1. Introduction and Reciprocity Non Commutative (NC) spaces are - PDF document

2015/11/14 Noncommutative Instantons 1. Introduction and Reciprocity • Non ‐ Commutative (NC) spaces are defined by noncommutativity of spatial coordinates: Masashi Hamanaka      [ , ]   x x i : NC parameter (real const.) (Nagoya U, Japan)  [ , ]  (cf. CCR in QM : ) q p i YI TP Workshop QFT2015 November 10th 2015 (  ``space ‐ space uncertainty relation’’  )  Resolution of singularity ~ ・ We give a proof of one ‐ to ‐ one correspondence between moduli space of instantons and moduli space of ADHM (  new physical objects) Com. space NC Space data in noncommutative spaces. ・ MH&Toshio Nakatsu(Setsunan), NC Instantons and Reciprocity, Ex) Resolution of small instanton singularity to appear, [cf. arXiv:1311.5227] (  U(1) instantons) ・ MH&TN, work in progress. [Nekrasov-Schwarz] ASDYM eq. in 4 ‐ dim. with G=U(N) NC ASDYM eq. with G=U(N) in Moyal • ASDYM eq. (real rep.)    • NC ASDYM eq. (real rep.) , 1 , 2 , 3 , 4       , , F F       F F :          12 34 F A A A A A A 01 23 ( : ) F A A A A A A                         , F F Field strength F F 13 42 , 02 31   1  0  O      1     0 F F      F F : 14 23 . A Gauge field  0  2  03 12   O    2 0 (N × N anti-Hermitian)   (Spell:All products are Moyal products.)     Under the spell, ( 0 , 0 ( . .)) F F F cpx rep     z z z z z z 1 1 2 2 1 2 i       ( ) ( ) : ( ) exp   ( ) we can calculate : f x g x f x g x    2  • There are two descripitions of NC extension:          ( ) ( ) ( ) ( ) ( 2 ) ‐ Moyal ‐ product formalism (deformation quantization) f x g x i f x g x O   2 Note: Coordinates and functions themselves ‐ Operator formalism (Connes’ theory) are c-number-valued usual ones i i                 [ , ] : ( ) x x x x x x  2 2    i G=U(N) NC ASDYM in operator formalism 2. Atiyah ‐ Drinfeld ‐ Hitchin ‐ Manin Construction based on duality for the instanton moduli space • Take coordinates as operators (in 2dim): ˆ               [ ˆ , ˆ ] complex [ ˆ , ] 2 rescale [ ˆ , ˆ ] 1 4dim. ASDYang-Mills eq. ADHM eq. (≒0dim. ASDYM) x y i z z a a (Difficult) (Easy) field (infinite matrix): ann op. cre op.   [ ,  ]  [ ,  ]      0  0 F F B B B B I I J J  ˆ 1 1 2 2 ˆ acting on Fock space: ( ˆ , )  z z z z 1 1 2 2 F z z F m n   [ , ] 0  B B IJ mn 0 F 1 2 1:1 N × N PDE ,    0 , 1 , 2 ,... z z m n 1 2 H C n n k × k Matrix eqs. integration Occupation number basis Sol.= instantons Sol.=ADHM data ˆ ˆ  2 ( ˆ , ) Tr F z z (G=U(N), C =k) (G=`U(k)’) H 2   ˆ  , , F F m m n n : , : , : • NC ASDYM eq. (real rep.) , , , 1 2 1 2     m m n n : B I J 1 2 1 2 A N N k k k N N k 1 , 2 m , m , n , n  1 2 1 2 ˆ ˆ       1 , 0 F F H 01 23  O  Gauge trf.: Gauge trf.: 1  1 0   ˆ ˆ    ~ ~ ~         , 1 1 1 , ( ) F F A   g A g g g    B g B g g U k  0 2  02 31   1 , 2 1 , 2 H  O  ~ ~ ˆ ˆ  ( )     2 2 1 , 0 g U N     I g I J J g F F 03 12 1

2015/11/14 Fourier ‐ Mukai ‐ Nahm transformation Fourier ‐ Mukai ‐ Nahm transformation Beautiful duality between instanton moduli on 4 ‐ tori Beautiful duality between instanton moduli on 4 ‐ tori and instanton moduli on the dual tori and instanton moduli on the dual tori 4dim. ASDYang-Mills eq. 4dim. ASD Yang-Mills eq. 4dim. ASDYang-Mills eq. 4dim. ASD Yang-Mills eq. on a 4-torus on the dual torus on a 4-torus on the dual torus ~ ~ ~ ~ G   0   0     F F 0 F F 0 F F F F z z z z z z z z 1 1 2 2     1 1 2 2     1 1 2 2 1 1 2 2  ~  0 ~ 0 F F   N × N PDE 0 N × N PDE 1:1 0 F F F z z k × k PDE z z k × k PDE 1 2   1 2   1 2 1 2 Sol.=instantons 1:1 Sol.=the dual instantons Sol.=instantons Sol.=the dual instantons (G=U(N), C =k) (G=U(k), C =N) (G=U(N), C =k) (G=U(k), C =N) 2 2 2 2 map F (Dirac eq.) ~ ~   :   ( ) ,   : A x V V ( ) : A N N A k k A k k       Define the maps F & G,  N × N ~            ( ) 0 : ( , 1 ) V e A ix V e i   2 & G ◦ F=id. & F ◦ G=id.    a : 2  V k N  On a 4-torus On the dual 4-torus On a 4-torus : On the dual 4-torus : x Family index thm.   Fourier ‐ Mukai ‐ Nahm trf. (radii of the torus  ∞ ) Fourier ‐ Mukai ‐ Nahm transformation Beautiful duality between instanton moduli on 4 ‐ tori Duality between instanton moduli on R and instanton moduli on the dual tori and instanton moduli on ``1pt.’’ [cf. van Baal, hep ‐ th/9512223] 4dim. ASDYang-Mills eq. 4dim. ASD Yang-Mills eq. 4dim. ASDYang-Mills eq. 0dim. ASD Yang-Mills eq. ~ ~ ~ ~ ~ on a 4-torus on the dual torus      : [ , ] F A A A A        ~ ~ ~ ~     0   0   F F 0 F F 0 F F F F z z z z     z z z z     1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 Matrix eq.！  ~  ~ 0 0 F F   1:1 0 1:1 0 N × N PDE F N × N PDE F z z k × k PDE z z k × k PDE 1 2 1 2     1 2 1 2 Sol.=instantons Sol.=the dual instantons Sol.=instantons Sol.=``dual instantons’’ (G=U(N), C =k) (G=U(k), C =N) (G=U(N), C =k) (G=U(k), ``C =N’’) 2 2 2 2 map G (Dirac eq.) map F (0dim Dirac eq.) ~ ~ ~     ( ) :       A x N N ( ) , A V V : A A k k          ~ k × k             (    )   0 ( ) 0 V e A ix V e D e A i            x Matrix eq. !  N   : 2 : 2 k V k N 4  On the dual 4-torus  1 pt. On a 4-torus : On the dual 4-torus : On a 4-torus  R x Linear alg. Family index thm.   D ‐ brane interpretation of ADHM Construction Atiyah ‐ Drinfeld ‐ Hitchin ‐ Manin (ADHM) Construction based on the following duality [Witten, Douglas] 4dim. ASDYang-Mills eq. ADHM eq.. (≒0dim. ASDYM) 4dim. ASDYang-Mills eq. ADHM eq. (≒0dim. ASDYM) [ , ]  [ , ] RHS is in fact z z z z (G=U(N), C =k) (G=`U(k)’) 2 1 1 2 2 G(4dim D.eq.) G(4dim D.eq.)   [ ,  ]  [ ,  ]      0   [ ,  ]  [ ,  ]      0 0 0 F F B B B B I I J J F F B B B B I I J J 1 1 2 2 1 1 2 2 z z z z z z z z 1 1 2 2 1 1 2 2     [ , ] 0 [ , ] 0  B B IJ  B B IJ 0 0 F F 1 2 1 2 N × N PDE F(0dim D.eq.) N × N PDE F(0dim D.eq.) z z z z 1 2 1 2 k × k matrix eq. k × k matrix eq. 1:1 Sol.=instantons Sol.=ADHM data SUSY trf. of gaugino D-term conditions (G=U(N), C =k) (G=`U(k)’) 2 0-0 strings  k × k: B       Proved in the F :  ,  0-4 strings  k × N ： I,J B  k k : A N N 1 , 2 ● same way as        ( , )  diag F F   ● SD ASD : , :   I J the Nahm trf. k N N k ● N D4 branes k D0 branes 2