2011 年 4 月 27 日 1 吉田豊 Y. Yoshida arXiv: [hep-th]

2011 年 4 月 27 日 2 Moore, Nekrasov & Shatashivli (1998), Nekrasov(2002) Instanton partition function in N =2 4-dim SYM k -Instanton partition function by Localization formula ex) G=U(N) vector multiplet Instanton number

2011 年 4 月 27 日 3 Instanton partition function with surface operator in N =2 SYM Alday et al(2009), Alday & Tachikawa, Bruzzo et al(2010) Instanton numberThe first Chern number Dimofte, Gukov & Hollands (2010) : Vortex partition function in N =(2,2) 2dim SQED ?

2011 年 4 月 27 日 4 The moduli space of abelian vortex with vortex number k in two dimension is isomorphic to Jaffe & Taubes(1980) Equivariant character k -vortex partition function for N =(2,2) SQED with single chiral multiplet contour integral representation

2011 年 4 月 27 日 5 Contribution from a vector multiplet Contribution from a chiral multiplet vortex partition function of N =(2,2) SQED with chiral multiplet ？ twisted mass

2011 年 4 月 27 日 6 5d Nekrasov partition(K-theoretic instanton counting) Introduction of Surface operator Introduction of A-brane Closed A-model on toric CY G=U(1) 4-dim pure N =2 SYM ex) Theory induced on the surface operator is N =(2,2) U(1) SQED with single chiral mutiplet string sidegauge theory side Kozcaz, Pasquetti & Wyllard(2010)

1. Introduction  2.Vortices in 2d super Yang-Mills theories  3. Localization of vortex in N =(2,2) SYM  4. Vortex partition and equivariant character  5. Relation to geometric indices  6. Summary 2011 年 4 月 27 日 7

8 Vortex equation (Bogomol’nyi equation) with G=U(N) 1.This equation preserves half of the supersymmetry. 2. On-shell action. Vortex number is defined by the first Chern number complexified FI-parameter

2011 年 4 月 27 日 9 Super YM theory with 8 SUSY (2-dim N =(4,4) SYM) The vector multiplet in N =(4,4) SYM consists of Hypermultiplets in N=(4,4) theory consists of matter content of N =(4,4) theory N =(2,2) vector multiplet N =(2,2) adjoint chiral multiplet N =(2,2) fundametnal chiral multiplet N =(2,2) anti-fundametnal chiral multiplet

2011 年 4 月 27 日 10 Vacuum (Higgs branch) r:FI-parameter Symmetry group of Vacuum Bosonic part of Lagrangian Global gauge group Flavor group twisted mass

2011 年 4 月 27 日 11 k -vortex moduli space in ( p+2 )-dim U(N) SYM with 8 SUSY by k D p - N D( p+2 ) brane construction(Hanany & Tong 2002) NS5 o o o o o o D2 o o o D0 o vortex partition function(zero mode theory) in N =(4,4) SYM from brane system

2011 年 4 月 27 日 12 D0-D0 D0-D2 I : orientational moduli B : translational moduli DRED of vector with gauge group DRED of adjoint chiral multiplet DRED of chiral malutiplet

2011 年 4 月 27 日 13 :k-vortex partition functions Chen and Tong (2006)  Mass deformation D-term condition The moduli space of k -vortexEto et al(2005) Hanany & Tong(2002) We consider mass deformation N =(4,4) theory. Taking large mass limit, we obtain N =(2,2) SYM with N chiral multiplets. Edalati & Tong (2007)

2011 年 4 月 27 日 14 DRED of 2d (0,2) chiral multipet DRED of 2d (0,2) fermi multipet In the presence of the mass term, vortex partition function is deformed multiplets decouple from the vortex theory heavy mass limit

2011 年 4 月 27 日 15  k -vortex partition function for N =(2,2) U(N) SYM with N -fundamental matter with This action is expressed in Q-exact form

2011 年 4 月 27 日 16 SUSY transformation generates the following vector field on Nekrasov (2002) Bruzzo et al (2002) Superdeterminant

2011 年 4 月 27 日 17 k -vortex parition function in G=U(N) N =(2,2) SYM N -flavor Vortex partition function in G=U(1) N =(2,2) SQED This agree with the result from the equivariant character

2011 年 4 月 27 日 18 We introduce the following torus action  Vortex moduli space

2011 年 4 月 27 日 19 At the fixed points, we can decompose the representation space as Gauge transformation Restriction map  Fixed point condition

2011 年 4 月 27 日 20 2d partition (Young diagram) 1d partition In the case of 4-dim instanton… In the case of 2-dim vortex

2011 年 4 月 27 日 21 character of each spaces Infinitesimal gauge transformation Tangent space of k -vortex moduli space

2011 年 4 月 27 日 22 equivariant character 3d vortex partition function Replacement

2011 年 4 月 27 日 23 -genus of complex manifold M Equivariant case The fixed points The weight at the point

2011 年 4 月 27 日 24 3d vortex partition function This corresponds to geometric genus This corresponds to Euler number N =(2,2) case N =(4,4) case

 We have obtained N =(2,2) vortex partition function from the mass deformation of N =(4,4) vortex partition function.  N =(2,2) vortex partition function can be written with Q-exact form ⇒ We can apply Localization formula ・ especially we reproduce abelian vortex from open BPS state counting or equivariant character of  Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram). 3d vortex partition is related to certain geometric indices of the k -vortex moduli space  Future direction Relation to integrable structure( KP hierarchy, spin chain), etc… 2011 年 4 月 27 日 25