L.J.Mason- The twistor programme and twistor strings: From twistor strings to quantum gravity?

8/3/2019 L.J.Mason- The twistor programme and twistor strings: From twistor strings to quantum gravity?

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The twistor programme and twistor stringsFrom twistor strings to quantum gravity?

L.J.Mason

The Mathematical Institute, [email protected]

Based on JHEP10(2005)009 (hep-th/0507269),hep-th/0606272 with Mohab Abou-Zeid & Chris Hull, and jointwork with Rutger Boels and Dave Skinner.
http://goforward/http://find/http://goback/


2/21

Outline

1 The twistor programmeTwistor spacePenrose-Ward transformThe non-linear graviton

2 Twistor strings

3 A proof of the Berkovits-Witten conjectureString eld theoryTwistor actions

4 Einstein (Super-) gravity


3/21

Twistor Correspondences

Quantum gravity requires a pregeometry for space-time.Penroses Proposal:Twistor space is the fundamental arena for physics.Flat correspondence:

Complex space-time M = C4, coords x

AA, A= 0,1 ,A = 0 ,1at metric ds 2 = dx AA dx BB AB A B , AB = [AB ] etc..

Twistor space T = C 4 , coords Z = ( A, A ), =( A,A ).Projective twistor spacePT = {T { 0}}/ {Z Z , C } = CP 3 .

Incidence relationA = ix AA A .

{Point x M } {Lx = CP 1 PT }, hgs coords A .


4/21

Penrose transform

Massless elds cohomology on PT = PT L = {A = 0}(x )A1 A2 ... An = Lx A1 A2 . . . An f B dB , f H 1(PT , O( n 2))(x )A1 A2 ... An =

Lx

n f

A

1 A

2 . . . A

n

B dB , f H 1 (PT , O(n 2))

homogeneous fns O(n ) are f (Z ), f (Z ) = n f (Z ), C . Cech cohomology for open cover {U i }

H 1(O) = {

hol. fns f ij

on U i

U j ,

f ij +

f jk +

f ki =

0}/ {

f ij =

g i

g j , }

Dolbeault cohomology

H 1 (O(n )) = {f (0,1)(n )| f = 0}/ {f = g }

n = 0 , 4 for ASD, SD Maxwell n = 2 , 6 for linearized gravity (note p a rity as ymme try).
http://find/http://goback/


5/21

Ward transform

H 1 (O): ASD Maxwell elds holomorphic line bundles.

For vector bundles:

Theorem (Ward)ASD Yang-Mills elds, D = d + A with D 2 = F with F + = 0, on E M are in 1 : 1 correspondence with holomorphic Bundles E PT trivial on each L x .

Easiest proof is in Euclidean signature where p : PT M ;Can pull back (E , D ) M to give (E , ) PT .

For such bundles 2E = F + , so

bundle is holomorphic F + = 0 .

[Reverse direction requires some complex analysis.]


6/21

The non-linear graviton

Theorem (Penrose)

Deformations of complex structure, PT ; PT

1 1asd deformations of conformal structure (M , ) ;(M , [g ]).

For Ricci at g [g ], PT must have

A holomorphic bration p : PT CP 1 A Poisson structure {, }I , on bres of p, valued in p O( 2).

Main ideas : We deform PT by plate tectonics or changing .

Ricci-at linearised deformations H 1

(O(2)){,}I

H 1

(T 1,0 PT

).The CP 1s in PT survive deformation. Dene

M = {moduli space of degree-1 CP 1s PT }.

x , y M connected by a light ray Incidence CP 1x

CP 1y

= . ; ASD conformal structure, [g ], Weyl + = 0 on M .


7/21

Supersymmetric extensions

Super-twistor space [Ferber]: PT [N ] = CP3 |N

,homogeneous coords (Z , i ), i = 1 , . . . , N , i anticommuting.

Correspondence: PT [N ] CP1 (x AA , iA ) M +[N ],

M +[N ] = chiral super-Minkowski space. Incidence relation

A = x AA A , i = iA A

Penrose-Ward transforms extends, e.g. a s H 1(PT [4], O)expands:

a s = a + i i + i j ij + i j k ijkl l + 1234 b

susy multiplet As := ( A, i , ij , i , B ) helicity 1 to 1 on M +[4].


8/21

Supersymmetric Ward transform

SUSY Ward transform:For a s 0,1(ad G C ), take d-bar operator a s = 0 + a s

{Hol vector bundle E PT [4]} {N = 4 SYM multiplet on M +[4].}

But, the interactions are an ASD truncation:cf Chalmers-Siegel action for A1 (ad G ) , B 2+ (ad G )

S asd [A, B ] = M R B F + , ; F + = 0 , [d + A, B ] = 0 .To extend to full YM:

S [A, B ] = M R B F + 2 B B , ; F + = B , [d + A, F + ] = 0.


9/21

Supersymmetric Ward transform

SUSY Ward transform:For a s 0,1(ad G C ), take d-bar operator a s = 0 + a s

{Hol vector bundle E PT [4]} {N = 4 SYM multiplet on M +[4].}

But, the interactions are an ASD truncation:cf Chalmers-Siegel action for A1 (ad G ) , B 2+ (ad G )

S asd [A, B ] = M R B F + , ; F + = 0 , [d + A, B ] = 0 .To extend to full YM:

S [A, B ] = M R B F + 2 B B , ; F + = B , [d + A, F + ] = 0.


10/21

Wittens conjectureThe connected prescription, after Roiban, Spradlin & Volovich

On-shell generating function for amplitudes with fullYang-Mills interactions is

A[As ] =

d = 1 M d

[4]

Det ( a s |C ) d

M d [4] = contour in moduli space of connected algebraic curves

C degree d in PT [4], C M d [4], d some measure.Selection rules:

M d [4] contribution processes with d + 1 l SD gluons. l := Number of loops, g := genus of C , then g l .

; concrete algebraic formulae for all tree amplitudes [RSV].


11/21

The Cachazo-Svrcek-Witten formulationThe disconnected prescription

If C = d lines then M d [4] = ( M+[4])

d / Sym d , but must includepropagators from holomorphic Chern-Simons action

S asd [a s ] = PT s CS (a s )[4] , [4] = Z dZ dZ dZ 4

i = 1

di

; diagrammatic formalism with MHV diagrams glued togetherwith Chern-Simons propagators.Gluing is done on-shell.Gukov, Motl & Nietzke argue connected version.Path integral formulation: The above can be expressed as

A[As ] = Da s eR PT [4] CS (a s )[4]

d = 1 M d [4] Det ( a s |C ) dHere a

s are off-shell, but a

s , on-shell, at .

; twistor action, see Boels talk.


12/21

Underlying string theory

String models: PT [4] is Calabi-Yau Witten: B-model in twistor space coupled to D1-instantons

(the holomorphic curves C of M d [4]).

Berkovits: Open string action on Riem-surface C with bdy,Z I : C T [4], Y I : C 1,0 (C ) T [4], A0,1(C )

S [Z I , Y I , A] = C Y I Z I + AY I Z I + Complex conjugateelds are real on boundary ( split signature for M ).(C M d [4] are worldsheet instantons.)


13/21

Conformal Supergravity

But: [Berkovits, Witten] Amplitudes contain N = 4 conformalsupergravity; wont decouple from loops.

Theories depend on C -structure J of PT [4] and:Witten: B-eld b 1,1(PT [4]) coupled in via C b Berkovits: 1-form g I (Z ) Z I coupled in by C g I Z

I .(J , g I ) or (J , b ) spectrum of N = 4 conf sugra.

This is a problem for gauge theory applications, but anopportunity for quantum gravity (although wrong theory....).


14/21

Heterotic twistor-stringsSee talk by D.Skinner

Optimal formulation should be as a twisted (0, 2)-model. Model depends only on C -str, and b -eld, on PT [4] and

bundle E PT [4]. Gives a Dolbeault formulation that allows off-shell elds. Allows C 2 (E ) = 0 to incorporate instantons Gives interpretation of b as twisting of Courant bracket in

the context of Hitchins generalised complex structures. Witten, hep-th/0504078 gives correspondence with

Berkovits type models as sheaves of chiral algebras. Directly gives generating functions of amplitudes as

integrals of det |C over instanton moduli spaces.


15/21

String eld theorydegree 0; Perturbative effects

We prove Berkovits-Wittens conjecture for conformal Sugra:Off-shell theory depends on

PT [4], with Calabi-Yau almost complex structure J

b (1,1)J (PT [4])

Best guess (Berkovits-Witten) for string eld theory is action

S [J , b ] = PT [4](N (J ) b ) [4] (1)where N (J ) =

2J T (

1,0

) (0

,2

) , is Nijenhuis tensor of J .Field equations: N (J ) = 0 = b ; so J is integrable.Gauge freedom: b b + + . ; PT [4] is a complex manifold, b H 1 (PT [4], d

1).

Gives: Spectrum of N = 4 conf sugra with ASD couplings.


16/21

String eld theoryInstanton effects

Instantons are pseudo-holomorphic maps Z : C PT [4] and

contribute

d M d [4] d e(R C b ) (2)to string eld theory action, where C is an instanton and M d [4] a

contour in the moduli space of instantons of degree- d .Disconnected prescription: C = d i = 1Lx i , Lx i degree-1,x i M [4] = real space of degree-1 instantons soM d

[4] = ( M [4])d / Sym d . Thus instanton contribution is

d M d [4] d eR i Lx i b =

d (M [4])d / Sym d d

i = 1

eR Lx i b d4 |8x i

=d

1

d ! M [4]eR Lx b

d

= expM [4]

eR Lx b


17/21

Twistor action

Thus path integral becomes

DJ Db exp PT [4](N b ) [4] + M [4] eR Lx b d4 |8x giving a string-eld theory action (incorporating instantons)

S [J , b ] = PT [4](N b ) [4] + M [4] eR Lx b d4 |8x (3)

Theorem (M, hep-th/0507269)

Let the real slice of M [4] arise from Euclidean signature reality conditions, and assume that only spin-2 parts of N = 4 csugra spectrum are present Then: ( 3 ) is equivalent to conformal supergravity action

M [4]|C |2 on (Euclidean signature) space-time.

[See Rutger Boels talk for gauge theory tw istor ac tions .]

B k i


18/21

Back to twistor programmeQuantum gravity?

Returning to the twistor programme, this resolves oldquestions:

Quantize metrics on M ; well dened events but fuzzylightconesbad (elds dont know how to propagate),

Quantize on twistor space ; fuzzy events but well dened

light raysgood.Note: we only have ASD (leg break) data.For physics, we need the SD (googly) part also.

Lets quantize anyway!

But, do we quantize the CP1s or the complex structures?

Answer: In string theory, quantization of the CP 1s leads toquantization of the complex structures.

But: Above shows that this leads to conf sugra. How do we get

Einstein (super-) gravity?

Ei i S i


19/21

Einstein SupergravityGauging symmetries; work with Abou-Zeid & Hull

Basic idea: Conformal gravity contains Einstein gravity;eliminate superuous elds by imposing symmetries.Berkovits string on T [4]:Consider strings on T

[4], action S =

Y

I Z I ;

reduce to PT [4] by gauging symmetry along = Z I / Z I byincluding eld A0,1 (C ) in the action

S =

C Y I Z I + A J , J = Y I Z I ;

gauge freedom A A + gives (Y I , Z I ) (e Y I , e Z I ).


20/21

Reduction to Einstein:We gauge symmetries 1-forms pulled back from CP 1 orCP 1 |N for projections PT [N ] CP 1 |N CP 1 .

For a general choice of such symmetries there is ananomaly.

For CP R |N CP 1 |N , we require R = 3, no restriction on N !. In particular we obtain N = 4 and h , h above give spectrum

of N = 4 supergravity. For PT [4] CP

1 we obtain a theory with spectrum ofN = 8 supergravity.

We obtain correct interactions for ASD couplings from

string perturbation theory. It is an open question as to whether the instanton

contributions give couplings of the full theory as they do forconformal supergravity.Perhaps just ASD truncation, so back to square one???

T i t ti lf d l


21/21

Twistor action, self-dual casehep-th/0706.??; joint with Martin Wolf

linear gravitational eld h H 1 (PT , O(2))For N = 8, i.e., on CP 3 |8 , this gives full multiplet. ; inf. deformation {h , }I = Hamiltonian vector eld using {, }I .In the full nonlinear ASD case, h can be regarded as a nitedeformation of a contact form and determines the C -structure.Full non-linear ASD eld equations become

0h + {h , h } = 0

with Chern-Simons-like action

S [h ] = PT (h h + 23 {h , h }h ) = natural wt -4 hol. volume form.It remains to relate this to (part of) a twisto r-st ring the ory.

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L.J.Mason- The twistor programme and twistor strings: From twistor strings to quantum gravity?

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