Affine SU(N) algebra from wall-crossings

Transcription

1 Affine SU(N) algebra from wall-crossings Takahiro Nishinaka ( KEK ) arxiv: : T. N. and Satoshi Yamaguchi 12 Aug. 2011

2 Counting BPS D-branes on CY3 Microstates of black holes in R 4 Instanton counting on D-branes Topological string Quiver gauge theory, Brane tilings Wall-crossing phenomena

3 Counting BPS D-branes on CY3 Microstates of black holes in R 4 Instanton counting on D-branes Topological string Quiver gauge theory, Brane tilings Wall-crossing phenomena

4 D4-D2-D0 on CY3 D4-D2-D0 bound states on CY3 # D-brane bound states? 6-dim ( Calabi-Yau ) The degeneracy of BPS D-branes depends on the moduli parameters = D2/D0 branes on a single D4 sizes and B-fields of compact cycles 4-dim ( R 4 ) BPS particle Wall-crossing phenomena N =2 supersymmetry

5 D4-D2-D0 on CY3 Large radii limit D0-brane D2-brane instanton on D4 magnetic flux on D4 Chern-Simons interaction on D4 D4 D0-charge = # instantons C (1) F F Q 0 = 1 F F, 8π 2 D4 D2-charge = # magnetic flux C (3) F Q 2 = 1 F k, 2π k : unit volume form on 2-cycle

6 ALE instantons If a D4-brane is wrapped on ALE space... A N 1 -ALE space: minimal resolution of C 2 /Z N (z 1,z 2 ) blowup (0, 0) N 1 (z 1,z 2 ) (ωz 1, ωz 2 ) where ω = e 2πi/N The instanton partition function on a D4-brane wrapping A N 1 ALE space = character of affine SU(N) algebra [Nakajima] [Vafa-Witten]

7 ALE instantons Affine SU(N) character q Q : Boltzmann weight of D0 : Boltzmann weight of D2 χ csu(n) 1 r = 1 η(q) N 1 n 1,,n N 1 Z q P N 1 i=1 n2 i P N 2 i=1 n in i+1 +n 1 r+ r2 2 N 1 N N 1 j=1 n Q j + N i j N r ex.) affine SU(2) χ csu(2) 1 r = 1 η(q) n= q n2 +nr+ r2 4 Q n+ r 2

8 What in this talk? Relation between ALE instantons and D4-D2-D0 wall-crossings

9 ALE C A N 1 -ALE C D A N 1 -ALE space (D) β 1 β 2 β N N 1 N A single D4-brane wrapped on D moduli D2-branes wrapped on β 1 β N 1 : N 1 blowup two-cycles D0-branes localized in CY3 = B-fields and radii of β 1 β N 1

10 Outline of the rest of this talk 1. Wall-crossings in d=4, N=2 SUSY theory 2. Apply it to our case. 3. Summary

11 Wall-crossings in d=4, N=2 SUSY theory

12 Wall-crossing of BPS states BPS index The trace is taken over all one-particle states with charge. Γ Ω(Γ; z) := 1 2 Tr Γ[( 1) 2J (2J) 2 ] This index is piecewise constant in the moduli space of vacua, but not globally constant. z ( : vacuum moduli parameters ) = # bosonic BPS multiplets # fermionic BPS multiplets Wall-crossing phenomena moduli space z discrete change Ω(Γ; z 1 ) Ω(Γ; z 2 ) Some decay BPS BPS BPS Γ Γ 1 + Γ 2 occurs. wall of marginal stability

13 Wall-crossing of BPS states So the moduli space is divided into chambers by the walls of marginal stability. z jump jump jump jump jump Ω(Γ,z 2 ) Ω(Γ,z 3 ) Ω(Γ,z 4 ) Ω(Γ,z 5 ) Ω(Γ,z 1 ) Ω(Γ,z 6 ) wall wall wall wall wall The BPS index is exactly constant in each chamber.

14 Wall-crossing of BPS states Semi-primitive wall-crossing formula Γ : primitive There is no positive integer that can divide out Γ. ( except for one ) Suppose the wall-crossing is associated to a BPS decay channel Γ Γ 1 + nγ 2, n N where Γ 1, Γ 2 are primitive. Then the BPS partition function behaves as Z Z (1 + ( 1) jγ2,γ e jγ 2 ) ±jγ 2,ΓΩ(jΓ 2 ) [Denef-Moore] j=1 [Kontsevich-Soibelman] through the wall-crossing. Here we defined Z = Γ Ω(Γ; z)e Γ, e Γ 1 e Γ 2 = e Γ 1+Γ 2 partition function Boltzmann weight

15 Apply it to our case!

16 ALE C (D) Wall-crossing? β 1 β 2 β N 1 For Γ = D4 + k i D2 (i) + ld0, possible decays are 1 2 N 1 N Γ Γ 1 + Γ 2 Γ 2 = m i D2 (i) + nd0 (Γ 1 = Γ Γ 2 ) (m i = ±1) Γ 2, Γ =0 There is no wall-crossing!! cf) wall-crossing formula Z Z (1 + ( 1) jγ2,γ e jγ 2 ) ±jγ 2,ΓΩ(jΓ 2 ) j=1

17 Add dummy cycle Dummy cycle and flops We add a dummy two-cycle. In the large radius limit of β 0, we should recover the original CY3. β 0 β 1 β 2 (D) β N 1 Now, we have a non-vanishing intersection product: β 0, D =1 1 2 N 1 N Non-trivial wall-crossings!

18 Add dummy cycle ex.) -ALE A 1 C D4 is wrapped on A 1 -ALE space β 1 1 2

19 Add dummy cycle Im z 1 0 Im z 0 ex.) -ALE A 1 C (0 < Im z 0 ) moduli parameters β 0 z 0 β 1 z 1 z 0,z 1 C 1 2 Im z i area of i-th cycle

20 Add dummy cycle Im z 1 0 Im z 0 ex.) -ALE A 1 C (0 < Im z 0 ) moduli parameters z 0,z 1 z 0 z 1 Im zi area of i-th cycle C 1 2

21 Add dummy cycle Im z 1 0 Im z 0 ex.) -ALE A 1 C 1 2 moduli parameters z 0,z 1 C Im z i area of i-th cycle

22 Add dummy cycle Im z 1 0 Im z 0 ex.) -ALE A 1 C moduli parameters D4 is wrapped on O( 1) P 1 z 0 z 1 2 Im z i z 0,z 1 C area of i-th cycle 1 z 0 = z 0, z 1 = z 0 + z 1

23 Add dummy cycle Im z 1 0 Im z 0 ex.) -ALE A 1 C moduli parameters D4 is wrapped on O( 1) P 1 z 0 z 1 2 z 0,z 1 C Im z i area of i-th cycle 1 z 0 = z 0, z 1 = z 0 + z 1

24 Add dummy cycle Im z 1 0 Im z 0 ex.) -ALE A 1 C moduli parameters z 0,z 1 C 2 Im z i area of i-th cycle 1

25 Add dummy cycle Im z 1 0 Im z 0 ex.) -ALE A 1 C D4 is wrapped on C 2 z 0 z moduli parameters Im z i z 0,z 1 C area of i-th cycle z 0 = z 1, z 1 = z 0 z 1

26 Im z 0 =+ (large radii) Z + z 0 Walls of MS Γ Γ 1 + Γ 2 Γ = D4 + l i D2 (i) + kd0 0 W m0,m 1 n : Γ 2 = i=0,1 m i D2 (i) + nd0 Im z 0 = (large radii) Z

27 Im z 0 =+ (large radii) Z + W 1, 1 0 W 1, 1 1 W 1, 1 2 W 1, W +1,0 2 W 1,0 0 W +1,0 1 W +1,0 0 W 1,0 2 z 0 W +1,0 1 W m0,m 1 n : Γ 2 = Walls of MS Γ Γ 1 + Γ 2 Γ = D4 + l i D2 (i) + kd0 i=0,1 m i D2 (i) + nd0 W +1,+1 1 W +1,+1 0 W +1,+1 1 Im z 0 = (large radii) Z

28 Results of wall-crossings Z + Z β 0 β 1 Z = n,m 0,m 1 Ω(D4 + m i D2 (i) + nd0)q n Q 0 m 0 Q 1 m 1 Result of wall-crossing formula Z + = Z n=0 1 n=1 1 q n (1 q n Q 0 )(1 q n Q 0 Q 1 ) m=1 (1 q n Q 0 1 )(1 q m Q 1 0 Q 1 1 )

29 Results of wall-crossings Z + Z Z = β 1 n,m 0,m 1 Ω(D4 + m i D2 (i) + nd0)q n Q 0 m 0 Q 1 m 1 Result of wall-crossing formula Z + = Z n=0 1 n=1 1 q n (1 q n Q 0 )(1 q n Q 0 Q 1 ) m=1 (1 q n Q 0 1 )(1 q m Q 1 0 Q 1 1 ) omit β 0 Z + Q0 independent = 1 1 q n n=1 3 m= q m2 Q 1 m = q1/8 η(q) 2 χcsu(2) 1 0 (q, Q 1 )

30 We can generalize it to A N 1 ALE space!!

31 Generalization Multiple flops We now have multiplet flop transitions. 1 2 N 1 N

32 Generalization Multiple flops We now have multiplet flop transitions. 1 2 N 1 N

33 Generalization Multiple flops We now have multiplet flop transitions. 1 2 N 1 N

34 Generalization Multiple flops We now have multiplet flop transitions. 1 2 N 1 N

35 Generalization Multiple flops We now have multiplet flop transitions. 1 2 N 1 N

36 Generalization Multiple flops We now have multiplet flop transitions. 1 2 N 1 N

37 Generalization Multiple flops We now have multiplet flop transitions. N 1 N 1 2

38 Generalization Multiple flops We now have multiplet flop transitions. 1 2 N 1 N Instantons on C 2

39 Generalization Affine SU(N) character from wall-crossings wall-crossings Q 0 Z (q, Q) Z + (q, Q) 1 = 1 q n n=1 From the wall-crossing formula, we find N 1 Z + (q, Q) = Z (q, Q) (1 q n Q 0 Q k ) k=0 n=0 m=1 (1 q m Q 1 0 Q 1 k ) = =1 1 1 q N+1 n 0,,n N 1 Z q P N 1 i=0 n i (n i 1) 2 N 1 k=0 ( Q 0 Q k ) n k

40 Generalization Z + (q, Q) = =1 1 1 q N+1 n 0,,n N 1 Z q P N 1 i=0 n i (n i 1) 2 N 1 k=0 ( Q 0 Q k ) n k After removing dummy cycle... Z + (q, Q) Q0 independent = n=1 1 1 q n 2 χ csu(n) 1 0 (q, Q) where χ csu(n) 1 0 (q, Q) = 1 η(q) N 1 n 1,,n N 1 Z q P N 1 N 1 i=1 n2 i P N 2 i=1 n in i+1 j=1 Q j n j Affine SU(N) character is correctly reproduced by wall-crossings!

41 Summary We study the relation between ALE instantons and wall-crossings of D4- D2-D0 on CY3. By adding a dummy cycle, the wall-crossings interpolate the instanton partition functions on ALE space and C 2. Our analysis relies on the wall-crossing formula of d=4, N=2 theories. Future work r χ csu(n) 1 The parameter of is left for future work. r Generalizations to other instanton partition functions on Vafa-Witten theories are to be studied.

42 That s all for my presentation. Thank you very much.