Bifiltered Affine Scheme in 4d N=2 SCFTs

• Bifiltered affine scheme is an algebro-geometric framework that encodes protected data of 4d N=2 SCFTs via a bifiltration on coordinate rings.
• It constructs the scheme from a bigraded polynomial ring and an arbitrary ideal, enabling the recovery of key indices like the Macdonald and Schur indices along with the Higgs branch.
• The methodology leverages jet schemes and Betti-number extremization to uniquely select physical theories by matching algebraic syzygies with OPE decoupling relations.

A bifiltered affine scheme is an algebro-geometric structure designed to encode protected data of four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs), particularly the operator product expansion (OPE) decoupling relations. Constructed from a polynomial ring with a bigrading and an (a priori arbitrary) ideal, the resulting affine scheme's coordinate ring is endowed with a bifiltration whose structure is central to recovering the Macdonald and Schur indices, as well as the Higgs branch, of the theory. Key features of the construction include the generalization to non-bihomogeneous ideals, a correspondence between algebraic data and physical observables, and a Betti-number extremization principle that uniquely selects the scheme corresponding to a given SCFT (Kang et al., 4 Feb 2026).

1. Formal Definition and Properties

Given a polynomial ring $S = \mathbb{C}[x_1, ..., x_m]$ with a prescribed bigrading $\deg(x_j) = (a_j, b_j) \in \mathbb{Z}^2$, one considers an ideal $I \subset S$ (not required to be bihomogeneous). The quotient $R = S/I$ serves as the coordinate ring of an affine scheme $X = \operatorname{Spec} R$. Although $R$ need not inherit a bigrading, it admits a bifiltration defined by the surjection $\pi: S \to R$: $$F^{p,q}R = \pi\left( \bigoplus_{p' \leq p,\, q' \leq q} S_{p',q'} \right)$$ These subspaces $F^{p,q}R$ satisfy:

• $F^{p,q}R \subset F^{p',q'}R$ for $p \leq p'$, $q \leq q'$
• $\displaystyle\bigcup_{p,q} F^{p,q}R = R$
• $F^{p,q}R \cdot F^{p',q'}R \subset F^{p+p',q+q'}R$

The triple $(X, R, F^{\bullet,\bullet})$ is referred to as a bifiltered affine scheme. In practice, the term "bifiltered" is often attributed to $R$ alone when $X$ is understood.

2. OPE Decoupling, Ideals, and Bifiltration

In the context of 4d $\mathcal{N}=2$ SCFTs, operator decoupling—where specific composite operators vanish in OPEs—naturally gives rise to algebraic relations. For example, a relation $T^{n+1} \sim 0$ for the stress-tensor superfield $T$ leads to $R = \mathbb{C}[x]/(x^{n+1})$ with $x \leftrightarrow T$. More generally, the vanishing of all components of $\partial_{x_d}P(x_2,...,x_k)$, derived from the associated vertex operator algebra, yields the Jacobian ideal $$I_P = (\partial_{x_2}P, ..., \partial_{x_k}P) \subset \mathbb{C}[x_2, ..., x_k]$$, and thus $R = S/I_P$.

The bigrading is chosen as:

• $q$ corresponds to the difference of conformal weight and $R$-charge
• $T$ represents the second $U(1)$ filtration intrinsic to $\mathcal{N}=2$ short multiplets

This algebro-geometric encoding of OPE decoupling relations allows extraction of protected information about the SCFT.

3. Extraction of the Macdonald Index and Higgs Branch

To recover protected indices, the construction employs the arc space (infinite jet space) of $X$. For each $n \geq 0$, the $n$-th jet algebra is defined as

$$J_nR = \frac{\mathbb{C}[x_j^{(0)}, ..., x_j^{(n)}\,|\,j=1...m]}{(f_i^{(0)}, ..., f_i^{(n)})}$$

where $f_i^{(\alpha)}$ is the $t^{\alpha}$ coefficient in the expansion of each generator $f_i$ of $I$. The direct limit $J_\infty R = \varinjlim_n J_n R$ is naturally trifiltered by $\deg(x_j^{(\alpha)}) = (\alpha, a_j, b_j)$.

The associated graded $$\operatorname{gr}(J_\infty R) = \bigoplus_{p,q,T} \operatorname{gr}^{p,q,T}(J_\infty R)$$ is tri-graded, and its Hilbert series is

$$HS_{p,q,T}(\operatorname{gr}(J_\infty R)) = \sum_{p,q,T} (\dim_\mathbb{C} \operatorname{gr}^{p,q,T}(J_\infty R))\, p^p\, q^q\, T^T$$

Specializing to $p = q$ produces the Macdonald index: $$I_{\mathrm{Mac}}(q,T) = HS_{q,q,T}(\operatorname{gr}(J_\infty R))$$ with the Schur index obtained by taking $T \to 1$.

The Higgs branch is given by the reduced scheme $X_{\mathrm{red}} = \operatorname{Spec}(R/\sqrt{0})$. In all Argyres–Douglas series analyzed, $X_{\mathrm{red}}$ is a point, as expected for SCFTs with a trivial Higgs branch (Kang et al., 4 Feb 2026).

4. Extremization via Betti Numbers and Syzygies

The Jacobian ideal $I_P$ generally depends on continuous moduli (coefficients in $P$). To select the SCFT corresponding to physical protected data, the construction imposes Betti-number constraints, specifically using the minimal free resolution of $J_1 R$. Employing the syzygy matrix $M$, the conditions are as follows: $$\sum_{d=2}^k \left( g_{d,1} f_{d,0} + g_{d,0} f_{d,1} \right) = 0$$ with exactly one minimal syzygy of bidegree $(k+N+2, 1)$ and, in the complete intersection (CI) case, also one of bidegree $(k+N+1,1)$. Explicitly, the Betti-number conditions are: $$R=S/I_P\text{ is CI,} \quad \beta_{2,(k+N+1,1)}(J_1R)=1,\quad \beta_{2,(k+N+2,1)}(J_1R)=1$$ These integer constraints isolate a unique point in the moduli space of $P$, corresponding to the physical theory labeled by $(k-1, N-1)$ and ensuring the bifiltered affine scheme yields the correct Macdonald index.

5. Illustrative Examples

Several examples demonstrate the effectiveness of the bifiltered affine scheme formalism in encoding SCFT data:

Theory	Coordinate ring $R$ or Potential $P$	Macdonald index (via Hilbert series)	Higgs branch
$(A_1,A_{2n})$	$\mathbb{C}[x_2]/(x_2^{n+1})$, $\deg(x_2)=(2,1)$	$\mathrm{PE}\left[\sum_{j=1}^n q^{2j} T^j\right]$	$\operatorname{Spec}\mathbb{C}$ (point)
$(A_2,A_9)$	$P = \frac{1}{21}x_2^7 + x_2^4 x_3^2 + x_2 x_3^4$	$HS_{q,q,T}(\operatorname{gr}(J_\infty R))$ agrees with known result	point
$(A_3,A_4)$	$$P = \frac{1}{10}x_2^5 + x_2^2 x_3^2 + x_2 x_4^2 + x_3^2 x_4$$	as above	point

In each case, the unique point in the moduli space selected by the Betti-number extremization precisely reproduces the protected operator spectrum as encoded in the Macdonald index. The Higgs branch coincides with the scheme-theoretic support as expected for the theory.

6. Significance and Outlook

The bifiltered affine scheme formalism provides a systematic algebro-geometric framework for encoding OPE decoupling relations and recovering protected information of 4d $\mathcal{N}=2$ SCFTs (Kang et al., 4 Feb 2026). The connection to arc space and jet schemes enables the calculation of the Macdonald index from first principles, while the geometric extremization principle—via Betti-number constraints—selects the physically relevant point within the continuous moduli of possible schemes. This construction suggests a route to a finer algebro-geometric classification of $\mathcal{N}=2$ SCFTs by identifying possible extremal bifiltered affine schemes corresponding to known and conjectural theories.

A plausible implication is that this extremization approach could distinguish all physically consistent SCFTs within a broad class of algebro-geometric models constructed from decoupling relations, although the generality of this result beyond the presented series remains to be investigated. The formalism also unifies several distinct protected quantities in a single geometric object, providing new tools for the study of SCFT moduli spaces and their deformation theory.