On the curious spectrum of duality-invariant higher-derivative gravitational field theories

Transcription

1 On the curious spectrum of duality-invariant higher-derivative gravitational field theories VIII Workshop on String Field Theory and Related Aspects ICTP-SAIFR 31 May 2016 Barton Zwiebach, MIT

2 Introduction Doubled α geometry. Expansion in fluctuations: Cubic amplitudes. Auxiliary fields vs. massive fields. O. Hohm, W. Siegel, and BZ, arxiv: O. Hohm and BZ, arxiv: , arxiv: O. Hohm and BZ, arxiv: , arxiv: U. Naseer and BZ, arxiv: O. Hohm, U. Naseer, and BZ, to appear

3 INTRODUCTION Both generalized geometry and doubled geometry (as in Double Field Theory) aim to find the proper framework for stringy gravity. Supergravity }{{} Riemannian geometry + α - corrections What is the geometry here? We have no good understanding of the geometry of string theory, not even that of classical string theory. 2

4 String theory on T D has an O(d,d,;Z) T-duality symmetry. The CSFT of this string theory is a fully consistent double field theory with infinite number of fields, momentum and winding. What can we see if we don t work with the full string theory? Classical string theory dimensionally reduced on T d leads to a LE-EFT (low-energy effective field theory) with an O(d, d,; R) duality symmetry. This is exact in α (but not in g s ). 3

5 The top-dimensional (D spacetime dimensions) classical LE-EFT theory for the massless fields is of the form S top = ( ) d D x L (2) +α L (4) +α 2 L (6) + Consider dimensional reduction on T d. Let D = d+n. We should get an n-dimensional LE-EFT with global O(d, d; R). Think of this as a constraint on the action S top. Very difficult to carry out in practice: Hard to do dimensional reduction of higher-derivative terms O(d,d) transformations acquire α corrections. Very complicated field redefinitions to display it. 4 If we work with alternative non-general covariant variables O(d, d) transformations may be left unchanged. This seems clear from CSFT and DFT (hints from Buscher rules).

6 5 How does T-duality work with α -corrections? p 1 ( S top = d D x (L (2) +α a 1,k L (4) 1,k +α L (6) 1,k +α 2 L (8) 1,k +α 3 L (10) 1,k +... ) k=1 p 2 +α 2 k=1 a 2,k ( L (6) 2,k +α L (8) 2,k +α 2 L (10) 2,k +... ) p 3 +α 3. k=1. a 3,k ( L (8) 3,k +α L (10) 3,k +... ) If L (2n) n,p built with familiar variables, T-duality has α corrections. p 1 duality invariants that begin with 4 derivatives p k duality invariants that begin with 2+2k derivatives a 2,1,,a 2,p2 label the duality invariants that begin with 6 derivatives Bosonic string theory O(α ): 8 possible terms up to field redefinitions. Only one linear combination is duality invariant (Meissner, Godazgar-Godazgar, Hohm-Zwiebach): p 1 = 1. No such analysis for O(α 2 )..

7 Guarantee T-duality symmetry by writing the top theory as a DFT x i = (x 0,,x D ), x i = ( x 0,, x D ) O(D, D; R) multiplets ) ) ) ( xi ( ( ξ X M = x i, M = i, ξ M = i i ξ i, M = 1,...,2D, O(D,D) index. O(D,D;R) metric: ( ) η MN 0 1 = η 1 0 MN Group elements: h t ηh = η Fields as O(D,D;R) tensors: Φ (X ) = h hφ(x), X = hx The doubled action is manifestly O(D, D; R) invariant: S top dft = d D xd D x L(Φ(X)). 6 It is also gauge invariant if we impose the strong constraint M M (...) = 0

8 SC means the theory depends on half the coordinates. Thus setting for all fields we must get S top dft S top d D x x i Φ(X) = 0 (1) O(D,D;R) breaks down to the subgroup that preserves (1): Geometric subgroup : GL(D) R D(D 1)/2. In dimensional reduction D = n+d we have and set (x 0,...,x n, x 0,..., x n ; y 1,...,y d, ỹ 1,...ỹ d ) x µ Φ(X) = 0, O(D,D;R) this time breaks down to ỹ Φ(X) = 0, Φ(X) = 0 (2) a ỹ a 7 O(d,d;R) GL(n) R n(n 1)/2. DFT guarantees the physical duality symmetry will arise!

9 There is no string theory in which we have control over the full set of classical α corrections. Would be nice to have an effective field theory with α corrections and exact duality invariance. 8

10 DOUBLED α GEOMETRY. Based on a chiral CFT (Siegel, Hohm, Siegel, Zwiebach): X M (z 1 )X N (z 2 ) = η MN ln(z 1 z 2 ) Action for a doubled metric M KL and a dilaton. M is an unconstrained version of the generalized metric H M and φ arise from tensor operators of the CFT and there is a CSFT like action S = e φ[ T S 1 6 T T T ]. In terms of the double metric and the dilaton: S = dxe φ[ 1 2 ηmn( M 1 3 M3) MN +cubic in M(2,4,6 deriv. ]. What does this action describe? 9 How is M related to conventional fields? M 2 = What is the action in terms of conventional fields?

11 Deformed brackets Physical subspace: gauge algebra is a deformed Courant bracket [ ξ1 + ξ 1, ξ 2 + ξ 1 ] = [ ξ 1, ξ 2 ] + Lξ1 ξ 2 L ξ2 ξ d( i ξ1 ξ 2 i ξ2 ξ 1 ) 1 2 ( ϕ(ξ1,ξ 2 ) ϕ(ξ 2,ξ 1 ) ) ϕ(v,w) tr ( d( V) W ) tr ( i V W ) dx i i k V l l W k dx i. Deformed inner product: V +Ṽ W + W = i V W +i W Ṽ ϕ(v,w), ϕ(v,w) tr( V W) Given a vector V i and a one-form Ṽ i the Lie derivative of the one-form along a vector ξ i is corrected: L ξ Ṽ = dϕ(ξ,v) 10

12 Transformations that generate [, ]? δ ξ+ ξ g = L ξg Exactly! δ ξ+ ξ b = L ξb +d ξ tr( d( ξ) Γ ). (Γ) k l Γ k il dxi, δ ξ+ ξ b ij = + [i p ξ q Γ p j]q δb needed to make the Chern-Simons corrected Ĥ diff. covariant: Ĥ(b,Γ) db Ω(Γ), Ω(Γ) = tr( Γ dγ+ 2 3 Γ Γ Γ) The theory to O(α ) is: S = d D x ( ) ge 2φ R+4( φ) ĤijkĤijk, 11 Contains O(α 2 ) terms, as required by gauge invariance.

13 S is exactly gauge invariant. Is it exactly duality invariant? Test T-duality by reduction to D = 1 (Hohm and BZ) 1. As expected, the O(α ) action is consistent with T-duality. 2. The O(α 2 ) action is not consistent with T duality. The action above cannot be the full theory. MUST INVESTIGATE THE THEORY FURTHER Connect to conventional fields. Some on-shell amplitudes. Exact quadratic theory. 12

14 DOUBLE METRIC TO GENERALIZED METRIC Hohm and BZ The generalized metric H satisfies (Hη) 2 = 1 and thus there are projectors If we write with F constrained to satisfy P = 1 2 (1 Hη), P = 1 2 (1+Hη) δh = PδHP T +P δh P T M = H+F F = PF P T + P F P T F is an auxiliary field: the equations of motion for M allow for an algebraic solution for F in terms of H and the dilaton. 13 This gives a systematic procedure to write the M DFT as a fully nonpolynomial H DFT with all numbers of derivatives.

15 CUBIC AMPLITUDES U. Naseer and BZ (2016) Use projected indices: A M = P Q M A Q, and A M = P Q M A Q to expand the double metric around a flat generalized metric M MN = H MN +m M N +m MN +a MN +ā M N Physical content: m MN e ij = h ij +b ij, a MN a ij, ā M N ā ij Gauge fixing conditions: i e ij = j e ij = 0. Three point functions: k (a) k (a) = 0, and k (a) k (b) = 0 14 The a fields look auxiliary and their action is, schematically, L = am+ a 2 +a 3 +a 2 m+am 2 The red term necessarily vanishes after gauge fixing. Eliminating a from the other terms yields quartic and higher order in m.

16 T ijk (k 1,k 2,k 3 ) η ij k k 12 +cyc., O( ) W ijk (k 1,k 2,k 3 ) 1 8 ki 23 kj 31 kk 12, O( 3 ) The on-shell three-point amplitudes A 3 for massless states are: (T +W) ijk (T +W) i j k bosonic string A 3 = e 1ii e 2ii e 3ii (T +W) ijk T i j k heterotic string (T +W) ijk (T W) i j k DFT The action therefore contains Riemann-cubed at order α 2 : S = d D x ) ge (R+4( φ) 2φ ĤijkĤijk 1 48 α 2 R αβ µν R ρσ µν αβ R ρσ. 15

17 THE MASS SPECTRUM (Hohm, Naseer, BZ, to appear) Study the exact quadratic action for e ij and the auxiliary fields L (2, 2) = 1 4 eij e ij ( ie ij ) ( je ij ) 2 +e ij i j φ φ φ 1 8 aij a ij 1 4 ( ia ij ) α a ij a ij 1 8āij ā ij 1 4 ( jā ij ) 2 1 2α ā ij ā ij The first line describes massless gravity, b-field and dilaton. The second and third line describe massive tensors. They have wrong-sign kinetic terms They are missing derivative terms 16 Do not have the Pauli-Fierz mass terms (a ij a ij a 2 ).

18 The field a ij in the two-derivative approximation propagates: 1. Ghost spin-two with m 2 = 4/α 2. Ghost scalar with m 2 = 4/α 3. Scalar tachyon with m 2 = 4/α ā ij propagates the same degrees of freedom but with opposite value of mass-squared. Consider the higher derivative contributions! 17

19 Define R i j e ij 2 φ L (4,2) = 1 16 ( i j a ij R) ( i j ā ij R) 2, L (6,2) = 1 64 ( i j ā ij + i j a ij 2R) 2 ( i j ā ij + i j a ij 2R). With the help of two scalar fields (and field redefinitions) one can write this as a two derivative theory!! L = 1 4 eij e ij aij a ij āij ā ij 1 4 ( i e ij) 2 ( + 1 j e ij) 2 4 +e ij i j φ φ φ ( i a ij) ia ij j ϕ+ 1 2 ϕ ϕ + 1 ( 1 α 2 aij a ij ϕ 2 ) ( i ā ij) iā ij j ϕ+ 1 2 ϕ ϕ 1 2āij ā ij + ϕ 2, The field a ij propagates: 1. Ghost spin-two with m 2 = 4/α Ghost scalar with m 2 = 4/α 3. Massive scalar with m 2 = 4/α

20 INTRIGUING OBSERVATIONS 1. A massive DFT for gravity, KR field and scalar: ( L mdft = 1 4 eij e ij + 1 Di e ij) ( D 4 j e ij) 2 +e ij D i D j φ φ D 2 φ 1 4 M2 (e ij e ij 4φ 2 ). 2. In the 2-derivative reformulation of the massive tensors one can take the tensionless limit α and gain gauge symmetries. 3. With slightly different factors of 2 in L (4) and L (6) one can find less DOF s than in the two-derivative action. Each a would propagate just a massive tensor and a massive scalar. 4. One can integrate the massive auxiliary fields and find a higher derivative (quadratic) theory for the massless graviton, b-field and dilaton Ghost condensation!

21 CONCLUSIONS AND OPEN QUESTIONS We begin to understand the physical content and interactions of an exactly a duality-invariant gravitational theory with derivative corrections. Develop a tensionless limit for the full theory. Is this some kind of string theory? No complete match with Siegel s chiral strings. Extract the lessons of duality covariance to all orders in α. Can we do bosonic strings? 20