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 Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones
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 CY 4 cones
Motivation and overview Part I Motivation and overview Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones
Motivation Motivation and overview Overview Recent progress in understanding the low energy dynamics on multiple M2-branes and AdS 4 / CFT 3 dualities. [Bagger, Lambert 07; Gustavsson 07; van Raamsdonk 08; Aharony, Bergman, Jafferis, Maldacena 08; . . . ] Gauge theories for membranes at conical singularities Freund-Rubin AdS 4 vacua of 11d supergravity (Warped) AdS 4 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 CY 4 cones
Motivation Motivation and overview Overview Aim Extend AdS 4 / CFT 3 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 AdS 4 × M 7 vacua ( M 7 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 CY 4 cones
Motivation Motivation and overview 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 CY 4 cone Top-down ( “stringy derivation” ) [Aganagic 09] (no D6) M2 at toric CY 4 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 CY 4 cones
Motivation Motivation and overview 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 X ij = 0 δ W = p ˆ ki X ij q j ˆ k Toric CY 4 cone reproduced as the Abelian geometric moduli space thanks to a quantum relation in the chiral ring: h ij T ˜ � T = X ij i , j Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones
Review Part II Review Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones
Brane tilings and toric quiver gauge theories Review “Stringy derivation” of quiver CS theories Brane tilings and 4d toric quiver gauge theories [Hanany, Kennaway 05; Franco, Hanany, Kennaway, Vegh, Wecht 05] W = A 1 B 1 A 2 B 2 − A 1 B 2 A 2 B 1 Mesonic moduli space is a toric CY 3 cone. Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones
Brane tilings and toric quiver gauge theories Review “Stringy derivation” of quiver CS theories Brane tilings and 3d toric quiver gauge theories [Hanany, Zaffaroni 08; Ueda, Yamazaki 08; Imamura, Kimura 08] W = A 1 B 1 A 2 B 2 − A 1 B 2 A 2 B 1 k = n 1 − n 2 + n 3 − n 4 Geometric moduli space is a toric CY 4 cone. Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones
Brane tilings and toric quiver gauge theories Review “Stringy derivation” of quiver CS theories Brane tilings with multiple bonds [Hanany, Vegh, Zaffaroni 08; Franco, Hanany, Park, Rodriguez-Gomez 08] W = C 13 C 32 B 1 A 2 B 2 − C 13 C 32 B 2 A 2 B 1 k 1 = n 1 − n 2 + n 3 − n 4 k 2 = n 2 − n 3 + n 4 − n 5 k 3 = − n 1 + n 5 Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones
Brane tilings and toric quiver gauge theories Review “Stringy derivation” of quiver CS theories Geometric moduli space of N = 2 quiver CS theories G U ( 1 ) G case for simplicity. � k i = 0. [Martelli, Sparks 08; Hanany, Zaffaroni 08] i = 1 Z = { X α , α = 1 , . . . , E | dW ( X ) = 0 } ⊂ C E F-flatness : | X α | 2 � G � 2 D i = k i σ i ∀ G ∀ E D-flatness : i = 1 , � g i [ X α ] σ i = 0 α = 1 , 2 π i = 1 E g i [ X α ] | X α | 2 D i ≡ � α = 1 Diagonal photon A diag ≡ � i A i dualised into a periodic scalar τ . i k i θ i , X α → e i g i [ X α ] θ i X α A i → A i + d θ i , τ → τ + 1 � Gauge : G G G Branch σ i = σ ∀ G � c i D i = 0 ∀ { c i } | � i = 1 : c i k i = 0 i = 1 i = 1 � k � 2 = � σ 1 i k 2 � i k i D i , 2 π = i � k � 2 Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones
Brane tilings and toric quiver gauge theories Review “Stringy derivation” of quiver CS theories Geometric moduli space of N = 2 quiver CS theories Geometric moduli space : M = ( Z � U ( 1 ) G − 2 ) / Z q , q = gcd { k i } Moduli spaces of 3d/4d TQGTs w/ same matter content and W : = Z � U ( 1 ) G − 1 = M 3 d � U ( 1 ) � M mes 4 d k [Jafferis, Tomasiello 08; Martelli, Sparks 08; Hanany, Zaffaroni 08] Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones
Brane tilings and toric quiver gauge theories Review “Stringy derivation” of quiver CS theories “Stringy derivation” of quiver CS theories [Aganagic 09] Toric CY 4 cone S 1 bundle over a 7-manifold, which is a toric CY 3 cone fibred over R . A Kähler parameter of CY 3 varies linearly along R . CY 3 = CY 4 � U ( 1 ) M − → quiver gauge theory on D2. ( Not quite... ) – S 1 parametrised by τ For an M2 probe: – CY 3 by mesonic full gauge invariants of the quiver (Kähler parameters are FI terms) – R parametrised by σ S 1 : M-theory circle in the reduction to IIA (after quotient). Curvature of the U ( 1 ) M bundle: RR F 2 , which induces CS terms on wv of fractional D2 probes. Fibration of CY 3 over R : scalar N = 2 partners of CS terms. Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones
New results Part III New results Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones
Top-down: from M-theory to IIA New results Bottom-up: from toric flavoured quiver gauge theories to toric CY 4 cones Singular reduction and extra objects in IIA The CY 4 � U ( 1 ) M reduction can always be done. However, if the fibre degenerates at some locus we expect extra objects in IIA. w M in Z 3 ambient space of 3d toric diagram. U ( 1 ) M : primitive vector � CY 4 � U ( 1 ) M : projection of 3d toric diagram along � w M . S 1 shrinks in CY 4 at codim C = 2 (external edge parallel to � w M in toric diagram): D6 at codim C = 3 (external face parallel to � w M in toric diagram): ? S 1 degenerates to S 1 / Z p 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 CY 4 cones
Top-down: from M-theory to IIA New results Bottom-up: from toric flavoured quiver gauge theories to toric CY 4 cones KK monopoles and D6 branes h + 1 adjacent external points in 3d toric diagram: h KK monopoles Local complex structure C 2 × C 2 / Z h 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 CY 3 : (collection of) toric divisor(s) Stefano Cremonesi 3d N = 2 flavoured quiver gauge theories and M2-branes at toric CY 4 cones
Top-down: from M-theory to IIA New results Bottom-up: from toric flavoured quiver gauge theories to toric CY 4 cones D6 branes and flavours Holomorphic embedding of D6 along collection of toric divisors in CY 3 : algebraic data . D2 on D6: massless flavour fields = ⇒ superpotential term δ W = pXq Subtlety: to a single toric divisor (or union of pairwise intersecting toric divisors) one associates Q bifundamentals, with same global charges. Z Q = π 1 (base of conical divisor) On each D6 at the conical CY 3 a Z Q -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 CY 4 cones
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