Wall Crossing and Quivers

Transcription

1 Wall Crossing and Quivers ddd (KIAS,dddd) dd 2014d4d12d Base on: H. Kim, J. Park, ZLW and P. Yi, JHEP 1109 (2011) 079 S.-J. Lee, ZLW and P. Yi, JHEP 1207 (2012) 169 S.-J. Lee, ZLW and P. Yi, JHEP 1210 (2012) 094 S.-J. Lee, ZLW and P. Yi, JHEP 1402 (2014) 047

2 Overview - Wall Crossing Phenomenon - Coulomb and Higgs phase: Particles V.S. Quivers - Intrinsic Higgs States and Quiver Invariants

3 Wall Crossing phenomenon Certain quantity changes discontinuously across a codimension-one wall in a space

4 Wall Crossing phenomenon BPS index changes discontinuous across the wall of marginal stability in the moduli space of a supersymmetric theory

5 BPS index - 4D N =2 SUSY: Q A α (A = 1, 2; α = 1, 2), ( Q A α) = Q A α, Q Aα = ε αβ Q A β, {Q A α, Q B β } = 2σ µ α β P µδ A B, {QAα, Q B β } = 2δα β εab Z. Bosonic symmetries: ISO(1, 3) SU(2) R U(1) R.

6 BPS index - 4D N =2 SUSY: Q A α (A = 1, 2; α = 1, 2), ( Q A α) = Q A α, Q Aα = ε αβ Q A β, {Q A α, Q B β } = 2σ µ α β P µδ A B, {QAα, Q B β } = 2δα β εab Z. Bosonic symmetries: ISO(1, 3) SU(2) R U(1) R. - For a massive state with mass M and central charge Z = e iθ Z, define ( R α ± = 1 2 e iθ/2 Qα 1 ± e iθ/2 σ µ α β P µm 1 Q ) 2 β. The non-vanishing anticommutator is {R α ±, R ± β } = σ µ α β P µm 1 (M Z ). ( ) Under SU(2) R, R α ±, σ µ α β P µm 1 R ± β transforms as a doublet R ± α cos φ 1 e iφ 3 R ± α ± sin φ 1 e iφ 2 σ µ α β P µm 1 R ± β.

7 BPS index - In the rest frame {R α ±, R ± β } = I α β (M Z ). ( ) R α ±, I α β R ± β transforms as a doublet Under SU(2) R.

8 BPS index - In the rest frame {R α ±, R ± β } = I α β (M Z ). ( ) R α ±, I α β R ± β transforms as a doublet Under SU(2) R. M> Z : Long-rep L j, [j] (2[0] + [ 1 2 ]) (2[0] + [ 1 2 ])

9 BPS index - In the rest frame {R α ±, R ± β } = I α β (M Z ). ( ) R α ±, I α β R ± β transforms as a doublet Under SU(2) R. M> Z : Long-rep L j, [j] (2[0] + [ 1 2 ]) (2[0] + [ 1 2 ]) M= Z : Short-rep Sj, [j] (2[0] + [ 1 2 ]), invariant under R+, R +

10 BPS index - In the rest frame {R α ±, R ± β } = I α β (M Z ). ( ) R α ±, I α β R ± β transforms as a doublet Under SU(2) R. M> Z : Long-rep L j, [j] (2[0] + [ 1 2 ]) (2[0] + [ 1 2 ]) M= Z : Short-rep Sj, [j] (2[0] + [ 1 2 ]), invariant under R+, R + - The index of BPS spectrum ) Ω γ = Tr H γ (( 1) 2J 3 = 1 2 Tr H γ (J3 2 ( 1)2J 3 )

11 BPS index - In the rest frame {R α ±, R ± β } = I α β (M Z ). ( ) R α ±, I α β R ± β transforms as a doublet Under SU(2) R. M> Z : Long-rep L j, [j] (2[0] + [ 1 2 ]) (2[0] + [ 1 2 ]) M= Z : Short-rep Sj, [j] (2[0] + [ 1 2 ]), invariant under R+, R + - The index of BPS spectrum ) Ω γ = Tr H γ (( 1) 2J 3 = 1 2 Tr H γ (J3 2 ( 1)2J 3 ) "Tr "dignore the center of mass half-hyper contribution

12 BPS index - In the rest frame {R α ±, R ± β } = I α β (M Z ). ( ) R α ±, I α β R ± β transforms as a doublet Under SU(2) R. M> Z : Long-rep L j, [j] (2[0] + [ 1 2 ]) (2[0] + [ 1 2 ]) M= Z : Short-rep Sj, [j] (2[0] + [ 1 2 ]), invariant under R+, R + - The index of BPS spectrum ) Ω γ = Tr H γ (( 1) 2J 3 = 1 2 Tr H γ (J3 2 ( 1)2J 3 ) "Tr "dignore the center of mass half-hyper contribution BPS multiplet Sj : ( 1) 2j (2j + 1)

13 BPS index - In the rest frame {R α ±, R ± β } = I α β (M Z ). ( ) R α ±, I α β R ± β transforms as a doublet Under SU(2) R. M> Z : Long-rep L j, [j] (2[0] + [ 1 2 ]) (2[0] + [ 1 2 ]) M= Z : Short-rep Sj, [j] (2[0] + [ 1 2 ]), invariant under R+, R + - The index of BPS spectrum ) Ω γ = Tr H γ (( 1) 2J 3 = 1 2 Tr H γ (J3 2 ( 1)2J 3 ) "Tr "dignore the center of mass half-hyper contribution BPS multiplet Sj : ( 1) 2j (2j + 1) No contributions from long-rep: e.g. 2S0 + S 1 2 L 0.

14 BPS index - In the rest frame {R α ±, R ± β } = I α β (M Z ). ( ) R α ±, I α β R ± β transforms as a doublet Under SU(2) R. M> Z : Long-rep L j, [j] (2[0] + [ 1 2 ]) (2[0] + [ 1 2 ]) M= Z : Short-rep Sj, [j] (2[0] + [ 1 2 ]), invariant under R+, R + - The index of BPS spectrum ) Ω γ = Tr H γ (( 1) 2J 3 = 1 2 Tr H γ (J3 2 ( 1)2J 3 ) "Tr "dignore the center of mass half-hyper contribution BPS multiplet Sj : ( 1) 2j (2j + 1) No contributions from long-rep: e.g. 2S0 + S 1 L 0. No contributions from multi-particle state 2

15 BPS index - The refined version: Protected spin character[d. Gaiotto, G.W. Moore and A. Neitzke, 10 ] Ω γ (y) = Tr H γ ( ( 1) 2J 3 y 2(J 3+I 3 ) ) It is an index since there is a Femionic operator Q = ε Aα Q Aα which is a singlet under J 3 + I 3, anticommutes with ( 1) F, and is invertible on long-rep. BPS multiplet Sj : ( 1) 2j yj+1 y j 1 y y 1 No contributions from long-rep since Q is invertible on long-rep. No contributions from multi-particle state

16 Moduli space - Moduli space: the space of vacua in the N = 2 theory, described by certain parameters, e.g. u = φ 2

17 Moduli space - Moduli space: the space of vacua in the N = 2 theory, described by certain parameters, e.g. u = φ 2 - If no states enter or leave the Hilbert space, the structure of the spectrum changes continuously under continuous change of the parameters.

18 Moduli space - Moduli space: the space of vacua in the N = 2 theory, described by certain parameters, e.g. u = φ 2 - If no states enter or leave the Hilbert space, the structure of the spectrum changes continuously under continuous change of the parameters. - Therefore, Ω is an invariant in the whole moduli space?

19 Where is the Wall? - Ω is invariant under any (continuous-)deformations of H 1 γ.

20 Where is the Wall? - Ω is invariant under any (continuous-)deformations of H 1 γ. - Ω could change when H 1 γ mixes with multiparticle spectrum, e.g. (2[0] + [ 1 2 ]) p (2[0] + [ 1 2 ]) p [1/2] (2[0] + [ 1 2 ])

21 Where is the Wall? - Ω is invariant under any (continuous-)deformations of H 1 γ. - Ω could change when H 1 γ mixes with multiparticle spectrum, e.g. (2[0] + [ 1 2 ]) p (2[0] + [ 1 2 ]) p [1/2] (2[0] + [ 1 2 ]) - Two inequalities: M M 1 + M 2 = Z γ1 + Z γ2 M = Z γ1 +γ 2 = Z γ1 + Z γ2 Z γ1 + Z γ2

22 Where is the Wall? - Ω is invariant under any (continuous-)deformations of H 1 γ. - Ω could change when H 1 γ mixes with multiparticle spectrum, e.g. (2[0] + [ 1 2 ]) p (2[0] + [ 1 2 ]) p [1/2] (2[0] + [ 1 2 ]) - Two inequalities: M M 1 + M 2 = Z γ1 + Z γ2 M = Z γ1 +γ 2 = Z γ1 + Z γ2 Z γ1 + Z γ2 - The wall of marginal stability locates at Z γ1 /Z γ2 R +

23 Wall Crossing phenomenon - N = 2 pure SU(2) SYM in 4D - How to describe the jump?

24 Kontsevich-Soibelman s Magic - Twisted torus algebra X γ1 X γ2 = ( 1) γ 1,γ 2 X γ1 +γ 2, γ 1, γ 2 : Schwinger product of charges

25 Kontsevich-Soibelman s Magic - Twisted torus algebra X γ1 X γ2 = ( 1) γ 1,γ 2 X γ1 +γ 2, γ 1, γ 2 : Schwinger product of charges - Automorphism of twisted torus algebra K γ : X γ X γ (1 X γ ) γ,γ,

26 Kontsevich-Soibelman s Magic - Twisted torus algebra X γ1 X γ2 = ( 1) γ 1,γ 2 X γ1 +γ 2, γ 1, γ 2 : Schwinger product of charges - Automorphism of twisted torus algebra K γ : X γ X γ (1 X γ ) γ,γ, - Co-dimension one locus: C γ = (u, arg( Z γ (u))

27 Kontsevich-Soibelman s Magic - Twisted torus algebra X γ1 X γ2 = ( 1) γ 1,γ 2 X γ1 +γ 2, γ 1, γ 2 : Schwinger product of charges - Automorphism of twisted torus algebra K γ : X γ X γ (1 X γ ) γ,γ, - Co-dimension one locus: C γ = (u, arg( Z γ (u)) - Wall-crossing formalism [M. Kontsevich and Y. Soibelman, 08 ] A(P) = C γ K ±Ω(γ) γ = 1

28 Kontsevich-Soibelman s Magic - Examples: γ 1, γ 2 = 1: Argyres-Douglas A 2 theory γ 1, γ 2 = 2: K γ2 K γ1 = K γ1 K γ1 +γ 2 K γ2 K (2, 1) K (0,1) = K (0,1) K (2,1) K (4,1)... K 2 (2,0)... K (6, 1) K (4, 1) K (2, 1) N = 2 pure SU(2) SYM in 4D

29 A Proof by Physicists[D. Gaiotto, G.W. Moore and A. Neitzke, 09, 10 ] - Adding a very heavy extenal BPS particle with arg Z = θ: [ ( ) ] L θ = exp i dt A 0 + e iθ φ + e iθ φ - The Hilbert space is modified: H H Lθ, Ω γ (L θ ) = Tr H Lθ,γ ( ) ( 1) 2J 3 - BPS bound: M Re(e iθ Z γ ) - Any BPS state is a bound state of L θ and BPS particles in the original theory. - The bound states appear/disappear when θ crosses arg Z γh. - The bound state configuration is decided by the balance of classical forces r h = γ h, γ c 2Im[e iθ Z γh ]

30 A Proof by Physicists - Let F Lθ = γ Ω γ(l θ )X γ - When θ crosses arg Z γh, F Lθ is transformed by X γ X γ (1 X γh ) ±Ω γ h γ,γ h It is K ±Ω γ h γ h! - Travel along any closed loop, F Lθ must come back to itself. - If the theory has "enough" line operators L θ, we will have K γ ±Ω(γ) = 1 C γ

31 Microscopic origin: - A BPS one-particle state is generically a bound state consisting of more than one charge centers, which are spatially distributed according to balance of classical forces. [K. Lee and P. Yi, 98 ]

32 Microscopic origin: - A BPS one-particle state is generically a bound state consisting of more than one charge centers, which are spatially distributed according to balance of classical forces. [K. Lee and P. Yi, 98 ] - Wall-crossing: the size of such bound states become infinitely large as a marginal stability wall is approached, e.g., R γ 1, γ 2 2Im[ Z γ1 Z γ2 ]

33 Low energy dynamics of BPS States - Can we computing Ω γ based on the bound states picture? - Starting from a set of known BPS states with charge γ i as a basis set The charge γ of any given BPS state can be written as γ = i n iγ i (n i Z + ) - The low energy dynamics of n i i-type BPS states is described by a Quantum Mechanics with four super charges, with SO(4) R- symmetry coming from SO(3) spatial rotation and SU(2) R in the 4D N = 2 theory. - A BPS bound state with charge γ = i n iγ i is related to a SUSY invariant vacuum of the system. - The corresponding refined BPS index is Ω(y) = Tr QM ( ( 1) 2J 3 y 2(J 3+I 3 ) )

34 The Coulomb phase dynamics - The Coulomb phase description: BPS particles interacting with Lorentz force - Written in N = 1 superspace Φ Aa = x Aa iθψ Aa, Λ A = iλ A + iθb A then add the N = 4 SUSY Constraints by hand - Kinetic term ( i L 1 = dθ 2 gab ABDΦ A Φ a B b 1 h 2 ABΛ A DΨ B ifab a Φ A aλ B + 1 3! cabc ABC DΦ A adφ B b DΦ C c ) + 1 2! nab ABC DΦ A adφ B b Λ C + 1 2! ma ABC Φ A aλ B Λ C + 1 l 3! ABC Λ A Λ B Λ C N = 4 SUSY Constraints: all the couplings are decided by a single function L, e.g. h AB = δ ab a A b B L

35 The Coulomb phase dynamics - Lorentz interaction between dyons: L 2 = dθ ( ik A (Φ)Λ A iw Aa (Φ)DΦ Aa) N = 4 SUSY constraints Aa K B = 1 2 ε abc ( Ab W Bc Bc W Ab ), ε abc Ab Bc K C = 0, Aa Ba K C = 0. - The N = 4 SUSY constraints implies W Aa = B γ A, γ B Wa Dirac ( x A x B ) 2 K A = Im [ e iθ Z A ] = Im [ e iθ Z A ] 1 2 B A γ A, γ B x A x B - The scalar potential V K 2 The moduli space M n = ( {x Aa K A = 0} R 3) /Γ

36 The Coulomb phase dynamics - The R symmetry is SO(4) = SU(2) L SU(2) R. SU(2) L is the rotation group, while SU(2) R is descendant of SU(2) R-symmetry of the underlying 4D N = 2 theory. The generators: J a I a = L a + A = A ( i 8 ε abc [ ˆψ Ab, ˆψ Ac ] i ) 4 [ ˆψ Aa, ˆλ A ] ( i 8 ε abc [ ˆψ Ab, ˆψ Ac ] + i ) 4 [ ˆψ Aa, ˆλ A ] x Aa : (3, 1); ψ Am = {ψ Aa, λ A } and super charge Q m : (2, 2). - MPS formula:[ J. Manschot, B. Pioline and A. Sen, 10 ] The index is given by a sum of fixed point contributions. Due to the y 2J 3 factor in the index, the fixed point configurations are the solutions of K A = 0 with all particles aligning on the z-axis. - An observation: all the known Coulomb branch BPS states are SU(2) R singlet.

37 The Higgs phase dynamics: Quiver Theory - The Higgs phase description: Quiver quantum mechanics [F. Denef, 02 ] - Quiver: Nodes+Arrows γ 31 U(N 3 ) U(N 1 ) γ 23 γ 12 U(N 2 ) W = c mnp Φ a 1ā 2 m Φ a 2ā 3 n Φ a 3ā 1 p Node v: a U(Nv ) vector multiplet (A v, Xv, i λ v, D v ); a FI parameter ζ v ; Arrow s(v w):a bifundamental chiral multiplet (φ s, ψ s, F s ), in the ( N v, N w ) of U(N v ) U(N w ); Number of arrows= γ v, γ w Closed loop: a superpotential

38 The Higgs phase dynamics: Brane picture - The N = 2 theory: Type II on a CY 3 - The BPS particles: The D-branes wrap on a supersymmetric cycle - γ 1, γ 2 : the intersecting number between two cycles - U(N i ) nodes: a basis of the cycles wrapped with N i D-branes - Bifundamental fields: open-strings attached between two D-branes on different cycles

39 The Higgs phase dynamics - The R symmetry is SO(4) = SU(2) L SU(2) R. (Q 1, Q 2 ) transforms as a doublet under the rotation group SU(2) L, and (Q 1, Q 1) transforms as a doublet under the SU(2) R which is descendant of SU(2) R-symmetry of the underlying 4D N = 2 theory. Especially, I 3 can be identified as the overall U(1) on Q α. - The moduli space: a Kähler manifold M = {φ a W φ a = 0, φ a φ a φ a φ a = ζ v }/ a: v a:v v - On the moduli space, the rotation SU(2) is identified as the SU(2) Lefschetz L 3 = (l d)/2, L + = K, L = K It is acting on the cohomology H(M) = l Hl (M) and d = dim C M. U(N v )

40 The Higgs phase dynamics - Acting on H p,q (M), the overall U(1) generator I 3 is identified as I 3 = (p q)/2 Obviously, [I 3, L 1,2,3 ] = 0. - The protected spin character is computed in the Higgs phase as Ω Higgs (y) = tr ( 1) 2L 3 y 2L 3+2I 3 = tr ( 1) l d y l d+p q = tr ( 1) p+q d y 2p d

41 Quivers with oriented closed loops - For quivers without oriented closed loop, Ω C = Ω H - For quivers with oriented closed loops, ΩC Ω H in general. [F. Denef, G.W. Moore, 07 ] The scaling solutions in Coulomb phase make the moduli space non-compact. The naive fixed point formulae is divergent at y = 1. The MPS formula with a minimal subtraction scheme which is consistent with wall-crossing was proposed.[ J. Manschot, B. Pioline and A. Sen, 10 ] Superpotential appears in Higgs phase. Both phases share the D term data ζ v = Im(e iθ Z v ), but only Higgs phase contains the data of superpotential. Coulomb phase index is related to the ambient space X = {φ a a: v φ a φ a a:v φ a φ a = ζ v }/ v U(N v ) of Higgs phase moduli space M? In the N = 2 supergravity, Ω H Ω c is related to the index of single centered BPS black holes which are always angular momentum singlet.[ J. Manschot, B. Pioline and A. Sen, 10 ]

42 Conjectures - Conjecture I: In the k-th branch of the moduli space Ω (k) Coulomb (y) = ( y) d k D k ( y) i M k (H(X k )): pull-back of the ambient cohomology; d k : thecomplex dimension of M k ; D k (x): the reduced Poincaré polynomial D k (x) [ ] x l dim i M k (H l (X k )) l - Conjecture II: The Intrinsic Higgs states in H(M k ) i M k (H(X k )) are essentially depend on the middle cohomology. The corresponding index ( y) d k χ ξ= y 2(M k ) ( y) d k D k ( y) is a branch independent invariant of the quiver. χ ξ = p q ( 1)q h p,q ξ p : the refined Euler character

43 Abelian Cyclic Quivers - Cyclic (n + 1)-Gon: Ζ 3 Z 2 1,..., Z 2 a2 Ζ 2 Ζ n Z 1 1,..., Z 1 a1 Z n 1,..., Z n an Ζ 1 Ζ n1 Z n1 1,..., Z n1 an1

44 Abelian Cyclic Quivers - D-term conditions - Superpotential a 1 W = β 1 =1 Z n+1 2 Z 1 2 = ζ 1, Z 1 2 Z 2 2 = ζ 2,. Z n 2 Z n+1 2 = ζ n+1, a n+1 β n+1 =1 - Branches: One of the Z i vanishing c β1 β 2 β n+1 Z (β 1) 1 Z (β 2) 2 Z (β n+1) n+1, 1. Generic cβ1 β 2 β n+1 generic F-term algebraic equations. 2. F-term conditions have scaling symmetries Zi λ i Z i. No solution to F-term conditions with all Z i nontrivial.

45 Abelian Cyclic Quivers: i M k (H(X k )) - k-th Branch: k i=i ζ i > 0, - Ambient space J i=k+1 ζ i < 0 X k = CP a 1 1 CP a k 1 1 CP a k+1 1 CP a n The a k F-terms Zk W = 0 define a complete intersecting. The complex dimension d k = n+1 i=1 a i 2a k n. - For the ambient space, H p,q (X k ) with p q are null, and i k P[X k ](x) = (1 x2a i ) (1 x 2 ) n = b 2l (X k ) x 2l - Lefschetz hyperplane theorem: H p,q (M k ) with p q, p + q < d k are null D k (x) = b dk (X k ) x d k + b 2l (X k ) (x 2l + x 2dk 2l ) 0 2l<d k

46 Abelian Cyclic Quivers: Ω (k) Coulomb (y) - MPS Formula Ω (k) Coulomb n+1 a i n ( 1) i=1 [ (y) = Gk (y) + ( 1) n G (y y 1 ) n k (y 1 ) + H k (y) + ( 1) n H k (y 1 ) ] n+1 G k (y) + ( 1) n G k (y 1 ) = a i sign[z i z i+1 ] s(p) y i=1, s(p) = sign[det M], p z i z i+1 M i,i = a i z i z i+1 + a z i+1 z i+2 3 i+1 z i+1 z i+2, 3 M i,i+1 = M i+1,i = a i+1 z i+1 z i+2 z i+1 z i The subtraction polynomial H k (y) = 0 l<n l n+1 i=1 a i 2Z λ l y l, the coefficients λ l are decided uniquely by requiring that (y) is finite when y = 1. Ω (k) Coulomb

47 Abelian Cyclic Quivers: Ω (k) Coulomb (y) - The index is invariant within each branch, so that we may pick a particularly convenient set of FI constants and simplify the problem. - At ζ k = ζ k+1 > 0, ζ i = 0(i k, k + 1) the fixed points are z k z k+1 = a k ρ, z i z i+1 = a i (i k), ρ + ζ k a i sign[z i z i+1 ] + sign[z k z k+1 ] a k ρ + ζ k ρ = (z i z i+1 ) = 0 i i k - The fixed points contribution is G k (y) = Θ(x) = {t i k =±1} [ ] ( ) t i Θ a i t i a k y i k 1 for x 0 0 for x < 0 i k a i t i a k i k,

48 Abelian Cyclic Quivers: Proof of conjecture I - Uniqueness of the H k was guaranteed by three requirements: regularity of index at y = 1, definite parity of G k, and parity of H k coinciding with that of G k. - The first conjecture is equivalent to y n G k (y 1 ( ) ) = y n G k (y 1 ) y d k 1 y 2a i nonpositive - Proof: y n G k (y 1 ) = y n = {t i k =±1} {t i k =±1} nonpositive i k i k i k [ ] ( ) t i Θ a i t i a k i k = y n Gk (y 1 ). i k y i k [ ] ( ) t i Θ a i t i a k n y a i t i +a k nonpositive nonpositive a i t i +a k +n i k nonpositive

49 Abelian Cyclic Quivers: H(X k ) - The Adjunction formula: td(t M k ) = ch ξ (T M k ) = p i k ( Ji 1 e J i ) ai ( ch( p T M k ) ξ p = ) ak 1 e i k J i i k J i (1 + ξe J i) a i 1 + ξ i k ( ξe i k J i where J i is the Kähler form from each CP a i 1 factor in X k. - Applying the Hirzebruch-Riemann-Roch formula χ ξ (M k ) = = = td(t M k ) ch ξ (T M k ) M k ( ) td(t M k ) ch ξ (T ak M k ) J i X k i k 1 ( 1 + ξe J ) ai ( i (1 + ξ) n J i 1 e i k J i X k 1 e J i i k 1 + ξe i k J i ) ak ) ak

50 Abelian Cyclic Quivers: Proof of conjecture II - Let ω i e J i, Ω (k) Higgs (y) = ( y) dk χ ξ= y 2(M k ) y ai y ai = ( 1) dk y ak y y 1 i k + ( y)n+2 i ai (y 2 1) n i ω i=1 dω i 2πi Ω (k) Higgs (y) ) Ω(k Higgs (y) = 1 a yak k ak +a y k ( 1)dk - From the expression of G k (y), we get [ ( 1 y 2 ) ai ] ω i 1 1 ω i 1 y 2 i ω i i y y 1 G k (y) + ( 1) n G k (y 1 ) G k (y) ( 1) n G k (y 1 ) (y y 1 ) n = ( 1) dk 1 a yak k ak +a y k yai y ai y y 1 y y 1 i k,k = Ω (k) Coulomb (y) Ω(k ) Coulomb (y) - Ω (k) Higgs (y) Ω(k) Coulomb (y) is independent of k. yai y ai y y 1 i k,k

51 Numerical illustration - 3-gon with a 1 = 4, a 2 = 5, a 3 = , ,

52 More general quivers - Ambient space: maximal reduced quiver without loop The branches for a multi-loop quiver are described by the non-empty branches of its maximal reduced quiver without loop, e.g. θ 1 θ 1 γ 14 γ 12 γ 14 γ 12 θ 4 γ 31 θ 2 θ 4 θ 2 γ 34 γ 23 γ 34 γ 23 θ 3 θ 3 Vanishing of the edge 31: θ3 > 0, θ 1 < 0. Four different branches depending on the sign of θ 2 and θ 4

53 More general quivers - Abelian quiver: Toric data, easy to deal with - How to deal with nonabelian gauge group?[bumsig Kim et.al] Fully Abelianize varieties X of the nonabelian varieties X U(1) U(1) γ 31 γ 12 γ 31 γ 12 U(2) γ 23 U(2) U(1) γ 31 γ 23 γ 12 U(1) γ 23 U(1) γ 23 γ 23 U(1) Additional insertion: e.g. for a single U(n): i<j X a = 1 â e( ) W X c( ) (J i J j ) 2 1 (J i J j ) 2

54 More general quivers - Coulomb phase computation: MPS type of partition sum γ 31 U(1) U(3) γ 23 γ 12 U(1) = γ 1 γ 1 γ 1 γ 3 γ 2 + γ 1 2γ 1 γ 3 γ 2 + γ 3 3γ 1 γ 2 - A one to one map for cases without intrinsic Higgs: (J i J j ) 2 1 (J i J j ) 2 = (J i J j ) 2 = 1 + δ ij e( ) c( ) = 1+(δ 12+δ 13 +δ 23 )+(δ 12 δ 13 +δ 12 δ 23 +δ 13 δ 23 +δ 12 δ 13 δ 23 )

55 Summary - The relation between PSC in 4D N = 2 theory and refined index in N = 4 QM. - Proof of the two conjecture about the Coulomb phase and the Higgs phase indices for cyclic quiver. - More general quivers: The basic way of computing Higgs cohomology is established Certain smart improvement is needed - More future directions: Can we directly compute ΩIntrinsic, e.g., by localization...? The relation between ΩIntrinsic and black hole entropy Whether and how Kontsevich-Soibelman algebra know about the quiver invariants?

56 T hank Y ou!