iia string in ads4 x cp3 superspaceiia型弦在ads4 cp3 superspace 课件

AdS4 CP3 superspace Dmitri Sorokin INFN, Sezione di Padova ArXivArXiv:0811.1566 :0811.1566 JaumeJaume GomisGomis, , LinusLinus WulffWulff and and D.S.D.S. S SQSQS0909, , DubnaDubna, 30 , 30 JulyJuly 2009 2009 ArXiv:0903.5407 ArXiv:0903.5407 P.A.GrassiP.A.Grassi, , L.WulffL.Wulff and D.S. and D.S. 1 1 AdS4/CFT3 correspondence uHolographic duality between 3d supersymmetric Chern-Simons-matter theories (BLG ; StefanskijStefanskij; ; DAuriaDAuria, , FrFr, Grassi D.S., Tkach & Volkov 1984) Geometrical groundGeometrical ground: : S S7 7 is the is the HopfHopf fibrationfibration over over CP CP 3 3 with with S S 1 1 fiber: fiber: S S7 7 vielbeinvielbein: : e em m a a = e e mm a a ( (y y) ) A Am m( (y y) ) 0 10 1 CPCP 3 3 vielbeinvielbein and and U U(1) connection (1) connection F F2 2 dA=JdA=JCP CP 3 3 (does not depend on (does not depend on S S1 1 fiber coordinate fiber coordinate z z) ) Our goal is to extend this structure toOur goal is to extend this structure to OSp(8|4)/SO(7)OSp(8|4)/SO(7) SO(1,3)SO(1,3) 1010 Hopf fibration of S7 as a coset space SU(4)SU(4) U(1)U(1) SU(3)SU(3) U(1)U(1) As a As a HopfHopf fibrationfibration, locally, , locally, S S 7 7 =CPCP 3 3 U(1) U(1) S S = = SO(8)SO(8) SO(7)SO(7) - symmetric space- symmetric space CPCP 3 3 = = SU(4)/SU(3)SU(4)/SU(3) U(1) symmetric spaceU(1) symmetric space SU(4)SU(4) U(1)U(1) SU(3)SU(3) U U d d (1)(1) SU(4)SU(4) U(1) U(1) SO(8) SO(8) CosetCoset representative and representative and CartanCartan forms: forms: K KS S7 7 (y,z)=K(y,z)=KCP CP 3 3 (y)(y) e ezT zT 1 1 =e e y y aa P Pa a e ezT zT 1 1 ( (T T1 1 is U(1) generator) is U(1) generator) K K -1-1 S S 7 7 dKdK S S 7 7 = = e ea a( (y y) ) P Pa a + + ( (y y) ) MMSU(3) SU(3)+A +A( (y y) ) T T 2 2 + + d dz z T T1 1 ( (T T 2 2 U U (1)(1) ) = = dydym m e e mm a a( (y y) ) P Pa a + +( (dzdz+ + dydym m A A mm ( (y y) ) P P 7 7 + ( + (d dz z - - A A( (y y) ) T Td d + + ( (y y) ) MMSU(3) SU(3) T Td d =1/2 (T=1/2 (T 1 1 -T-T 2 2 ) is U) is U d d (1) generator, (1) generator, P P 7 7 =1/2 (T1/2 (T 1 1 +T+T 2 2 ) translation along ) translation along S S1 1 fiber fiber 1111 Hopf fibration of OSp(8|4)/SO(7)SO(1,3) uK11,32 - D=11 superspace with the bosonic subspace AdS4 S7 and 32 fermionic directions K K 11,32 11,32 = M = M10,32 10,32 S S1 1 (locally) (locally) uM10,32 - D=10 superspace with the bosonic subspace AdS4 CP3 and 32 fermionic directions (it is not a coset space) MM10,32 10,32 is the superspace we are looking for is the superspace we are looking for! ! basebase fiber fiber 1212 1st step: Hopf fibration over K10,24 = OSp(6|4)/U(3)xSO(1,3) uK11,24 = K10,24 S1 (locally) AdS4 x S7 SOSO(1,3)(1,3) SUSU(3)(3) U U d d (1)(1) OSpOSp(6(6|4) |4) U U(1)(1) K K 11,24 11,24 = =OSp OSp(6(6|4)|4) U U(1) (1) OSpOSp(8(8|4)|4) K K 11,24 11,24 ( (x,yx,y, , , ,z z)=)= K K10,24 10,24 ( (x,y x,y, , ) ) e e zTzT 1 1 = = e ex xa aP Pa a e ey y aa P Pa a e e Q Q 2424 e e zTzT 1 1 K K -1-1dK dK = = E E a a ( (x,yx,y, , ) ) P P a a + + E Ea a( (x,y x,y, , ) ) P Pa a + + E E ( (x,yx,y, , ) ) Q Q +( +(dzdz+ + A A( (x,yx,y, , ) ) ) ) P P 7 7 + ( + (d dz z - - A A) ) T T d d + + ( (x,yx,y, , ) ) MMSU(3) SU(3) CartanCartan forms: forms: 1313 2nd step: adding 8 fermionic directions 8 K K 11,3211,32 ( (x,y x,y, , , ,z z, , ) ) =K=K11,24 11,24 ( (x,yx,y, , , ,z z) ) e e QQ8 8 = = K K10,24 10,24 ( (x,y x,y, , ) ) e e zTzT 1 1 e e QQ 8 8 QQ 8 8 are 8 are 8 susysusy generators which extend OSp(6|4) to OSp(8|4) generators which extend OSp(6|4) to OSp(8|4) OSp(6|4) OSp(6|4) superalgebrasuperalgebra: : QQ24 24, ,Q Q24 24= =MMSO SO(2,3)(2,3)+ +MMSU SU(4)(4) QQ 8 8 , ,QQ 8 8 =MMSO SO(2,3)(2,3)+ +T T 1 1 OSp(2|4) OSp(2|4) superalgebrasuperalgebra: : Extension to OSp(8|4)Extension to OSp(8|4): : QQ24 24, ,Q Q 8 8 =M=M ? ? SO SO(8)/(8)/SUSU(4)(4) U U(1)(1) CartanCartan forms of forms of OSpOSp(8|4)/(8|4)/SOSO(7)(7) SOSO(1,3):(1,3): K K -1-1dK dK = = E E a a ( (x,y,x,y, , , )+)+dzdz V V a a( ( ) ) P P a a + + E Ea a( (x,y x,y, , , , ) ) P Pa a + +dzdz ( ( )+ )+ A A( (x,yx,y, , , , ) ) P P 7 7 + E+ E24 24( (x,y, x,y, , , ) ) Q Q24 24 + E + E 8 8 ( (x,y,x,y, , , ) ) Q Q 8 8 + connection terms + connection terms 1414 3rd step: Lorentz rotation in AdS4 x S1 (fiber) tangent space K K -1-1dK dK = (= (E E a a ( (x,y,x,y, , , )+)+dzdz V V a a( ( ) ) ) P P a a + + E Ea a( (x,y x,y, , , , ) ) P Pa a +( +(dzdz ( ( )+ )+ A A( (x,yx,y, , , , ) ) ) ) P P 7 7 + E+ E24 24( (x,y, x,y, , , ) ) Q Q24 24 + E + E 8 8 ( (x,y,x,y, , , ) ) Q Q 8 8 + connection terms + connection terms E Ea a ( (x,yx,y, , , , ) ) = (= (E E b b ( (x,y,x,y, , , )+)+dzdz V V b b( ( ) ) ) b b a a ( ( ) ) + + ( (dzdz ( ( ) + ) + A A( (x,yx,y, , , , ) ) 7 7a a ( ( ) ) Lorentz transformation of the Lorentz transformation of the supervielbeinssupervielbeins: : S S1 1 : : dz dz ( ( ) )+ + A A ( (x,yx,y, , , , ) ) = ( (dzdz ( ( ) + ) + A A( (x,yx,y, , , , ) ) 7 77 7 ( ( ) ) + + ( (E E b b ( (x,yx,y, , , , ) + ) + dzdz V V b b( ( ) ) ) b b 7 7 ( ( ) ) E E aa( (x,y x,y, , , , ) ) = E E aa( (x,y x,y, , , , ) ) E E 2424( (x,y, x,y, , , ) ), , E E 8 8 ( (x,y,x,y, , , ) rotated 32 fermionic ) rotated 32 fermionic vielbeinsvielbeins MM10,32 10,32 P Pa a P P7 7 1515 Superstring action in AdS4 CP3 superspace g gij ij = =E E i i A A E E j j B B ABAB induced worldsheet metric induced worldsheet metric NS-NS field NS-NS field B B 2 2 is obtained by dimensional reduction of is obtained by dimensional reduction of A A 3 3 in D=11 in D=11 Kappa-symmetry gauge fixingKappa-symmetry gauge fixing ( (P.A.GrassiP.A.Grassi, , L.WulffL.Wulff and D.S. arXiv:0903.5407 and D.S. arXiv:0903.5407 ) ) (Killing spinor, (Killing spinor, supersolvablesupersolvable, or , or superconformalsuperconformal gauge, 1998) gauge, 1998) - parametrize 3d slice of AdS4 - “- “chiralitychirality” condition on the ” condition on the AdSAdS boundary boundary 1616 Guage fixed superstring action in AdS4 CP3 (upon a T-dualization), P.A.Grassi, L.Wulff and D.S. arXiv:0903.5407 =( ( 2424 , , 8 8 ),), =1/2(1+ 1/2(1+ 0 0 1 1 2 2 ) ) The action contains fermionic terms up to a quartic order onlyThe action contains fermionic terms up to a quartic order only AdS4 CP3 = = 0 0 1 1 2 2 3 3 1717 Conclusion uThe comple