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
Introduction Doubled α geometry. Expansion in fluctuations: Cubic amplitudes. Auxiliary fields vs. massive fields. O. Hohm, W. Siegel, and BZ, arxiv:1306.2970 O. Hohm and BZ, arxiv:1407.3803, arxiv:1407.0708. O. Hohm and BZ, arxiv:1510.00005, arxiv:1509.02930. 1 U. Naseer and BZ, arxiv:1602.01101 O. Hohm, U. Naseer, and BZ, to appear
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
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
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).
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 )..
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
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!
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
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 = 1+... What is the action in terms of conventional fields?
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 1 1 2 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
Transformations that generate [, ]? δ ξ+ ξ g = L ξg Exactly! δ ξ+ ξ b = L ξb +d ξ + 1 2 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 + 1 2 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) 2 1 12ĤijkĤijk, 11 Contains O(α 2 ) terms, as required by gauge invariance.
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
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.
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.
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 2 1 12ĤijkĤijk 1 48 α 2 R αβ µν R ρσ µν αβ R ρσ. 15
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 + 1 4 ( ie ij ) 2 + 1 4 ( je ij ) 2 +e ij i j φ φ φ 1 8 aij a ij 1 4 ( ia ij ) 2 + 1 2α 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 ).
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
Define R i j e ij 2 φ L (4,2) = 1 16 ( i j a ij R) 2 1 16 ( 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 + 1 4 1 8 aij a ij 1 4 1 8 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) 2 1 2 ia ij j ϕ+ 1 2 ϕ ϕ + 1 ( 1 α 2 aij a ij ϕ 2 ) ( i ā ij) 2 + 1 2 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 18 2. Ghost scalar with m 2 = 4/α 3. Massive scalar with m 2 = 4/α
INTRIGUING OBSERVATIONS 1. A massive DFT for gravity, KR field and scalar: ( L mdft = 1 4 eij e ij + 1 Di e ij) 2 4 + 1 ( 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. 19 5. Ghost condensation!
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