2d scft from m2-branes - caltech particle...

2d SCFT from M2-branes

Chan Y. Park

California Institute of Technology

Sep. 5, 2013 @ KIAS

K. Hori, CYP, Y. Tachikawa, to appear

Outline

1. 2d SCFT from the IR limit of 2d N = (2, 2) theories

2. Supersymmetric vacua

3. 2d BPS spectrum from spectral network

4. Chiral ring and S2 partition function

5. Summary and outlook

2d N = (2, 2) theory from M2-branesConfiguration of branes:

• k M2-branes between an M5-brane wrapping a curve t(v) andan M5′ at t = t0, where v = x4 + i x5 & t = exp(x7 + i x10).

x0 x1 x2 x3 v x6 t x8 x9M5 − − − − − · t(v) · ·M5′ − − · · − · t0 − −M2 − − · · vi − t0 · ·

• When t(v) = t0 + vN , the low-energy theory from theM2-branes, which we call M (N , k), is a 2d N = (2, 2) theoryon the Coulomb branch with twisted superpotential

Weff = trΣN+1T ,

where ΣT = diag(Σ1, . . . ,Σk) for T ' U(1)k .

1. 2d SCFT from the IR limit of 2d N = (2, 2) theories 4 / 31

2d N = (2, 2) gauge theory from M2-branes

• When the curve t(v) is such that P(v), defined as

exp(∂vP(v)) = t(v)/t0,

is a polynomial, the 2d theory from the k M2-branes is theU(k) gauge theory without matter field and with the tree leveltwisted superpotential

W = trP(Σ) + πi(k + 1)trΣ.

• When P(v) = vN+1, the low-energy effective theory has,among infinitely many others, a set of ground states at Σ = 0,and this “Σ = 0 sector” flows to a nontrivial conformal fieldtheory in the infra-red limit.


2d N = (2, 2) LG model from M2-branes

• The 2d theories from M2-branes is equivalent to aLandau-Ginzburg model with chiral fields X1, . . . ,Xk andsuperpotential W = W (X1, . . . ,Xk), where

W (x1, . . . , xk) =k∑

a=1σN+1

a ,

xb =∑

a1<···<ab

σa1 · · ·σab , b = 1, . . . , k.


2d N = 2 SCFT from M2-branes

• The Landau-Ginzburg model flows in the IR to an N = 2SCFT coset model with

GH = SU(N )1

S [U(k)×U(N − k)] .

[Kazama-Suzuki, 1988][Lerche-Vafa-Warner, 1989][Gepner, 1991]

• When k = 1, the coset model is an AN−1 minimal modelSCFT, which is the IR limit of LG model with W (X) = XN+1

N+1 .This is argued to describe the 2d theory from a singleM2-brane ending at the same M5-branes. [Tong, 2006]

• uj , which give the relevant perturbations∑

j

[uj σ

N+1−ja

]of

the Landau-Ginzburg model from the fixed point, have

∆(uj) = jN + 1 .


Outline






Supersymmetric vacua of the 2d gauge theory

• The gauge theory on the Coulomb branch gets a correction∆W to the twisted superpotential,

∆W = πi(k + 1)k∑

a=1Σa,

which cancels the tree level theta term and give

Weff =W|T + ∆W =k∑

a=1P(Σa).

• This gives the effective potential

Ueff =k∑

a,b=1

e2ab(σ)2 (P ′(σa)− 2πina)(P ′(σb)− 2πinb), na,b ∈ Z,

hence supersymmetric ground states satisfy P ′(σa) = 2πina.• When P ′u(σ) = σN +

∑j ujσ

N−j , The Witten index of then1 = · · · = nk = 0 sector is

(Nk).

2. Supersymmetric vacua 9 / 31

Supersymmetric vacua of M (N , k) and LG(N , k)

• The number of ground states of M (N , k; uj) is(N

k)due to the

s-rule. [Hanany-Witten, 1996][Hanany-Hori, 1997].• The space of supersymmetric ground states of the LG modelis naturally identified with the representation ∧kCN ofSU(N ), therefore its dimension is

(Nk). [Lerche-Vafa-Warner, 1989]

2. Supersymmetric vacua 10 / 31

Outline






More evidence for M (N , k) ≡ LG(N , k)

• k ↔ N − k duality• M (N , k; uj) has the k ↔ N − k duality, due to the

Hanany-Witten transition. [Hanany-Witten, 1996].• LG(N , k) also has the k ↔ N − k duality. When k > N − k,

we can re-express everything in terms of N − k chiral fields.[Gepner, 1989]

• KS(N , k) = KS(N ,N − k): GH = SU(N)1

S[U(k)×U(N−k)] .• We can match the BPS spectrum of

M ′(N , k, uN ) ≡ M (N , k; u2 = · · · = uN−1 = 0 6= uN )

with the BPS spectrum of LG ′(N , k, uN ), the deformation ofKS(N , k) with the most relevant term uN X1.[Fendley-Mathur-Vafa-Warner, 1990][Fendley-Lerche-Mathur-Warner, 1991][Lerche-Warner, 1991]

3. 2d BPS spectrum from spectral network 12 / 31

M ′(N , k, uN ): ground states & solitons

• Ground states: weights of k exterior power of the fundamentalrepresentation of AN−1.

• Solitons: roots connecting the weights.• Project the weight space onto the W -plane such that of

AN , vertices of N -simplex, is a Petrie polygon.• The ground states & the solitons of LG ′(N , k, uN ) have thesame structure. [Lerche-Warner, 1991]

N = 4, k = 1 N = 4, k = 2 N = 5, k = 1


S-walls and spectral network

• The low-energy effective theory of a 4d N = 2 gauge theory isdescribed by a Seiberg-Witten curve & a differential,

f (x, y) = 0, λ = λ(x, y)dx.

• When we see the curve as a multi-sheeted cover over thex-plane, we obtain an Sjk-wall of a spectral network by solving

∂λjk∂τ

= (λj(x, y)− λk(x, y)) dxdτ = eiθ,

where λj is the value of λ on the j-th sheet, and τ is a realparameter along the Sjk-wall. [Klemm-Lerche-Mayr-Vafa-Warner, 1996][Shapere-Vafa,

1999][Gaiotto-Moore-Neitzke, 2009,2010,2011,2012]

• The collection of the S-walls at a value of θ is called aspectral network. [Gaiotto-Moore-Neitzke, 2012]


S-walls around a branch point of ramification index N

H12L

H21L

H21L

N = 2

H13L

H32L

H21L

H21L

H31L

H23L

H31L

H12L

N = 3

H41LH32L

H41LH32L

H34LH21L

H12LH43L

H23LH14L

H31L

H31L

H42L

H24L

H13L

N = 4

For f (x, y) = x − yN and λ = y dx, there areN 2 − 1 Sjk-walls described by

xjk(τ) = (exp(iθ)/ωjk)N

N+1 τ,

where ωjk = ωj − ωk and ωk = exp(2πiN k

).


BPS joint of three S-walls

Three S-walls Sij , Sjk , and Sik of a spectral network from AN>1can form a joint, where λij + λjk = λik is satisfied.

H12L

H23L

H13L

spectral network Seiberg-Witten curve and S-walls


2d BPS states from spectral network

θ = 0

• A flat M2-brane that gives aground state of the 2d N = (2, 2)theory ends at a point (t0, vj) onthe Seiberg-Witten curve.

• A finite Sjk-wall from a branchpoint to t = t0 gives a 2d BPSsoliton that interpolates twoground states (t0, vj) and (t0, vk).[Gaiotto-Moore-Neitzke, 2011]

• The central charge of the 2d BPSstate is

Z =∫ τs

τbλjk(t) ∂t

∂τdτ =

∫ τs

τbeiθdτ,

where the branch point is at t(τb)and t(τs) = t0.



0 < θ < π/2




Z =∫ τs

τbλjk(t) ∂t

∂τdτ =

∫ τs

τbeiθdτ,




π/2 < θ < π




Z =∫ τs

τbλjk(t) ∂t

∂τdτ =

∫ τs

τbeiθdτ,




θ ≈ π




Z =∫ τs

τbλjk(t) ∂t

∂τdτ =

∫ τs

τbeiθdτ,



M (N = 3, k; u2, u3): varying θ


M (N = 3, k; u2 → 0, u3)


M ′(N = 3, k = 1, u3): ground states & solitons

1

2

3

On the v-plane (∼W -plane)

� Ground states: of A2. � Solitons: roots of A2.



2

1

3 3

1

2

1

2

3

k = 1

1

2

3

@12D

@23D

@13D

k = 2

� Ground states: = of A2. � Solitons: roots of A2.



spectral network

1

2

3

4

on the W -plane weight space of A3

� Ground states: of A3. � Solitons: roots of A3.



1

2

3

4

@12D@23D

@34D @41D

@13D

@24D

On the W -plane weight space of A3

� Ground states: = of A3. � Solitons: roots of A3.



1

2

3

4

k = 1

@412D

@123D

@234D

@341Dk = 3

� Ground states: = of A3 � Solitons: roots of A3.


Outline






Chiral ring of the gauge theory and the LG model

• For the LG model with superpotential Wu(x), its chiral ring

C[x1, . . . , xk ]/(∂x1Wu(x), . . . , ∂xk Wu(x))

is generated by the chiral variables x1, . . . , xk .• For the gauge theory with twisted superpotential

W = f (Σ) + πi(k + 1)trΣ,

the relations are ∂σa f (σ) = 2πina. For the na = 0 sector, thetwisted chiral ring is

C[σ1, . . . , σk ]Sk/If ,

where If is obtained from the relations.• The two rings are isomorphic under x 7→ x(σ) for generic f ,

Wu .4. Chiral ring and S2 partition function 26 / 31

Partition function of the gauge theory

• The S2 partition function of the U(k) gauge theory with thetwisted superpotential is [Benini-Cremonesi, Doroud-Gomis-Le Floch-Lee, 2012]

Zgauge = Λ−k2 ∑m∈Zk

∫Rk

∏a

d(τa)∏a<b

((τa − τb)2+

+ (ma −mb)2

4r2

)∑a

mae−ir [trP(Σ)+trP(Σ)]

• At the IR regime rΛ� 1, the sum turns into an integral(2r)k ∫

Rk∏

a dva for va = ma2r , and we have

Zgauge → Λ−k2rk∫

Ck

∏a

dσadσ̄a∏a<b|σa − σb|2 e−ir [trP(Σ)+trP(Σ)],

where σa = τa + iva and Σ = diag(σ1, . . . , σk).

4. Chiral ring and S2 partition function 27 / 31

Partition function of the LG model

• When we introduce variables Xa via

det(z − Σ) =∑

aXazk−a,

then the Jacobian between σa and Xa is∏

a<b(σa − σb), andthe partition function of the gauge theory is the same as theS2 partition function of the LG model[Gomis-Lee, 2012]

ZLG = (rΛ)k∫

Ck

∏a

dXadXae−ir [W (X)+W (X)],

with k variables Xa and superpotential W (X) if we identify

P(Σ) = W (X1, . . . ,Xk).

4. Chiral ring and S2 partition function 28 / 31

Outline






Summary

• We claim• 2d N = (2, 2) theory from k M2-branes between an M5-brane

and a ramified system of N M5-branes,• 2d N = 2, 2 U(k) gauge theory with low-energy twisted

superpotential Weff = trΣN+1T ,

• 2d N = 2 LG model with superpotential W (X) = trΣN+1T

flow in the IR to the same N = 2 SCFT described by aKazama-Suzuki coset model.

• As evidence, we compare• supersymmetric vacua of the theories,• the BPS spectrum of the 2d theory from the branes with that

of the LG model, and• the chiral rings and the S2 partition functions of the gauge

theory and the LG model.

5. Summary and outlook 30 / 31

Outlook

• Boundary states of Kazama-Suzuki models and 2d BPSspectrum from spectral network. [Nozaki, 2001]

• Generalization to other N = 2 coset models to betterunderstand both 2d physics and spectral network with generalgroups & representations.

• 2d coset model SCFT as a dual theory of supersymmetrichigher-spin theory in AdS3. [Gaberdiel-Gopakumar, Creutzig-Hikida-Ronne, 2011]

• 2d coset model SCFT as a boundary theory of M2-branes.

5. Summary and outlook 31 / 31

2d scft from m2-branes - caltech particle...

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