BPS-State Counting: Quiver Invariant, Abelianisation & Mutation - PowerPoint PPT Presentation

BPS-State Counting: Quiver Invariant, Abelianisation & Mutation S.-J.L. , Z.-L.Wang, and P.Yi ; 1205.6511 , 1207.0821 , 1310.1265 H.Kim, S.-J.L. , and P.Yi ; 1504.00068 Seung-Joo Lee Virginia Tech Southeastern regional mathematical string theory meeting Apr 11, 2015 Monday, April 13, 15

Outlin e Warm-Up - A topology Exercise Rudiments - index and Wall-Crossing - BPS Quivers Quiver Invariants - Characterisation of the Higgs Moduli Spaces Non-Abelian Quivers - Abelianisation - Mutation Monday, April 13, 15

Warm Up a topology exercise Monday, April 13, 15

• a 3 a 2 a 1 • a 3 a 2 • a 3 a 2 a 3 a 2 a 1 a 1 a 1 Chamber 1 Chamber 2 Chamber 3 P a 2 − 1 × P a 3 − 1 P a 3 − 1 × P a 1 − 1 P a 1 − 1 × P a 2 − 1 O (1 , 1) ⊕ a 1 O (1 , 1) ⊕ a 2 O (1 , 1) ⊕ a 3 Monday, April 13, 15

• a 3 = a 2 = • 4 5 a 1 = 6 • 4 5 • 4 5 4 5 6 6 6 Chamber 1 Chamber 2 Chamber 3 P 4 × P 3 P 3 × P 5 P 5 × P 4 O (1 , 1) ⊕ 4 O (1 , 1) ⊕ 6 O (1 , 1) ⊕ 5 Monday, April 13, 15

• 4 5 • 4 5 4 5 6 6 6 Chamber 1 Chamber 2 Chamber 3 • 1 · y -5 0 · y -4 • 1 · y -3 2 · y -3 0 · y -2 0 · y -2 2 · y -1 3 · y -1 1 · y -1 52 · y 0 52 · y 0 52 · y 0 1 · y 1 3 · y 1 2 · y 1 0 · y 2 0 · y 2 2 · y 3 1 · y 3 0 · y 4 1 · y 5 Monday, April 13, 15

Cahmber 1 Chamber 2 Chamber 3 H . , . H . , . H . , . • 1 0 0 • 1 0 2 0 0 0 0 0 0 0 0 2 0 1 0 0 3 0 0 0 26 26 0 26 26 0 0 26 26 0 0 0 2 0 1 0 0 3 0 0 0 0 0 0 0 0 1 0 2 0 0 0 1 Monday, April 13, 15

Rudiments BPS Index Monday, April 13, 15

N =2 Basics • We consider N =2 Abelian gauge theories. • States have integer charges: γ ∈ Z 2 r ≡ Γ • Poincare extends to N =2 super-Poincare: M gets bounded by |Z|. • M=|Z| case: “short” repre, S j =[ j ] r hh , ⊗ where r hh =2[0] [1/2] is the 4-dim l irrep of the odd alg. ⊕ • M>|Z| case: “long” repre, L j =[ j ] r hh r hh ⊗ ⊗ Monday, April 13, 15

BPS Index � � � � • For the Hilbert space , � � � � ⊕ n j ( γ ) ⊕ m l ( γ ) H 1 � ⊕ S j L l γ = � j ∈ 1 l ∈ 1 2 Z ≥ 0 2 Z ≥ 0 define the BPS index as: � ( − 1) 2 j (2 j + 1) n j ( γ ) Ω ( γ ) := j ∈ 1 2 Z ≥ 0 γ ( − 1) 2 J 3 = Tr � H 1 • Only genuine short reps contribute to . Ω ( γ ) • The little super-algebra contains su (2) R and hence one can define the refined index as: y =1 γ ( − 1) 2 J 3 y 2 I 3 +2 J 3 Ω ( γ ; y ) = Tr � γ ( − 1) 2 J 3 Ω ( γ ) = Tr � − − → H 1 H 1 Monday, April 13, 15

Rudiments Wall-crossing Monday, April 13, 15

Wall-Crossing • is invariant under arbitrary deformations of , H 1 Ω ( γ ) γ but may change under deformations of the theory. • The index is ill-defined when mixes with the multi-ptl H 1 γ spectrum, i.e., if can split into and s.t. γ 1 γ γ 2 , . Z 1 /Z 2 ∈ R + γ 1 + γ 2 = γ • Thus, in the parameter space, there appears a wall, across which the BPS index jumps. Monday, April 13, 15

Wall-Crossing • Generic BPS one-particle states as loose bound states of charge centers, balanced by classical forces. [Lee, Yi `98; Bak, Lee, Lee, Yi `99; Gauntlett, Kim, Park, Yi `99; Stern, Yi `00; Gauntlett, Kim, Lee, Yi `00] • The equilibrium distances become infinite as one approaches the wall [Denef `02] : R = � γ 1 , γ 2 � | Z 1 + Z 2 | Im[ ¯ 2 Z 1 Z 2 ] Monday, April 13, 15

Rudiments BPS Quivers Monday, April 13, 15

BPS Quivers • BPS states ~ D-branes wrapping various cycles. • Low-energy D-brane dynamics by a D =4, N =1 quiver gauge theory reduced to the eff. particle world-line. • E.g. IIB on CY 3 : one-particle BPS states seen as a D3-brane wrapping a SLag. • Two pictures arise for the same BPS bound state of branes: (1) Set of particles at equilibrium (2) Fusion of D-branes related via quiver quantum mechanics [Denef `02] Monday, April 13, 15

BPS Quivers • U(1) vectors include x v =(x v1 , x v2 ,x v3 ) and bi-fund. chirals include Z vwk=1,...,a vw , where a vw = � γ v , γ w � • x 2 • Two phases (1) Coulomb: x v , Z vwk γ 2 (2) Higgs: x v , Z vwk • Z 23 k=1,...,a 23 Z 12 k=1,...,a 12 x 3 x 1 γ 3 γ 1 Monday, April 13, 15

BPS Index • For large x v - x w , chirals are massive and eff. dynamics leads to � γ w , γ v � , with � θ v = 2 Im[ e − i α Z γ v ( u )] K v � | x w � x v | � θ v ( u ) = 0 for � v w � = v • By studying the sol n space , one M = { x v | K v = 0 , ∀ v } \ R 3 can obtain the Coulomb index Ω Coulomb ( { γ v } ; y ) [de Boer, El-Showk, Messamah, van Den Bleeken `09], [Manschot, Pioline, Sen `11] • Dialing the coupling to 0, one can describe the system as QM on the variety . � M H = { Z k vw | D v = θ v , ∀ v } / U (1) • The Higgs index is given as: v ( − 1) p + q − d y 2 p − d h p,q ( M H ) � Ω Higgs ( { γ v } ; y )= p,q Monday, April 13, 15

Coulomb vs Higgs • It has been shown [Denef `02; Sen `11] : Ω Coulomb = Ω Higgs • Multi-center picture has a smooth transition into the fused D-brane picture at a single point. • The two pictures might become very different if the quivers have a loop [Denef, Moore `07] : Ω Coulomb << Ω Higgs Monday, April 13, 15

Quiver Invariants Monday, April 13, 15

Intrinsic Higgs States • The Higgs phase might in general have more states than the Coulomb phase multi-center states. • We may call these additional ones “intrinsic” Higgs states. • Thus, the Higgs index can be written as: Ω Higgs = Ω Coulomb + “ Ω Inv ” Ω Inv • The intrinsic Higgs states are expected not to experience wall-crossing. Monday, April 13, 15

Cyclic Example • Consider a 3-node quiver with superpotential � C k 1 k 2 k 3 Z k 1 12 Z k 2 23 Z k 3 W ( { Z k 12 } , { Z k 23 } , { Z k 31 } ) = 31 • There arise 3 different quiver varieties, in each of which one set of chirals vanishes. γ 2 • The moduli space is embedded • Z 23 k=1,...,a 23 Z 12 k=1,...,a 12 by F-terms in D-term variety. i M H → A � γ 3 γ 1 Z 31 k=1,...,a 31 Monday, April 13, 15

Characterisation of Ω Inv Ω Inv • Embedding structure i M H → A � Naturally splits the Higgs phase states: ⇒ = H • ( M H ) = i ∗ ( H • ( A )) ⊕ [ H • ( M H ) /i ∗ ( H • ( A ))] ! ! ! Ω Higgs Ω Coulomb Ω Inv Ω Inv [ S.-J.L. , Z.-L.Wang, P.Yi `12] (cf.) [Bena, Berkooz, de Boer, El-Showk, van Den Bleeken `12] • Lefschetz Hyperplane Theorem implies that the Hodge diamond is of a cross shape. Monday, April 13, 15

= Ω Inv = Ω Coulomb 4 5 • 1 • 1 0 0 26 26 0 2 0 6 1 0 0 0 0 0 0 3 0 0 0 0 26 26 0 0 • 1 0 0 3 0 0 0 0 0 0 0 0 0 2 0 0 2 0 0 26 26 0 0 0 0 2 0 1 0 0 1 Monday, April 13, 15

Nonabelian Quivers Abelianisation Monday, April 13, 15

θ 2 n 2 b a n 3 n 1 θ 3 θ 1 Monday, April 13, 15

n 1 n 2 n 3 Monday, April 13, 15

Abelianisation ˜ X X Monday, April 13, 15

The Prescription in a Nutshell • [Martin, `00], [Ciocan-Fontanine, Kim, Sabbah, `06] • (cf.) [Hori, Vafa, `00] Monday, April 13, 15

Loopless Quivers π � a = ι � ˆ o a 1 � � Bridging : , where . . . W = Weyl( G ) a = ˆ a ∧ e ( ∆ ) o | W | ˜ X X � o ∆ = L α root α Monday, April 13, 15

Loopless Quivers � ye − x µ − y − 1 � �� � ω y ( T X ) ≡ x µ · 1 − e − x µ µ 1 X ) ∧ e ( ∆ ) � ω y ( T ˜ Index : , . . . Ω ( y ) = | W | ω y ( ∆ ) ˜ X x where . (1 − e − x ) · ( ye − x − y − 1 ) ω y ← f ω y ( x ) = Monday, April 13, 15

Quivers with a Potential X ) ∧ e ( ˜ N ) 1 ∧ e ( ∆ ) � ω y ( T ˜ Index : Ω ( y ) = ω y ( ∆ ) , . . . ω y ( ˜ | W | N ) ˜ X x where . ω y ← f ω y ( x ) = (1 − e − x ) · ( ye − x − y − 1 ) Monday, April 13, 15

Applications • Non-Abelian Quiver Invariant • Partition-sum Structure of the Index Monday, April 13, 15

Applications • Non-Abelian Quiver Invariant • Partition-sum Structure of the Index Monday, April 13, 15

Applications • Non-Abelian Quiver Invariant • Partition-sum Structure of the Index Chamber (a) Chamber (b) Higgs ( y ) = 1 y 2 + 7 + y 2 . Ω ( a ) Ω ( b ) Higgs ( y ) = 6 Coulomb ( y ) = 1 Ω ( a ) Ω ( b ) y 2 + 2 + y 2 Coulomb ( y ) = 1 Ω inv = 5 Monday, April 13, 15

Applications • Non-Abelian Quiver Invariant • Partition-sum Structure of the Index Monday, April 13, 15

Applications • Non-Abelian Quiver Invariant • Partition-sum Structure of the Index Monday, April 13, 15

Applications • Non-Abelian Quiver Invariant • Partition-sum Structure of the Index P 1 = ( { 1 , 1 , 1 } ; { 1 } ) P { } { } P 2 = ( { 1 , 2 } ; { 1 } ) P { } { P 3 = ( { 3 } ; { 1 } ) Monday, April 13, 15