Instantons in Deformed Supersymmetric Gauge Theories
Shin Sasaki (University of Helsinki)
Based on the work [hep[hep--th/th/0705.35320705.3532, JHEP 07 (2007) 068 ] (K.Ito, H.Nakajima and S.S) Closed string background and deformed SYM theories
Constant NS-NS 2-form ÆNoncommutative SYM theory in D-brane world- volume [Seiberg-Witten (1999)]
Constant R-R ÆSYM theory in deformed geometry (non(anti)commutative superspace)
[Ooguri -Vaf a, S eib erg (2003)]
We focus on deformed SYM theories defined on D-branes w.v. in the presence of R-R backgrounds
SUSY07 : 31 July@ Karlsruhe 2 R-R backgrounds in IIB string theory
SfSelf-dual R-R 5-ffform field strength Ædeffformation of superspace geometries N=2 non-singgyletly deformed harmonic N=1 superspace deformation superspace from open string amplitudes [Ooguri-Vafa, Seiberg (2003)] [Ito-S.S (2006)] N= 4 superspace deformation from open string amplitudes [Ito-Kobayashi-S.S (2006)]
R-R 1-form Æ deformation of superspace geometry with specific structure N=2 (N=4) singletly deformed harmonic superspaces (v.o. charge conservation)
R-R3R 3-form deformation? Æ [Ito-NkjiNakajima-S.S . (2007)] 3 D3/D(-1) system and instantons
ADHM construction = Systematic construction of instantons in gauge theories [Atiyah-Hitchin-Drinfeld-Manin (1978)] ADHM construction of instantons in SYM Æ D3/D-instanton brane configuration
U(N), k-instanton = k D(-1), N D3 D3/D(-1), D(-1)/D(-1) open string Æ ADHM moduli
k D(-1)-branes instanton effective action of 4D SYM theory Æ D(-1)-bfftitibrane effective action
N D3-branes SUSY07 : 31 July@ Karlsruhe 4 D-instanton effective action with SD R- R 3-form background
D3/D(-1) with self-dual R-R 3-form bkg. (N=2 ggpraviphoton bk g)g. )
D-instanton effective action = Deformed instaton effective action Æ -background deformation of N=2 instanton effective action [Billo-Frau-Fucito-Lerda (2006)] ÆBRST exact after topological twisting Ælocalization technique for moduli space integration of instanton partition function
Explicit calculation of instanton p.f. [Nekrasov-Okounkov (2003)]
R-R backgrounds naturally regularize the moduli space integration 5 Deformed SYM in the presence of SD 3-form R-R bkg. [Ito-Nakajima-S.S (()]2007)] Filed theory calculations of deformed instantons (consistency check) Deformed SYM theory in the presence of self-dual R-R3R 3-form background Æ Open string disk amplitudes with closed string vertex insertion (NSR-formalism)
R-R vertex operator deformed action open string vertex operator
RR correction
SUSY07 : 31 July@ Karlsruhe 6 N=4 SYM in the presence or self-dual R-R 3-form bkg.
Euclidean D3-branes in Æ N=4 U(N) SYM Self-dual R-R3R 3-form field strength background
N=2 graviphoton and vector-like backgrounds are unified into N=4 graviphoton background
AllifAll informa tion o fN2iif N=2 is inc lddithN4luded in the N=4
# of non-zero disk amplitudes to calculate is less than N=2 case
SUSY07 : 31 July@ Karlsruhe 7 N=4 oppgen/closed string vertex operators
note: closed string left and right mover are identified by boundary condition [Billo-Frau-Pesando-Lerda (2004)] Non-zero amplitudes— selection rules
(I) Charge cance lla tion com bina tion
(II) power counting Disk normalization + Zero-slope scaling vertex operators
(III) Lorentz and internal index structure Ex.
There are three non-vanishing
contributions SUSY07 : 31 July@ Karlsruhe 9 Deformed N=4 SYM action
Zero-slope limit Æ N=4 deformed action Suppyersymmetr y and consistec y with the Myers action
Remaining deformed SUSY
Myers action
Self-dual 3-form (and its dual 7-form)
Bosonic part is consistent with the deformed action derived from amplitudes calculation 11 N= 4 ADHM so lu tion [K.Ito, H.Nakajima, S.S, work in progress]
From the deformed action , instanton equation and its solution are obtained
N=4 ADHM solution : ADHM moduli
12 ADHM cons tra in t
ADHM constraint
Constraint on
SUSY07 : 31 July@ Karlsruhe 13 Comment on mass deformation (R-R 3-form with internal SD condition)
Mass deformation of the instanton effective action can be achieved by Another type of R-R3R 3-form backgorund (self-dual in internal space)
for appropriate mass matrix background
SUSY07 : 31 July@ Karlsruhe 14 N=4 Æ N=2 reduction
Set-up Orbifolding D3/D3 Vertex operator projection ÆU(N) N=2 SYM on the branes N (fractional) D3-branes at orbifold fixed point
CttConstant gravi ihtbkphoton backgroun d
Vector-like background (after orbifold projection)
N4N=4 Æ N2bN=2 bac kgroun ddd decompos ition graviphoton
Vector bkg. 15 N=4 Æ N=2 reduction
N=4 Æ N=2 orbifold projection for open string v.o.
SUSY07 : 31 July@ Karlsruhe 16 Amplitudes reduction
Ægraviphoton + vector-like background
SUSY07 : 31 July@ Karlsruhe 17 Deformed N=2 SYM : SUSY
Supersymmetry is deformed and the number of SUSY is depen den t on the ran k o f the bac kgorun d
SUSY07 : 31 July@ Karlsruhe 18 Deformed ADHM from deformed SYM [K.Ito, H.Nakajima, S.S, work in progress]
ADHM solution
Deformed constraint
19 Instanton effective action
Instanton effective action
[Billo-Frau-Fucito-Lerda (2006)] ÆPrecisely reproduces D-D-instantoninstanton effective action in the presence of Omega-background (up to term)
R-R deformed N=2 instanton Ù N=2 Omega-background effective action ? R-R deformed N=1,4 instanton Ù N=1,4 Omega-background effective action [K.Ito, H.Nakajima and S.S, work in progress] Summary and future work
Derived deformed N=4 SYM action that correctly reproduces NN2=2D3/D( D3/D(-1) system with self-dual R-R background and also omega-background deformation of instanton effective action ADHM solution in the deformed N=2 , 4 SYM is found Gravitational correction to instanton solutions
[future work] Perform complete calculation of N=4 instanton effective action an d fin d the s im ilar ity w ith N= 2 de forme d instanton calculus N=1 generalization
21 SUSY07 : 31 July@ Karlsruhe 22 Omega-background deformation
6D Omega-background
Omega-deformed N=2 SYM
Æ Deformed instanton effective action
BRST-exact after topological twisting ÆApplicable localization technique in moduli space integration ÆInstanton partition function can be calculated [Nekrasov-Okounkov (2004)] SUSY07 : 31 July@ Karlsruhe 23