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Localization of Vortex Partition F unction of N =(2,2) Super Yang-Mills Theory

Localization of Vortex Partition F unction of N =(2,2) Super Yang-Mills Theory. 吉田 豊. Y. Yoshida arXiv:1101.0872[ hep-th ]. Introduction. Instanton partition function in N =2 4-dim SYM. Instanton number. k - Instanton partition function by Localization formula

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Localization of Vortex Partition F unction of N =(2,2) Super Yang-Mills Theory

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  1. Localization of VortexPartition Function of N=(2,2) Super Yang-Mills Theory 吉田豊 Y. Yoshida arXiv:1101.0872[hep-th] 2011年4月27日　SAL@KEK

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

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

  4. Themoduli space of abelian vortex with vortex number k in two dimension is isomorphic to Jaffe & Taubes(1980) Equivariant character k-vortex partitionfunction for N=(2,2) SQED with single chiralmultiplet contour integral representation 2011年4月27日　SAL@KEK

  5. Contribution from a vector multiplet Contribution from a chiralmultiplet vortexpartition function of N=(2,2) SQEDwith chiralmultiplet？ twisted mass 2011年4月27日　SAL@KEK

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

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

  8. Vortices in 2d super Yang-Mills Theories Vortex equation (Bogomol’nyi equation) with G=U(N) 1.This equation preserves half of the supersymmetry. 2. On-shell action. complexifiedFI-parameter Vortex number is defined by the first Chern number 2011年4月27日　SAL@KEK

  9. Super YM theory with 8 SUSY (2-dim N=(4,4) SYM) matter content of N=(4,4) theory The vector multiplet in N=(4,4) SYM consists of N=(2,2) vector multiplet N=(2,2) adjointchiralmultiplet Hypermultiplets in N=(4,4) theory consists of N=(2,2) fundametnalchiralmultiplet N=(2,2) anti-fundametnalchiralmultiplet 2011年4月27日　SAL@KEK

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

  11. vortex partition function(zero mode theory) in N=(4,4) SYM from brane system k-vortex moduli spacein (p+2)-dim U(N) SYM with 8 SUSY by kDp- N D(p+2)brane construction(Hanany& Tong 2002) 0 1 2 3 4 5 6 7 8 9 NS5 o ooooo D2 o oo D0 o 2011年4月27日　SAL@KEK

  12. D0-D0 DRED of vector with gauge group DRED ofadjointchiralmultiplet B : translational moduli D0-D2 DRED of chiralmalutiplet I :orientationalmoduli 2011年4月27日　SAL@KEK

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

  14. DRED of 2d (0,2) chiralmultipet DRED of 2d (0,2) fermimultipet In the presence of the mass term, vortex partition function is deformed heavy mass limit multiplets decouple from the vortex theory 2011年4月27日　SAL@KEK

  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　SAL@KEK

  16. Localization of vortex partition functions in N=(2,2) SYM SUSY transformation generates the following vector field on Nekrasov (2002) Bruzzo et al (2002) Superdeterminant 2011年4月27日　SAL@KEK

  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　SAL@KEK

  18. Vortex partitionandequivariantcharacter • Vortex moduli space We introduce the following torus action 2011年4月27日　SAL@KEK

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

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

  21. characterof each spaces Infinitesimalgauge transformation Tangent space of k-vortex moduli space 2011年4月27日　SAL@KEK

  22. equivariant character Replacement 3d vortex partition function 2011年4月27日　SAL@KEK

  23. Relation to geometric indices -genus of complex manifold M Equivariant case The fixed points The weight at the point 2011年4月27日　SAL@KEK

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

  25. Summary • 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　SAL@KEK

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