Takahiro Nishinaka ( Ritsumeikan U. ) 1. Review of [ Beem, Lemos, - PowerPoint PPT Presentation

Chiral algebras for 4d superconformal field theories N ≥ 2 Takahiro Nishinaka ( Ritsumeikan U. ) 1. Review of [ Beem, Lemos, Liendo, Peelaers, Rastelli, van Rees ] arXiv: 1312.5344 thanks to : Matt Buican, Jaewang Choi, Kazuki Kiyoshige, Zoltan Laczko, Hironori Mori, Sanefumi Moriyama, Yuji Tachikawa, Seiji Terashima, Ruidong Zhu w/ Matt Buican, Zoltan Laczko 2. The last part is based on arXiv: 1706.03797 ( Queen Mary )

OPEs in 4d CFTs X O 1 ( x ) O 2 (0) = c 12 k ( x ) O k (0) k Q: What do we know about the OPEs when we have SUSY.

Introduction ( ) ⇥ ⇤ [ S α , O ] = = 0 S ˙ α , O Let us focus on chiral primary operators such that O or ⇥ ⇤ Φ = O + θψ + θ 2 F Q ˙ α , O = 0 The OPE of two chiral primaries is non-singular + O 1 ( x ) O 2 (0) = O 3 (0) chiral We can safely set to get x = 0 O 1 (0) O 2 (0) = O 3 (0) ( chiral ring )

⊃ Super Conformal Field Theory ∀ 4d N=1 SCFT O 1 (0) O 2 (0) = O 3 (0) ( 0d OPEs ) This is useful to study SUSY vacua of the theory.

⊃ x 3 , 4 ’13 [ Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees ] z 0 ∃ ∀ 2d chiral algebra 4d N=2 SCFT ( Vertex Operator Algebra ) Since here is a coordinate dependence, this algebra captures more than the SUSY vacua of the theory.

⊃ Goal of this talk ’13 [ Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees ] 1. I will review… • 4d N=2 SCFTs 2d chiral algebras Virasoro algebra ( ) c < 0 = • Its character 4d superconformal index 2. I will also talk about our recent work. ( exotic 4d N=2 SCFT whose detail is totally unclear ) ’17 [ Buican - Laczko - TN ]

Outline 6 slides 1. 2d chiral algebra 7 slides 2. Examples 3. What’s still to be understood 3 slides 4. Our recent work 7 slides

Outline 6 slides 1. 2d chiral algebra 7 slides 2. Examples 3. What’s still to be understood 3 slides 4. Our recent work 7 slides

2d chiral algebra N=2 supercharges I ˙ ± ± , S Q I 4d N=2 SCFT ± , Q I ˙ ± , S I I = 1 , 2 2 ˙ Schur operators : annihilated by 2 Q 1 − , S 1 − , S − O − , Q ˙ ∆ − ( j 1 + j 2 ) − 2 R = 0 scaling dim. su(2) R charge so(4) spins 2 ˙ {Q , Q † } = ∆ − ( j 1 + j 2 ) − 2 R Q ≡ Q 1 − + S − ⇒ = = Schur ops. Ker � Ker / Im Q {Q , Q † } Q ' cohomology

2d chiral algebra ∆ − ( j 1 + j 2 ) − 2 R = 0 Schur operators scaling dim. su(2) R charge so(4) spins e.g. ) j 1 = j 2 = 1 SU(2) R current J µ σ µ ∆ = 3 , 2 , R = 1 + ˙ + Higgs branch op. O ∆ = 2 R, j 1 = j 2 = 0 j 1 = j 2 = 1 σ µ derivative + ∂ µ ∆ = 1 , + ˙ 2 The spectrum of Schur operators is generally highly non-trivial.

2d chiral algebra ’13 [Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees] x 3 , 4 z = x 1 + ix 2 4d 2d x 3 = x 4 = 0 O 2 (0) O 1 ( z, ¯ z ) Schur Schur Twisted translation z b z b L − 1 O 1 (0) e − zL − 1 − ¯ z ) ≡ e zL − 1 +¯ L − 1 O 1 ( z, ¯ SU(2) R lowering op. ⇥ ⇤ Q , L − 1 = 0 L − 1 ≡ P 1 − iP 2 � ⇒ = 1 b Q , Q L − 1 = b ˙ L − 1 ≡ P 1 + iP 2 + R − − ( -exact ) Q ✓ − ◆ 2 ˙ Q ≡ Q 1 − + S

2d chiral algebra ’13 [Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees] z b z b L − 1 O 1 (0) e − zL − 1 − ¯ z ) ≡ e zL − 1 +¯ L − 1 4d 2d O 1 ( z, ¯ -exact Q Then the 4d OPE implies the following “2d OPE” + c 12 k X -exact O 1 ( z, ¯ z ) O 2 (0 , 0) = z h 1 + h 2 − h k O k (0 , 0) Q k ( ) h = ∆ − R In the sense of the -cohomology, Q c 12 k 2d chiral algebra X O 1 ( z ) O 2 (0) = z h 1 + h 2 − h k O k (0) k

2d chiral algebra ’13 [Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees] Virasoro sub-algebra x 3 , 4 SU(2) R current + ≡ J µ σ µ J + ˙ + ˙ + z 0 z b z b T ( z ) ≡ e zL − 1 +¯ L − 1 J + ˙ + (0) e − zL − 1 − ¯ L − 1 4 ⇡ 4 � IJ x 2 g µ ν − 2 x µ x ν ⇡ 2 ✏ IJK x µ x ν x ρ · J ρ ν (0) ∼ − 3 c 4d + 2 i K (0) 4d J I µ ( x ) J J + · · · x 8 x 6 T ( z ) T (0) ∼ − 6 c 4d + 2 T (0) 2d + · · · z 4 z 2 Virasoro algebra w/ c 2d = − 12 c 4d < 0

2d chiral algebra 4d Schur op. O k twisted translation O ( z ) ∆ = j 1 + j 2 + 2 R cohomology h = ∆ − R 4d 2d SU(2) R current stress tensor T ( z ) J + ˙ + SUSY flavor G F affine G F current J A M A J A ( z ) current µ Higgs branch Virasoro primary O ( z ) O operators

Outline 1. 2d chiral algebra 7 slides 2. Examples 3. What’s still to be understood 3 slides 4. Our recent work 7 slides

Examples ’13 [ Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees ] Schur ! χ free hypermultiplet q SU(2) R doublet φ q φ † ψ 1 1 4d q † ( x ) q (0) ∼ φ † ( x ) φ (0) ∼ x 2 , x 2

Examples ’13 [ Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees ] Schur ! χ free hypermultiplet q SU(2) R doublet φ q φ † ψ 1 1 x 3 , 4 4d q † ( x ) q (0) ∼ φ † ( x ) φ (0) ∼ z , z ¯ z ¯ z z 2d q twisted ( z ) = q ( z, ¯ z φ † ( z, ¯ z ) + ¯ z ) q twisted ( z ) φ (0) ∼ 1 z ( 2d symplectic boson ) T = 1 2( q ∂φ − φ∂ q )

The same analysis for a free vector mult. is easy. But, for interacting theories, it is not straightforward to identify the corresponding 2d chiral algebra. Many guessworks have been done by using the equivalence: 4d 2d = Tr 4d local ops. ( − 1) F q ∆ − R Tr chiral alg. ( − 1) F q L 0 ( superconformal index ) ( character of chiral alg. )

Examples ’95 [ Argyres - Douglas ] ’95 [ Argyres - Plesser - Seiberg -Witten ] ’96 [ Eguchi - Hori - Ito - Yang ] H 0 Argyres-Douglas theory massless monopole, dyon H 0 Argyres-Douglas N=2 pure SU(3) ( N=2 SCFT ) deep IR Tr 4d local ops. ( − 1) F q ∆ − R ’15 [ Cordova - Shao ] = 1 + q 2 + q 3 + q 4 + q 5 + 2 q 6 + 2 q 7 + 3 q 8 + 3 q 9 + 4 q 10 + 4 q 11 + 6 q 12 + · · · q − 27 ✓ ◆ (20 ` − 3)2 (20 ` +7)2 120 X = 40 40 q − q Q ∞ n =1 (1 − q n ) ` ∈ Z identical to the character of Virasoro algebra w/ c 2d = − 22 ( = − 12 c 4d ) 5

Examples H 0 Argyres-Douglas theory ’15 [ Cordova - Shao ] This result strongly suggests that, for this AD theory, c 2d = − 22 = 2d chiral algebra Virasoro algebra w/ 5 This immediately implies the absence of fermionic Schur ops. Moreover, Hall-Littlewood 2 ˙ annihilated by 2 Q 1 − , S 1 − , S − O − , Q operators : ˙ 2 ˙ 2 + Q S ˙ + When mapped to 2d, they cannot be generated by T ( z )

Examples H 0 Argyres-Douglas theory ’15 [ Cordova - Shao ] This result strongly suggests that, for this AD theory, c 2d = − 22 = 2d chiral algebra Virasoro algebra w/ 5 This immediately implies the absence of fermionic Schur ops. Moreover, Hall-Littlewood 2 ˙ annihilated by 2 Q 1 − , S 1 − , S − O − , Q operators : ˙ Absence of such ops in this thy! 2 ˙ 2 + Q S ˙ + When mapped to 2d, they cannot be generated by T ( z )

We can learn much about the spectrum of 4d Schur ops. from the 2d chiral algebra. It is more powerful than the superconformal index. Combining it w/ other 4d data will give us more information. + e.g.) 2d stress tensor correlator 4d fusion rules ’15 [ Liendo - Ramirez - Seo ] new 4d unitarity bound ⇒ = c 4 d ≥ 11 c 2d ≤ − 22 ⇐ ⇒ 30 5 ( for interacting N=2 SCFTs )

Examples ’13 [ Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees ] ’16 [ TN - Tachikawa ] N > 2 SCFT ’16 [ Lemos, Liendo, Meneghelli, Mitev ] ✓ − ◆ = 2 ˙ � Schur ops. Ker {Q , Q † } Q ≡ Q 1 − + S • 4d N=2 superconformal algebra (SCA) has no supercharge that commutes w/ . {Q , Q † } 2d N=0 chiral alg. ⇒ = • If you have N>2 SCA, there are such supercharges : 2d N=2 4d N=3 SCFT 4d N=4 SCFT 2d N=4

In summary, • For interacting theories, the chiral algebra can be guessed w/ help of = Tr chiral alg. ( − 1) F q L 0 Tr 4d local ops. ( − 1) F q ∆ − R ( superconformal index ) ( character of chiral alg. ) e.g.) SU(2) N f = 4, MN’s E 6 , E 7 , E 8 , AD theories, … • 2d chiral algebra tells much about the spectrum of Schur ops. and more.

Outline 1. 2d chiral algebra 2. Examples 3. What’s still to be understood 3 slides 4. Our recent work 7 slides

The full set of relevant chiral algebras? • Not all 2d chiral algebras are related to 4d ∃ ∀ 2d chiral algebra 4d N=2 SCFT ( The converse is not true. ) • Which class of 2d chiral algebras is related to 4d N=2 SCFTs? c 2d ≤ − 22 ( if the 4d is interacting ) c 2d < 0 5 There are perhaps more constraints, which are NOT fully understood.

The full set of relevant chiral algebras? ( 2 weeks ago ) ’17 [ Beem - Rastelli ] Recent conjecture operators involving { { chiral algebra V ⊃ C 2 ( V ) ≡ a derivative 4d Higgs branch ⇣ ⌘ . { nilpotent elements { V/C 2 ( V ) ' chiral ring conjecture [ Zhu ] [ Arakawa ] This immediately implies that T k + ϕ is null. ( ) ∃ k > 0 , ϕ ∈ C 2 ( V )

∃ ∀ 2d chiral algebra 4d N=2 SCFT

∀ 2d chiral algebra ∀ 4d N=2 SCFT in a class not established yet We might be able to classify 4d N=2 SCFT in terms of 2d chiral algebras.

Outline 1. 2d chiral algebra 2. Examples 3. What’s still to be understood 4. Our recent work 7 slides