Deformation quantizations from vertex operator algebras

In this note we address the question whether one can recover from the vertex operator algebra associated with a four-dimensional N=2 superconformal field theory the deformation quantization of the Higgs branch of vacua that appears as a protected subsector in the three-dimensional circle-reduced theory. We answer this question positively if the UV R-symmetries do not mix with accidental (topological) symmetries along the renormalization group flow from the four-dimensional theory on a circle to the three-dimensional theory. If they do mix, we still find a deformation quantization but at different values of its period.


Introduction and summary
Conformal field theories famously possess a convergent, associative operator product algebra of local operators. Unfortunately, an analytic handle on this algebraic structure appears well out of reach in dimensions larger than two. If the symmetry algebra of the theory additionally includes supersymmetry, however, one can attempt to perform a cohomological truncation with respect to one of its nilpotent supercharges and aim to study the necessarily simpler, still associative algebra of cohomology classes instead. This approach has met with great success. In the seminal paper [1], it was shown that a construction of precisely this type defines a correspondence between four-dimensional N = 2 superconformal field theories (SCFTs) and vertex operator algebras (VOAs): 1,2 V : {4d N = 2 SCFTs} −→ {VOAs} . (1.1) Similarly, three-dimensional N = 4 superconformal field theories can be mapped to topological algebras [34,35]. What's more, these topological algebras have been shown to be a deformation quantization of the ring of holomorphic functions over the Higgs branch of vacua M H [35]. 3 Three-dimensional N = 4 supersymmetry dictates that the Higgs branch is a hyperkähler cone, so we have It is important to remark that the image of DQ is a C * -equivariant even deformation quantization satisfying two additional properties. The first one states that the deformed multiplication truncates sooner than C * -equivariance demands, while the second one implements the physical condition of unitarity. The aim of this note is to investigate the following question Question 1. Can one recover from the vertex operator algebra associated with a fourdimensional N = 2 superconformal field theory T 4d the deformation quantization corresponding to the dimensional reduction T 3d of that SCFT?
An obvious first clue that this may be possible is the fact that the Higgs branch of  [21], while free-field realizations mirroring the effective field theory description of the SCFT in a Higgs vacuum allow one to reconstruct the vertex operator algebra itself [24,29]. All in all, we thus have interconnections as in figure 1, and we aim to fill in the question mark in this figure.
Our strategy to answer question 1 was suggested in our previous paper [23] and is as follows. An important entry of the SCFT/VOA dictionary states that the vacuum character of the VOA equals a particular limit of the superconformal index of the SCFT-the fr e e f i e l d c o n s t r.

DQ
? classical limit Figure 1: Interconnections between various objects associated with the four-dimensional theory T 4d and its three-dimensional dimensional reduction T 3d .
so-called Schur limit. This index can be defined as the partition function of T 4d placed supersymmetrically on a manifold with topology S 3 × S 1 . In [23,27], this equality of the torus partition function of the vertex operator algebra and the Schur index of the SCFT was made explicit using supersymmetric localization techniques. Indeed, for Lagrangian theories it was proved that the path integral can be localized to a slice of field configurations describing the VOA living on a torus T 2 ⊂ S 3 × S 1 . What's more, the computation was extended to efficiently compute torus correlation functions of fields of the vertex operator algebra. Similarly, in [36] a supersymmetric localization computation of correlation functions defining the topological algebra was performed on the three-sphere S 3 . 4 The key observation is now that, roughly speaking, upon dimensionally reducing the former computation along the circle S 1 , one lands on the latter. Moreover, although localization techniques are not available for non-Lagrangian theories, there is no a priori obstruction to formalizing this procedure and to apply it to non-Lagrangian theories as well. It amounts to, still roughly speaking, taking the high-temperature limit β → 0 of torus correlation functions of the vertex operator algebra. The qualifiers "roughly speaking" refer to various subtleties. First of all, the fields of the vertex operator are strictly larger in number than those participating in the deformation quantization. To remove the spurious fields, we accompany the hightemperature limit with a particular rescaling of the fields of the VOA. In detail, if a(z) is a field of conformal weight h a , we rescale as a(z) → β ha a(z). Second, the results on the threesphere suffer from operator mixing, which can be disentangled by a Gram-Schmidt process 4 See [37,38] for a localization computation of the deformation quantization of the Coulomb branch.
that aims to diagonalize the matrix of two-point functions. Third, by moving outside the realm of Lagrangian theories, we have opened the door for the IR R-symmetries of the threedimensional SCFT being realized as a mixture of the four-dimensional UV R-symmetries and accidental (topological) symmetries emerging along the renormalizaton group flow from the four-dimensional UV SCFT to the three-dimensional IR SCFT. This type of mixing leaves an imprint on the three-dimensional theory on the three-sphere that takes the form of imaginary Fayet-Iliopoulos parameters [39]. 5 These directly affect the deformation quantization and ultimately spell an end to any hopes of answering our question in the positive in all Note added: when this note was being finalized, the paper [41] appeared on the arXiv, which addresses the same question.

Algebraic structures in 4d and 3d SCFTs
The aim of this note is to relate vertex operator algebraic structures one can carve out in four-dimensional N = 2 superconformal field theories to topological algebras appearing in their three-dimensional N = 4 superconformal dimensional reduction. In this section we start by briefly reviewing how these algebras emerge. 5 Recall that Fayet-Iliopoulos parameters are mass parameters for topological symmetries and that on the three-sphere imaginary parts of mass parameters encode R-symmetry admixtures [40].

SCFT/VOA correspondence
Four-dimensional N = 2 superconformal field theories, i.e., quantum field theories whose symmetry algebra contains the superalgebra su(2, 2|2), take part in a remarkable correspondence with vertex operator algebras [1,9]: The vertex operator algebra arises as a cohomological reduction of the operator product algebra of local operators of the SCFT. The cohomology is computed with respect to a nilpotent supercharge É ∈ su(2, 2|2) that takes the schematic form of the sum of a Poincaré supercharge and a special conformal supercharge: É = Q+S. At the origin of space, harmonic representatives of É-cohomology classes are characterized by a linear relation among their quantum numbers. They were dubbed "Schur operators", because they are precisely the operators counted by the Schur limit of the superconformal index [42]. Concretely, where E denotes the conformal dimension of the operator O, (j 1 , j 2 ) its su(2) 1 × su(2) 2 rotational quantum numbers, and R the value of its SU(2) R Cartan charge. Unitarity of the theory implies that these operators automatically satisfy r + j 1 − j 2 = 0. They can be transported away from the origin while remaining in cohomology by É-closed translation operators.
It is an algebraic fact that only motion in a complex plane C [z,z] is possible, and, moreover, that the (twisted) translation in thez-direction is É-exact. 6 Therefore, cohomology classes depend only on the holomorphic coordinate z. The operator algebra of the four-dimensional SCFT endows these holomorphic cohomology classes with the structure of a vertex operator algebra. Note that the (holomorphic) conformal weight of a Schur operator is measured by the eigenvalue of L 0 , the Cartan generator of the sl(2) z algebra of Möbius transformations of z. Upon embedding this algebra in the four-dimensional conformal algebra, one finds on any Schur operator O(0) that The subalgebra of the four-dimensional conformal algebra so(4, 2) that preserves the complex plane C [z,z] set-wise is sl(2) z × sl(2)z × u(1) ⊥ . The sl(2) factors generate the standard Möbius transformations on the respective complex coordinate and u(1) ⊥ contains rotations orthogonal to C [z,z] . While the É-closed translation of z is simply generated by L −1 , the É-exact translation of the coordinatez is generated bȳ where R − is the su(2) R lowering operator. Due to the presence of the SU (2) R generator, one often speaks of "twisted translation." To arrive at the second equality, we used the relation among quantum numbers of (2.2).
The set of Schur operators is non-empty for any local SCFT. Indeed, the four-dimensional stress-energy tensor, which is guaranteed to exist in a local quantum field theory, resides in a superconformal multiplet of su(2, 2|2) that contains a Schur operator -a certain component of the Noether current of the SU(2) R symmetry. Its twisted translation defines a holomorphic cohomology class whose operator product expansions identify it as the Virasoro stress tensor of the VOA. The Virasoro central charge c 2d is related to the Weyl anomaly coefficient c 4d of the above-lying superconformal field theory by the universal relation and their multiplication is obtained from the coincident limit of their standard operator product expansion. Note that Higgs branch chiral ring operators are automatically Lorentz scalars j 1 = j 2 = 0 and U(1) r neutral, and thus are special instances of Schur operators.
What's more, one can prove that all Higgs branch chiral ring operators give rise to Virasoro primary operators in the vertex operator algebra and that the generators of the ring are 7 By no means do the operators associated with Higgs branch chiral ring operators exhaust the full set of Schur operators. See [1] for a complete dissection of the body of Schur operators. 8 Recall that a hyperkähler manifold features three complex structures J i , i = 1, 2, 3, that anticommute with one another, satisfy J 2 i = −1 and J 1 J 2 = J 3 , and are compatible with the metric g(X, Y ) = g(J i X, J i Y ). The three complex structures are rotated as a triplet by an SU (2) isometry. Their associated Kähler forms ω i are closed dω i = 0. The conical structure implies that the metric takes the form ds 2 = dr 2 + r 2 ds 2 base . 9 To define the coordinate ring C[M H ], we have singled out one of the complex structures, say J 3 . Thanks to the SU (2) isometry acting on the complex structures, all choices are equivalent. The other two Kähler forms organize themselves into a holomorphic symplectic form Ω (2,0) = ω 1 + iω 2 , turning the coordinate ring C[M H ] into a commutative, associative Poisson algebra. Note that the SU (2) isometry of the hyperkähler cone is identified with the R-symmetry SU (2) R . The choice of complex structure is correlated with the Cartan decomposition of this SU (2) group. Finally, for future purposes, it is worth mentioning that the coordinate ring admits a C * grading, which is the complexification of the dilatation symmetry with the U (1) R ⊂ SU (2) R Cartan subalgebra. strong generators of the VOA.
Of particular importance in this class of Schur operators are the moment map operators associated with flavor symmetries G F of the SCFT, or, equivalently, with hyperkähler isometries of the Higgs branch. Their twisted-translated counterparts satisfy meromorphic operator product expansions defining an affine current algebraĝ F . The level k 2d of this Kac-Moody algebra is related to the four-dimensional flavor central charge k 4d via It is clear from the preceding discussion that the vertex operator algebra is intimately related to the geometry of the Higgs branch of vacua. This relationship was fleshed out and made precise in [21] (see also [43,44]). In that paper, it was conjectured that the Higgs branch can be recovered (as a holomorphic symplectic variety) from the vertex operator algebra as the so-called associated variety of the VOA. This variety is defined as the spectrum of the ring one obtains by performing a certain quotient of the ring of VOA operators whose multiplication is the standard normal ordered product. Vice versa, strong evidence has been obtained in [24,29] that the VOA can be given a (generalized) free-field construction mirroring the effective field theory description of the SCFT in a Higgs vacuum. 10 Schematically, the situation is thus as follows fr e e f i e l d c o n s t r.
We conclude this subsection by expressing in formulae the statement made implicitly above that the vacuum character of the vertex operator algebra χ 0 (q) equals the Schur limit of the superconformal index I S (q) Here STr denotes the supertrace over the space of states of the VOA and H(S 3 ) is the Hilbert space of states on the three-sphere of the four-dimensional SCFT. In the trace over the latter, many cancellations take place and ultimately only Schur operators contribute. As 10 See also [12] for a different kind of free-field realizations.
we will review in more detail below, both sides of this identification can be decorated with (Schur) operator insertions. For Lagrangian theories, the Schur index dressed with additional insertions can be computed explicitly using supersymmetric localization techniques as a correlation function of the four-dimensional SCFT placed on S 3 × q S 1 [23] (see also [27]). Very similarly to the SCFT/VOA correspondence of the previous subsection, threedimensional N = 4 superconformal field theories admit a cohomological truncation to a one-dimensional topological algebra, i.e., an associative algebra additionally endowed with an evaluation map [34,35]. As was uncovered in [35], this algebra is a noncommutative deformation of the coordinate ring C[ M H ]. 12 Moreover, it was shown in that same paper that the leading term of this deformation is determined by the Poisson bracket: the topological 11 Recall that three-dimensional N = 4 superconformal field theories are defined by the requirement that their symmetry algebra contain the superalgebra osp(4|4, R). Its bosonic subalgebra is given by (su(2) C ⊕ su(2) H ) × sp(4, R), where one recognizes the R-symmetry algebras and the three-dimensional conformal algebra sp(4, R) ≃ so(3, 2). 12 Indeed, harmonic representatives of cohomology classes at the origin are characterized by the requirement that their quantum numbers satisfy E = R H , i.e., that they are Higgs branch chiral ring operators. While remaining in cohomology, these operators can only be (twisted) translated away from the origin along a line, and the coordinate dependence along this line is exact. Thus, only the ordering of the twisted-translated cohomology classes along the line is retained. Their algebraic properties descend from the three-dimensional operator product algebra: they form an associative, but not necessarily commutative, algebra, and taking vacuum expectation values provides the above-mentioned evaluation map. algebra thus defines a deformation quantization: 13

SCFT/deformation quantization correspondence
Various other properties of this deformation have been established in [35] and can be summarized as follows. Let f ∈ A p , g ∈ A q be elements of the pth and qth C * -graded component In other words, f and g are holomorphic functions over the Higgs branch that correspond to Higgs branch chiral ring operators of SU(2) H charge R H = p 2 and q 2 respectively. Then the multiplication of their twisted-translated cohomology classes ordered along the line, which we denote by ⋆, reads where f ·g is simply the multiplication in the ring A and {f, g} P B denotes the Poisson bracket.
Here ζ is an immaterial book-keeping device that keeps track of one unit of SU(2) H -charge.
Moreover, the ⋆-product satisfies the following properties: A fifth property is related to unitarity. Let ρ be a rotation over π in SU(2) H followed by complex conjugation, then for for an infinite-dimensional group of gauge equivalences. They were conjectured to be perfect in [35]. All in all, the deformation quantization thus depends on a finite number of intrinsic parameters, the so-called period of the quantization. Up to the action of the Weyl group of the Coulomb branch flavor symmetry, they are in one-to-one correspondence with real mass parameters of topological Coulomb branch symmetries, i.e., Fayet-Iliopoulos terms, one could in principle turn on in the theory [35]. Exciting progress towards proving this conjecture has recently been made in the mathematics literature [45]. 13 Recall from footnote 9 that the coordinate ring of a hyperkähler cone is naturally a Poisson algebra.

From VOA to deformation quantization
As we have explained in the previous section, both the vertex operator algebras associated with four-dimensional N = 2 SCFTs and the deformation quantizations corresponding to three-dimensional N = 4 SCFTs are intimately related to the geometry of their Higgs branches of vacua -the latter in an obvious manner, as they are the quantizations of the coordinate ring of the Higgs branch, while the former in a slightly less manifest manner.
Moreover, it is a well-known fact that the Higgs branch of vacua remains invariant under (trivial) dimensional reduction. A natural question to ask is then if one can fill in the question mark in the following diagram: In this section, we will attempt to answer this question by formalizing the dimensional reduction of the four-dimensional theory T 4d placed supersymmetrically on the (warped) product-manifold S 3 × q S 1 . The partition function on this background computes the Schur limit of the superconformal index: In other words, using (2.7), it computes the torus partition function of the vertex operator algebra associated with the SCFT. The relevant torus T 2 ⊂ S 3 × q S 1 arises as the pointwise fixed set of a certain spatial U(1) rotation; its complex structure τ is determined by q = e 2πiτ . Moreover, it was shown in [23] (see also [27]) that one can enrich the computation with (twisted-translated) Schur operators inserted at points on said torus T 2 . 14 The resulting If T 3d admits a UV Lagrangian description in terms of vector and hypermultiplets, it was shown in [36] that the correlation functions defining the one-dimensional topological algebra associated with (the Higgs sector of) T 3d are accessible via a localization computation on the three-sphere. Note however that operators on the three-sphere may mix, see [36,47].
The correct flat-space basis can be determined by diagonalizing the matrix of two-point functions. As we suggested in [23], and will show in detail in section 4, dimensionally reducing the Lagrangian setup on S 3 × q S 1 along S 1 allows one to make direct contact with these computations. Moreover, by observing that this S 1 -reduction is nothing but the high- it can be phrased purely vertex operator algebraically. This formulation is helpful because the VOA is often known even in the absence of a Lagrangian description of T 4d .
Simply considering the high-temperature limit of torus correlators cannot be the complete story, since, as explained above, the set of Schur operators is strictly bigger than the collection of Higgs branch chiral ring operators. Taking a cue from the notion of contracting algebras, we propose a rescaling of the four-dimensional Schur operators by powers of β that ensures that only correlation functions of Higgs correlation functions survive in the β → 0 limit.
Finally, one should observe that the R-symmetries of the three-dimensional theory in the IR may be realized as a mixture of the four-dimensional UV R-symmetries and accidental (topological) symmetries. A simple criterion for when this must be the case was put forward in [39], namely if the Coulomb branch chiral ring has generators of U(1) r charges that are not quantized in half-integer units. Such mixing is encoded in the three-sphere partition function as imaginary parts of the Fayet-Iliopoulos parameters. Such parameters directly modify the period of the resulting quantization, thus taking us away from the specific superconformal deformation quantization DQ(T 3d ).

Torus correlation functions
We start by putting the vertex operator algebra V[T 4d ] on the torus T 2 = C/(Z + τ Z).
The modular parameter τ takes values in the upper half plane H; it is acted on by the where w j = e 2πiz j relates the coordinates z j on the torus to coordinates w j on the plane. The normalization is provided by the unnormalized one-point function of the identity operator O ½ = ½, in other words, the vacuum character: In these equations we have used the nome q defined as q := e 2πiτ . It will be useful later on to also recall that one-point functions are position independent and easily expressed in terms If the field is not integer-moded or has odd statistics the result is zero. In particular, for the stress-energy tensor T , which transforms as Let us introduce some notation for the quantities of our most direct interest, namely the the torus two-, and three-point functions . We also use the notation a I (τ ) = b I½ (z 1 , z 2 ; τ ) for the one-point functions.

High temperature limit and deformation quantization
Implementing the intuition outlined above, we would like to take the high-temperature is one unit of SU(2) R charge larger than that of a three-dimensional one E = R. 16 In total we thus put forward the rescaling of Schur operators to be Quite elegantly, this rescaling can be phrased in terms of the conformal weight of the fields of the VOA, which are known as soon as the vertex operator algebra itself has been identified.
Applying this rescaling to one-point functions, we define and trivially find from (3.4) Next, we define, for Re(z 1 ) > Re(z 2 ), which constitutes a matrix of real numbers. Using Zhu's recursion relations [48], we can indeed prove explicitly that the position dependence drops out in our limit: where and we used the standard notation for the various terms in the operator product expansion Using the high-temperature behavior (A.8) of the functions P m+1 and of normalized one- it is now easy to convince oneself that the second term does not contribute in the limit (3.10). We thus find Note that this equality for the high-temperature limit of two-point functions is in some sense a trivial extension of (3.9). Finally, we define for Re(z 1 ) > Re(z 2 ) > Re(z 3 ) which we expect to be explicitly calculable as 17 The computation of the limit (3.16) defining the coefficients c IJK (and its special cases involving one or two identity operators (3.10) and (3.9)) can be further simplified. Let us start by considering the vacuum character χ 0 (τ ). Under an S-transformation τ →τ = −1/τ it can be written as where the sum runs over the elements of the vector-valued modular form in which the vacuum character resides. 18 The characters χ j (τ ) start off as χ j (τ ) = e 2πiτ (−c/24+h j ) (1 + . . .), but possibly also involve logarithmic terms. The coefficients S 0j are rational numbers. The high-temperature limit of the vacuum character is then easily derived   20 We leave a more detailed study of this structure for future work.
Having computed the one-, two-, and three-point coefficients a I , b IJ , and c IJK , our next task is to find a new basis of operatorsÔ I that diagonalizes b IJ . This Gram-Schmidt process is a recursive task that can be easily performed. Note that the new basis still respects the R-filtration, i.e., the new operator can only be different from the original one by operators of lower R-charges. In particular the diagonalization ensures that in the new basis the onepoint functionsâ I =b I½ = 0, for I = ½. In fact, even stronger, we claim that in the new diagonal basisb Implementing the change of basis on the constants c IJK , which we denote in the new basis asĉ IJK , we further claim that  20 This product is defined as for two states a, b. It is associative, but non-commutative. Zhu's algebra additionally involves a quotient with respect to a certain ideal.
We do not currently know how to prove these claims in all generality. For Lagrangian theories, however, it is easy to convince oneself that they are true. See the next section for more details. Together, the inverse of the (non-zero part of the diagonal) matrixb IJ and the collection of numbersĉ IJK allow one to definê Here the indices can be understood to only range over V H . Other operators have non-zero three-point constantsĉ IJL anyway. At last, these define the desired algebrâ The algebra defined by (3.25) is a C * -equivariant deformation quantization. These properties immediately follow from the construction presented so far. In particular the R-filtration guarantees C * -equivariance, and the antisymmetric part of the leading term is the Poisson bracket as follows from an easy computation: whose leading p = 0 term gives a −ha+1 b = {a, b} PB if a, b ∈ V H [21] . The other properties we would like this deformation quantization to possess are less obvious. For Lagrangian theories, they follow indirectly from the arguments presented in the next section. We will assume that they all hold also for non-Lagrangian cases, but it would clearly be desirable to prove this statement.
Our next task is to answer the question if this algebra is some deformation quantization or precisely the one that follows from the SCFT/Deformation quantization correspondence.
Note that even if the algebra satisfies all five properties listed in subsection 2.2, and therefore all gauge freedom is (conjecturally) fixed, it still has finitely many free parameters in its

Lagrangian proof of proposal
Our proposal to extract from the vertex operator algebra associated with a four-dimensional N = 2 SCFT the deformation quantization of the Higgs branch of vacua of its dimensional reduction is in essence based on taking the high-temperature limit of torus correlation functions of the VOA. Computing torus correlators is a difficult task in general, although tools like Zhu's recursion relations are often helpful [48]. There is, however, a subset of VOAs whose torus correlation functions can be computed relatively easily, namely those associated with four-dimensional Lagrangian theories. As mentioned before, supersymmetric localization techniques applied to the four-dimensional theory placed on S 3 × q S 1 lead one to a computational recipe of torus correlators in terms of explicit contour integrals [23,27]. Their integrands have a transparent structure that allows one to extract the deformation quantization of the dimensionally reduced theory. In particular, in this section we will show in detail that the high-temperature limit of these torus correlation functions directly reduces to twisted Higgs branch correlation functions of the three-dimensional theory on the three-sphere. From these one can straightforwardly deduce the desired deformation quantization [36]. In fact, a detailed understanding of the Lagrangian procedure is what allowed us to formulate our general proposal.

Lagrangian proof
Recall from (2.7) that the unflavored Schur index I (T 4d ) S (q) of any four-dimensional N = 2 superconformal field theory T 4d equals the vacuum character of the associated vertex operator algebra V(T 4d ). Standard arguments further identify the Schur index with the partition function of T 4d placed supersymmetrically on a S 3 × q S 1 background, see (3.1). 21 Concretely, for an N = 2 superconformal gauge theory with gauge algebra g and hypermultiplets trans- 21 As explained in detail in [23], the supersymmetric background is described by the metric ds 2 = ℓ 2 cos 2 θ(dϕ − iβ + dt) 2 + ℓ 2 sin 2 θ(dχ − iβ − dt) 2 + ℓ 2 dθ 2 + (−iℓτ + ℓβ + ) 2 dt 2 , for some constants β ± and τ such that Re τ + iβ + = 0. To ensure the background preserves supersymmetry one should also turn on an SU (2) R and U (1) r background gauge field. The locus θ = 0 defines a torus T 2 ⊂ S 3 × q S 1 . On this torus one can insert (almost-)BPS operators: the curved-space counterpart of the twisted translated Schur operators on flat space. Correlation functions of these operators admit a localization computation [23], leading to the integral formula we present in the main text.
forming in the representation R of g, the Schur index I S /vacuum character of the associated VOA χ 0 /partition function on S 3 × q S 1 can be written as a contour integral 22 . technique provides an explicit contour integral expression for these correlation functions [23], Here the coordinates z = ϕ + τ t parametrize the torus T 2 with complex modulus τ . As such, they are doubly-periodic with periods 1 and τ . Most importantly, O GT is the correlation function of the vertex operators in a Gaussian theory defined in terms of a bcβγ system.
In more details, the corresponding twisted-translated Schur operators are gauge-invariant composites of the vector multiplet gaugini λ,λ and hypermultiplet scalars Q A , A = 1, 2.
Note that these letters transform in the adjoint representation and the matter representation R of the gauge algebra g respectively. Using Wick's theorem, the Gaussian theory is fully defined by specifying the propagators of these letters: As before, a is defined by a = e 2πia , while w, w ′ are weights in the adjoint representation.
With these ingredients we can compute any torus correlation function in the vertex 22 See appendix A for details on Jacobi theta and Dedekind eta functions.
operator algebra associated with the Lagrangian SCFT. According to our proposal, we should consider their high-temperature limit. Let us thus take q = e −2πβ and a = e −2πβσ . Then β parametrizes the radius of the temporal circle of S 3 × q S 1 . Sending β → +0 while keeping σ fixed shrinks that circle and effectively dimensionally reduces T 4d to a three-dimensional theory T 3d on the three-sphere. Indeed, one easily computes the high-temperature limit of the contour integral in (4.1): , (4.5) where we recognize the prefactor as capturing the Cardy behavior (3.18). Indeed, π 6β (dim R− dim g) = 4π c 4d −a 4d β which matches the expected effective central charge in (3.19). 23 The integral itself is precisely the S 3 partition function Z S 3 T 3d of a three-dimensional N = 4 supersymmetric theory T 3d with UV Lagrangian description in terms of a gauge theory with hypermultiplets transforming in representation R of the gauge group G with Lie algebra g.
More generally, we can take the high-temperature limit of any correlation function computed as in (4.2). It clearly suffices to analyze the behavior of the propagators. We find In the presence of derivatives, for example, we can compute Clearly, if we rescale Q → (βℓ) 1/2 Q, then the Q A propagator reproduces on the nose the two-point function of the twisted-translated Higgs branch operators of [36]. Let us give a few more details. The authors of [36] performed a localization computation of N = 4 supersymmetric theories on the three-sphere S 3 that allows additional insertions of twistedtranslated Higgs branch chiral ring operators on a circle S 1 ⊂ S 3 . Their result mirrors the expression (4.2) in the obvious manner

Lagrangian examples
Before moving on to non-Lagrangian tests in the next section, let us quickly summarize a few simple Lagrangian examples where the relevant vertex operator (sub)algebras are current algebras. We perform the general analysis of current algebras g k in appendix B; the final result of the current-current star product is given by 24 where f ab c denotes the structure constants and κ ab the Killing form. Furthermore, we used the effective central charge defined in (3.19), h ∨ is the dual coxeter number of g, and dim g denotes its dimension. Let us introduce a convenient parameter capturing the ζ 2 coefficient: For su(2) k current algebras, it is customary to reorganize the currents as 12) 24 We use conventions in which the longest root has length squared ψ 2 = 2.
leading to the star products A first elementary example is su(2) − 1 2 , the Z 2 -invariant vertex operator subalgebra of the associated VOA of a free four-dimensional hypermultiplet. In this case the vacuum character reads , (4.14) whose high-temperature behavior can be easily computed using the formulas in appendix A, resulting in c eff = 2 and thus yielding µ = − 3 16 . 25 This indeed agrees with the deformation quantization of the Z 2 gauge theory of the free hypermultiplet [35]. Note that no diagonalization is required in this example, and that both the su(2) − 1 2 torus correlation functions and the star products can also be analyzed directly using the Lagrangian machinery introduced above.
A second slightly less trivial example concerns the current algebra su(2) − 3 2 , which appears as a subalgebra of the VOA associated with four-dimensional N = 4 super Yang-Mills theory with gauge group SU (2). This VOA is the small N = 4 superconformal algebra at c = −9 [1].
The reason we can focus on the currents only is obvious: we have already proved that all VOA correlators involving fermionic letters (or derivatives) would vanish in our high-temperature limit, which effectively removes the supercurrents, and similarly the canonical stress-energy tensor has vanishing correlators in our high-temperature limit. 26 The VOA's character reads Noting its trivial Cardy behavior, since of course c 4d = a 4d , we have c eff = 0. Computing 25 The value of c eff could also have been obtained by using footnote 19 in combination with formula (3.19), as of course the quaternionic dimension of the relevant Higgs branch is one. 26 Recall that the stress tensor is cohomologous to the Sugawara stress tensor in the small N = 4 superconformal algebra at c = −9. Its status as zero in the deformation quantization thus implies that the Higgs branch relation κ ab J a J b = 0 indeed holds. µ defined in (4.11) then results once again in the value µ = − 3 16 . Let us verify that this deformation quantization matches with the three-dimensional computation. The relevant three-dimensional theory is of course N = 8 SU(2) SYM and its S 3 partition function, viewed as an N = 4 SU(2) SYM theory coupled to an adjoint hypermultiplet, is Correlation functions of twisted Higgs branch operators can be easily computed by where the Gaussian correlator O GT can be computed via Wick contractions and using the (slightly renormalized) propagator (see (4.6)) Reorganizing the su(2) F flavor currents (4.20) We thus exactly recover the star product (4.10) predicted from the vertex operator subalgebra su(2) −3/2 , with µ = −3/16.

Non-Lagrangian examples
In this section, we turn our attention to non-Lagrangian SCFTs with known VOAs. In particular we analyze a class of examples with trivial Higgs branches to probe the decoupling of non-Higgs branch operators, and study various instances of theories whose associated VOAs are current algebras. We compute various star-products of the resulting deformation quantization and confirm that their period takes different values than their superconformal values if the R-symmetries mix with accidental symmetries.

(A 1 , A 2n ) Argyres-Douglas theories
As a first non-Lagrangian test of our proposal we consider the (A 1 , A 2n ) Argyres-Douglas theories. Their associated vertex operator algebras are the Virasoro minimal models Vir 2,2n+3 [2,21]. As these theories have a trivial Higgs branch, we should find a trivial deformation quantization. In other words, this example will test if the Virasoro stress tensor is removed properly in our high-temperature limit.
The vertex operator algebra Vir 2,2n+3 is strongly generated by the stress tensor T (z).
Its central charge is c = − 2n(6n+5) 2n+3 . On the torus, the stress tensor always has a non-zero one-point function t := T (z) T 2 , which can be computed in terms of the derivative of the vacuum character, see (3.5). Combined with the Cardy behavior of (3.18), we find a non-zero high-temperature limit: and thus a non-diagonal b IJ , as clearly b T ½ = 0. A Gram-Schmidt procedure is thus needed to diagonalize the matrix b IJ , instructing us in particular to study correlation functions of From (3.14) and (3.16), the high-temperature limit of the rescaled stress tensor correlation functions reads (n = 2, 3 for our purposes) where we have used that ∂ τ = 2πiq∂ q . Combined with the Cardy behavior of χ 0 , we obtain This equality immediately leads to a trivial high-temperature limit of all correlation functions ofT , thanks to the identity (C.2). Moreover, the vanishing behavior remains true when including composites ofT , constructed as the regular part of the coincident limit of products ofT , or in the presence of spatial derivatives. We conclude that any Virasoro algebra results in a trivial deformation quantization and in particular so does Vir 2,2n+3 .
In fact, we can prove the stronger statement that the stress tensor T of any VOA trivializes when considering our high-temperature limit and thus decouples from the resulting deformation quantization. Indeed, let us first write as before Additionally inserting a collection of stress tensors m l=1 β 2 T (z l ) simply produces an m-fold derivative with respect to τ on the supertrace [21,48] lim β→+0 m l=1 The high-temperature behavior of the latter can be inferred from its properties under Stransformation. A reasoning similar to the one above then allows us to conclude that all In appendix C we present an alternative analysis of the decoupling of the stress tensor based on Ward identities.
Their associated VOAs are given by su(2) k=− 4n 2n+1 current algebra [2,21]. Their vacuum characters read which have Cardy behavior determined in terms of Using the results of appendix B on the deformation quantization to which current algebras reduce, and the redefinition of currents as in (4.12), we find the elementary star products These results can be compared against a three-dimensional computation. Let us start with identifying the three-dimensional N = 4 supersymmetric theory to which the (A 1 , D 2n+1 ) theory flows. The circle reduction of the vacuum characters χ (n) 0 can be easily performed: . (5.10) where we removed the Cardy behavior and set a = βm. Up to irrelevant numerical prefactors, this result can be recognized as the S 3 -partition function of three-dimensional N = 4 SQED with 2 flavors in the presence of an imaginary FI-parameter (see, e.g., [49]), . (5.11) Note that the appearance of an imaginary FI-parameter is the hallmark of the mixing of the UV R-symmetries with accidental Coulomb branch flavor symmetries [39] along the flow from four dimensions to three. 27 This is guaranteed to occur whenever the four-dimensional theory contains a Coulomb branch chiral ring operator of non-half-integer U(1) r -charge.
Denoting the twisted translated Higgs branch operator as Q Ai , we define the moment map operators of the su(2) flavor symmetry Turning off the mass parameters, these operators have vanishing one-point functions, while their two-point and three point functions can be computed with ease following the methods of [36], (5.13) 27 The paper [39] analyzed in detail the case n = 1.
The resulting elementary star-products take the form, As expected, once we substitute in ξ = i 2n+1 , this is precisely the same star product as obtained from the VOA. Note that, for example, for n = 1 the value ξ = 0 would have reproduced the star-product DQ[dim. red. of (A 1 , D 3 )], see [35]. We conclude that the nonzero value of the Fayet-Iliopoulos parameter has modified the period of the quantization.

(A 1 , D 4 ) Argyres-Douglas theories
The analysis of an arbitrary member of the series of (A 1 , D 2n+2 ) Argyres-Douglas theories is beyond the scope of this paper, as their associated vertex operator algebras (the so-called W n+1 algebras [50]) do not easily lend itself to the computation of torus correlation functions.
However, the n = 1 member of this series, the (A 1 , D 4 ) theory, is amenable to a detailed analysis as its associated VOA is simply su(3) −3/2 [1,2,14]. Using the results of appendix B and the vacuum character we immediately find the elementary star product We can verify this result directly against the computations of the relevant star-products in [51]. Alternatively, we can use the dimensional reduction of this theory, which is N = 4 SQED with three flavors. Note that in this case there is no mixing between R-symmetries and accidental symmetries [39], thus we expect to find the correct period. In fact, as was shown in [35], the minimal nilpotent orbit of SU(3) admits only a unique even, C * -equivariant deformation quantization, so there was no room to modify the period to start with. The star product in the three-dimensional setting was computed in [36], Of course it agrees with the results of [51], and it also matches (5.19) by identifying J a ≡ J m n (T a ) m n with properly chosen su(3) generators T a in the fundamental representation, such that [T a , T b ] = if abc T c and tr T a T b = δ ab . and transforms elegantly under the S-transformation We can easily deduce its high-temperature behavior Another useful series of functions P m (z|τ ) is related to ϑ 1 (z|τ ) as They have high-temperature asymptotics (assuming Re z > 0) Note the exponential suppression as β → +0 for P m>2 .
The Weierstrass ℘-function is defined as It is an elliptic function with double poles at z = m + nτ for m, n ∈ Z, which is manifest from its alternative definition (A.10) In particular, its expansion in z around the origin reads without constant term. The high-temperature behavior of ℘ is To conveniently write down torus correlation functions of current algebras, we also define The functions S i are referred to as genus-one Szegö kernels [52]. They are related to the Weierstrass ℘-function by thanks to the standard identity ϑ 2 (0|τ )ϑ 3 (0|τ )ϑ 4 (0|τ ) = 2η(τ ) 3 . We thus have

B Current algebras and star product
In this appendix we present explicit expressions for torus correlation functions ofĝ k currents, derive their high-temperature behavior, and compute the resulting ⋆-products. An affine current algebra g k is defined by the basic operator product expansion where f ab c are the structure constants of the underlying Lie algebra g, i.e., [T a , T b ] = if ab c T c , and κ ab is the Killing form.
Two-and three-point current correlation functions on a torus T 2 with complex modulus τ , for algebras that do not possess a cubic casimir, read [52] 28 Here χ 0 denotes the vacuum character of g k and we used the genus-one Szegö kernels, see (A.13). The ℘(z, τ ) function in the two-point function provides the z −2 pole at the origin as 28 Note that by g-symmetry J a (z) = 0. mandated by the OPE (B.1), while the second term accounts for the Sugawara relation Here we also used (3.5) expressing the one-point function of the stress tensor in terms of the derivative of the vacuum character. The three functions W i (τ ) are defined by solving the equations leaving one immaterial degree of freedom undetermined. To avoid clutter, we have also To extract a star product, we consider the high-temperature limit of the rescaled correlation functions of the currents. To this end, we assume the Cardy behavior captured by the effective central charge (see also (3.18) and (3.19)) which holds for all examples we consider. We also commute the high-temperature asymptotic with ∂ τ at will. In the end, we find Note that using (3.14) and (3.16), the fact that the two-point and three-point function are controlled by the same combination of quantities could also be seen from the results in appendix C of [53].
Finally, defining J ≡ ζ π J, the above correlation function leads to the star product We expect this result to remain valid even if the algebra has a cubic casimir. Note that for a current algebra, the central charge is given by the Sugawara central charge, The star product can thus be rewritten as (B.12)

C Stress tensor and Virasoro VOA
In this appendix we analyze torus correlation functions of stress tensors making use of Ward identities [54,55]. In particular, we provide an alternative proof that in the high- First of all, we note thatT -correlation functions can be reorganized as Hence, the high-temperature limit of n i=1T (z i ) can be deduced from the high-temperature behavior of t n−p and the combinations p i=1 T (z i ) − t p with p ≤ n. The latter piece of information can be extracted by carefully analyzing the relation [55] (slightly rearranged as factors to avoid clutter) Apparently, terms in T (z)T (z 1 ) − t 2 either have at most β −3 = β −(2×2−1) singularity or are exponentially suppressed; once rescaled by β 4 , the correlation function vanishes as β → +0.
This observation, combined with (C.4), initiates a recursive argument ensuring that higher correlation functions β 2(p+1) ( T (z) p j=1 T (z j ) − t p+1 ), for p ≥ 2, also vanish in the high-temperature limit. Indeed, assuming the high-temperature behavior of p j=1 T (z j ) −t p has at most β −(2p−1) singularity, terms on the right hand side of (C.4) either have at most a β −(2p+1) singularity, like ∂ τ t p , or are exponentially suppressed, like the ℘ ′′ (z − z j ) terms.
Once rescaled by β 2p+2 , they go away in the high-temperature limit. Finally, this observation is to be applied back to (C.2), which clearly shows that β n n i=1T (z i ) vanishes in the same limit. Furthermore, inserting spatial derivatives or forming composites of T by taking the regular part of coincident limits will not make the correlation functions nonzero.
To conclude, the set of correlation functions only involving the stress-energy tensor decouples. Moreover, slightly modified, we expect the above argument to extend to all correlation functions containing a stress energy tensor.

D Characters
In this appendix we collect characters of various VOAs used in the main text.
The Virasoro minimal models Vir p,p ′ are labeled by two coprime natural numbers p, p ′ ≥ 2 with central charge c = 1 − 6 (p−p ′ ) 2 pp ′ . The spectrum of Vir p,p ′ contains a finite number of primaries of conformal dimensions h r,s = (pr−p ′ s) 2 −(p−p ′ ) 2 where λ r,s ≡ pr − p ′ s. For each Vir p,p ′ , χ r=1,s=1 is the vacuum character.