplet.

60 Potential The potential for the scalar from the metric can be nicely expressed in terms of torsion classes κ S pot = d y g τ τ1 τ 3 τ 2 W 1 8κ = ± 2 ϕdϕ

61 Potential The potential for the scalar from the metric can be nicely expressed in terms of torsion classes κ S pot = d y g τ τ1 τ 3 τ 2 Get contributions from the superpotential W 1 8κ = ± 2 ϕdϕ

62 Potential The potential for the scalar from the metric can be nicely expressed in terms of torsion classes κ S pot = d y g τ τ1 τ 3 τ 2 Get contributions from the superpotential W 1 8κ = ± 2 ϕdϕ This result agrees precisely with the superspace result!

63 Tensor Hierarchy and Chern-Simons Actions in Superspace References: K. B., M. Becker, W. D. Linch and D. Robbins, , ) In the first paper we embedded the tensor hierarchy consisting of all fields descending from the M-theory three-form C C, C, C, C MNP abc abµ aµν µνρ and the corresponding abelian gauge transformations into superspace. δ C = dλ

64 We explicitly constructed the supersymmetrized Chern-Simons action Re[ (2 α 2 α θ α α ) S = i d xd Φ EG + Φ W W + Σ EW 12κ θ ˆ ˆ 2 α α d xd [ 2 Φ UH + VEH + ( VD U D VU ) Wα + 12κ α! α α! ( VD U D VU ) W Σ UD U Σ UD U XEU ˆ ] α! α! α α!

65 We explicitly constructed the supersymmetrized Chern-Simons action Re[ (2 α 2 α θ α α ) S = i d xd Φ EG + Φ W W + Σ EW 12κ θ ˆ ˆ 2 α α d xd [ 2 Φ UH + VEH + ( VD U D VU ) Wα + 12κ α! α α! ( VD U D VU ) W Σ UD U Σ UD U XEU ˆ ] α! α! α α!

66 We explicitly constructed the supersymmetrized Chern-Simons action Re[ (2 α 2 α θ α α ) S = i d xd Φ EG + Φ W W + Σ EW 12κ θ ˆ ˆ 2 α α d xd [ 2 Φ UH + VEH + ( VD U D VU ) Wα + 12κ α! α α! ( VD U D VU ) W Σ UD U Σ UD U XEU ˆ ] α! α! α α! 2) In the second paper we coupled this system to the non-abelian gauge field arising from the metric.

67 Stay Tuned! More To Come...