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3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones Stefano Cremonesi (Tel-Aviv University) Cambridge Univ., DAMTP March 18, 2010 based on: JHEP 1002 , 036 (2010) [arXiv:0911.4127] with F . Benini, C. Closset Stefano

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3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

Stefano Cremonesi

(Tel-Aviv University)

Cambridge Univ., DAMTP March 18, 2010 based on: JHEP 1002, 036 (2010) [arXiv:0911.4127] with F . Benini, C. Closset

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Outline of the talk

Motivation and overview Review: 4d and 3d toric quiver gauge theories New results: 3d toric flavoured quiver gauge theories Examples Conclusions

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Motivation and overview

Part I Motivation and overview

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Motivation and overview Motivation Overview

Recent progress in understanding the low energy dynamics on multiple M2-branes and AdS4/CFT3 dualities.

[Bagger, Lambert 07; Gustavsson 07; van Raamsdonk 08; Aharony, Bergman, Jafferis, Maldacena 08; . . . ]

Gauge theories for membranes at conical singularities Freund-Rubin AdS4 vacua of 11d supergravity (Warped) AdS4 vacua of type IIA supergravity 3d quiver gauge theories (w/ Chern-Simons terms): toy models for condensed matter systems

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Motivation and overview Motivation Overview

Aim Extend AdS4/CFT3 dualities to N ≥ 2 3d gauge theories with fundamental and antifundamental matter (flavours). M2 (and KK monopoles) in M-theory, D2/D6 in IIA. Gauge duals to infinitely many AdS4 × M7 vacua (M7 Sasaki-Einstein), alternative to gauge theories w/o 4d parent

[Hanany, Vegh, Zaffaroni 08; . . . ]

Quantum effects are crucial (’t Hooft monopole operators) Introducing and integrating out chiral flavours generates CS terms Useful for condensed matter applications? Previous works: Cherkis, Hashimoto 02; Gaiotto, Jafferis 09. Contemporary work: Jafferis 09.

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Motivation and overview Motivation Overview

Framework: toric geometry and quiver gauge theories from brane tilings. Toric geometry is a natural arena for studying KK monopoles. Bottom-up

[Martelli, Sparks 08; Hanany, Zaffaroni 08; . . . ] (no flavours)

3d N = 2 toric quiver gauge theory (CS, flavours) − → toric CY4 cone Top-down (“stringy derivation”)

[Aganagic 09] (no D6)

M2 at toric CY4 cone − → 3d N = 2 toric quiver gauge theory (CS, flavours) Caveats...

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Motivation and overview Motivation Overview

Type IIA brane tiling models without 4d/IIB parent lack a stringy derivation of the gauge theory so far. Flavour symmetries? + data on CS levels Replaced by flavoured quiver theories consistent with M → IIA reduction: D6 brane embedding Xij = 0 ⇒ δW = pˆ

kiXijqjˆ k

Toric CY4 cone reproduced as the Abelian geometric moduli space thanks to a quantum relation in the chiral ring: T ˜ T =

  • i,j

X

hij ij

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Review

Part II Review

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Review Brane tilings and toric quiver gauge theories “Stringy derivation” of quiver CS theories

Brane tilings and 4d toric quiver gauge theories

[Hanany, Kennaway 05; Franco, Hanany, Kennaway, Vegh, Wecht 05]

W = A1B1A2B2 − A1B2A2B1 Mesonic moduli space is a toric CY3 cone.

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Review Brane tilings and toric quiver gauge theories “Stringy derivation” of quiver CS theories

Brane tilings and 3d toric quiver gauge theories

[Hanany, Zaffaroni 08; Ueda, Yamazaki 08; Imamura, Kimura 08]

W = A1B1A2B2 − A1B2A2B1 k = n1 − n2 + n3 − n4 Geometric moduli space is a toric CY4 cone.

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Review Brane tilings and toric quiver gauge theories “Stringy derivation” of quiver CS theories

Brane tilings with multiple bonds

[Hanany, Vegh, Zaffaroni 08; Franco, Hanany, Park, Rodriguez-Gomez 08]

W = C13C32B1A2B2 − C13C32B2A2B1 k1 = n1 − n2 + n3 − n4 k2 = n2 − n3 + n4 − n5 k3 = −n1 + n5

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Review Brane tilings and toric quiver gauge theories “Stringy derivation” of quiver CS theories

Geometric moduli space of N = 2 quiver CS theories

U(1)G case for simplicity.

G

  • i=1

ki = 0.

[Martelli, Sparks 08; Hanany, Zaffaroni 08]

F-flatness: Z = {Xα, α = 1, . . . , E | dW(X) = 0} ⊂ CE D-flatness: Di = ki σi

2π

∀G

i=1 ,

|Xα|2 G

  • i=1

gi[Xα] σi 2 = 0 ∀E

α=1 ,

Di ≡

E

  • α=1

gi[Xα] |Xα|2 Diagonal photon Adiag ≡

i Ai dualised into a periodic scalar τ.

Gauge: Ai → Ai + dθi, τ → τ + 1

G

  • i ki θi, Xα → ei gi [Xα]θi Xα

Branch σi = σ ∀G

i=1: G

  • i=1

ci Di = 0 ∀ {ci} |

G

  • i=1

ciki = 0

σ 2π = 1 k2

  • i ki Di ,

k2 =

i k 2 i

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Review Brane tilings and toric quiver gauge theories “Stringy derivation” of quiver CS theories

Geometric moduli space of N = 2 quiver CS theories

Geometric moduli space: M = (Z U(1)G−2)/Zq , q = gcd{ki} Moduli spaces of 3d/4d TQGTs w/ same matter content and W: Mmes

4d

= Z U(1)G−1 = M3d U(1)

k

[Jafferis, Tomasiello 08; Martelli, Sparks 08; Hanany, Zaffaroni 08]

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Review Brane tilings and toric quiver gauge theories “Stringy derivation” of quiver CS theories

“Stringy derivation” of quiver CS theories

[Aganagic 09]

Toric CY4 cone S1 bundle over a 7-manifold, which is a toric CY3 cone fibred over R. A Kähler parameter of CY3 varies linearly along R. CY3 = CY4 U(1)M − → quiver gauge theory on D2. (Not quite...) For an M2 probe: – S1 parametrised by τ – CY3 by mesonic full gauge invariants of the quiver (Kähler parameters are FI terms) – R parametrised by σ S1: M-theory circle in the reduction to IIA (after quotient). Curvature of the U(1)M bundle: RR F2, which induces CS terms

  • n wv of fractional D2 probes.

Fibration of CY3 over R: scalar N = 2 partners of CS terms.

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results

Part III New results

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

Singular reduction and extra objects in IIA

The CY4 U(1)M reduction can always be done. However, if the fibre degenerates at some locus we expect extra objects in IIA. U(1)M: primitive vector wM in Z3 ambient space of 3d toric diagram. CY4 U(1)M: projection of 3d toric diagram along wM. S1 shrinks in CY4 at codimC = 2 (external edge parallel to wM in toric diagram): D6 at codimC = 3 (external face parallel to wM in toric diagram): ? S1 degenerates to S1/Zp out of the tip can lead to Gaiotto-Witten’s non-Lagrangian sectors. N = 4 example: NS5, (p, q)5 intersecting D3 on a circle.

[Jafferis 09]

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

KK monopoles and D6 branes

h + 1 adjacent external points in 3d toric diagram: h KK monopoles Local complex structure C2 × C2/Zh SU(h) gauge symmetry at non- isolated singularity in the bulk (wv gauge symmetry on h D6 branes) Flavour SU(h) global symmetry in the boundary CFT Holomorphic embedding of D6 in CY3: (collection of) toric divisor(s)

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

D6 branes and flavours

Holomorphic embedding of D6 along collection of toric divisors in CY3: algebraic data. D2 on D6: massless flavour fields = ⇒ superpotential term δW = pXq Subtlety: to a single toric divisor (or union of pairwise intersecting toric divisors) one associates Q bifundamentals, with same global charges. ZQ = π1(base of conical divisor) On each D6 at the conical CY3 a ZQ-valued connection, flat everywhere but at the tip, specifies which bifundamental is coupled to a flavour pair (p, q). Location of D6 along R: real mass parameter for flavours. It lifts to a blow-up parameter in M-theory.

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

D6 branes and flavours (II)

D6 source RR fields. Jump of fluxes in CY3 when crossing D6: δ

  • C2

F2 = #(C2 ∩ D6) , δ

  • C4

F4 = #(C4 ∩ D6 ∩ C(Fwv)

4

) Flavours shift CS levels. Jump of CS levels when integrating out positive vs. negative mass flavours. Comment: 3d vs. 4d flavoured quiver gauge theories No continuous gauge anomalies: no RR C1 tadpole. Message Complex codim. 2 locus of fixed pts. of U(1)M = ⇒ D6 branes in IIA = ⇒ quiver corresponding to CY3 = CY4 U(1)M is flavoured.* *Caveat: as for CS levels, projection is dangerous.

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

Flavouring 3d toric quiver gauge theories

Starting point: 3d toric quiver gauge theory with bifundamentals X. W = W0(X) Flavouring: couple hα flavours (pα, qα) to bifundamentals Xα W = W0(X) +

E

  • α=1

hα

  • iα=1

(pα)iαXα(qα)iα Goal: geometric moduli spaces of toric flavoured quiver gauge theories. U(1)G (1 M2-brane): toric CY4 cones C(M7) (necessary for N = 2 CFT). U(N)G (N M2): SymNC(M7) expected. Large N duals: AdS4 × M7. Large CS levels or number of flavours allows reduction to IIA. Superpotential terms for flavours consistent with embedding of D6 in IIA.

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

W = A1B1A2B2 − A1B2A2B1 + pA1q

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

Half-integral shifts of CS levels (parity anomaly)

Chiral flavours: complex representations of unitary gauge groups. No continuous gauge anomalies in 3d. But Z2 gauge anomaly of fermion determinant, cancelled by half-integral bare CS levels: ki + 1 2

  • ψ
  • gi[ψ]

2 ∈ Z Couple a flavour pair (pα, qα) to bifundamental Xα: δki = ±1 2 gi[Xα] Couple hα flavour pairs to bifundamental Xα: δki = hα 2 − γα

  • gi[Xα] ,

with an integer γα ∈ [0, hα].

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

Real masses

Give real masses to flavour superfields:

  • d4θ Z †e

˜ mθ ¯ θZ

Flavour symmetry promoted to a background gauge symmetry, VEV of σF. Flavour fermion ψ w/ charge qψ under U(1) ∈ U(h) flavour group: Mψ = qψσF. Integrating massive fermions out shifts CS levels: ki → ki + 1 2

  • massiveψ
  • gi[ψ]

2 Sgn(Mψ) Decoupling a flavour pair (pα, qα) via real mass ∼ σF: δki = 1 2gi[Xα] Sgn(σF)

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

Complex masses

Coupling flavours to a string of bifundamentals Oα = n

β=1 Xβ by integrating

  • ut massive flavours (via superpotential mass term).

W = W0 + p1X1q1 + p2X2q2 − → W = W0 − 1 m p1X1X2q2 IIA: VEV for higher dimensional fields localised at intersection of D6 branes. M-theory: VEV for higher dim’l fields localised at non-isolated singularities.

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

’t Hooft monopole operators

Puncture in Euclidean theory, inserting quantised magnetic flux 1 2π

  • S2 F = n

CS: monopole operator acquires electric charge: S ⊃ k 4π

  • A ∧ dA = k n
  • A0 dt

Diagonal monopole operators T (n)

  • Magnetic flux n in each U(1)
  • Electric charges from CS terms: (nk1, nk2, . . . , nkG)
  • BPS (with suitable background of σ)

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

’t Hooft monopole operators (II)

Monopole charges under any U(1) in the theory are quantum corrected by fermionic (zero) modes:

[Borokhov, Kapustin, Wu 02; Gaiotto, Witten 08; Benna, Klebanov, Klose 09]

δQ[T (n)] = −|n| 2

  • ψ Q[ψ]

Toric quiver gauge theories w/o flavours: quantum corrections cancel. Flavouring introduces quantum corrections to charges: δQ[T (n)] = |n| 2

  • α hαQ[Xα]

Conjectured quantum relation in the chiral ring T (n) T (−n) =

α X hα α

|n| We will use unit flux monopole operators: T (1) ≡ T, T (−1) ≡ ˜ T.

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

Geometric moduli space, chiral ring

Use T, ˜ T and bifundamentals to construct gauge invariants under U(1)G. Toric quiver gauge theories without flavours ˜ Z = {Xα, T, ˜ T | dW0 = 0, T ˜ T = 1} ⊂ CE+2 , M = ˜ Z U(1)G−1 . Gauge charges of monopoles: gi[T] = −gi[˜ T] = ki. Flavoured toric quiver gauge theories Branch: pα = qα = 0 ∀ α. Same classical F-terms, different quantum relation. ˜ Zflav = {Xα, T, ˜ T | dW0 = 0, T ˜ T =

  • α X hα

α } ⊂ CE+2 ,

Mflav = ˜ Zflav U(1)G−1 . Gauge charges of monopoles: gi[T (±1)] = ±ki + 1

2

  • α hαgi[Xα]

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

Finding the toric diagram

Mmes

4d

= {Xα|dW0 = 0} U(1)G−1 toric CY3 cone Mflav = {Xα, T, ˜ T | dW0 = 0, T ˜ T =

α X hα α } U(1)G−1

toric CY4 cone Construction of toric diagram of Mflav: take the 3d toric diagram of the quiver theory without flavours add columns of points above and below the original points according to CS levels and number of flavours. Xα =

c

  • i=1

tPαi

i

, α = 1, . . . , E ti: perfect matchings (points in original toric diagram) Pαi: perfect matching matrix

  • column of
  • α(hα − γα)Pαi

points above ti

  • column of
  • α γαPαi

points below ti (up to multiplicities)

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

Finding the toric diagram (II)

Trick to solve for Mflav: consider a different (auxiliary) theory whose geometric moduli space is a toric CY4 cone that can be found algorithmically prove that such a geometric moduli space is the same as Mflav For hα flavours (pα, qα) coupled to Xα ≡ Xij: Contributions of (Ci1, C12, . . . , Chαj) to CS levels (via incidence matrix): (nij − γα, nij − γα + 1, . . . , nij − γα + hα) or permutation thereof.

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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New results Top-down: from M-theory to IIA Bottom-up: from toric flavoured quiver gauge theories to toric CY4 cones

Flavours vs. multibonds

Are they dual? Same CY4, same CY4/Zq: not geometric dual (ordinary mirror symmetry) If there is a duality, local in quiver/tiling Reduction to IIA leads to D6-branes (→ flavoured quiver) Flavour symmetries in 3d gauge theories without 4d parent? “Dual ABJM”: superconformal index, supersymmetry, global symmetries?

[Choi, Lee, Song 08; Rodriguez-Gomez 09]

Other branches do not seem to match (preliminary) Further study still required.

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Examples

Part IV Examples

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Examples

Flavouring N = 8

W = Φ1[Φ2, Φ3] +

h1

  • i=1

p1,iΦ1q1,i +

h2

  • j=1

p2,jΦ2q2,j +

h3

  • l=1

p3,lΦ3q3,l . IIA: D2 at z1 = z2 = z3 = 0, hi D6 at zi = 0 in C3, all at x9 = 0. M-theory: CY4 cone T ˜ T = Φh1

1 Φh2 2 Φh3 3

in C5

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Examples

Flavouring N = 8: (h1, h2, h3) = (h, 0, 0)

C2/Zh × C2: T ˜ T = Φh

1

Φ2 Φ3 ‘Dual ABJM’ is nothing but the D2-D6 system of Cherkis, Hashimoto 02: when h = 1, mirror symmetric to ABJM at level 1.

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Examples

Flavouring N = 8: (h1, h2, h3) = (1, 1, 0)

C × C: T ˜ T = Φ1Φ2 Φ3 (h1, h2, h3) = (a, b, 0): CY4 cone is C × C(Laba).

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Examples

Flavouring ABJM at levels (1, −1)

W = A1B1A2B2 − A1B2A2B1 + pA1q CS levels ( 1

2, − 1 2): C × C

CS levels ( 3

2, − 3 2): C(Y 2,1(CP2))

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Examples

Flavouring ABJM at levels (1, −1)

W = A1B1A2B2 − A1B2A2B1+ + p1A1q1 + p2A2q2 CS levels (0, 0): C(Q1,1,1)

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Conclusions

Part V Conclusions

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones

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Conclusions

Conclusions

Summary: Added flavours to 3d N = 2 toric quiver gauge theories New gauge theories for M2-branes at toric CY4 singularities Consistent with D6 embedding in KK reduction of M-theory to IIA Conceptual way of generating CS terms Future directions: Higgs branches and 3d mirror symmetry Non-abelian moduli space Romans mass

[Gaiotto-Tomasiello 09]

Other degenerations of U(1)M

Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY4 cones