Noncommutative (=NC) Instantons and Reciprocity · Anti‐Self‐Dual Yang ‐Mills (ASDYM) eqs....
Transcript of Noncommutative (=NC) Instantons and Reciprocity · Anti‐Self‐Dual Yang ‐Mills (ASDYM) eqs....
Noncommutative (=NC) Instantons and Reciprocity
・MH&T.Nakatsu, ADHM Construction and Group Actions for NC Instantons, to appear (Moyal‐product formalism)
・MH&T.Nakatsu, Gauge instantons in NC space, work in progress, … (operator formalism)
cf.MH&Nakatsu, ADHM const. of NC Instantons, arXiv:1311.5227,…
Masashi Hamanaka (Nagoya U, Japan)
Based on
Group30, July17th 2014 in [email protected]μSv/h
1. Introduction• Non‐Commutative (NC) spaces are defined bynoncommutativity of spatial coordinates:
(cf. CCR in QM : )( ``space‐space uncertainty relation’’)
Resolution of singularity( new physical objects)
Ex) Resolution of small instanton singularity( U(1) instantons)
ipq ],[
~
Com. space NC Space
: NC parameter (real const.)
[Nekrasov-Schwarz]
ixx ],[
Anti‐Self‐Dual Yang‐Mills (ASDYM) eqs. play important roles in elementary particle theory, geometry and integrable systems.• Finite‐action solutions (instantons) reveal non‐perturbative effects in QFT ADHM is essential.
• Noncommutative instantons are used to obtain the Nekrasov partition fcn.
• We want to understand the ADHM fully.• (NC extension background flux)
1. Introduction
ASDYM eq. in 4‐dim. with G=U(N)• ASDYM eq. (real rep.)
• There are two descripitions of NC extension:‐Moyal‐product formalism (deformation quantization)
‐ Operator formalism (Connes’ theory)
.2314
,4213
3412 ,
FFFFFF
AAAAAAF :
4,3,2,1,
:A Gauge field(N×N anti-Hermitian)
Field strength
NC ASDYM eq. with G=U(N) in Moyal• NC ASDYM eq. (real rep.)
1203
3102
2301
,
,
FF
FF
FF ):( AAAAAAF
00
00
2
2
1
1
O
O
(Spell:All products are Moyal products.)
)()()(2
)()(
)(2
exp)(:)()(
2
Oxgxfixgxf
xgixfxgxf
Under the spell, we geta theory on NC spaces:
i
xxxxxx
:],[
,)()()(),( )2(2)1()0( xAxAxAxA Under the spell, the solution is deformed:
Note: Coordinates and functions themselvesare c-number-valued usual ones
2. Atiyah‐Drinfeld‐Hitchin‐Manin Constructionbased on reciprocity for the instanton moduli space
0
0
21
2211
zz
zzzz
F
FF
kNNkkk JIB :,:,:2,1
ADHM eq. (≒0dim. ASDYM)(Easy)
4dim. ASDYang-Mills eq.(Difficult)
Sol.=ADHM data(G=`U(k)’)
Sol.= instantons(G=U(N), C =k)
1:1
NNA :
0],[0],[],[
21
2211
IJBBJJIIBBBB
k × k Matrix eqs.N × N PDE
2
Gauge trf.:
)(
11
NUggggAgA
Gauge trf.:
gJJIgI
kUggBgB~,~
)(~,~~
1
2,11
2,1
Fourier‐Mukai‐Nahm transformationBeautiful reciprocity between instanton moduli on 4‐
tori and instanton moduli on the dual tori
0
0
21
2211
zz
zzzz
F
FF
4dim. ASD Yang-Mills eq.on the dual torus
4dim. ASDYang-Mills eq.on a 4-torus
Sol.=the dual instantons(G=U(k), C =N)
Sol.=instantons(G=U(N), C =k)
1:1
NNA :
k × k PDEN × N PDE
2 2
0~0~~
21
2211
F
FF
kkA :~
On a 4-torus On the dual 4-torus
Define the maps F & G,& G◦F=id. & F◦G=id.
F
G
Fourier‐Mukai‐Nahm transformationBeautiful reciprocity between instanton moduli on 4‐
tori and instanton moduli on the dual tori
0
0
21
2211
zz
zzzz
F
FF
4dim. ASD Yang-Mills eq.on the dual torus
4dim. ASDYang-Mills eq.on a 4-torus
Sol.=the dual instantons(G=U(k), C =N)
Sol.=instantons(G=U(N), C =k)
1:1
VVxA ,)(
k × k PDEN × N PDE
2 2
0~0~~
21
2211
F
FF
kkA :)(~
map F (Dirac eq.)
On a 4-torus On the dual 4-torusx: :
0)~(
VixAeV
NkV 2:Family index thm.
Fourier‐Mukai‐Nahm trf. (radii of the torus∞)Reciprocity between instanton moduli on R
and instanton moduli on ``1pt.’’ [cf. van Baal, hep‐th/9512223]
0
0
21
2211
zz
zzzz
F
FF
0dim. ASD Yang-Mills eq.4dim. ASDYang-Mills eq.
Sol.=``dual instantons’’(G=U(k), ``C =N’’)
Sol.=instantons(G=U(N), C =k)
1:1
VVA
k × k PDEN × N PDE
2 2
0~0~~
21
2211
F
FF
kkA :~
map F (0dim Dirac eq.)
On a 4-torusR On the dual 4-torus1 pt.
0)~(
VixAeV
NkV 2:Linear alg.
4
]~,~[~~:~ AAAAF
Matrix eq.!
Matrix eq. !
Atiyah‐Drinfeld‐Hitchin‐Manin (ADHM) Constructionbased on the following reciprocity
0
0
21
2211
zz
zzzz
F
FF
kNNk
kk
JIB
:,:,:2,1
ADHM dim. (≒0dim. ASDYM)4dim. ASDYang-Mills eq.
Sol.=ADHM data(G=`U(k)’)
Sol.=instantons(G=U(N), C =k)
NNA :
0],[0],[],[
21
2211
IJBBJJIIBBBB
k × k matrix eq.N × N PDE
2Proved in the same way as the Nahm trf.
],[],[ 2211 zzzz RHS is in fact
1:1
F(0dim D.eq.)
G(4dim D.eq.)
ADHM(Atiyah‐Drinfeld‐Hitchin‐Manin) constructionEx.) Commutative BPST instanton(N=2, k=1)
0
0
21
2211
zz
zzzz
F
FF
ADHM eq.(≒0dim. ASDYM)4dim. ASDYang-Mills eq.
Sol.=ADHM data(G=`U(1)’)
BPST instanton(G=U(2), C =1)
0],[0],[],[
21
2211
IJBBJJIIBBBB
k × k matrix eq.N × N PDE
2
)(222
2
22
)(
))((2
22,)()(
ziF
zbxi
A
0( ),0,
11,2,12,1
JI
B
i
position
size0 : singular
M0
singularity
C
R
ADHM(Atiyah‐Drinfeld‐Hitchin‐Manin) constructionEx.) NC BPST instanton(N=2, k=1)
0
0
21
2211
zz
zzzz
F
FF
NC ADHM eq.NC ASDYang-Mills eq.
Sol.:ADHM data(G=`U(1)’)
NC BPST instanton(G=U(2), C =1)
0],[],[],[
21
2211
IJBBJJIIBBBB
k × k marix eq.N × N PDE
2
i
position
size0 : regular!
02( ),0,
,2,12,1
JI
B
Fat by ζ !
FA , :exact sol.
M0
Resolution of the singurarity!
C
R
ADHM(Atiyah‐Drinfeld‐Hitchin‐Manin) constructionEx.) NC BPST instanton(N=2, k=1)
0
0
21
2211
zz
zzzz
F
FF
NC ADHM eq.NC ASDYang-Mills eq.
Sol.=ADHM data(G=`U(1)’)
NC BPST instanton(G=U(2), C =1)
0],[],[],[
21
2211
IJBBJJIIBBBB
k × k matrix eq.N × N PDE
2
i
position
size0 : regular!
02( ),0,
,2,12,1
JI
B
Fat by ζ!
FA , :exact sol.By calculation of TrFΛF
Do k×k ADHM data giveInstanton number k in general ? (We prove this.)
C
R
3.Origin of instanton number in ADHM construction
We prove the formula:
Then:
kTrd
ffTrxdFFTrxdC
kk
kN
116
816
116
1
2
1242
**422
[Corrigan-Goddard-Osborn-Templeton]
comes from the size of the ADHM data!
)(12 rrf kkk
1)(: f
22
11
11
22 )(BzBz
Bz
BzJI :determined by the
ADHM data only:f always exists.
[Nakajima, Maeda-Sako]
)detlog( 220
124**4
fFFTr
ffTrxdFFTrxd
N
kN
)1),2((''1`` 22
kUr
f
Ex)
4. Proof of the reciprocity: (inst)↔(ADHM)
NC instanton NC ADHM
NNA :
kNNk
kk
JIB
:,:,:2,1
(i) ASD (ASDYM)(ii) C_2=k(iii) D^2 has inverse
(i) ASD (ADHM eq.)(ii) matrix size = k, N(iii) ∇^2 has inverse
F
G
Proof of the one-to-one ⇔ Define the maps F & G,& G◦F=id. & F◦G=id.
(iii) is automatically satisfied in the noncommutative situation [Maeda-Sako] [Nakajima]
F:(ADHM)→(inst):ADHM constructionNC instanton NC ADHM
NNVVA :
kNNk
kk
JIB
:,:,:2,1
NVVV 1,0
22
11
11
22 )(BzBz
Bz
BzJI
kkNopDirac 2)2.(dim0
0dim.Dirac eq.
(i) ASD (ASDYM)(ii) C_2=k
(i) ASD (ADHM eq.)(ii) matrix size= k, N←[MH Nakatsu]
[Nekasov-Schwarz]
G:(inst)→(ADHM): inverse constructionNC instanton NC ADHM
NNA :
kN
kk
rJI
zxdB
:,
,:
3
2,14
2,1
kxdDe 1,0 4
.dim4: opDiracDe
(i) ASD (ASDYM)(ii) C_2=k
(i) ASD (ADHM eq.)(ii) matrix size= k, N
4dim.Dirac eq.
[Maeda-Sako2009] proves the existenseof the Dirac zero-mode be a formalpower expansion of θ recursively. ,),(
,),()2(2)1()0(
)2(2)1()0(
AAAxAx
G◦F=id:(ADHM)→(inst)→(ADHM)NC instanton NC ADHM
??,?2,12,1
JJIIBB
4dim.Dirac eq.
The answerCfVVJIB
),,,(
0dim.Dirac eq.NC ADHM
VJIB ,,2,1 VVA
kxdDe 1,0 4
33
2,12,14
2,1
,,r
JIr
JI
BzxdB
[Maeda-Sako] are shown
find in terms of
the original B, I, J, Vto give the new data
F◦G=id:(inst)→(ADHM)→(inst)NC instanton
0dim.Dirac eq.
),( AVV
4dim.Dirac eq. NC ADHM
VJIB ,,2,1A
[Maeda-Sako] assume the existence of V.[MH-Nakatsu] prove it
NC instanton
? AA
CVD 42 :the answer
AVVA
NVVV 1,0
are shown (existence proof is also made by us)
find in terms of
the original instanton Aμto give the new instanton
5. Conclusion and Discussion• We prove the one‐to‐one correspondence between NC instantons and NC ADHM data in the Moyal‐product formalism. [to appear soon…]
• This is valid only in the region that the θ‐expansions converge.
• We proceed to reveal the reciprocity in operator formalism. (mostly completed [work in progress] )
• We are also interested in non‐finite action solutions (soliton‐like) of NC ASDYM (twistor theory and Ward’s conejecture) in relation to lower‐dimensional integrable systems such as KdV. [See e.g. MH, arXiv:1101.0005] (and perhaps the AGT corresp.)