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

BPS-State Counting:

Quiver Invariant, Abelianisation & Mutation

Seung-Joo Lee

Virginia Tech Southeastern regional mathematical string theory meeting

Apr 11, 2015

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

Monday, April 13, 15

Outline

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

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Warm Up

a topology exercise

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• a3 a2

a1 a3 a2 a1

• a3 a2

a1

• a3 a2

a1

Pa2−1 × Pa3−1 Pa3−1 × Pa1−1 Pa1−1 × Pa2−1 O(1, 1)⊕a1 O(1, 1)⊕a2 O(1, 1)⊕a3

Chamber 1 Chamber 2 Chamber 3

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O(1, 1)⊕4 O(1, 1)⊕5 O(1, 1)⊕6 P4 × P3 P3 × P5 P5 × P4

• 4 5

6

• a3 = a2 =

a1 =

• 4 5

6 4 5 6

Chamber 1 Chamber 2 Chamber 3

• 4 5

6

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1· y-1 52· y0 1· y1

• 1· y-3

0· y-2 2· y-1 52· y0 2· y1 0· y2 1· y3

• 1· y-5

0· y-4 2· y-3 0· y-2 3· y-1 52· y0 3· y1 0· y2 2· y3 0· y4 1· y5

• 4 5

6 4 5 6

Chamber 1 Chamber 2 Chamber 3

• 4 5

6

Monday, April 13, 15

H.,.

1 26 26 1

• 1

0 0 0 2 0 0 26 26 0 0 2 0 0 0 1

• 1

0 0 0 2 0 0 0 0 0 0 0 3 0 0 0 0 26 26 0 0 0 0 3 0 0 0 0 0 0 0 2 0 0 0 1

H.,. H.,.

Cahmber 1 Chamber 2 Chamber 3

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Rudiments

BPS Index

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• We consider N=2 Abelian gauge theories.
• States have integer charges:
• Poincare extends to N=2 super-Poincare:

M gets bounded by |Z|.

• M=|Z| case: “short” repre, Sj=[j] rhh,

where rhh=2[0] [1/2] is the 4-diml irrep of the odd alg.

• M>|Z| case: “long” repre, Lj=[j] rhh rhh

γ ∈ Z2r ≡ Γ ⊕ ⊗ ⊗ ⊗

N=2 Basics

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• For the Hilbert space ,

define the BPS index as:

• Only genuine short reps contribute to .

H1

γ =

• j∈ 1

2 Z≥0

Sj

⊕nj(γ)

• ⊕
• l∈ 1

2 Z≥0

Ll

⊕ml(γ)

• Ω(γ) :=
• j∈ 1

2 Z≥0

(−1)2j(2j + 1) nj(γ) = Tr

H1

γ(−1)2J3

Ω(γ)

BPS Index

• The little super-algebra contains su(2)R and hence one can

define the refined index as:

Ω(γ; y) = Tr

H1

γ(−1)2J3y2I3+2J3

y=1

− − → Ω(γ) = Tr

H1

γ(−1)2J3

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Rudiments

Wall-crossing

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• is invariant under arbitrary deformations of ,

but may change under deformations of the theory.

• The index is ill-defined when mixes with the multi-ptl

spectrum, i.e., if can split into and s.t. , .

• Thus, in the parameter space, there appears a wall, across

which the BPS index jumps. γ1

γ2

γ1 + γ2 = γ

Z1/Z2 ∈ R+

Wall-Crossing

H1

γ

Ω(γ) H1

γ

γ

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• 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 = γ1, γ2 2 |Z1 + Z2| Im[ ¯ Z1Z2]

Wall-Crossing

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Rudiments

BPS Quivers

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• 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 CY3: 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]

BPS Quivers

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• U(1) vectors include xv=(xv1, xv2,xv3) and bi-fund. chirals

include Zvwk=1,...,avw , where avw =

• Two phases

γ1 γ2 γ3

• Z23k=1,...,a23 Z12k=1,...,a12
• x2

x3 x1

(1) Coulomb: xv , Zvwk (2) Higgs: xv , Zvwk

γv, γw

BPS Quivers

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• For large xv-xw, chirals are massive and eff. dynamics leads to

, with

• By studying the soln space , one

can obtain the Coulomb index

[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
• n the variety .
• The Higgs index is given as:

Kv

• w=v

γw, γv |xw xv| θv(u) = 0 for v

θv = 2 Im[e−iαZγv(u)]

M = {xv | Kv = 0 , ∀v} \ R3

BPS Index

ΩCoulomb({γv}; y)

MH = {Zk

vw | Dv = θv , ∀v}/

• v

U(1)

ΩHiggs({γv}; y)=

• p,q

(−1)p+q−d y2p−d hp,q(MH)

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• It has been shown [Denef `02; Sen `11]:
• 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]:

Coulomb vs Higgs

ΩCoulomb = ΩHiggs ΩCoulomb << ΩHiggs

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Quiver Invariants

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• 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:
• The intrinsic Higgs states are expected not to experience

wall-crossing.

Intrinsic Higgs States

ΩHiggs = ΩCoulomb + “ ΩInv” ΩInv

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γ1 γ2 γ3

• Z23k=1,...,a23 Z12k=1,...,a12

Z31k=1,...,a31

MH

i

• → A

W({Zk

12}, {Zk 23}, {Zk 31}) =

• Ck1k2k3 Zk1

12 Zk2 23 Zk3 31

Cyclic Example

• Consider a 3-node quiver with superpotential
• There arise 3 different quiver varieties, in each of which
• ne set of chirals vanishes.
• The moduli space is embedded

by F-terms in D-term variety.

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• Embedding structure

Naturally splits the Higgs phase states:

[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.

!

Characterisation of ΩInv

ΩHiggs ΩCoulomb ΩInv

! !

ΩInv

ΩInv

MH

i

• → A

= ⇒ H•(MH) = i∗ (H•(A)) ⊕ [H•(MH)/i∗ (H•(A))]

Monday, April 13, 15

4 5 6

= ΩCoulomb = ΩInv

• 1

26 26 1

• 1

0 0 0 2 0 0 0 0 0 0 0 3 0 0 0 0 26 26 0 0 0 0 3 0 0 0 0 0 0 0 2 0 0 0 1

• 1

0 0 0 2 0 0 26 26 0 0 2 0 0 0 1

Monday, April 13, 15

Nonabelian Quivers

Abelianisation

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b

θ1 θ2 θ3

a

n2 n3 n1

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n2 n1 n3

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X ˜ X

Abelianisation

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• [Martin, `00], [Ciocan-Fontanine, Kim, Sabbah, `06]
• (cf.) [Hori, Vafa, `00]

The Prescription in a Nutshell

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Loopless Quivers

• X

a = 1 |W|

• ˜

X

ˆ a ∧ e(∆)

Bridging : , where . . .

πa = ιˆ a

W = Weyl(G)

∆ =

• root α

Lα

• Monday, April 13, 15

Loopless Quivers

Index :

ωy(T X) ≡

• µ
• xµ ·

ye−xµ − y−1 1 − e−xµ

• , . . .

where .

Ω(y) = 1 |W|

• ˜

X

ωy(T ˜ X) ∧ e(∆) ωy(∆)

ωy ← fωy(x) = x (1 − e−x) · (ye−x − y−1)

Monday, April 13, 15

Quivers with a Potential

Index : Ω(y) =

1 |W|

• ˜

X

ωy(T ˜ X) ∧ e( ˜ N) ωy( ˜ N) ∧ e(∆) ωy(∆), . . .

where .ωy ← fωy(x) =

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

• Non-Abelian Quiver Invariant
• Partition-sum Structure of the Index

Applications

Monday, April 13, 15

• Non-Abelian Quiver Invariant
• Partition-sum Structure of the Index

Ω(a)

Higgs(y) = 6

Ω(a)

Coulomb(y) = 1

Ω(b)

Higgs(y) = 1

y2 + 7 + y2 . Ω(b)

Coulomb(y) = 1

y2 + 2 + y2

Ωinv = 5

Chamber (a) Chamber (b)

Applications

Monday, April 13, 15

• Non-Abelian Quiver Invariant
• Partition-sum Structure of the Index

Applications

Monday, April 13, 15

• Non-Abelian Quiver Invariant
• Partition-sum Structure of the Index

Applications

Monday, April 13, 15

• Non-Abelian Quiver Invariant
• Partition-sum Structure of the Index

Applications

P1 = ({1, 1, 1}; {1}) P { } { } P2 = ({1, 2}; {1}) P { } { P3 = ({3}; {1})

Monday, April 13, 15

• Non-Abelian Quiver Invariant
• Partition-sum Structure of the Index

Ω Ω Ω[

] [ ] [ ]

Ω[

](y)

(y) (y) (y)

c1(P1; y) · c2(P2; y) · c3(P3; y) ·

=

+ +

c(P; y) ≡ 1 |Γ(P)|

N

Y

v=1 lv

Y

av=1

1 rv,av y − y−1 yrv,av − y−rv,av

Applications

Monday, April 13, 15

• Non-Abelian Quiver Invariant
• Partition-sum Structure of the Index

Applications

Monday, April 13, 15

• Non-Abelian Quiver Invariant
• Partition-sum Structure of the Index

Applications

• Asymptotic behavior? cf. [Cordova, Shao `15], [Kim `15]
• Another path towards Non-Abelian Quivers?

· · · Works in principle for any quivers but practically hard · · ·

Monday, April 13, 15

Nonabelian Quivers

Mutation

Monday, April 13, 15

Q = ({Ni} ; [bij])ζi − → Q = ({ Ni} ; [ ])

ζi

• bij

µ µL

k

µR

k

γ

is

• Relate the index of a complicated quiver to that of a

simpler one via mutation:

• With respect to a node k, either Left or Right: or
• The action on charges characterises the mutation:
• Thus, in the parameter space, there appears a wall, across

which the BPS index jumps.

Left and Right Mutations

Monday, April 13, 15

Q = ({Ni} ; [bij])ζi − → Q = ({ Ni} ; [ ])

ζi

• bij

µ µL

k

µR

k

γ

is

• Relate the index of a complicated quiver to that of a

simpler one via mutation:

• With respect to a node k, either Left or Right: or
• The action on charges characterises the mutation:
• Thus, in the parameter space, there appears a wall, across

which the BPS index jumps.

Left and Right Mutations

Monday, April 13, 15

Q = ({Ni} ; [bij])ζi − → Q = ({ Ni} ; [ ])

ζi

• bij

µ µL

k

µR

k

γ

is

• Relate the index of a complicated quiver to that of a

simpler one via mutation:

• With respect to a node k, either Left or Right: or
• The action on charges characterises the mutation:
• Thus, in the parameter space, there appears a wall, across

which the BPS index jumps.

Left and Right Mutations

Monday, April 13, 15

Q = ({Ni} ; [bij])ζi − → Q = ({ Ni} ; [ ])

ζi

• bij

µ µL

k

µR

k

γ

is

• Relate the index of a complicated quiver to that of a

simpler one via mutation:

• With respect to a node k, either Left or Right: or
• The action on charges characterises the mutation:
• Thus, in the parameter space, there appears a wall, across

which the BPS index jumps.

Left and Right Mutations

Monday, April 13, 15

Triangle Quiver with

N = (1, 1, N)

Monday, April 13, 15

Triangle Quiver with

µL

3

µR

3

• N = (1, 1, N)
• Trade off between vectors and chirals could be made.
• Would all mutations preserve the Witten index?

Monday, April 13, 15

Mutation as a viewpoint change in how BPS particles are distinguished from anti-BPS particles

[Alim, Cecotti, Cordova, Espahdodi, Rastogi, Vafa `11]

q

• q
• p

p

Z

Monday, April 13, 15

• IV

III I II

Monday, April 13, 15

• µL

3

µR

3

• IV

III I II

Monday, April 13, 15

• I

II IV III

Monday, April 13, 15

Monday, April 13, 15

Can we mutate with respect to node 3?

µL

3

µR

3

Thus, ( ) must reproduce and ( and ).

(Q)

µL

3

ΩQ(II)

ΩQ(III)

ΩQ(I)

ΩQ(IV)

µR

3(Q)

Monday, April 13, 15

µL

3

µR

3

For example, take a=7, b=5, c=4 and N=2.

7 13 4 5 4 5 4 5 2 2 3 13 Thus, ( ) must reproduce and ( and ).

(Q)

µL

3

ΩQ(II)

ΩQ(III)

ΩQ(I)

ΩQ(IV)

µR

3(Q)

ˆ Q ≡

Monday, April 13, 15

µL

3

µR

3

7 13 4 5 4 5 4 5 2 2 3 13

• Ω(I)

Ω(II) Ω(III) Ω(IV) − − Ω(b I) Ω(b II) Ω(c III) Ω(c IV) − − Ω(b I) Ω(b II) Ω(c III) Ω(c IV)

=? =? =? =? =? =? =? =? =? =? =? =?

Thus, ( ) must reproduce and ( and ).

(Q)

µL

3

ΩQ(II)

ΩQ(III)

ΩQ(I)

ΩQ(IV)

µR

3(Q)

ˆ Q ≡

Monday, April 13, 15

µL

3

µR

3

7 13 4 5 4 5 4 5 2 2 3 13 In principle one can compute all these indices via the Abelianisation. But the toric varieties involved here are of dimension 20-ish, meaning that one needs to deal with such high-rank lattices. Furthermore, the analytical structure for Witten index is encoded only implicitly as one needs to extract the intersection numbers in a combinatorial manner. Thus, ( ) must reproduce and ( and ).

(Q)

µL

3

ΩQ(II)

ΩQ(III)

ΩQ(I)

ΩQ(IV)

µR

3(Q)

ˆ Q ≡

Monday, April 13, 15

• Compact expression has been obtained:

where are the zero modes of Cartan part, and the “integrand” is with and g(I)

chiral(u) = −sinh qI(u)+( RI

2 −1)z

2

sinh qI(u)+ RI

2 z

2

Index of d=1 GLSM via Path Integral

[K.Hori, H.Kim, P.Yi `14]

• (cf.) [Benini, Eager, Hori, Tachikawa `13], [Cordova, Chao `14], [Hwang, Kim, Kim, Park `14]

g(A)

vector(u) =

• 1

2sinh z

2

rA

α∈∆A

sinh α(u)

2

sinh α(u)−z

2

u = x3 + iA0 |zero-mode

g(u) =

• A

g(A)

vector(u)

• I

g(I)

chiral(u)

Ω(y; ζ) = 1 |W|JK-Resζ [g(u)dru]

Monday, April 13, 15

• Compact expression has been obtained:

where are the zero modes of Cartan part, and the “integrand” is

• The JK-Res is a sum over all co-dim “r” singularities in , defined as

intersection of hyperplanes via :

Index of d=1 GLSM via Path Integral

[K.Hori, H.Kim, P.Yi `14]

• (cf.) [Benini, Eager, Hori, Tachikawa `13], [Cordova, Chao `14], [Hwang, Kim, Kim, Park `14]

u = x3 + iA0 |zero-mode

g(u) =

• A

g(A)

vector(u)

• I

g(I)

chiral(u)

Ω(y; ζ) = 1 |W|JK-Resζ [g(u)dru]

(C)r

{Qi1, · · · , Qir}

1 |det(Q)| if

• therwise

ζ Span+ Qi1, · · · , Qir

JK-Resζ:{Qi1,··· ,Qir } dru (Q1 · u) · · · (Qr · u) =

Monday, April 13, 15

• Ω(I)

= 50 , Ω(II) = 1/y4 + 2/y2 + 87 + 2y2 + y4 , Ω(III) = 1/y6 + 2/y4 + 4/y2 + 89 + 4y2 + 2y4 + y6 , Ω(IV) = 1/y6 + 2/y4 + 4/y2 + 54 + 4y2 + 2y4 + y6 .

− − Ω(b I) = 1/y6 + 2/y4 + 4/y2 + 89 + 4y2 + 2y4 + y6 , Ω(b II) = 35 , Ω(c III) = 1/y4 + 2/y2 + 37 + 2y2 + y4 , Ω(c IV) = 1/y4 + 2/y2 + 87 + 2y2 + y4 . Ω(b I) = 1/y10 + 2/y8 + 4/y6 + 6/y4 + 8/y2 + 58 + 8y2 + 6y4 + 4y6 + 2y8 + y10 , Ω(b II) = 1/y6 + 2/y4 + 4/y2 + 54 + 4y2 + 2y4 + y6, Ω(c III) = 50 , Ω(c IV) = 50 .

µL

3

µR

3

7 13 4 5 4 5 4 5 2 2 3 13 Thus, ( ) must reproduce and ( and ).

(Q)

µL

3

ΩQ(II)

ΩQ(III)

ΩQ(I)

ΩQ(IV)

µR

3(Q)

ˆ Q ≡

Monday, April 13, 15

µL

3

µR

3

The JK-Res approach leads to the desired links even analytically.

• Ω(I)

Ω(II) Ω(III) Ω(IV) − − Ω(b I) Ω(b II) Ω(c III) Ω(c IV) − − Ω(b I) Ω(b II) Ω(c III) Ω(c IV)

Thus, ( ) must reproduce and ( and ).

(Q)

µL

3

ΩQ(II)

ΩQ(III)

ΩQ(I)

ΩQ(IV)

µR

3(Q)

ˆ Q ≡

Monday, April 13, 15

Summary and Outlook

• d=4 N=2 BPS states were studied via d=1 N=4 Quiver GLSM
• Wall-crossing-sensitive indices have wall-crossing-safe invariants
• The quiver invariants of an abelian cyclic quiver theory are naturally

characterised as the “middle” cohomology;

non-abelian generalisation of the geometric interpretation?

• The moduli space geometry for a non-abelian quiver can be tackled via

abelianisation and/or path integral

• Mutation of d=1 quiver theory can only be selectively performed to

preserve Witten index

• Asymptotics in the large-rank limit and d=4 N=2 BPS black-hole microstates?

Monday, April 13, 15

Summary and Outlook

Thank you!

• d=4 N=2 BPS states were studied via d=1 N=4 Quiver GLSM
• Wall-crossing-sensitive indices have wall-crossing-safe invariants
• The quiver invariants of an abelian cyclic quiver theory are naturally

characterised as the “middle” cohomology;

non-abelian generalisation of the geometric interpretation?

• The moduli space geometry for a non-abelian quiver can be tackled via

abelianisation and/or path integral

• Mutation of d=1 quiver theory can only be selectively performed to

preserve Witten index

• Asymptotics in the large-rank limit and d=4 N=2 BPS black-hole microstates?

Monday, April 13, 15