Free field constructions of VOAs
from the Higgs branch

Leonardo Rastelli
Yang Institute for Theoretical Physics, Stony Brook
Based on upcoming work(s) withC. Beem, F. Bonetti and C. Meneghelli.

String Math, Sendai
June 21 2018

To any N “ 2 SCFT, one can canonically associate a vertex operator algebraχ : 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 algebraof 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 the precise connection has so far been elusive.

Leonardo RastelliFree field constructions of VOAs from the Higgs branch

The category of VOAs is unwieldy. No classification remotely in sight.
VOAs that arise from 4d SCFTs are however very special.
Today, I will describe a striking property that holds 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 may lead to a tractable classification program.
Our free field constructions have mathematical interest in their own right.
They vastly generalize examples found by Adamovic.

Leonardo RastelliFree field constructions of VOAs from the Higgs branchWe are interested in N “ 2 4d SCFTs from the viewpoint of the OPE algebraof their local operators,
Oi pxqOj pyq “ÿckij
Ok pyq ,Ei `Ej ´Ek|x ´ y|
kx , y P R4

tOi pxqu can be organized in representation of the sup2, 2|2q superalgebra.
Bosonic subalgebra: sup2, 2q ˆ sup2qR ˆ up1qr .
sup2, 2q – sop4, 2q: conformal algebra in d “ 4,sup2qR ˆ up1qr R-symmetry
Fermionic generators: Poincaré supercharges QIα , Q̃I α9Conformal supercharges SIα , S̃ I α9We label operators by the Cartan quantum numbers pE, R, r, j1 , j2 q.

Leonardo RastelliFree field constructions of VOAs from the Higgs branch

While for operators in generic representations the OPE algebra is extremelycomplicated, simplifications occur for operators in shortened representations.
Some interesting commutative subalgebras are obtained by considering suitablecohomologies with respect to Poincaré supercharges:§Coulomb chiral ring, built with operators with E “ r§Higgs chiral ring, built with operators with E “ 2R§Hall-Littlewood chiral ring, built with operators with E “ 2R ` r.

A much richer non-commutative subalgebra is defined by passing to the9́
cohomology of Q “ Q1´ ` S̃ 2 .
It has the structure of a vertex operator algebra (VOA).

Leonardo RastelliFree field constructions of VOAs from the Higgs branch

VOA basics

Informally, a VOA is a meromorphic OPE algebra in two dimensions,O1 pzqO2 pwq „kc12Ok pwq
h1 `h2 ´hkpz´wqk

ÿEach operator can be expanded in a Laurent series,
Opzq “8ÿz ´n´h On ,On P EndpVqn“´8The modes are conserved charges acting on the state space V.
V is identified with the space tOi p0qu of local operators at the origin,
B n Op0q ô O´n´h |Ωy .

Leonardo RastelliFree field constructions of VOAs from the Higgs branch

The normal ordered product of operators a and b is defined at the origin asNOpa, bqp0q :“ a´ha b´hb |Ωy .
The bracket t, u is defined by picking out the simple pole in the OPE¿dzta, bupwq “apzqbpwq
2πiA strong generator of a VOA is an slp2q primary that cannot be written asN.O. product. All operators of a VOA can then be written as N.O. products ofthe strong generators and their holomorphic derivatives.

Leonardo RastelliFree field constructions of VOAs from the Higgs branchSchur operators
Our state space V is the space of Schur operators of the SCFT, defined as theoperators that satisfy
R“E ´ j1 ´ j2,
2r “ j2 ´ j1

ñ L0 “ h “ E ´ R ,where E is the conformal dimension and j1 , j2 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 c4d {2 STrHrS 3 s pq E´R q “ STrV pq L0 ´c2d {24 q
Leonardo RastelliFree field constructions of VOAs from the Higgs branch

Away from the origin, Q -closed operators are “twisted-translated”,´´Opz, z̄q “ ezL´1 `z̄pL̄´1 `R q OSch p0q e´zL´1 ´z̄pL̄´1 `R q .
In terms of a basis OpI1 ...I2R q of the sup2qR representation,Opz, z̄q “ Op`¨¨¨`q pz, z̄q ` z̄Op`¨¨¨´q pz, z̄q ` ¨ ¨ ¨ ` z̄ 2R Op´¨¨¨´q pz, z̄q .
The z̄ dependence is in fact exact at the level of Q cohomology,
so correlators are meromorphic.
The holomorphic conformal weight ish “ E ´ R “ R ` j1 ` j2
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.

Leonardo RastelliFree field constructions of VOAs from the Higgs branch

Structural properties§

Both Bose and Fermi statistics are allowed: VO super algebra.

§The VOA is either Z graded or Z{2 graded by L0 , but no spin-statistics§Virasoro enhancement of slp2q, with c2d “ ´12 c4d ,where c4d is one of the conformal anomaly coefficient.

§Affine enhancement of global flavor symmetry, with k2d “ ´ k24d .

§MHiggs “ XV (still a conjecture) Beem LR
MHiggs = Higgs branch of the 4d SCFT, XV = associated variety of V.
In particular, the associated variety must be symplectic. This has deep
consequences. E.g., modular behavior of the Schur index.
See C. Beem’s talk at String Math 2017.

Leonardo RastelliFree field constructions of VOAs from the Higgs branch

The associated variety XVArakawa

There is a natural commutative algebra associated to V,RV :“ V{C2 rVs
C2 rVs :“ Spanta´ha ´1 b , a , b P Vu .
Normal ordering descends to a commutative product on RV .
In words: to determine RV , remove states containing holomorphic derivatives.
For example, for an affine current algebra at generic level k,
A1
An
RV “ SpantJ´1. . . J´1|Ωyu ,Ai.e., the polynomial ring CrA s in the ”zero-modes” A ” J´1|Ωy.

RV is also endowed with a Poisson structure induced by the bracket t, u.
XV :“ MaxSpec RV

Leonardo RastelliFree field constructions of VOAs from the Higgs branch

Example: H1 Argyres-Douglas and slp2q´4{3Beem Meneghelli LR, to appear

The H1 SCFT can be realized by a single D3 brane probing the A1 singularity.
Its Higgs branch is the one-instanton moduli space for g “ slp2q,ĘMHiggs “ Omin pgq :“ G-orbit of the highest root of g .
Writing
M “ˆ 3´`´3˙ˆP slp2q ,M2 “p3 q2 ` ` ´00p3 q2 ` ` ´˙we recognize
MHiggs “ Spanr3 , ` , ´ s{xp3 q2 ` ` ´ y – C2 {Z2 .
Symplectic holomorphic structure encoded in the bracketst` , ´ u “ 23 ,t3 , ` u “ ` ,Leonardo Rastellit3 , ´ u “ ´´ .

Free field constructions of VOAs from the Higgs branch!

“ 0The VOA associated to the H1 SCFT is the affine current algebra slp2q´4{3 .
The generators are J A , A “ 1, 2, 3, withJ A pzqJ B pwq „kδ AB
ABC J C pwq`,2pz ´ wqz´wk“´43For 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 qand thus
RV “ Cr1 , 2 , 3 s{xA p1 1 ` 2 2 ` 3 3 qy .
Note that¸2˜ÿ A A
 “0in RV ,Aso RV is not reduced. The nilradical is simply the quadratic Casimir, sopRV qred “ Cr1 , 2 , 3 s{x1 1 ` 2 2 ` 3 3 y .
Finally, the associated variety
XV :“ MaxSpecpRV q “ Spanr1 , 2 , 3 s{x1 1 ` 2 2 ` 3 3 y – MHiggs .
Leonardo RastelliFree field constructions of VOAs from the Higgs branchWith hindsight, we can retrace the above steps and recover the VOA by an“affine uplift” of XV – MHiggs .
The starting point isMHiggs “ Spanr3 , ` , ´ s{xp3 q2 ` ` ´ y .
Let’s solve the relation in the open patch ` ‰ 0,
´ “ ´p3 q2.
`The symplectic structure is then encoded in the single bracket t3 , ` u “ ` .
Define the affine uplift in terms of two chiral bosons ϕ and δ:
3 Ñ J 3 “1Bϕ ,xϕ, ϕy` Ñ J ` “ eδ`ϕt3 , ` u “ ` ÝÑ J 3 pzqJ ` p0q „Leonardo RastelliJ ` p0qz

Free field constructions of VOAs from the Higgs brancht` , j ` u “ 0 Ñ J ` J ` „ reg ñ xδ, δy “ ´xϕ, ϕy
For j ´ Ñ J ´ , consider the ansatz
J ´ “ pc1 pBδq2 ` c2 pBϕq2 q e´δ´ϕc1 and c2 are uniquely fixed, and finally J ´ J ´ „ reg only if
14k P t´2, ´ , ´ u .
23This 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 slp2qk , which needs three.

Leonardo RastelliFree field constructions of VOAs from the Higgs branchPhysical interpretation: EFT on the Higgs branchThis is the first example of a seemingly universal phenomenon:
free field construction mimics the EFT description at a generic point of MHiggs .
In the H1 SCFT, the low energy degrees of freedom on the Higgs branch
consist of a single free hyper (= two half hypers), in correspondence with thetwo 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.

Leonardo RastelliFree field constructions of VOAs from the Higgs branchFree field construction of Deligne-Cvitanović series
Beem Meneghelli LR, to appear

slp2q´4{3 generalizes to a remarkable sequence of AKM algebras ĝk ,_with g P tA1 , A2 , G2 , D4 , F4 , E6 , E7 , E8 u and k “ ´ h6 ´ 1 .
[See Harvey’s talk for an appearance of the same series, but at k “ 1]With the exceptions of G2 and F4 (whose 4d interpretation in unclear), theseare the VOAs are associated to the rank one SCFTs that arise on a single D3
brane probing the corresponding F-theory singularity.
ĘMHiggs “ O
min pgq :“ G-orbit of eθ (highest root)ĘA calculation shows that XV “ O
min pgq – MHiggs . Arakawa Moreau

Leonardo RastelliFree field constructions of VOAs from the Higgs branchLet slθ p2q “ xeθ , fθ , hθ y Ă g the slp2q triple associated to the highest root.
Decomposing g with respect to slθ p2q and its commutant g6 ,´¯´¯g “ g6 ‘ sl2 ‘ pR, 2q “ Cfθ ‘ R´ ‘ g6 ‘ Chθ ‘ R` ‘ Ceθ .

gg6

Rh_kA1A2G2D4F4E6E7E8´CA1A1 ‘ A1 ‘ A1
C3A5D6E7´1` ‘ 1´
4p2, 2, 2q
141
2032
56

23469121830

´ 43
´ 32
´ 53
´2
´ 52
´3
´4
´6”Broken” generators (that don’t commute with eθ ): fθ , hθ , R´
Hence dimC pOmin q “ 2 ` dimpRq “ 2ph_ ´ 1q .
R is a pseudoreal representation of dimension 2ph_ ´ 2q.
The commutant g6 Ă sppRq.
Leonardo RastelliFree field constructions of VOAs from the Higgs branchBasic idea: introduce δ, ϕ to construct slθ p2q, as before
To capture Goldstones in R, introduce a system Vξ of symplectic bosons,ΩAB,z12ξA pz1 qξB pz2 q „A, B “ 1, . . . h_ ´ 2 .

As before, hθ „ Bϕ.
{ 1 Ă Vξ .
Affine subalgebra pg6k6 Ă sppRq´2“Easy” generators are the highest weights,eθ “ eδ`ϕ ,δ`ϕeA “ ξ A e 2 .

Lowest weights more involved. For example,´¯B 2 δ e´pδ`ϕqfθ pzq “ T 6 ´ p k2 Bδq2 ` kpk`1q2
where the extra term T 6 P Vξ .

Leonardo RastelliFree field constructions of VOAs from the Higgs branchThis is again a free field construction of the simple quotient.
For g ‰ A1 , 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 freemassless hypers. In this class of examples, they are just the Goldstone bosonsof the broken flavor symmetry.
r “ R ` hθ .
In the (free) infrared SCFT, new assignment of R-charge, RThis explains the shifted scaling dimension (=1/2) of the ξA .

Leonardo RastelliFree field constructions of VOAs from the Higgs branchN “ 4 super Yang-Mills: unbroken U p1qs

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.
MHiggs “C2 b Vg,Weylg

Vg » Rrankpgq .

At a generic point on the Higgs branch, an unbroken U p1q 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 Weylg .
Beem Lemos Liendo Peelaers LR van Rees

Leonardo RastelliFree field constructions of VOAs from the Higgs branch

For g “ sup2q, the VOA is just VirN “4 with c “ ´9.
MHiggs “ C2 {Z2 .
We expect, and find, a construction of the simple quotient in terms of:
the usual ϕ, δ ‘ additional fermionic fields λ, λ̃ for the unbroken U p1q,
λpz1 qλ̃pz2 q „12z12

The slp2q´3{2 subalgebra is realized as in the examples above, with T 6 “ Tλλ̃ ,1while the fermionic generators contain exponentials e˘ 2 pδ`ϕq .
Closely related (by field redefinition) to a construction of Adamovic:
Virc“´91 q.
,´ 2
N “4 realized in terms of pβ, γqp1,0q and pb, cqp 3
2
All generators are written as N.O. products, with no exponentials.
Notably, J ` “ β, G` “ b.

Leonardo RastelliFree field constructions of VOAs from the Higgs branch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` q “ p` {2,hpb` q “ pp` ` 1q{2, where tp` u are the degrees of the fundamental invariants.

´ 3c2n ´1X

p1 , . . . , prank

An´1
BnDn
E6E7
E8
F42, 3, . . . , n2, 4, . . . , 2n2, 4, . . . , 2pn ´ 1q; n2, 5, 6, 8, 9, 12
2, 6, 8, 10, 12, 14, 18
2, 8, 12, 14, 18, 20, 24, 30
2, 6, 8, 12

np2n ` 1q
np2n ´ 1q7813324852

H3
H4I2 ppq

2, 6, 10
2, 12, 20, 302, 2p33
1242pp ` 1q

Curiously, the construction extends from Weyl to general Coxeter groups.
There is no obvious 4d interpretation of the non-crystallographic cases H3 , H4 , I2 ppq.

Leonardo RastelliFree field constructions of VOAs from the Higgs branch

Residual Interacting SCFTs on the Higgs branch
Another interesting generalization is to SCFTs whose Higgs EFT cannot be
described purely in terms of free fields.
The simplest examples are the Argyres-Doublas theories ADpA1 ,D2n`1 q .
χrADpA1 ,D2n`1 q s “ slp2q ´4n2n`1

For each n,MHiggs “ XV “ C2 {Z2 .
For n “ 1, we recover the H1 SCFT, studied in detail above.
For n ą 1, the Higgs EFT includes, in addition to one free hyper,
an interacting SCFT, ADpA1 ,A2n´2 q . One has χrADpA1 ,A2n´2 q s “ Vir2,2n`1 .
We then expect to be able to construct slp2q ´4n in terms of:2n`1

the usual δ, ϕ ‘ an additional ”abstract” Vir2,2n`1 .
In particular, the correction term in fθ should be T 6 “ TVir .
Precisely such a construction was found by Adamovic.
Leonardo RastelliFree field constructions of VOAs from the Higgs branch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 thestandard Drinfeld-Sokolov reduction.
Standard DS reduction imposes a quantum constrain Jα´ “ 1 on an affine
current associated to a negative root.
We have also found examples of free field constructions that are “inverting” amore general reduction, where the operators set to 1 are not affine currents.
A nice example is N “ 2 Super QCD, i.e., SU pN q gauge theory with 2N
fundamental hypers. The operators set to 1 are the baryons.

Leonardo RastelliFree field constructions of VOAs from the Higgs branch

The R-filtration

All these free field constructions come equipped with a natural 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 with E “ 2R) and what doesn’t.
For example, for N “ 4 SYM, we assign the “F-weights”
β`γ`b`c`1p
2 `1 ´ 12 p`

1p2 `1 ´ 12 p`

F of a polynomial is the maximum F of its monomials.
In examples, refining the character by F reproduces the Macdonald index.

Leonardo RastelliFree field constructions of VOAs from the Higgs branchGeneral picture

Emerging picture:
The VOAs associated to 4d SCFTs can be “built” in terms of “irreducible”
C2 -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 C2 -cofinite blocks cannot be arbitrary, because 4d unitarity
is very restrictive.
For example, if one assumes full VOA = Virasoro:
are Virp2, 2k ` 1q the only models that can arise from unitary 4d theories?
Towards a classification program.

Leonardo RastelliFree field constructions of VOAs from the Higgs branch