Introduction Intersecting surface defects Summary Open problems Intersecting surface defects and 2d Conformal Field Theory Yiwen Pan Uppsala University [1610.03501] Gomis, Le Floch, YP, Peelaers [1612.xxxxx] YP, Peelaers Nov 17 2016 Oviedo Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 1 / 27
Introduction Intersecting surface defects Summary Open problems Outline Introduction: class- S , CFT, partition functions, AGT Surface defects and their intersection Construction Two simplest intersecting defect systems Partition functions, correlators, dualities Higgsing Summary, conjectures Open problems Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 2 / 27
Introduction Intersecting surface defects Summary Open problems Introduction class- S theories Liouville/Toda AGT surface defect (class- S construction) AGT with surface defect Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 3 / 27
Introduction Intersecting surface defects Summary Open problems Motivations QFTs are well studied on smooth spaces ( S n , S n × S 1 , ...), spaces with boundaries ( D n , R k , . . . ) Explore QFTs on intersecting spaces, e.g., R 2 x 1 ,x 2 =0 ∪ R 2 x 3 ,x 4 =0 ⊂ R 4 , R 2 x 1 ,x 2 =0 ∩ R 2 x 3 ,x 4 =0 = (0 , 0 , 0 , 0) Enrich the family of surface defects in four dimensions Generalize AGT correspondence to include intersecting surface defects Explore new dualities Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 4 / 27
Introduction Intersecting surface defects Summary Open problems Class S of type A n f [Gaiotto] 4d N = 2 theories on M 4 Labeled by punctured Riemann surfaces Σ g,n ⇒ T g,n on M 4 M 4 unrelated to Σ Some of them in weak coupling regime ⇒ quiver gauge theories Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 5 / 27
Introduction Intersecting surface defects Summary Open problems Class S of type A n f [Gaiotto] 4d N = 2 theories on M 4 Labeled by punctured Riemann surfaces Σ g,n ⇒ T g,n on M 4 M 4 unrelated to Σ Some of them in weak coupling regime ⇒ quiver gauge theories Examples: consider Riemann spheres n f n 2 f Free hypers • n f n f n f SU ( n f ) SQCD • • n f Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 5 / 27
Introduction Intersecting surface defects Summary Open problems Partition functions Z M For QFTs T on space M who have Lagrangians Defined formally as path integral � D [fields] e − S M [fields] Z M ( T ) ≡ For some theories of class- S , simplified to ordinary integrals/sums � Z M ( T ) = � Z cl (Φ) Z 1 − loop (Φ) Z instanton (Φ) Examples will be shown later Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 6 / 27
Introduction Intersecting surface defects Summary Open problems Liouville/Toda CFT [Teschner, ‘95; Zamolodchikov, Zamolodchikov, ‘96;] Liouville theory: 2d CFT on Σ g,n ; Toda, the generalized version Depend on a param b ( ↔ central charge) Liouville ↔ W 2 ∼ V irasoro , Toda ↔ W n f Vertex operators V α ( x ) : Location: x Momentum: α Special ones: degenerate vertex op. V α deg ( x ) Pick R : irrep of su ( n f ) deg. momentum α deg ∝ Ω R Insert V α ( x ) at the punctures Correlation functions � V α 0 (0) V β 1 ( x 1 ) ...V β n ( x n ) V α 1 (1) V α ∞ ( ∞ ) � Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 7 / 27
Introduction Intersecting surface defects Summary Open problems AGT relation [Alday, Gaiotto, Tachikawa] S 4 b -partition functions (of T g,n ) = Liouville/Toda correlators (on Σ g,n ); n f Vα ∞ ( ∞ ) � � � � Z S 2 ⊂ S 4 = | x | 2 γ 0 | 1 − x | 2 γ 1 Vα 0 (0) b Vα 1 (1) • n f n f Vα ∞ ( ∞ ) � � � � Z S 2 ⊂ S 4 = | x | 2 γ 0 | 1 − x | 2 γ 1 Vα 0 (0) q b n f Vα 1 (1) Vα ( q ) • • n f Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 8 / 27
Introduction Intersecting surface defects Summary Open problems Surface defect (class- S construction) [Gomis, Le Floch; Gadde, Gukov; Gaiotto, Kim, ...] Insert degenerate puncture(s)/vertex operator(s); Labeled by a representation R of su ( n f ) with highest weight Ω R ; Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 9 / 27
Introduction Intersecting surface defects Summary Open problems Surface defect (class- S construction) [Gomis, Le Floch; Gadde, Gukov; Gaiotto, Kim, ...] Insert degenerate puncture(s)/vertex operator(s); Labeled by a representation R of su ( n f ) with highest weight Ω R ; Examples (before) n f n 2 f Free hypers • n f n f n f SU ( n f ) SQCD • • n f Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 9 / 27
Introduction Intersecting surface defects Summary Open problems Surface defect (class- S construction) [Gomis, Le Floch; Gadde, Gukov; Gaiotto, Kim, ...] Insert degenerate puncture(s)/vertex operator(s); Labeled by a representation R of su ( n f ) with highest weight Ω R ; Examples (after), R determines 2d quiver (inside dashed box) n f 2d n ν . . . n 2 f Free hypers + defect R • × n f n f 2d . . . R n f SU ( n f ) SQCD + defect × n ν • • n f Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 9 / 27
Introduction Intersecting surface defects Summary Open problems AGT relation with one defect [Gomis, Le Floch; ...] S 2 ⊂ S 4 b -partition function = Liouville/Toda deg. correlators; Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 10 / 27
Introduction Intersecting surface defects Summary Open problems AGT relation with one defect [Gomis, Le Floch; ...] S 2 ⊂ S 4 b -partition function = Liouville/Toda deg. correlators; Example: α deg = − b Ω symm n � = − nbh 1 , x ∝ e − 2 πξ FI , n f 2d, ξ FI Vα ∞ ( ∞ ) � � � � Z S 2 ⊂ S 4 = | x | 2 γ 0 | 1 − x | 2 γ 1 Vα 0 (0) b n Vα deg ( x ) Vα 1 (1) • × n f Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 10 / 27
Introduction Intersecting surface defects Summary Open problems Intersecting surface defects One surface defect (QFT construction) Intersecting surface defect (QFT construction) Couplings The example Higgsing Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 11 / 27
Introduction Intersecting surface defects Summary Open problems One surface defect: as 4d-2d system QFT Construction 4d bulk space M 4 , N = 2 theory T 4d . Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 12 / 27
Introduction Intersecting surface defects Summary Open problems One surface defect: as 4d-2d system QFT Construction 4d bulk space M 4 , N = 2 theory T 4d . 2d subspace D ⊂ M 4 , N = (2 , 2) theory T 2d . T 4d , T 2d couple supersymmetrically . Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 12 / 27
Introduction Intersecting surface defects Summary Open problems One surface defect: as 4d-2d system QFT Construction 4d bulk space M 4 , N = 2 theory T 4d . 2d subspace D ⊂ M 4 , N = (2 , 2) theory T 2d . T 4d , T 2d couple supersymmetrically . D, T 2d M 4 , T 4d Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 12 / 27
Introduction Intersecting surface defects Summary Open problems One surface defect: as 4d-2d system QFT Construction 4d bulk space M 4 , N = 2 theory T 4d . 2d subspace D ⊂ M 4 , N = (2 , 2) theory T 2d . T 4d , T 2d couple supersymmetrically . D, T 2d M 4 , T 4d Physical quantity: the partition function of the 4d-2d coupled system Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 12 / 27
Introduction Intersecting surface defects Summary Open problems Intersecting surface defects: as 4d-2d-0d system QFT Construction bulk M 4 , N = 2 theory T 4d . Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 13 / 27
Introduction Intersecting surface defects Summary Open problems Intersecting surface defects: as 4d-2d-0d system QFT Construction bulk M 4 , N = 2 theory T 4d . D L ⊂ M 4 , N = (2 , 2) theory T 2d . L D R ⊂ M 4 , N = (2 , 2) theory T 2d . R Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 13 / 27
Introduction Intersecting surface defects Summary Open problems Intersecting surface defects: as 4d-2d-0d system QFT Construction bulk M 4 , N = 2 theory T 4d . D L ⊂ M 4 , N = (2 , 2) theory T 2d . L D R ⊂ M 4 , N = (2 , 2) theory T 2d . R P ≡ D L ∩ D R ⊂ M 4 , 0d N = 2 theory T 0d (0d N = 2 Fermi or chiral). Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 13 / 27
Introduction Intersecting surface defects Summary Open problems Intersecting surface defects: as 4d-2d-0d system QFT Construction bulk M 4 , N = 2 theory T 4d . D L ⊂ M 4 , N = (2 , 2) theory T 2d . L D R ⊂ M 4 , N = (2 , 2) theory T 2d . R P ≡ D L ∩ D R ⊂ M 4 , 0d N = 2 theory T 0d (0d N = 2 Fermi or chiral). L , R and T 0d couple supersymmetrically. T 4d , T 2d Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 13 / 27
Introduction Intersecting surface defects Summary Open problems Intersecting surface defects: as 4d-2d-0d system QFT Construction D R , T 2d R P, T 0d D L , T 2d L M 4 , T 4d Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 13 / 27
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