Exact Operator Map from Strong Coupling to Free Fields: Beyond Seiberg-Witten Theory

In quantum field theory (QFT) above two spacetime dimensions, one is usually only able to construct exact operator maps from the ultraviolet (UV) to the infrared (IR) of strongly coupled renormalization group (RG) flows for the most symmetry-protected observables. Famous examples include maps of chiral rings in 4d $\mathcal{N}=2$ supersymmetry. In this letter, we construct the first non-perturbative UV/IR map for less protected operators: starting from a particularly"simple"UV strongly coupled non-Lagrangian 4d $\mathcal{N}=2$ QFT, we show that a universal non-chiral quarter-BPS ring can be mapped exactly and bijectively to the IR. In particular, strongly coupled UV dynamics governing infinitely many null states manifest in the IR via Fermi statistics of free gauginos. Using the concept of arc space, this bijection allows us to compute the exact UV Macdonald index in the IR.


Introduction
In order to gain insight into strongly coupled QFT, it is useful to construct universal and calculable observables.However, there is often tension: the less calculable an observable is, the more interesting the dynamics it can probe.
In the case of 4d N = 2 QFTs, the half-BPS chiral ring is a calculable space of operators maximally protected by supersymmetry.Through the celebrated machinery of Seiberg-Witten (SW) theory [1,2], it can be followed exactly along strongly coupled RG flows to the IR, where it gives the two-derivative effective theory on a moduli space of vacua called the "Coulomb branch." One longstanding open question in strongly coupled QFT in d > 2 is to give an exact UV/IR map of nonchiral observables less protected by supersymmetry.In this letter, we solve this problem for a ring arising from normal-ordered products of superpartners of the energymomentum tensor.Unlike the SW ring, this ring is non-chiral, quarter-BPS, and hence "half" as protected by supersymmetry.Geometrically, these results give an infinite-dimensional generalized tangent space of the Coulomb branch.
Our approach is to first focus on the closest and simplest strongly coupled 4d analog of an exactly solvable 2d QFT: the original or "minimal" Argyres-Douglas (MAD) superconformal field theory (SCFT) [3].Indeed, from the point of view of the Coulomb branch effective theory, this SCFT is maximally simple.It also has the simplest symmetry structure of any 4d N = 2 SCFT.Finally, parts of the local operator algebra are maximally simple for a unitary theory with a vacuum moduli space [4-7][35].
This "closeness" of the MAD theory to the Coulomb branch effective theory and certain exact spectroscopic results [7] prompted us to conjecture the local operator algebra is generated as follows [7] O ∈ Ē×m 6/5 × E ×n −6/5 , ∀ O ∈ H L . ( Here O is any local operator of the SCFT (H L is the corresponding Hilbert space), and the righthand side of the inclusion represents the (m, n)-fold operator product expansion (OPE) of Ē6/5 and E −6/5 .In the language of [9], Ē6/5 is the multiplet housing the dimension 6/5 chiral primary whose vev parameterizes the Coulomb branch (E −6/5 houses the conjugate anti-chiral primary).Turning on a vev for the corresponding primary initiates an RG flow to the Coulomb branch and, in the deep IR, to free super-Maxwell theory.Since the multiplets generating the MAD operator algebra are, in this sense, "Coulombic," we refer to the above conjecture as the "Coulombic generation" of the spectrum.
Given (1), it is natural to try relating all nondecoupling parts of the MAD spectrum to super-Maxwell operators.A first step is to consider the generating multiplets (1).As described above, the RG map in this case follows from the SW construction [3] Ē6/5 −→ DFree 0(0,0) , where the righthand side is the free vector multiplet housing the chiral ϕ primary [36].
Another natural representation to consider is the stress tensor multiplet, which appears in the m = n = 1 OPE in (1).Since the RG flow preserves N = 2, we have where the multiplet on the righthand side is the stress tensor multiplet of free super-Maxwell theory [37].
The MAD Schur ring only has ĈR(j,j) multiplets [5].Moreover, it has an "extremal" subsector.These are Schur operators and multiplets that, for a given SU (2) R weight, R, have lowest spin, j.The stress tensor multiplet is the case R = j = 0.More generally, extremal Schur operators, O R,Ext ∈ ĈR( 12 R(R+2), 1   2 R(R+2)) , map as follows [5] where we have used Fermi statistics to rearrange the gauginos in a fashion of use below.
Given this discussion, it is natural to expect a general relation between the UV and IR Schur rings.However, there are potential obstacles: Regarding (a), (2) implies the UV origin of the gauginos is in the MAD chiral sector, not the Schur sector [38].Moreover, because the IR is free, it has higher spin symmetries which are absent in the UV [11,12].The breaking of these symmetries in the flow back to the UV is encoded as follows [6] On the righthand side, we have emergent complex higher spin current multiplets, while, on the lefthand side, we have "longer" protected multiplets that include nonvanishing divergences of would-be MAD higher-spin currents.For real higher-spin currents [13] On the lefthand side, we have certain UV long multiplets.Therefore, a main task is to carve out the subsector of IR Schur operators corresponding to UV Schur operators.This discussion is summarized in Fig. 1.
Regarding (b), note it is common for Schur operators to decouple in Coulomb branch flows.For example, on a genuine Coulomb branch consisting of free vectors at generic points, flavor symmetries decouple.Since flavor symmetry Noether currents lie in Schur multiplets, Schur operators can decouple.More generally, decoupling is unrelated to flavor.
Given the "closeness" of the MAD SCFT to the Coulomb branch, it is reasonable to expect both obstacles are irrelevant.We will soon see this is the case.
A useful feature of the UV Schur ring is its simplicity.Indeed, as explained in the Supplemental Material, it is generated by the where ":• • • :" denotes the normal-ordered product [39].
Consistency with (3) suggests looking for an IR null state related to (8).Indeed, using (3), the non-trivial UV dynamics leading to (8) maps to an IR constraint enforced by Fermi statistics [40] Ĉ1 Given this discussion, we propose the following map: Main statement: An arbitrary monomial in the MAD Schur ring is mapped as follows to the IR Here S Free Vector is the set of all IR Schur operators [41].
On the other hand, Fermi statistics naively looks more constraining than (9).Indeed, (λ 1 + ) 2 = ( λ1 2 = 0 implies (9), not vice versa.Therefore, we should make sure there are as many null states on one side of (10) as on the other.
Using results on "leading ideals," [15,26] we will show that, for operators in (10), Fermi statistics is equivalent to (9).Combined with the fact that the ∂ i + J subject to (8) generate the UV Schur ring, we establish (10).As a byproduct, we show that the Macdonald index, an observable that counts Schur operators, is exactly computable in the IR.
We have avoided discussing the relation of 4d Schur rings to 2d vertex operator algebras (VOAs) [17].The main reason is our discussion is inherently 4d, and the twisting in [17] somewhat obscures this (we will return to the 2d free field construction of [18,19] in section III).However, as we discuss, the 4d/2d map is useful in deriving (8).Moreover, results on arc spaces [20] imply the UV Schur ring is characterized as claimed around ( 8) [21].
The plan of the paper is: in section I we briefly review the MAD theory and its Schur sector.In section II, we show Fermi statistics does not lead to additional constraints spoiling (10).We conclude with a general discussion in section III.

I. THE MAD THEORY'S SCHUR SECTOR
We briefly review the construction of the Schur ring, describe its counting by the Macdonald index, and discuss the example of the MAD theory.Finally, we explain how the map in [17] can be used to derive (8) and explain how the UV Schur ring is generated (details appear in the Supplemental Material).
A Schur operator, O, satisfies where the lefthand side is the scaling dimension, R is the SU (2) R weight, r is the U (1) r charge, and j, j denote spin weights.Operators carrying these quantum numbers are counted by the Macdonald index where the trace is over the space of Schur operators, q and T are fugacities, and (−1) F is fermion number.
The MAD Macdonald index was computed via TQFT in [22], but the elegant expression in [14] is particularly useful Here, ∂ i + J contributes q i+2 T , and products of operators give products of contributions.
To interpret the physical states contributing to (14), we briefly recall the Schur ring to VOA map [17] (see [17] for further details).The idea is to perform an SU (2) R twist of right-moving sl(2, R) transformations on a plane inside R 4 .Then, the algebraic constraints in (11) imply that Schur operators are non-trivial cohomology elements Moreover, twisting guarantees that planar translations by ∂ − + are cohomologically trivial while those generated by ∂ + are not.As a result, we can map twisted-translated Q i cohomology classes in (15) to operators that only depend on a holomorphic planar coordinate, z.These latter operators are members of a VOA.Particularly relevant for us are the maps where [J] Q is the cohomology class of the SU (2) R current [42], T 2d is the VOA stress tensor, c 2d is the corresponding central charge (twisting leads to 2d non-unitarity), h is the holomorphic scaling dimension, and χ is the 4d/2d map.
Using this construction (specifically the T → 1 limit of ( 14), which becomes the VOA vacuum character) and some orthogonal arguments we will return to in the discussion section, the authors of [23] argued that the VOA corresponding to the MAD theory is the Lee-Yang vacuum module [43] This VOA is built from normal-ordered products of ∂ i T 2d for arbitrary i.
Famously, Lee-Yang has an h = 4 null state This null relation is the 2d incarnation of ( 8) (e.g., see [4]).Indeed, from the general construction in [17], we can work out the terms that do not vanish in the z → 0 limit of the T 2d (z)T 2d (0) OPE by considering all SU (2) R components of the 4d J i1i2 In particular, the null state in (18) corresponds to a 4d null state with h = 4, and multiplet selection rules imply this operator has R = 2 [44].It therefore corresponds to the vanishing normal-ordered product This equation is a non-trivial UV dynamical constraint.
Given that the VOA in ( 17) is strongly generated by T 2d , it is natural to conjecture that the 4d Schur ring is generated by normal-ordered products of ∂ i J subject to (19) [45].Let us call this ring R MAD ∞ and define Indeed, as explained in the Supplemental Material, recent results on arc spaces imply the counting of operators in R MAD ∞ matches (14).More precisely, where the lefthand side is the Hilbert series of R MAD ∞ .Since all operators involved are bosonic, this result is a highly non-trivial check of the claim that the 4d Schur ring is generated by products of ∂ i J subject to (19).
Next we apply the RG map (4) and reproduce the Macdonald index in terms of the IR degrees of freedom and Fermi statistics.

II. IR FERMI STATISTICS
When flowing to the IR, J → λ 1 + λ1 +, and, as explained around ( 9), the UV dynamics that lead to (19) manifest as IR Fermi statistics.Therefore, our goal is to apply the map in (10) and reproduce (21) in the IR.
Therefore, we must show Fermi statistics doesn't imply additional constraints.Intuitively, we expect this not to be an issue since the IR operators we consider do not probe the full emergent Schur ring.For example, they are blind to accidental higher-spin symmetries.
To make our discussion precise, we first write a UV basis of operators and make contact with the extremal Schur operators (5).As explained in the Supplemental Material, we can use results in algebraic geometry to show that a suitable basis consists of The extremal case (5) has n i+1 − n i = 2 and n 1 = 0.
Applying the RG map (4) to (22), we get a composite operator made of fermions.Due to Fermi statistics, which is generally stronger than (19), one may worry the operator vanishes.To show it does not, we pick a representative non-vanishing term.In general, there are multiple non-vanishing terms after distributing derivatives.We simply should make a consistent choice.To that end, we choose Clearly there is a one-to-one correspondence (not equality) between ( 23) and ( 22) after setting At the level of operators, In particular, since n i+1 − n i ≥ 2, we see ⌊ ni 2 ⌋ ̸ = ⌊ nj 2 ⌋ as long as i ̸ = j.Therefore, the fermions do not annihilate.It is also obvious that operators in (23) are linearly independent for different sets of m i , m ′ i .
As a result, (22) gives an IR basis.We see that Fermi statistics does not over-constrain our subring of observables, and we reproduce (21) in the IR.

III. DISCUSSION
FIG. 1: RG maps to the IR Schur sector, SFree Vector.We describe the flow between the UV MAD Schur sector, SMAD, and a closed subsector of the IR Schur operators, SFree Vector (yellow shading).IR Schur operators in the complement of SFree Vector (blue shading) come from non-Schur UV operators.
As far as we are aware, (10) is the first exact map of non-chiral quarter-BPS observables along a strongly coupled RG flow.UV dynamics giving rise to null relations are reduced to IR Fermi statistics (it would be interesting to derive these relations via UV defect endpoint operators).Noting that the IR gauginos are related by supersymmetry to the coordinates on the Coulomb branch, and thinking of an arc space as an infinite-dimensional generalization of a tangent space, we see that our results constitute a certain geometrical completion of Seiberg-Witten theory for the MAD SCFT.
It is surprising that a Coulomb branch flow knows so much about the Schur sector (this sector is typically associated with the Higgs branch).At the same time, this fact strengthens our conjecture (1) and shows that Coulomb branch and Schur sector physics unify into a deeper structure (see also [30,31]).
The above phenomena are indirectly related to those in [23].There the authors computed a less refined limit of the superconformal index by summing over massless and massive Coulomb branch BPS states.We instead keep track of the Schur operators along the RG flow.In so doing, we recover additional 4d quantum numbers (SU (2) R charges).
When does the above construction generalize to other Coulomb branch flows?A reasonable conjecture is that it generalizes whenever the UV "hidden" symmetries of the Schur ring (Virasoro here) are all related to symmetries of the full 4d theory that are not explicitly broken along the RG flow and do not decouple (SU (2) R in the present case).Indeed, as we show in the Supplemental Material, (A 1 , A 2r ) SCFTs have similar IR embeddings of their Schur sectors.These theories have purely Virasoro hidden symmetry related to unbroken SU (2) R .
On the other hand, consider Coulomb branch flows for theories with W N >2 symmetry.For example, the (A 2 , A 3 ) SCFT has (hidden) W 3 symmetry [23].Using the Macdonald index [14,22], it is easy to argue that the W 3 current sits in a Ĉ1(0,0) multiplet.It is simple to check that the corresponding Schur operator cannot be built from gauginos and derivatives.In this case, we expect the W 3 symmetry to decouple along flows to generic points on the Coulomb branch [46] Let us also discuss how our work is related to known free field constructions [18,19].There the authors studied Higgs branch RG flows and focused on massless degrees of freedom (in N > 2 SUSY, such moduli spaces embed in larger structures that include Coulomb branches).In these cases, some of the symmetries are spontaneously broken, but one can construct UV 2d VOA operators in terms of IR 2d VOA degrees of freedom (see also related work in [32]) [47].
We have instead followed 4d operators along Coulomb branch RG flows.Understanding such flows from the Schur sector perspective is crucial, since the Coulomb branch is the most universal moduli space of an interacting 4d N = 2 SCFT [48].A more closely related 2d version of our discussion in the spirit of [18,19] is to fermionize the Coulomb gas construction of the Lee-Yang theory (along the lines of [33,34]).However, this would require us to express the IR version of the UV stress tensor as a composite not built purely out of 2d avatars of IR gauginos (see (2.1) of [34]) [49].
As emphasized in (2), ( 6), (7), and Fig. 1, the full IR Schur sector is connected via RG flow to various UV sectors.It will be interesting to use these maps to further constrain the UV (from our Coulombic generation conjecture, we expect the corresponding UV operators generate the MAD theory).For example, we can consider products of operators in (10) with other operators and infer aspects of the C spectrum [13].
Finally, it is tempting to take our results and search for a geometrical completion of Seiberg-Witten theory in more general 4d N = 2 QFTs.
If (A.27) holds, then I MAD M can be obtained from words built out of J and the derivative ∂ + , subject to the condition J 2 = 0.This is because R MAD ∞ defined in (20) of the main text contains all such operators.Therefore, we would like to check whether Here V n,k is the set of operators built from k J's and n derivatives ∂ + (subject to J 2 = 0), which, in general, take the form (we have suppressed complex coefficients in front of each term in the sum for simplicity) and dim V n,k is the dimension of each such linearly independent subspace.
As a result, to prove (A.27), we need to show that Interestingly, the RHS can be identified with a qhypergeometric series [50] ∞ k=0 for arbitrary r and a i , where we have also explicitly written the perturbative expansion for the first few orders.
It is easy to compute dim V n,k numerically to high order and verify this statement.Below, we will also prove it analytically.
To do so, it is useful to introduce the concept of an arc space.An arc space is a special kind of topological space that is intimately connected with the singularities of algebraic varieties.In the context of QFT, such spaces have appeared in various places (e.g., see [18,20,27]).In our case, the arc space encodes the operators in the Schur sector, and one can characterize the Schur spectrum from the associated arc space Hilbert series.
Here we follow [20] and start with the affine scheme From this structure, we have the jet scheme, X m , which can be thought of as a generalization of the notion of a tangent space.It is given by 33) In writing the above ideals, we introduced a derivation, D, such that D(x Given this discussion, we can consider the inverse limit and obtain the arc space 34) In [18,20,27], the above construction arises in the context of 2d VOAs, and R = R V is the associated Zhu's C 2 algebra.Roughly speaking, this is a commutative 2d algebra obtained by getting rid of all operators containing derivatives in the VOA V R V = V/C 2 (V) , C 2 := Span {a −ha−1 b|a, b ∈ V} .
(A.35) When the 4d SCFT has a Higgs branch, Zhu's C 2 algebra enables one to reconstruct this moduli space [27].In the case of the MAD theory, there is no Higgs branch.However, Zhu's C 2 algebra still contains important information about this theory.Indeed, from (18) of the main text, it is easy to see that R Vir c=−22/5 = C[x]/⟨x 2 ⟩ . (A.36) Constructing the arc space associated with (A.36) and showing that its operators are counted as in (A.28) is strong evidence for the fact that the arc space undoes the twisting of the MAD theory that led to the Lee-Yang VOA.At a physical level, the arc space therefore provides an inverse map from 2d to 4d for the case at hand (and also for the generalizations we discuss in section III of the Supplemental Material).
To prove (A.28), we consider N = 1 and x 1 = J in (A.34).The derivation, D, can be regarded as the derivative acting on local operators.Then we can identify x The ideal is generated by f (0) 1 = J 2 and, more generally, all f (i) ).As a result, the above arc space is exactly the space describing operators made out of J and derivatives in (A.29) subject to the constraint J 2 = 0.In other words, R ∞ → R MAD ∞ , and we need to consider .37) (a) All IR Schur operators need not come from UV Schur operators.(b) In general SCFTs, UV Schur operators can decouple along flows to the Coulomb branch.
) Numerical indices are SU (2) R weights, and signs indicate spin weights.These relations imply looking for Schur operators with h ≤ 4.
This definition then specifies the action ofD on all C[x (i) 1 , • • • , x (i) N ].In particular, f (i) j := D i (f j ) is also a polynomial.