Double Field Theory at SL(2) angles

Transcription

1 Double Field Theory at SL(2) angles Adolfo Guarino Université Libre de Bruxelles Iberian Strings 207 January 7th, Lisbon Based on arxiv: & arxiv:

2 Duality covariant approaches to strings - Different strings related by dualities: IIA/IIB T-duality, IIB S-duality, - String dualities realised as global symmetries in lower-dimensional SUGRA lower-dimensional phenomenon VS higher-dimensional phenomenon gaugings, embedding tensor, non-geometry, ß-supergravity, Today s talk : Extended Field Theories [ extend internal coords to transform under duality ] Double Field Theory (DFT) Orthogonal groups O(d,d) [ half-max SUGRA (T-duality) ] Exceptional Field Theory (EFT) Exceptional groups E d+(d+) [ max SUGRA (U-duality) ] [ Siegel 93] [ Hull & Zwiebach (Hohm) 09 0] [ Hohm & Samtleben 3 ] 2

3 Dualities in SUGRA and Extended Field Theory D Maximal sugra / EFT Half-maximal sugra DFT 9 R + SL(2) R + O(, +n) R + O(, +n) 8 SL(2) SL(3) R + O(2, 2+n) R + O(2, 2+n) 7 SL(5) R + O(3, 3+n) R + O(3, 3+n) 6 SO(5, 5) R + O(4, 4+n) R + O(4, 4+n) 5 E 6(6) R + O(5, 5+n) R + O(5, 5+n) 4 E 7(7) SL(2) O(6, 6+n) R + O(6, 6+n) 3 E 8(8) O(8, 8+n) R + O(7, 7+n) Duality groups of half-maximal SUGRA and DFT differ for D<5 * n = additional vector multiplets 3

4 in this talk we will look at D=4 : EFT with E 7(7) duality group [ Hohm & Samtleben 3 ] SL(2)-DFT with SL(2) x O(6,6+n) duality group [ arxiv: ] DFT with R + x O(6,6+n) duality group [ Siegel 93] [ Hull & Zwiebach 09] [ Hohm, Hull & Zwiebach 0] [ Hohm & Kwak ] 4

5 E 7(7) -EFT - Space-time : external ( D=4 ) + generalised internal ( coordinates in 56 of E 7(7) ) y M [ momentum, winding, ] Generalised diffs = ordinary internal diffs + internal gauge transfos Generalised Lie derivative built from an E 7(7) -invariant structure Y-tensor L U M = N U M U N M + Y MN N P U Q Closure requires a section constraint : Y PQ Q =0 Two maximal solutions : M-theory ( 7 dimensional ) & Type IIB ( 6 dimensional ) [ massless theories ] Massive IIA arises as a deformation of EFT [ Romans 86 ] [ Hohm & Kwak (sec const violated) ] [ Ciceri, A.G. & Inverso 6 ] 5

6 E 7(7) -EFT - E 7(7) -EFT action [ D µ µ L Aµ ] S EFT = Z d 4 xd 56 ye ˆR + 48 gµ D µ M MN D M MN 8 M MN F µ M F µ N + e L top V EFT (M,g) with field strengths & potential term given by F µ M = 2@ [µ A ] M A µ,a M E + two-form terms ( tensor hierarchy ) V EFT (M,g) = 48 M M N M KL + 2 M M L M NK 2 M N M MN 4 MMN M N g 4 M g N g µ - Two-derivative potential : ungauged N=8 D=4 SUGRA when 6 (x, y) = (x)

7 From E 7(7) -EFT to SL(2)-DFT - Halving EFT with E 7(7) symmetry to obtain SL(2)-DFT with SL(2) x O(6,6) symmetry E 7(7)! SL(2) SO(6, 6) 56! (2, 2)+(, 32) y M! y M + y A EFT SL(2)-DFT α = ( +, - ) vector index of SL(2) M vector index of SO(6,6) A M-W spinor index of SO(6,6) [ see Dibitetto, A.G. & Roest for SUGRA ] via a Z 2 truncation ( vector = +, spinor = - ) on coordinates, fields, etc. - SL(2)-DFT generalised Lie derivative [ DFT corresponds to an α = + orientation ] L U M = N U M U N M + MN N P U Q N [M U N] - SL(2)-DFT section constraints : N N] =0 7

8 - SL(2)-DFT action [ D µ µ L Aµ ] S SL(2)-DFT = SL(2)-DFT with SL(2) x O(6,6) symmetry Z d 4 xd 24 ye ˆR + 4 gµ D µ M D M + 8 gµ D µ M MN D M MN M 8 M MN F µ M F N µ + e L top V SL(2)-DFT (M,g) with field strengths & potential term given by F µ M = 2@ [µ A ] M A µ,a M S + two-form terms ( tensor hierarchy ) V SL(2)-DFT (M,g) = M αβ M MN[ 4 ( αmm γδ )( βn M γδ ) 8 ( αmm PQ )( βn M PQ ) + 2 ( αmm γδ )( δn M βγ )+ 2 ( αmm PQ )( βq M NP ) ] + 2 M MN M PQ ( αm M αδ )( δq M NP )+ 2 M αβ M γδ ( αm M MQ )( δq M βγ ) 4 M αβ M MN [ g ( αm g) g ( βn g)+( αm g µν )( βn g µν ) ] 2 g ( αm g) βn (M αβ M MN ), - Two-derivative potential : ungauged N=4 D=4 SUGRA when (x, y) = (x) 8

9 Section constraints & SL(2) angles - 6 dimensional solution of sec. constraints : N= SUGRA in D=0 [ as in DFT ] - Scherk-Schwarz (SS) reductions with SL(2) x O(6,6) twist matrices yield N=4, D=4 gaugings [ Schön & Weidner 06 ] U M N = e e U M N f MNP = 3 e e Q[M U N R U P ] R U S Q M = 2U M N (e e ) [ de Roo & Wagemans 85 ] - Moduli stabilisation requires gaugings G = G x G 2 at relative SL(2) angles f G 2 ( sec. constraint violated ) N] 6=0 [ not possible in DFT ] f + 9

10 Example : SO(4) x SO(4) gaugings and non-geometry - SS with U(y M ) 2 O(6, 6) : Half of the coords of type + & half of type - - SL(2)-superposition of two chains of non-geometric fluxes ( H,!, Q, R ) ± f + f +abc = H (+) abc, f +ijk = H (+) ijk, f +ab c =! (+) ab c, f +ij k =! (+) ij k f +ā bc = Q (+)ab c, f +ī jk = Q (+)ij k, f +ā b c = R (+)abc, f +ī j k = R (+)ijk f f ijk = H (-) ijk, f abc = H (-) abc, f ij k =! (-) ij k, f ab c =! (-) ab c f ī jk = Q (-)ij k, f ā bc = Q (-)ab c, f ī j k = R (-)ijk, f ā b c = R (-)abc Most general family (8 params) of SO(4) x SO(4) gaugings of N=4 SUGRA - SO(4) x SO(4) SUGRA : AdS 4 & ds 4 vacua ( sphere/hyperboloid reductions) [ de Roo, Westra, Panda & Trigiante 03 ] [ Dibitetto, A.G. & Roest 2 ] - ``Hybrid ± sources to cancel flux-induced tadpoles : SL(2)-dual NS-NS branes 0

11 Summary & Future directions - SL(2)-DFT captures the duality group of N=4 SUGRA in D=4 - SL(2)-DFT sec. constraints : N= SUGRA in D=0 & N=(2,0) SUGRA in D=6 - SL(2)-DFT action extendable to SL(2) x SO(6,6+n) and deformable as EFT [ Ciceri, A.G. & Inverso 6 ] - Non-geometric gaugings at non-trivial SL(2) angles : full moduli stabilisation [ not possible in DFT ] - Flux formulation of SL(2)-DFT : sec. cons violating terms & dual NS-NS branes [ Aldazabal, Graña, Marqués & Rosabal 3 ] - Cosmological applications of SL(2)-DFT ( de Sitter, inflation, ) [ Hassler, Lüst & Massai 4 ]

12 Muito obrigado!! Thanks a lot!! 2

13 Extra material 3

14 Dualities in SUGRA and Extended Field Theory D Maximal sugra / EFT Half-maximal sugra DFT 9 R + SL(2) R + O(, +n) R + O(, +n) 8 SL(2) SL(3) R + O(2, 2+n) R + O(2, 2+n) 7 SL(5) R + O(3, 3+n) R + O(3, 3+n) 6 SO(5, 5) R + O(4, 4+n) * R + O(4, 4+n) 5 E 6(6) R + O(5, 5+n) R + O(5, 5+n) 4 E 7(7) SL(2) O(6, 6+n) R + O(6, 6+n) 3 E 8(8) O(8, 8+n) R + O(7, 7+n) Duality groups of half-maximal SUGRA and DFT differ for D<5 * There is also the chiral N=(2,0) SUGRA in D=6 with R + x O(5,n) duality group 4

15 SO(4) x SO(4) twist matrices - O(6,6) twist : U M N (y M )= I6 0 6 I6 b u 06 I I u t = um n b mp (u t ) p n mp u p n (u t ) m n + mp b pq (u t ) q n where mn = ( ()) ab ( (2) ) ij!,b mn = (b ()) ab (b (2) ) n m = (u b ()) a 0 3 j 0 3 e (u (2) ) i! A u (),(2) = b (),(2) = (),(2) = (cos Y (),(2) +cos e Y (),(2) ) 2 (sin Y (),(2) +sin e Y (),(2) ) 0 2 (sin Y (),(2) +sin e Y (),(2) ) 2 (cos Y (),(2) +cos e Y (),(2) ) sin(y 2 (),(2) 0 sin(y 2 (),(2) ey (),(2) ) tan (Y 2 (),(2) 0 tan (Y 2 (),(2) ey (),(2) ) 0 ey (),(2) ) C A, ey (),(2) ) C A, C A, 0 e Y () = ( c 0 a 0 0)(y + y + )+( d 0 b 0 0)(y y ) ey () = ( c 0 + a 0 0)(y + + y + )+( d 0 + b 0 0)(y + y ) Y (2) = ( c 0 2 a 0 3)(y +4 y + 4 )+( d 0 2 b 0 3)(y 4 y 4 ) ey (2) = ( c a 0 3)(y +4 + y + 4 )+( d b 0 3)(y 4 + y 4 ) e

16 Deformed EFT ( XFT ) - Generalised Lie derivative [ no density term ] L U M = N U M U N M + Y MN N P U Q in terms of an E n(n) -invariant structure Y-tensor. Closure requires sec. constraint - Deformed generalised Lie derivative el U M = N U M U N M + Y MN N P U Q X NP M N U P in terms of an X deformation which is E n(n) -algebra valued non-derivative - Closure & triviality of the Jacobiator require ( together with sec. constraint ) X MN P =0 X MP Q X NQ R X NP Q X MQ R + X MN Q X QP R =0 X constraint Quadratic constraint (gauged max. supergravity) 6 [ X deformation vs embedding tensor ]

17 X deformation : background fluxes & Romans mass Y PQ Q =0 section constraint X P P =0 X constraint [ algebraic system ] M-theory ( n coords ) Type IIB ( n- coords ) SL(n) orbit Freund-Rubin param. ( n = 4 and n = 7 ) massless IIA (subcase) + SL(n-) orbit p-form fluxes compatible with SL(n-) SL(2)-triplet of -form flux ( includes compact SO(2) ) New massive Type IIA ( n- coords ) Massive Type IIA described in a purely geometric manner!! SL(n-) orbit p-form fluxes compatible with SL(n-) dilaton flux Romans mass parameter ( kills the M-theory coord ) [ QC = flux-induced tadpoles ] 7

18 E 7(7) -XFT action e LAµ - E 7(7) -XFT action [ D µ µ ] [ y M coords in the 56 of E 7(7) ] S XFT = Z d 4 xd 56 ye ˆR + 48 gµ D µ M MN D M MN 8 M MN F µ M F µ N + e L top V XFT (M,g) with field strengths & potential given by ( deformed tensor hierarchy ) F µ M = 2@ [µ A ] M + X [PQ] M A µ P A Q A µ,a M E + two-form terms V XFT (M,g,X) = V EFT (M,g)+ 2 MMN M KL X MK N M PL + V SUGRA (M,X) cross term gauged max. sugra - Two-One-Zero-derivative potential : gauged 4D max. sugra when (x, y) = (x) 8

19 Extended (super) Poincaré superalgebra - Central charges (internal symmetries) Z IJ =(a a IJ ) T a - The algebra : [P µ, P ]=0 [M µ, M ]=i ( M µ M µ µ M + µ M ) [P µ, M ]=i ( µ P µ P ) T a, T b = if ab c T c [T a, P µ ]=[T a, M µ ]=0 Q I, P µ = Q I, P µ = 0 Q I, T a =(b a ) I J Q J n Q I, Q J o Q I, T a = Q I, M µ = 2 ( µ ) Q I Q I, M µ = 2 Q I ( µ ) n n Q Q = 2 Z IJ Q I, Q J = 2 Z IJ Z o o n o Q I, Q J Q J (b a ) I J = 2 IJ ( µ ) P µ