2d SCFT from M2-branes

Transcription

1 2d SCFT from M2-branes Chan Y. Park California Institute of Technology Sep. 5, KIAS K. Hori, CYP, Y. Tachikawa, to appear

2 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 S 2 partition function 5. Summary and outlook

3 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 S 2 partition function 5. Summary and outlook

4 2d N = (2, 2) theory from M2-branes Configuration of branes: k M2-branes between an M5-brane wrapping a curve t(v) and an M5 at t = t 0, where v = x 4 + i x 5 & t = exp(x 7 + i x 10 ). x 0 x 1 x 2 x 3 v x 6 t x 8 x 9 M5 t(v) M5 t 0 M2 v i t 0 When t(v) = t 0 + v N, the low-energy theory from the M2-branes, which we call M (N, k), is a 2d N = (2, 2) theory on the Coulomb branch with twisted superpotential W eff = trσ N+1 T, where Σ T = diag(σ 1,..., Σ k ) for T U(1) k. 1. 2d SCFT from the IR limit of 2d N = (2, 2) theories 4 / 31

5 2d N = (2, 2) gauge theory from M2-branes When the curve t(v) is such that P(v), defined as exp( v P(v)) = t(v)/t 0, is a polynomial, the 2d theory from the k M2-branes is the U(k) gauge theory without matter field and with the tree level twisted superpotential W = trp(σ) + πi(k + 1)trΣ. When P(v) = v N+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 field theory in the infra-red limit. 1. 2d SCFT from the IR limit of 2d N = (2, 2) theories 5 / 31

6 2d N = (2, 2) LG model from M2-branes The 2d theories from M2-branes is equivalent to a Landau-Ginzburg model with chiral fields X 1,..., X k and superpotential W = W (X 1,..., X k ), where W (x 1,..., x k ) = k a=1 σ N+1 a, x b = σ a1 σ ab, b = 1,..., k. a 1 < <a b 1. 2d SCFT from the IR limit of 2d N = (2, 2) theories 6 / 31

7 2d N = 2 SCFT from M2-branes The Landau-Ginzburg model flows in the IR to an N = 2 SCFT coset model with G H = 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 A N 1 minimal model SCFT, which is the IR limit of LG model with W (X) = X N+1 N+1. This is argued to describe the 2d theory from a single M2-brane ending at the same M5-branes. [Tong, 2006] u j, which give the relevant perturbations [ ] j u j σa N+1 j of the Landau-Ginzburg model from the fixed point, have (u j ) = j N d SCFT from the IR limit of 2d N = (2, 2) theories 7 / 31

8 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 S 2 partition function 5. Summary and outlook

9 Supersymmetric vacua of the 2d gauge theory The gauge theory on the Coulomb branch gets a correction W to the twisted superpotential, k W = πi(k + 1) Σ a, a=1 which cancels the tree level theta term and give k W eff = W T + W = P(Σ a ). This gives the effective potential U eff = k a,b=1 a=1 eab 2 (σ) (P (σ a ) 2πin a )(P 2 (σ b ) 2πin b ), n a,b Z, hence supersymmetric ground states satisfy P (σ a ) = 2πin a. When P u(σ) = σ N + j u jσ N j, The Witten index of the n 1 = = n k = 0 sector is ( N k ). 2. Supersymmetric vacua 9 / 31

10 Supersymmetric vacua of M (N, k) and LG(N, k) The number of ground states of M (N, k; u j ) is ( N k ) due to the s-rule. [Hanany-Witten, 1996][Hanany-Hori, 1997]. The space of supersymmetric ground states of the LG model is naturally identified with the representation k C N of SU(N ), therefore its dimension is ( N k ). [Lerche-Vafa-Warner, 1989] 2. Supersymmetric vacua 10 / 31

11 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 S 2 partition function 5. Summary and outlook

12 More evidence for M (N, k) LG(N, k) k N k duality M (N, k; u j ) 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): G H = SU(N) 1 S[U(k) U(N k)]. We can match the BPS spectrum of M (N, k, u N ) M (N, k; u 2 = = u N 1 = 0 u N ) with the BPS spectrum of LG (N, k, u N ), the deformation of KS(N, k) with the most relevant term u N X 1. [Fendley-Mathur-Vafa-Warner, 1990][Fendley-Lerche-Mathur-Warner, 1991][Lerche-Warner, 1991] 3. 2d BPS spectrum from spectral network 12 / 31

13 M (N, k, u N ): ground states & solitons Ground states: weights of k exterior power of the fundamental representation of A N 1. Solitons: roots connecting the weights. Project the weight space onto the W -plane such that A N, vertices of N -simplex, is a Petrie polygon. The ground states & the solitons of LG (N, k, u N ) have the same structure. [Lerche-Warner, 1991] of N = 4, k = 1 N = 4, k = 2 N = 5, k = d BPS spectrum from spectral network 13 / 31

14 S-walls and spectral network The low-energy effective theory of a 4d N = 2 gauge theory is described 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 the x-plane, we obtain an S jk -wall of a spectral network by solving λ jk τ = (λ j(x, y) λ k (x, y)) dx dτ = eiθ, where λ j is the value of λ on the j-th sheet, and τ is a real parameter along the S jk -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 a spectral network. [Gaiotto-Moore-Neitzke, 2012] 3. 2d BPS spectrum from spectral network 14 / 31

15 S-walls around a branch point of ramification index N N = 2 N = 3 N = 4 For f (x, y) = x y N and λ = y dx, there are N 2 1 S jk -walls described by x jk (τ) = (exp(iθ)/ω jk ) N N+1 τ, where ω jk = ω j ω k and ω k = exp ( 2πi N k ). 3. 2d BPS spectrum from spectral network 15 / 31

16 BPS joint of three S-walls Three S-walls S ij, S jk, and S ik of a spectral network from A N>1 can form a joint, where λ ij + λ jk = λ ik is satisfied spectral network Seiberg-Witten curve and S-walls 3. 2d BPS spectrum from spectral network 16 / 31

17 where the branch point is at t(τ b ) and t(τ s ) = t d BPS spectrum from spectral network 17 / 31 2d BPS states from spectral network θ = 0 A flat M2-brane that gives a ground state of the 2d N = (2, 2) theory ends at a point (t 0, v j ) on the Seiberg-Witten curve. A finite S jk -wall from a branch point to t = t 0 gives a 2d BPS soliton that interpolates two ground states (t 0, v j ) and (t 0, v k ). [Gaiotto-Moore-Neitzke, 2011] The central charge of the 2d BPS state is Z = τs τ b λ jk (t) t τ dτ = τs τ b e iθ dτ,

18 where the branch point is at t(τ b ) and t(τ s ) = t d BPS spectrum from spectral network 17 / 31 2d BPS states from spectral network 0 < θ < π/2 A flat M2-brane that gives a ground state of the 2d N = (2, 2) theory ends at a point (t 0, v j ) on the Seiberg-Witten curve. A finite S jk -wall from a branch point to t = t 0 gives a 2d BPS soliton that interpolates two ground states (t 0, v j ) and (t 0, v k ). [Gaiotto-Moore-Neitzke, 2011] The central charge of the 2d BPS state is Z = τs τ b λ jk (t) t τ dτ = τs τ b e iθ dτ,

19 where the branch point is at t(τ b ) and t(τ s ) = t d BPS spectrum from spectral network 17 / 31 2d BPS states from spectral network π/2 < θ < π A flat M2-brane that gives a ground state of the 2d N = (2, 2) theory ends at a point (t 0, v j ) on the Seiberg-Witten curve. A finite S jk -wall from a branch point to t = t 0 gives a 2d BPS soliton that interpolates two ground states (t 0, v j ) and (t 0, v k ). [Gaiotto-Moore-Neitzke, 2011] The central charge of the 2d BPS state is Z = τs τ b λ jk (t) t τ dτ = τs τ b e iθ dτ,

20 where the branch point is at t(τ b ) and t(τ s ) = t d BPS spectrum from spectral network 17 / 31 2d BPS states from spectral network θ π A flat M2-brane that gives a ground state of the 2d N = (2, 2) theory ends at a point (t 0, v j ) on the Seiberg-Witten curve. A finite S jk -wall from a branch point to t = t 0 gives a 2d BPS soliton that interpolates two ground states (t 0, v j ) and (t 0, v k ). [Gaiotto-Moore-Neitzke, 2011] The central charge of the 2d BPS state is Z = τs τ b λ jk (t) t τ dτ = τs τ b e iθ dτ,

21 M (N = 3, k; u 2, u 3 ): varying θ 3. 2d BPS spectrum from spectral network 18 / 31

22 M (N = 3, k; u 2, u 3 ): varying θ 3. 2d BPS spectrum from spectral network 18 / 31

23 M (N = 3, k; u 2, u 3 ): varying θ 3. 2d BPS spectrum from spectral network 18 / 31

24 M (N = 3, k; u 2, u 3 ): varying θ 3. 2d BPS spectrum from spectral network 18 / 31

25 M (N = 3, k; u 2 0, u 3 ) 3. 2d BPS spectrum from spectral network 19 / 31

26 M (N = 3, k; u 2 0, u 3 ) 3. 2d BPS spectrum from spectral network 19 / 31

27 M (N = 3, k; u 2 0, u 3 ) 3. 2d BPS spectrum from spectral network 19 / 31

28 M (N = 3, k = 1, u 3 ): ground states & solitons On the v-plane ( W -plane) Ground states: of A 2. Solitons: roots of A d BPS spectrum from spectral network 20 / 31

29 M (N = 3, k = 2, u 3 ): ground states & solitons k = 1 k = 2 Ground states: = of A 2. Solitons: roots of A d BPS spectrum from spectral network 21 / 31

30 M (N = 4, k = 1, u 4 ): ground states & solitons 3 1 spectral network on the W -plane weight space of A 3 Ground states: of A 3. Solitons: roots of A d BPS spectrum from spectral network 22 / 31

31 M (N = 4, k = 2, u 4 ): ground states & solitons On the W -plane weight space of A 3 Ground states: = of A 3. Solitons: roots of A d BPS spectrum from spectral network 23 / 31

32 M (N = 4, k = 3, u 4 ): ground states & solitons k = 1 k = 3 Ground states: = of A 3 Solitons: roots of A d BPS spectrum from spectral network 24 / 31

33 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 S 2 partition function 5. Summary and outlook

34 Chiral ring of the gauge theory and the LG model For the LG model with superpotential W u (x), its chiral ring C[x 1,..., x k ]/( x1 W u (x),..., xk W u (x)) is generated by the chiral variables x 1,..., x k. For the gauge theory with twisted superpotential W = f (Σ) + πi(k + 1)trΣ, the relations are σa f (σ) = 2πin a. For the n a = 0 sector, the twisted chiral ring is C[σ 1,..., σ k ] S k /I f, where I f is obtained from the relations. The two rings are isomorphic under x x(σ) for generic f, W u. 4. Chiral ring and S 2 partition function 26 / 31

35 Partition function of the gauge theory The S 2 partition function of the U(k) gauge theory with the twisted superpotential is [Benini-Cremonesi, Doroud-Gomis-Le Floch-Lee, 2012] Z gauge = Λ k2 m Z k + (m a m b ) 2 4r 2 R k a ) a d(τ a ) a<b ( (τ a τ b ) 2 + m a e ir[trp(σ)+trp(σ)] At the IR regime rλ 1, the sum turns into an integral (2r) k R a k dv a for v a = ma 2r, and we have Z gauge Λ k2 r k dσ a d σ a σ a σ b 2 e ir[trp(σ)+trp(σ)], C k a a<b where σ a = τ a + iv a and Σ = diag(σ 1,..., σ k ). 4. Chiral ring and S 2 partition function 27 / 31

36 Partition function of the LG model When we introduce variables X a via det(z Σ) = a X a z k a, then the Jacobian between σ a and X a is a<b (σ a σ b ), and the partition function of the gauge theory is the same as the S 2 partition function of the LG model[gomis-lee, 2012] Z LG = (rλ) k dx a dx a e ir[w (X)+W (X)], C k a with k variables X a and superpotential W (X) if we identify P(Σ) = W (X 1,..., X k ). 4. Chiral ring and S 2 partition function 28 / 31

37 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 S 2 partition function 5. Summary and outlook

38 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 W eff = trσ N+1 T, 2d N = 2 LG model with superpotential W (X) = trσ N+1 T flow in the IR to the same N = 2 SCFT described by a Kazama-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 S 2 partition functions of the gauge theory and the LG model. 5. Summary and outlook 30 / 31

39 Outlook Boundary states of Kazama-Suzuki models and 2d BPS spectrum from spectral network. [Nozaki, 2001] Generalization to other N = 2 coset models to better understand both 2d physics and spectral network with general groups & representations. 2d coset model SCFT as a dual theory of supersymmetric higher-spin theory in AdS 3. [Gaberdiel-Gopakumar, Creutzig-Hikida-Ronne, 2011] 2d coset model SCFT as a boundary theory of M2-branes. 5. Summary and outlook 31 / 31