Kähler representations for twisted supergravity. Laurent Baulieu LPTHE. Université Pierre et Marie Curie, Paris, France. Puri, January 6th, 2011

Transcription

1 Kähler representations for twisted supergravity Laurent Baulieu LPTHE Université Pierre et Marie Curie, Paris, France Puri, January 6th,

2 Introduction The notion of twisting a supersymmetry algebra is very useful to understand (i) its structure and (ii) the properties of the corresponding quantum field theory. A generic definition of the supersymmetric twist is that it is a mere linear change of field variables that reduces in a controlable way the size of the global symmetries, but keeps it large enough to uniquely determine the same theory as the superpoincaré symmetry

3 One can for instance covariantly select among the 16 generators of maximal supersymmetry n 9 susy generators that close off-shell. Then many of all known finiteness properties can be proven within the context of local quantum field theory. One finds that maximal supersymmetry is overdetermining, - 3 -

4 The twisted fields look like those of a Topological Field Theory : for instance, in d=4, the 8 components of a couple of Majorana spinors are expressed as a vector, a selfdual 2-form and a scalar. The spin-statistics relation is recovered after untwisting, and the full Poincaré supersymmetry appears as an effective phyical symmetry. The missing (eg 7=16-9 for N=4 SYM) symmetry generators pop-up for free, without having been invited. Analogous features occur for conformal supersymmetry

5 Supersymmetry and holomorphic twist The idea is of considering manifolds with SU(n) SO(d = 2n) holonomy, and of using holomorphic representations of spinors. This gives a strikingly simple description of N=1 SYM supersymmetric theories, up to d=10. The trick is that the 2 n components of a spinor naturally decompose on holomorphic and antiholomorphic forms. The new result is for N=1, d=4 supergravity. We first explain the d=4 Yang Mills case

6 The action is d 4 xtr{ 1 2 F µνf µν i λ Dλ h2 } d 4 xtr{ 1 2 ϕ D µ D µ ϕ i λ Dλ X X} When the 4-manifold has holonomy SU(2), both balanced twisted multiplets (3,4,1) and (2,4,2) of the N=1,d=4 gauge theory are A m, A m, Ψ m, κ m n, κ, h Φ, Φ, Ψ m, κ mn, κ, B m n, B mn The complex coordinate notation is z m, z m with m = 1, 2 and m = 1, 2 for the euclidean space

7 To build invariants, one uses the complex structure tensor J. The notation X m Y m, m=1,2, means X m Y m X m Y n J m n where the tensor J m n is the complex structure of the Kähler manifold with SU(2) U(1) SO(4) holonomy

8 One has F µν F µν + d(..) = F m n F mn + (F m m ) 2 F mn F mn + hfm m h2 Moreover, selfduality of a 2-form is F m n = F mn = 0 Fm m = J m n F m n = 0 and antiselfduality is F m n + J m n F p p = 0-8 -

9 Twisted spinors In Euclidean 4-space, the analytical continuation of a Majorana spinor is defined by (anti) holomorphic forms λ (Ψ m, κ m n, κ) that is, ( 1 2, 1 2 ) = (1, 0) (0, 2) (0, 0). This generalizes in higher dimensions. Indeed, [lawson], on a complex spin manifold the complex spinors can be identified with forms S ± C Ω 0,even odd, so that we can identify Ψm as a left handed Weyl spinor λ α and (κ m n, κ) as a right handed Weyl spinor λ α. A reality condition implies that the fields Ψ m, κ m n, κ count for 4 (real) field degrees of freedom. This is what one enforces in the path integral formula

10 What is going on is the following change of field variables (Ψ m, κ m n, κ), with Ψ m =λ α σ µ α 1 eµ m, κ m n = λ α σ α κ=δ α 2 λ α µν 2 eµ me ν n, We have defined the euclidean σ matrices as σ µ = (iτ c, 1), τ c, c = 1, 2, 3 being the Pauli matrices. There is nothing very mysterious. In a Weyl basis (where γ 5 is diagonal), one has something like Ψ m λ α κ m n = λ 1 + λ 2 κ = λ 1 λ

11 Thus the 4d-Dirac action for a Majorana spinor satisfies λγ µ D µ λ = κ m n D [m Ψ n] + κd m Ψ m It can be added to the Yang-Mills action F mn F mn + hf m m h2 These decomposition is most useful for twisting YM theories with only N=1 susy (Johansen, Losev, Anselmi, Witten, Baulieu, Kanno, Singer, Bossard, Tanzini, etc...)

12 3 nilpotent symmetry generators exist (one scalar δ and one vector δ p ), acting on the fields of the balanced multiplets, with δ 2 =0 {δ p, δ q }=0 {δ, δ p }= p + δ gauge (A p ) The δ and δ p transformation laws can be determined, as when one solves cohomological BRST equations. This reduces the question of finding a supersymmetric theory to a cohomological problem

13 When one writes an invariant Lagrangian, it turns out to be δ-exact. Then, the vector symmetry δ p is associated to the conservation law of the δ-antecedent of the energy momentum tensor of the δ-exact action. So, N=1,d=4 supersymmetry is fully fixed by the invariance under 3 generators. The fourth symmetry is a consequence of these 3 symmetries. Its generator δ mn is also nilpotent

14 In practice, the 3 relevant supersymmetry generators δ and δ p can be determined either directly, by TQFT gauge symmetry considerations, as a generalization of Baulieu Singer method, or by trial and error, by enforcing the nilpotency properties. The scalar supersymmetry of N=1,d=4 possesses a very simple cohomology. More fields are needed to enrich this cohomology, as in N=2, or N=4. The action of δ and δ p reads as follows, in a TQFT typical way :

15 δa m = Ψ m δa m = 0 δψ m = 0 δκ = h δh = 0 δκ m n = F m n δ p A m =J pm κ δ p A m = κ p m δ p Ψ m = F pm J pm h δ p κ = 0 δ p h = D p κ δ p κ m n = 0 δ Φ = κ δ κ = 0 δκ mn = B mn δb mn = 0 δb m n = 2D [ m Ψ n] + [κ m n, Φ] δψ m = D m Φ δφ = δ p Φ = 0 δ p κ mn = 2J p[m D n] Φ δ p B mn = 2J p[m D n] κ + D p κ mn 2J p[m [Ψ n, Φ] δ p B m n = 0 δ p Ψ m = B p m δ p Φ = Ψ p δ p κ = D p Φ

16 Lagrangian : The δ and δ p invariant action is made of two independent terms L = δ(κ m n F mn + κ(h + F m m )) + δ(κ mn B m n + ΦD m Ψ m ) The δ p symmetry fixes the δ-antecedents of both terms, and, moreover, L is δδ p -exact, L = δδ p (...)

17 Notice that one has the symmetry δ mn, which pops up as a providential symmetry of both terms of L = δδ p (...), and satisfies {δ mn, δ mn } = 0 {δ, δ mn } = 0 {δ mn, δ p } = J p[m n] Once the existence of δ mn has been noticed, the 4 generators δ, δ p, δ mn can be untwisted as the 4 superpoincare generators Q α, Q α

18 Quantum field theory applications (not the subject here): Only 3 generators are needed, for a complete analysis of all N=1,d=4 theories, including finiteness theorems, and the relationship between ABBJ anomalies and supersymmetry anomalies. This generalizes, when on extends the supersymmetry, introducing R- symmetries. For instance, to prove the vanishing of the β function in N=4, or various N=2, only the invariance under 5 generators is needed to provide the necessary Ward identities, within the context of local quantum field theory. A lot of this extends for superconformal symmetry in d=4; and also in d=8, using octonionic or SU(4) type instantons

19 I = N=1,d=4 supergravity revisited ɛ abcd e c e d R cd (ω) + λγ a e a γ 5 Dλ + B 2 da + G 3G 3 A 1 and B 2 (Sokatchev and all; Stelle and West) are propagating auxiliary fields, with gauge invariances, giving an off-shell balanced multiplet(6,12,6) (e a (6 = ), λ(12 = 16 4), A(3 = 4 1), B 2 (3 = ) = (6, 12, 6) A gauges chirality. D = ω ab γ ab + Aγ 5 is the covariant derivative and G 3 = db λγ a λe a

20 The spin connection is constrained by T a = de a + ω ab λγ a λ = G a cde b e c so ω ab = ω ab (e, λ, B 2 ) = ω ab (e, λ) + G ab c e c The gravitino λ = λ µ dx µ is a 1-form Majorana spinor. It will be twisted accordingly

21 Following a 1986 work of Baulieu and Bellon, call s the BRST operator of reparametrization (ξ), local susy(χ), Lorentz(Ω), chiral U(1) (c), 2- form gauge symmetry (B1). 1 We can subtract from s the reparametrization, by defining ŝ ŝ = s L ξ with sξ µ = ξ ν ν ξ µ χγµ χ The off-shell closure relation s 2 = 0 is equivalent to ŝ 2 = i χγχ Reparametrization invariance is decoupled by the operation exp i ξ, when classical and ghost fields are unified into graded sums

22 The action of the oeprator ŝ is as follows ŝe a = Ω ab e b λγ a χ ŝλ= Dχ Ω ab γ ab λ cγ 5 λ ŝb 2 = db 1 1 λγ a χe a ŝa= dc χγ a γ 5 X a ŝω ab = DΩ ab + χγ a γ 5 X a (1) The spinor X a turns out to be proportional to equations of motions, X a = ρ ab e b (G abc γ bc + ɛ abcd γ 5 )λ The property s 2 = 0 is warranteed by the ghost transformation laws

23 There is unification between classical fields and ghosts going on. It is as the one that occurs when analyzing anomalies by descent equations. Here it is : ω ω + Ω, A A + c, λ λ + χ, B 2 B 2 + B1 1 + B0, 2 d d + ŝ + i χγχ. The symmetry of supergravity is given by distorted horizontality conditions on the curvatures of unified fields, such as ˆF (d + ŝ + i χγχ )(A + c), and so on. Off-shell closure is equivalent to the consistency of Bianchi identities. The existence and the determination of the invariant action are almost obvious from the Bianchi identities

24 One has in fact ˆT a ˆde a + (ω + Ω) ab e b ( λ + χ)γ a (λ + χ) Ĝ 3 ˆd(B 2 + B B 2 0) ( λ + χ)γ a (λ + χ)e a ˆρ ( ˆd + ω + Ω + A + c)(λ + χ) = G a cde b e c = G abc e a e b e c = ρ ab e a e b ˆR ab ˆd(ω + Ω) + (ω + Ω) 2 = R ab + χγ [a X b] χγc χg ab c ˆF ˆd(A + c) = F + χγ a γ 5 X a κγc χɛ abcd G bcd (2) where ˆd = d + ŝ + i χγχ. The horizontality distorting spinor functional X a = ρ ab e b (G abc γ bc + ɛ abcd γ 5 )λ is uniquely determined by Bianchi identities

25 Novel feature : Twisted expression of N=1, d=4 supergravity The twisted gravitino is made of the 3 one-forms Ψ m, κ m n, κ The Rarita Schwinger action is κ m n e n DΨ m + κ e m DΨ m The Einstein action will turn out to be very simple in twisted variables. Does it mean that the link between topological gravity and supergravity is as simple as that between NSR superstrings and the topological sigma model?

26 The twisted supergravity fields are f ieldsgh.n.m.d. f ields :gh.n.m.d. e m 0 / A 0 / e m 0 / B 2 0 / ψ m 1 / ω m n 0 / κ mn 1 / ω mn 0 / κ 1 / ω m n 0 / (3) The needed twisted spinorial field-equation dependent functions X a that we found for the distortion of horizontality equations of supergravity with M. Bellon in our old untwisted work are now (LB, M. Bellon V. Rey) X p, X p, X p,m, X p,m, X p, m n, X p, m n, (4)

27 For what follows, the important observations (see Landau, R = ΓΓ) is that the Einstein Lagrangian, modulo a boundary term, is the square of the self-dual (or antiselfdual) part of the connection L E = ω ab+ e b e c ω ac+ (5) This comes from L E = ɛ abcd e a e b R cd = e a e b (R ab+ R ab ) e a e b R ab = e a e b (R ab+ + R ab ) = d(...) We will decompose this in (anti)holomorphic components

28 For any given 2-form F, one has (F + ) mn = F mn (F + ) m n = F m n (F + ) mn = J mn (J pq F pq ) One thus finds the yet unfamiliar expression of the Einstein action ( ) L E = ω m n e n e p ω pm + ω (e n e m ω m n + e n e m ω mn where ω J pq ω pq. The analogy with YM is striking ; It suggests us how to decompose the BRST symmetry operator s into scalar, vector and (irrelevant) antiselfdual parts

29 δ δ q δ mn e p ψ p J qp κ 0 e m 0 κ q m J m[n ψ m] ψ n 0 ω qn + J qn ω 0 κ ω + A 0 ω mn κ m n ω m n 0 J m[n ω m] n + J m[n J m] n ω (6) B 2 Ψ m e m κ q n e n + J qn κe n Ψ m e n A X q,p X q + X m, m q X m,n ω rs X r,s J qr X s 0 ω r s 0 X q, r s + J qq X q, r s J r[m X s,n] ω r s X s,r + J s,r X q,p X q,r s + J qr X s 0 The four twisted supersymmetry operators δ, δ p and δ mn are given by this table

30 Asking δ, δ p invariances, one finds that the supergravity action is thus localized around gravitational instantons, in a typically TQFT invariant way I E = ( ) δ κ m n e n e q ω pm + κ(e n e m ω nm + e n e m ω m n + e m e m ω ) That is I E = ( ɛ abcde a e b e c e d R cd (ω) + λγ a e a γ 5 Dλ + B 2 da + G 3G 3 ) where T a = G a bc eb e c, that is ω ab = ω ab (e, λ) + G ab c e c. It is remarkable that everything has been fixed by the δ and δ p symmetries and the demand of a scaling dimension of the action identical to that of the Einstein Lagrangian. The underlying localization around gravitational instantons could help for computing certain topological quantities

31 (I expect similar results in d=8, which, in turn, can be dimensionally reduced, SU(4) SU(2).)

32 Overall conclusion The SU(N) decompositions of spinors enlighten quite well the construction of twisted supersymmetric theories and the questions of non-closure of supersymmetry. It gives efficient tools for proving supersymmetric QFT properties. The trick is T wisted (more geometrical) supersymmetry P oincare supersymmetry Poincaré (as well as conformal) supersymmetry has a lot of troublesome redundancy. It may appear as a kind of physical effective symmetry, which sometimes emerges after untwisting a TQFT, whenever it is possible