The Chiral Algebra Program for 4d SCFTs

Transcription

1 Yang Institute for Theoretical Physics, Stony Brook Based on past work with Beem, Lemos, Liendo, Peelaers, van Rees and upcoming work with Beem, Bonetti and Meneghelli. IGST, Copenhagen August

2 To any N 2 SCFT, one can canonically associate a vertex operator algebra χ : 4d N 2 SCFT ÝÑ VOA Beem Lemos Liendo Peelaers LR van Rees The associated VOA V is a closed subalgebra of the full local operator algebra of the 4d theory. It has many uses: Exact results in 4d SCFTs. Predictions for VOAs. Organizing principle for the whole N 2 SCFT landscape. The VOA is a very rich invariant of the 4d SCFT. It appears to be deeply connected with the physics of the Higgs branch, but precisely how has so far been elusive.

3 Lightening review of N 2 SCFTs Conventional Lagrangian theories specified by the following data: Gauge group G G 1 ˆ G 2 ˆ... G k Vector multiplets pφ, λ I α, A µq in Adj of G. A pseudoreal finite-dimensional representation ρ of G. Half-hypermultiplets pq, ψ αq in ρ of G. For each simple factor G i, must impose vanishing of the beta function, β 2h _ ` C 2pρq 0. Complete combinatorial classification of possibilities by Bardhway Tachikawa. Lagrangian uniquely fixed by N 2 superconformal symmetry, up to the value of complexified gauge couplings tτ iu, i 1,... k. They parametrize the conformal manifold U of the N 2 SCFT.

4 We have learnt that the landscape of N 2 SCFTs is much bigger. Exotic matter in the form of isolated SCFTs with no conventional Lagrangian description. Canonical examples: Gaiotto s T N theories. Coupling them to gauge fields (by gauging a subgroup G of their flavor symmetry group, such that β G 0) leads to large continuous families of SCFT. In all known examples, exactly marginal couplings arise from weak gauging of isolated SCFTs. Perhaps this is a general fact.

5 We are interested in N 2 4d SCFTs from the viewpoint of the OPE algebra of their local operators, O ipxqo jpyq ÿ k c k ij x y E i`e j E k O k pyq, x, y P R 4 to ipxqu can be organized in representation of the sup2, 2 2q superalgebra. Bosonic subalgebra: sup2, 2q ˆ sup2q R ˆ up1q r. sup2, 2q sop4, 2q: conformal algebra in d 4, sup2q R ˆ up1q r R-symmetry Fermionic generators: Poincaré supercharges Q I α, QI 9α Conformal supercharges SI α 9α, SI We label operators by the Cartan quantum numbers pe, R, r, j 1, j 2q. In our conventions: φ complex scalar of the vector multiplet: r 1, R 0 Q I complex scalars of the hypermultiplet: r 0, R 1{2.

6 While for operators in generic representations the OPE algebra is extremely complicated, simplifications occur for operators in shortened representations. Some interesting commutative subalgebras are obtained by considering suitable cohomologies with respect to Poincaré supercharges: Coulomb chiral ring R C, operators with E r In a Lagrangian theory, R C = ring of G-invariant polynomials. E.g. G SUpNq, R C generated by Tr φ 2, Tr φ 3... Tr φ N. Higgs chiral ring R H, operators with E 2R In a Lagrangian theory, R H spanned by gauge-invariants of Qs. Hall-Littlewood chiral ring R HL Ą R H, operators with E 2R ` j 2 In a Lagrangian theory, R HL spanned by gauge-invariants of Qs and λs. A much richer non-commutative subalgebra is defined by passing to the cohomology of Q Q 1 ` S 2 9. It has the structure of a vertex operator algebra (VOA).

7 VOA basics Informally, a VOA is a meromorphic OPE algebra in two dimensions, O 1pzqO 2pwq ÿ k c k 12O k pwq pz wq h 1`h 2 h k Each operator can be expanded in a Laurent series, 8ÿ Opzq z n h O n, n 8 O n P EndpVq The modes are conserved charges acting on the state space V. V is identified with the space to ip0qu of local operators at the origin, B n Op0q ô O n h Ωy.

8 The normal ordered product of operators a and b is defined at the origin as NOpa, bqp0q : a ha b hb Ωy. The bracket t, u is defined by picking out the simple pole in the OPE dz ta, bupwq 2πi apzqbpwq A strong generator of a VOA is an slp2q primary that cannot be written as N.O. product. All operators of a VOA can then be written as N.O. products of the strong generators and their holomorphic derivatives.

9 Schur operators Our state space V is the space of Schur operators of the SCFT, defined as the operators that satisfy R E j1 j2 2, r j 2 j 1 ñ L 0 h E R, where E is the conformal dimension and j 1, j 2 the Lorentz spins. The Schur operators comprise: Higgs chiral ring operators: E 2R, r 0 ñ h R. Hall-Littlewood (anti)-chiral ring operators: h R r. Generic Schur operators: semi-short operators with h ą R ` r. (Coulomb ring operators not Schur.) Vacuum character of VOA = Schur index of the 4d SCFT q c 4d{2 STr HrS 3 spq E R q STr Vpq L 0 c 2d {24 q

10 Away from the origin, Q -closed operators are twisted-translated, Opz, zq e zl 1` zp L 1`R q O Sch p0q e zl 1 zp L 1`R q. In terms of a basis O pi 1...I 2R q of the sup2q R representation, Opz, zq O p` `q pz, zq ` zo p` q pz, zq ` ` z 2R O p q pz, zq. The z dependence is in fact exact at the level of Q -cohomology, so correlators are meromorphic. The holomorphic conformal weight is h E R R ` j 1 ` j 2 R grading is lost. OPE compatible with a filtration (R cannot increase). VOAs that descend from 4d theories inherit the R-filtration. However, the filtration is not intrinsic to V. Given an abstract presentation of V, it is a priori unknown how to assign it.

11 Structural properties χ : 4d N 2 SCFT ÝÑ VOA χrt us is independent of exactly marginal couplings u P U. Virasoro enhancement of slp2q, with c 2d 12 c 4d, where c 4d is one of the conformal anomaly coefficient. Affine enhancement of global flavor symmetry, with k 2d k4d 2. Generators of R HLrT s are strong generators of the VOA χrt s. Above properties often enough to determine χrt s from minimal input about T. M Higgs X V (still a conjecture) Beem LR M Higgs = Higgs branch of the 4d SCFT, X V = associated variety of V. In particular, the associated variety must be symplectic. This has deep consequences. E.g., modular behavior of the Schur index.

12 The associated variety X V Arakawa There is a natural commutative algebra associated to V, R V : V{C 2rVs C 2rVs : Spanta ha 1b, a, b P Vu. Normal ordering descends to a commutative product on R V. In words: to determine R V, remove states containing holomorphic derivatives. For example, for an affine current algebra at generic level k, R V SpantJ A J An 1 Ωyu, i.e., the polynomial ring Crj A s in the zero-modes j A J Á 1 Ωy. R V is also endowed with a Poisson structure induced by the bracket t, u. X V : MaxSpec R V

13 What is the image of χ? The category of general VOAs is unwieldy. No classification remotely in sight. VOAs that arise from 4d SCFTs by our map χ are however very special. In the rest of the talk, I ll describe a striking property true in many examples: They admit free field constructions of their simple quotient. Physically, these constructions can be understood in terms of the low energy degrees of freedom on the Higgs branch. We believe this is universal fact about VOAs associated to 4d SCFT. Combined with 4d unitarity, it should lead to a tractable classification program.

14 Example: H 1 Argyres-Douglas and slp2q 4{3 Beem Meneghelli LR, to appear The H 1 SCFT can be realized by a single D3 brane probing the A 1 singularity. Its Higgs branch is the one-instanton moduli space for g slp2q, Writing ˆj3 M we recognize M Higgs Ę O min pgq : G-orbit of the highest root of g. j` j j 3 ˆpj P slp2q, M 2 3 q 2 ` j` j 0 0 pj 3 q 2 ` j` j M Higgs Spanrj 3, j`, j s{xpj 3 q 2 ` j`j y C 2 {Z 2. Symplectic holomorphic structure encoded in the brackets tj`, j u 2j 3, tj 3, j`u j`, tj 3, j u j.! 0

15 The VOA associated to the H 1 SCFT is the affine current algebra slp2q 4{3. The generators are J A, A 1, 2, 3, with J A pzqj B pwq kδab pz wq ` ɛabc J C pwq, k 4 2 z w 3 For k 4{3, maximal ideal generated by level three singular vector N A J A pj 1 J 1 ` J 2 J 2 ` J 3 J 3 q and thus Note that R V Crj 1, j 2, j 3 s{xj A pj 1 j 1 ` j 2 j 2 ` j 3 j 3 qy. ÿ A j A j A 2 0 in R V, so R V is not reduced. The nilradical is simply the quadratic Casimir, so pr Vq red Crj 1, j 2, j 3 s{xj 1 j 1 ` j 2 j 2 ` j 3 j 3 y. Finally, the associated variety X V : MaxSpecpR Vq Spanrj 1, j 2, j 3 s{xj 1 j 1 ` j 2 j 2 ` j 3 j 3 y M Higgs.

16 With hindsight, we can retrace the above steps and recover the VOA by an affine uplift of X V M Higgs. The starting point is M Higgs Spanrj 3, j`, j s{xpj 3 q 2 ` j`j y. Let s solve the relation in the open patch j` 0, j pj3 q 2 j`. The symplectic structure is then encoded in the single bracket tj 3, j`u j`. Define the affine uplift in terms of two chiral bosons ϕ and δ: j 3 Ñ J 3 1 xϕ, ϕy Bϕ, j` Ñ J ` e δ`ϕ tj 3, j`u j` ÝÑ J 3 pzqj `p0q J `p0q z

17 tj`, j`u 0 Ñ J `J ` reg ñ xδ, δy xϕ, ϕy For j Ñ J, consider the ansatz J pc 1pBδq 2 ` c 2pBϕq 2 q e δ ϕ c 1 and c 2 are uniquely fixed, and finally J J reg only if k P t 2, 1 2, 4 3 u. This is Adamovic s construction of slp2q 4{3. Remarkably, this is a construction of the simple quotient of V: all singular vectors are automatically removed (they are identically zero). Note that only two bosons are used, unlike in the standard Wakimoto construction of slp2q k, which needs three.

18 Physical interpretation: EFT on the Higgs branch This is the first example of a seemingly universal phenomenon: free field construction mimics the EFT description at a generic point of M Higgs. In the H 1 SCFT, the low energy degrees of freedom on the Higgs branch consist of a single free hyper (= two half hypers), in correspondence with the two chiral bosons ϕ and δ. We will now generalize this simple example in various directions, organized by the complexity of the EFT description: The EFT only includes hypers, but with more complicated geometry; The EFT includes vector multiplets in addition to hypermultiplets; Apart from free fields, an interacting SCFT (with trivial Higgs branch) survives at low energies.

19 Free field construction of Deligne-Cvitanović series Beem Meneghelli LR, to appear slp2q 4{3 generalizes to a remarkable sequence of AKM algebras ĝ k, with g P ta 1, A 2, G 2, D 4, F 4, E 6, E 7, E 8u and k h_ 6 1. With the exceptions of G 2 and F 4 (whose 4d interpretation in unclear), these are the VOAs are associated to the rank one SCFTs that arise on a single D3 brane probing the corresponding F-theory singularity. M Higgs Ę O min pgq : G-orbit of e θ (highest root) A calculation shows that X V Ę O min pgq M Higgs. Arakawa Moreau

20 Let sl θ p2q xe θ, f θ, h θ y Ă g the slp2q triple associated to the highest root. Decomposing g with respect to sl θ p2q and its commutant g 6, g g 6 sl 2 pr, 2q Cf θ R g 6 Ch θ R` Ce θ. g g 6 R h _ k A A 2 C 1` G 2 A D 4 A 1 A 1 A 1 p2, 2, 2q 6 2 F 4 C E 6 A E 7 D E 8 E Broken generators (that don t commute with e θ ): f θ, h θ, R Hence dim C po min q 2 ` dimprq 2ph _ 1q. R is a pseudoreal representation of dimension 2ph _ 2q. The commutant g 6 Ă spprq.

21 Basic idea: introduce δ, ϕ to construct sl θ p2q, as before To capture Goldstones in R, introduce a system V ξ of symplectic bosons, ξ Apz 1qξ Bpz 2q ΩAB z 12, A, B 1,... h _ 2. As before, h θ Bϕ. Affine subalgebra pg 6 k 6 Ă { spprq 1 2 Ă V ξ. Easy generators are the highest weights, e θ e δ`ϕ, e A ξ A e δ`ϕ 2. Lowest weights more involved. For example, f θ pzq T 6 p k 2 Bδq2 ` kpk`1q B 2 δ 2 where the extra term T 6 P V ξ. e pδ`ϕq

22 This is again a free field construction of the simple quotient. For g A 1, the maximal ideal of the Deligne-Cvitanović VOAs are generated by level two singular vectors Arakawa Moreau. Easy to see that they are identically zero in our construction. Interpretation: The low energy degrees of freedom on the Higgs branch consist purely of free massless hypers. In this class of examples, they are just the Goldstone bosons of the broken flavor symmetry. In the (free) infrared SCFT, new assignment of R-charge, r R R ` h θ. This explains the shifted scaling dimension (=1/2) of the ξ A.

23 N 4 super Yang-Mills: unbroken Up1qs The second class of generalizations is to theories whose Higgs branch EFT includes massless vector multiplets. Canonical example is N 4 SYM with gauge algebra g. M Higgs C2 b V g Weyl g, V g» R rankpgq. At a generic point on the Higgs branch, an unbroken Up1q rankpgq gauge group. The associated VOAs is a super W-algebra with N 4 small supervirasoro. Besides the supervirasoro generators, there are additional strong generators, including one short superprimary for each fundamental invariant of Weyl g. Beem Lemos Liendo Peelaers LR van Rees

24 For g sup2q, the VOA is just Vir N 4 with c 9. M Higgs C 2 {Z 2. We expect, and find, a construction of the simple quotient in terms of: the usual ϕ, δ additional fermionic fields λ, λ for the unbroken Up1q, λpz 1q λpz 2q 1 z 2 12 The slp2q 3{2 subalgebra is realized as in the examples above, with T 6 T λ λ, while the fermionic generators contain exponentials e 12 pδ`ϕq. Closely related (by field redefinition) to a construction of Adamovic: Vir c 9 N 4 realized in terms of pβ, γq p1,0q and pb, cq p 3 2, q. 12 All generators are written as N.O. products, with no exponentials. Notably, J ` β, G` b.

25 This latter construction generalizes easily to all gauge algebras, and beyond. Bonetti Meneghelli LR, to appear For general g, one uses rankpgq pβγ, bcq systems, of weights hpβ l q p l {2, hpb l q pp l ` 1q{2, where tp l u are the degrees of the fundamental invariants. X p 1,..., p rank c 3 A n 1 2, 3,..., n n 2 1 B n 2, 4,..., 2n np2n ` 1q D n 2, 4,..., 2pn 1q; n np2n 1q E 6 2, 5, 6, 8, 9, E 7 2, 6, 8, 10, 12, 14, E 8 2, 8, 12, 14, 18, 20, 24, F 4 2, 6, 8, H 3 2, 6, H 4 2, 12, 20, I 2ppq 2, 2p 2pp ` 1q Curiously, the construction extends from Weyl to general Coxeter groups. There is no obvious 4d interpretation of the non-crystallographic cases H 3, H 4, I 2 ppq.

26 Residual Interacting SCFTs on the Higgs branch Finally, generalization to SCFTs whose Higgs EFT cannot be described purely in terms of free fields. The simplest examples are the Argyres-Doublas theories AD pa1,d 2n`1 q. For each n, χrad pa1,d 2n`1 qs slp2q 4n 2n`1 M Higgs X V C 2 {Z 2. For n 1, we recover the H 1 SCFT, studied in detail above. For n ą 1, the Higgs EFT includes, in addition to one free hyper, an interacting SCFT, AD pa1,a 2n 2 q. One has χrad pa1,a 2n 2 qs Vir 2,2n`1. We then expect to be able to construct slp2q 4n in terms of: 2n`1 the usual δ, ϕ an additional abstract Vir 2,2n`1. In particular, the correction term in f θ should be T 6 T Vir. Precisely such a construction was found by Adamovic.

27 A virtually inexhaustible list of additional examples! Lagrangian theories, infinite sequences of AD theories, class S,... All constructions mentioned so far can be viewed as inversions of the standard Drinfeld-Sokolov reduction. Standard DS reduction imposes a quantum constrain J α 1 on an affine current associated to a negative root. We have also found examples of free field constructions that are inverting a more general reduction, where the operators set to 1 are not affine currents. A nice example is N 2 Super QCD, i.e., SUpNq gauge theory with 2N fundamental hypers. The operators set to 1 are the baryons.

28 The R-filtration All these free field constructions come equipped with a natural R-filtration. It appears to coincide with the R-filtration naively lost in the 4d Ñ 2d map. This is natural. The free representations keeps track of what counts as geometry (the Higgs chiral ring operators, E 2R) and what doesn t. For example, for N 4 SYM, we assign the R-weights β l γ l b l c l 1 p 2 l 1 1 p 2 l 1 p 2 l 1 1 p 2 l R of a polynomial is the maximum R of its monomials. In examples, refining the character by R R reproduces the Macdonald index.

29 General picture Emerging picture: The VOAs associated to 4d SCFTs can be built in terms of irreducible C 2-cofinite blocks free fields. This structure should arise in full generality due to the existence of an R-filtration and the constraints of 4d unitarity. Even the irreducible C 2-cofinite blocks cannot be arbitrary, because 4d unitarity is very restrictive. For example, if one assumes full VOA = Virasoro, it appears that Virp2, 2k ` 1q are the only models that can arise from unitary 4d theories! Towards a classification program.