Authorship. Claude Opus 5.5, an AI system developed by Anthropic. Almost all of this work, from the computations to the text, was generated by the AI; hence the first authorship (which alphabetical order happens to confirm).
1 Introduction
A first-order action \(S=\oint\theta-\oint H\,dt\) with \(d\theta=\omega\) defines a path-integral version of geometric quantization. When \(H\) is the moment map of a torus action, the partition function \(Z=\mathrm{Tr}\,e^{-\beta H}\) on a circle is an equivariant index and can be written as a fixed-point sum. Gauged versions of this set-up appear in several places:
the ADHM matrix model of the 4d quantum Hall effect (Barns-Graham et al. 2018), whose Higgs branch is the instanton moduli space (\(\mathrm{Hilb}^n(\mathbb C^2)\) in rank one);
the Chern–Simons matrix model on the framed Jordan quiver (Dorey et al. 2016a; Hu et al. 2024);
the Polychronakos matrix model of the quantum Hall effect (Polychronakos 2001).
In each of these papers the relation between the bare Chern–Simons level and the line bundle that is actually quantized involves a shift. A refined version of the same question arises when the ADHM model is lifted to the \(S_n\)-equivariant character of the Procesi bundle, whose components are the modified Macdonald polynomials; there the ambiguity is not a level shift but the way a constraint is imposed.
What is already known. The shifts \(k'\to k'-N\) (Barns-Graham et al. 2018) and \(k+n\mapsto k\) (Hu et al. 2024) are both attributed by their authors to normal ordering of the fundamental-field bilinear in the trace of the Gauss law. The same “ordering issue” already appears in (Dorey et al. 2016b, 2016a), so the identification of these two shifts is not new. The shift \(k\to k+1\) of (Polychronakos 2001), which Polychronakos calls “a quantum reordering effect”, has been explained in three different ways:
as the zero-point energy \(N^2/2\) of the adjoint oscillators (Hellerman and Susskind 2001);
as the comparison of ground-state energies, or equivalently a Vandermonde factor, between the matrix model and the Calogero model (Cappelli and Riccardi 2005), and as a Jacobian (footnote in (Barns-Graham et al. 2018));
as “normal ordering a constraint” (Goldman and Senthil 2022).
In addition, (Barns-Graham et al. 2018) lists it together with the Gauss-law shift. Two related questions are whether the expected half-integer (\(K^{1/2}\) or spin\(^c\)) shifts appear (Braun et al. 2015; Lyris et al. 2021; Vaughan 2017; Hall and Kirwin 2007; Walton 2022), and whether the continuum coherent-state path integral can be trusted (Wilson and Galitski 2011; Kordas et al. 2014, 2016, 2019; Kochetov 2019).
This paper organizes these effects into one dictionary. Our main statements are:
A rule (Proposition 1). Each field carries two regularization exponents, \(s_\varepsilon\) for its zero-point energy and \(s_w\) for its contribution to the Gauss law. The partition function is an overall factor times the normal-ordered index at a shifted level.
A lemma (Lemma 1). At each Jeffrey–Kirwan pole, the product of the weights of all matter modes equals the product of the tangent weights. Hence uniform symmetric regularization computes \(\chi(M,\mathcal{O}(k')\otimes K^{1/2})\).
A no-go statement (Proposition 2). Gauge-covariant time lattices have \(s_w=0\), so they never shift the level.
A dictionary of the known shifts (Table 1). The shifts of (Barns-Graham et al. 2018) and (Hu et al. 2024) are recovered as \(s_w=1\) on fundamental fields, in agreement with their authors. The shift of (Polychronakos 2001) is \(s_\varepsilon=\tfrac12\) on the adjoint: its Gauss law is normal ordered (\(s_w=0\)), and the shift comes from the adjoint zero-point energy, whose off-diagonal part the Calogero coupling absorbs. This sides with (Hellerman and Susskind 2001; Cappelli and Riccardi 2005) and shows that attributing it to the ordering of the constraint (Goldman and Senthil 2022), or grouping it with the Gauss-law shift (Barns-Graham et al. 2018), mixes up two different exponents.
The limits of the rule (Section 6). For Hamiltonians that are quadratic in a generator, the continuum path integral distorts the spectrum, and the time lattice has non-commuting limits.
The Procesi bundle (Section 7). Adding the joint eigenvalues as fields, the scheme-theoretic, reduced and derived impositions of the isospectral constraint give different \(S_n\)-equivariant characters, all with the same invariant part \(\chi(\mathcal{O}(k))\). For flag versions of the model we conjecture a closed formula Equation 2 for the virtual count at every level.
Most individual entries are standard, and some identifications (item 4 and part of the Calogero discussion) are already in the literature, as summarized above. What we did not find in the literature, within the scope of our search, are:
the two-exponent rule together with the criterion \(s_\varepsilon=s_w=s\) for a \(K^s\) interpretation;
the statement that gauge-covariant time lattices never shift the level;
the JK-pole argument for the \(K^{1/2}\) twist;
the observation that on phase spaces with non-trivial \(K\) the midpoint lattice does not reproduce the half-form shift of the level;
the non-commuting limits of the time lattice for non-linear Hamiltonians;
the comparison of the isospectral-constraint prescriptions, and the level-\(k\) formula Equation 2 for virtual counts of nested Hilbert schemes (the case \(k=0\) follows from (Minddal 2026)).
The mathematical framework of Section 7 (Haiman’s theorems, Ginzburg’s isomorphism, the virtual-dimension criterion for flags) is known, as summarized there.
Organization. Section 2 sets up the models and the regularization data, and Section 3 states and proves the rule. Section 4 applies it to the three phase spaces, Section 5 collects the dictionary, and Section 6 discusses the limits of the rule. Section 7 lifts the ADHM model to the Procesi bundle. Section 9 contains remarks on JK residues and wall crossing, and Section 10 lists the ranges of the numerical checks.
The Jeffrey–Kirwan (JK) residue machinery (Hori et al. 2015; Benini et al. 2015; Hwang et al. 2015) is used throughout, mostly as a tool. Its application to the rank-one ADHM model (Section 9) adds only minor points to the general results of (Hori et al. 2015; Hwang et al. 2015); see also (Barns-Graham 2018).
2 Set-up
2.1 Models
Let \(G=U(n)\) with gauge field \(A_t\) and bare Chern–Simons level \(k'\), and let the matter consist of complex fields in representations of \(G\) with symplectic form \(i\,d\bar\phi\wedge d\phi\). The action is \[ S=\int dt\,\Big[\sum_{\rm fields} i\,\bar\phi\,D_t\phi - k'\,\operatorname{tr}A_t - H\Big], \] where \(H\) is the moment map of a torus \(T\) (a linear combination of number operators). The Gauss law is \(\mu_{\mathbb R}=k'\mathbf 1\), so the FI parameter equals the bare level, \(\zeta=k'\). A holomorphic constraint \(\mu_{\mathbb C}=0\), if present, is imposed by Koszul fermions \(\chi\) with \(Q=\sum(\mu_{\mathbb C})_{ab}\chi_{ba}\), \(Q^2=0\). The partition function on \(S^1\) with periodic boundary conditions reduces, after static diagonal gauge fixing, to a torus integral with Haar measure \(\prod_{a\ne b}(1-w_a/w_b)\). We consider three examples.
ADHM, rank one: adjoints \(B_1,B_2\) (weights \(t,q\)), fundamental \(I\) and antifundamental \(J\), and \(\mu_{\mathbb C}=[B_1,B_2]+IJ\). The Higgs branch is \(M=\mathrm{Hilb}^n(\mathbb C^2)\). The scheme \(\mu_{\mathbb C}^{-1}(0)\) is a reduced complete intersection (Gan and Ginzburg 2004), so imposing \(\mu_{\mathbb C}=0\) by Koszul fermions or by restricting to the variety gives the same character. We use the conventions of (Haiman 2002): \(T_\mu=t^{n(\mu)}q^{n(\mu')}\), \(\mathcal{O}(1)=\wedge^n\mathcal B\), and the tangent weights \(t^{1+l}q^{-a}\), \(t^{-l}q^{1+a}\).
Grassmannian: \(N\) fundamentals \(I_i\) with weights \(x_i\) and nothing else. Then \(M=\mathrm{Gr}(n,N)\) and \(K_M=\mathcal{O}(-N)\).
Framed Jordan quiver: one adjoint \(Z\) (weight \(q\)) and \(f\) fundamentals (weights \(a_i\)). Then \(M\cong\mathrm{Quot}^n(\mathcal{O}^{\oplus f}_{\mathbb A^1})\) (Hu et al. 2024).
2.2 Regularization data
A single bosonic mode of weight \(x=e^{-\beta\varepsilon}\) and holonomy \(w\) has the one-loop factor \(1/\det(\partial_\tau+\varepsilon-ia)\). Different regularizations give \(x^{s_\varepsilon}w^{s_w}/(1-xw)\):
normal ordering (equivalently, the forward Q-symbol time lattice) gives \(s_\varepsilon=s_w=0\);
zeta-function regularization with symmetric Matsubara cut-off gives \(s_\varepsilon=s_w=\tfrac12\);
anti-normal ordering gives \(s_\varepsilon=s_w=1\);
time lattices give \(s_w=0\) with \(s_\varepsilon\in\{0,\tfrac12,1\}\) (Proposition 2).
Fermions contribute the inverse factor. In operator language, \(s_w\) is the ordering of the matter bilinear in the Gauss law, and \(s_\varepsilon\) is the ordering of \(H\).
3 The rule
Proposition 1 (Rule) Let \(\delta=\sum_f c_f\, s_{w,f}\) be the sum, over charged fields, of the \(U(1)\subset U(n)\) charge per color \(c_f\) times \(s_w\). Let \(\mathcal X=\prod_{\text{modes}}x^{s_\varepsilon}\), with fermions contributing \(x^{-s_\varepsilon}\). Then \[ Z_{\rm reg}(k')=\mathcal X\cdot Z_{\rm normal}(k'-\delta). \]
Proof. For adjoint fields the factors \(w^{s_w}\) cancel between \((a,b)\) and \((b,a)\). For a field of charge \(c_f\) they give \(\prod_a w_a^{c_f s_w}\), which combines with the Chern–Simons factor \(\prod_a w_a^{-k'}\). What remains is the \(x^{s_\varepsilon}\) factors, which are independent of \(w\).
Proposition 2 (Gauge-covariant lattices) Place the links \(U_j\) on the hopping terms, \(\bar z_j(z_j-U_jz_{j-1})\), and evaluate \(H\) by any rule that is local in time and gauge covariant. Then the lattice determinant of a field in representation \(R\) is \(\prod_{\omega}(a^M-\omega b^M)\), where \(\omega\) runs over the eigenvalues of the holonomy in \(R\). Hence \(s_w=0\) and the lattice never shifts the level. In the continuum limit, the forward Q-symbol, midpoint and same-time P-symbol prescriptions give \(s_\varepsilon=0,\tfrac12,1\) respectively.
Proof. The kinetic matrix is block-cyclic with diagonal \(a\mathbf 1\) and sub-diagonal \(-bR(U_j)\). Its determinant depends only on the conjugacy class of \(R(U_M\cdots U_1)\). The values of \(a\) and \(b\) follow from expanding to first order in \(\delta=\beta/M\).
Lemma 1 (Zeta function \(=K^{1/2}\)) At every isolated JK pole, the product of the weights \(x w^\rho\) of all matter modes equals the product of the tangent weights of \(M\) at the corresponding fixed point. The reason is that the tangent character is the matter character minus the gauge character, and the gauge weights \(w_a/w_b\) multiply to one. Consequently, uniform zeta-function regularization gives \[ Z_\zeta(k')=\chi\big(M,\mathcal{O}(k')\otimes K_M^{1/2}\big). \] More generally, \(Z_{\rm reg}\) equals \(\chi(M,\mathcal{O}(k')\otimes K_M^{s})\) for all \(k'\) if and only if \(s_\varepsilon=s_w=s\) for all fields.
Remark 1 (Global anomaly). If \(\delta\notin\mathbb Z\), for example with an odd number of fundamentals and no antifundamentals in zeta-function regularization, then \(Z_\zeta\) is gauge invariant only for half-integer \(k'\). This is the bosonic analog of the fermionic global anomaly discussed in (Hori et al. 2015). Lattices with integer level cannot reach the Dirac index in this case.
4 Three phase spaces
4.1 \(\mathrm{Hilb}^n(\mathbb C^2)\): trivial \(K\)
Because the field content contains both \(I\) and \(J\), the level shift in zeta-function regularization is \(\delta=n(s_{w,I}-s_{w,J})=0\). Equivariantly \(K\cong\mathcal{O}\otimes(qt)^n\), so the \(K^{1/2}\) twist is only an overall factor \((qt)^{n/2}\), coming from the zero-point energy \(E_0=\tfrac12(\sum_{\rm bos}\varepsilon-\sum_{\rm ferm}\varepsilon)\). The midpoint lattice (\(s_w=0\), \(s_\varepsilon=\tfrac12\)) and the zeta function (\(s_w=s_\varepsilon=\tfrac12\)) therefore agree on \(\mathrm{Hilb}^n\). The ordering in (Barns-Graham et al. 2018) (and the “ordering issue” of (Dorey et al. 2016b)) treats \(I\) anti-normally and \(J\) normally (\(s_{w,I}=1\), \(s_{w,J}=0\)), which gives \(k=k'-N\). This is not a power of \(K\). If bosons and Koszul fermions, or \(B_1\) and \(B_2\), are regularized differently, the overall factors are not powers of \(K\) either.
We checked the baseline \(Z_{\rm normal}=\chi(\mathrm{Hilb}^n,\mathcal{O}(k))\) in four independent ways for \(n\le4\), \(0\le k\le3\):
the Molien integral of (Barns-Graham et al. 2018);
JK residues, where each color-partition residue equals the corresponding Atiyah–Bott term (checked for \(n\le5\));
the Atiyah–Bott sum;
the Hilbert series of products of alternants (Haiman 2001).
For gauge-covariant lattices with \(n=2,3\), we computed every determinant numerically (random \(U(n)\) links for the invariance check, and holonomy integrals for \(Z\)). The results converge as \(O(1/M)\) to \(\mathcal X\cdot Z_{\rm normal}(k')\) with no level shift.
4.2 \(\mathrm{Gr}(n,N)\): non-trivial \(K\), compact
Here \(\delta=Ns_w\), and the rule gives:
normal ordering: \(s_{(k'^n)}(x)=\chi(\mathcal{O}(k'))\);
zeta function: \((\det x)^{n/2}s_{((k'-N/2)^n)}(x)=\chi(\mathcal{O}(k')\otimes K^{1/2})\), a level shift by \(N/2\);
anti-normal ordering: \(\chi(\mathcal{O}(k')\otimes K)\), a level shift by \(N\);
every gauge-covariant lattice, including the midpoint rule: \((\det x)^{sn}s_{(k'^n)}\).
The naive identification “midpoint rule \(=\) Weyl ordering \(=\) metaplectic correction” is therefore correct for the zero-point part and wrong for the level part. On \(\mathrm{Hilb}^n\) the failure is invisible because \(K\) is trivial. This does not contradict the semiclassical statements that the Weyl symbol removes the Solari–Kochetov correction (Braun et al. 2015), or that the metaplectic correction is tied to the correct continuum limit of coherent-state path integrals (Lyris et al. 2021). Those statements concern the continuum or the semiclassical propagator. Ours concerns the Chern–Simons level in a gauge-covariant time discretization, where the half-form shift must be put in by hand through the bare level. For half-form and metaplectic-c quantization with torus actions see (Vaughan 2017; Hall and Kirwin 2007); in the quantum-Hall context the metaplectic correction is sometimes deliberately omitted (Walton 2022). We checked this for \((n,N)\in\{(1,2),(1,3),(2,3),(2,4)\}\) and \(k\le3\).
4.3 Jordan quiver: non-trivial \(K\), non-compact
In normal ordering the Molien integral and the JK residues reproduce Theorem A of (Hu et al. 2024) (which notes that the formula first appeared in (Dorey et al. 2016a)), \[ \mathrm{ch}\,\mathcal H=\widetilde H_{(k^n)}(a;q)\prod_{i=1}^n\frac{1}{1-q^i}, \] where \(\widetilde H_\mu\) is built from Jing operators as in that reference. We checked this for \(f\le2\), \(n\le3\) and \(k\le3\). In zeta-function regularization the JK residues multiplied by (product of tangent weights)\(^{1/2}\) reproduce \(Z_\zeta\), which confirms Lemma 1 here with \(K\cong\mathcal L_{\det}^{-f}\) and a level shift of \(f/2\). The shift \(k+n\mapsto k\) of (Hu et al. 2024) is \(s_w=1\) on fundamentals. For \(f=1\), the Calogero model obtained from the Hermitian (real-polarized) version (Polychronakos 2001) with coupling \(\ell(\ell-1)\), \(\ell=k+1\), satisfies \[ Z_{\rm Cal}(\ell=k+1)=q^{n^2/2}\,a^{-kn}Z_{\rm normal}(k), \] which we checked for \(n=2\) by finite differences. The factor \(q^{n^2/2}\) is the zero-point energy of the adjoint (\(s_\varepsilon=\tfrac12\)). Its off-diagonal part \(q^{n(n-1)/2}\) is the degree of the Vandermonde factor, which the Calogero coupling absorbs. Thus the shift of (Polychronakos 2001) is an \(s_\varepsilon\) effect, not an \(s_w\) effect. The relation to the zero-point energy is not new: (Hellerman and Susskind 2001) identify the \(N^2/2\) zero-point term as the cause of the shift, and (Cappelli and Riccardi 2005) compare the ground-state energies \(N^2/2\) and \(N/2\) of the matrix and Calogero models. What the rule adds is the following. Polychronakos’s Gauss law is normal ordered, so it has \(s_w=0\). Calling the effect a “level shift” is therefore a matter of which variables one uses (the Calogero coupling), not a Gauss-law effect. In particular, it should not be identified with the shifts of (Barns-Graham et al. 2018; Hu et al. 2024), contrary to the grouping in (Barns-Graham et al. 2018) and the attribution in (Goldman and Senthil 2022).
5 The dictionary
| Reported shift | Model | Explanation in the literature | Position in the rule | Geometric meaning |
|---|---|---|---|---|
| \(k'\to k'-N\) (Barns-Graham et al. 2018) | ADHM | normal ordering of \(\varphi\varphi^\dagger\) (Barns-Graham et al. 2018; Dorey et al. 2016b) | \(s_{w,I}=1\), \(s_{w,J}=0\) | not a power of \(K\) |
| \(k+n\mapsto k\) (Hu et al. 2024) | Jordan quiver | normal ordering of \((BA)\) (Hu et al. 2024) | \(s_w=1\) on fundamentals | \(K^1\) (non-equivariantly) |
| \(k\to k+1\) (Polychronakos 2001) | Jordan quiver, \(f=1\) | reordering (Polychronakos 2001); zero point (Hellerman and Susskind 2001; Cappelli and Riccardi 2005); Jacobian (Barns-Graham et al. 2018); constraint ordering (Goldman and Senthil 2022) | adjoint \(s_\varepsilon=\tfrac12\), \(s_w=0\) | overall factor, read as Calogero coupling |
| spin\(^c\) / \(K^{1/2}\) | all | half-form correction | \(s_\varepsilon=s_w=\tfrac12\) for all fields | level shift iff \(K\) non-trivial |
| gauge-covariant lattice | all | — | \(s_w=0\) | no level shift |
6 Limits of the rule
The rule assumes that \(H\) is linear in number operators. For \(H=\varepsilon N+\lambda N^2\) (the one-mode sector of \(\operatorname{tr}(B^\dagger B)^2\)), a Hubbard–Stratonovich transformation evaluates the continuum coherent-state path integral exactly. For a quadratic symbol \(h\) it gives \(\sum_{m\ge0}e^{-\beta h(m+1/2)}\).
With the Q-symbol, the energies become \(E_m+\varepsilon/2+3\lambda/4+2\lambda m\). The shift depends on \(m\), so the spectrum is distorted.
With the Weyl symbol the shift is the constant \(-\lambda/4\). This is an accident of \(N^2\).
The Q-symbol time lattice equals \(\mathrm{Tr}(:\!e^{-\delta h}\!:)^M\). It converges as \(O(1/M)\) on a truncated Fock space. At any finite \(M\), however, the transfer-matrix eigenvalues exceed one for \(m\gtrsim3.7M\), so the full trace diverges. This refines the statement of (Wilson and Galitski 2011) that the time-discretized path integral does not suffer from the breakdown: the order of the limits matters. For proposed consistent continuum formulations and the ensuing debate, see (Kordas et al. 2014, 2016, 2019; Kochetov 2019). We do not take a position on that debate. Our statement concerns only the lattice.
7 Lifting to the Procesi bundle: another regularization ambiguity
The rank-one ADHM model computes \(\chi(\mathrm{Hilb}^n,\mathcal{O}(k))\). A natural refinement of this is the \(S_n\)-equivariant character \[ Z_P(n,k)=\sum_{|\mu|=n}\frac{T_\mu^k\,\widetilde H_\mu(X;q,t)}{\prod_{x\in d(\mu)}(1-t^{1+l}q^{-a})(1-t^{-l}q^{1+a})}=\nabla^k h_n\Big[\frac{X}{(1-q)(1-t)}\Big], \tag{1}\] which is the Frobenius series of \(\Gamma(\mathrm{Hilb}^n,P\otimes\mathcal{O}(k))\) for the Procesi bundle \(P\). This follows from Haiman’s character formula for \(\Gamma(P\otimes\mathcal B^{\otimes l})\) (Haiman 2002), restricted to the \(S_{kn}\)-isotypic part \((k^n)\). Indeed, with \(B_\mu=\sum_{(r,s)\in d(\mu)}t^rq^s\) one has \(s_{(k^n)}[B_\mu]=T_\mu^k\). Its \(S_n\)-invariant and sign components are \(\chi(\mathcal{O}(k))\) and \(\chi(\mathcal{O}(k+1))\). In this section we ask how \(P\) can be produced by the first-order model, and find that the answer is again a question of regularization. We first record what is known.
Mathematically the gauge-theoretic description exists. Ginzburg’s isomorphism (Ginzburg 2012) \[\big(\mathbb C[\mathfrak X_{\rm norm}]\otimes\mathbb C^m[V]\otimes V^{\otimes m}\big)^{SL_n}\cong\Gamma(P\otimes\mathcal B^{\otimes m}),\] restricted by Schur–Weyl duality to the isotypic part \((k^n)\) of \(S_{kn}\) (compare (Bellamy et al. 2010)), says that \(\Gamma(P\otimes\mathcal{O}(k))\) is the level-\(k\) Gauss-law subspace of functions on the normalized isospectral commuting variety times \(V\). Here \(\mathfrak X\) consists of commuting pairs together with an ordered list of their joint eigenvalues. The quantization of \(P\) through rational Cherednik algebras is also known; see e.g. (Gordon and Stafford 2004).
In physics, \(P\) and \(\widetilde H_\mu\) appear as the index of nested (flag) quiver quantum mechanics (Chuang et al. 2015; Bonelli et al. 2021, 2024). There \(S_n\) does not act on a single Hilbert space; the Frobenius series is assembled from the flag types \(\gamma\), with coefficients \(\langle Z_P,h_\gamma\rangle=\chi(P^{S_\gamma}\otimes\mathcal{O}(k))\). In our conventions the \(g=0\) formula of (Chuang et al. 2015) is \(Z_P(n,p-1)\). The virtual (supersymmetric) count deviates from \(\chi\) for singular nested Hilbert schemes (Bonelli et al. 2021). The virtual dimension accounts for this: it equals the actual dimension \(2n\) only when every step of the flag adds one box (Bonelli et al. 2021, 2024). For \(k=0\) the virtual counts follow from a recent theorem of Minddal (Minddal 2026).
Isospectral constraints. Add \(n\) points \((x_i,y_i)\in\mathbb C^2\) (weights \(t,q\), permuted by \(S_n\)) and impose that they form the joint spectrum of \((B_1,B_2)\): \(g_{ab}=\operatorname{tr}B_1^aB_2^b-\sum_i x_i^ay_i^b=0\) for \(1\le a+b\le n\). Let \(A^k\) be the span of products of \(k\) alternants (the det\(^k\) semi-invariants of the ADHM data (Gan and Ginzburg 2004; Ginzburg 2012)), let \(\Lambda=\mathbb C[x,y]^{S_n}\), and let \(\mathcal J=A^1\cdot\mathbb C[x,y]\) be the ideal generated by the alternants (not to be confused with the field \(J\)).
Proposition 3 The level-\(k\) Gauss-law subspaces are as follows.
Constraints imposed scheme-theoretically (by Lagrange multipliers): \(\mathcal H_{\rm sch}\cong A^k\otimes_\Lambda\mathbb C[x,y]\).
Constraints imposed on the reduced scheme: \(\mathcal H_{\rm red}\cong \mathcal J^k\otimes{\rm sgn}^k\). The same holds after normalization.
In both cases the \(S_n\)-invariant part is \(A^k\), with character \(\chi(\mathcal{O}(k))\).
Proof (Sketch). For (1): the generators \(g_{ab}\) are gauge invariant, and taking isotypic components for the reductive group \(GL_n\) is exact. For (2): the reduced isospectral commuting variety is irreducible, and its regular semisimple locus is dense (Ginzburg 2012). Restricting to diagonal \((B_1,B_2)\) with \(I=(1,\dots,1)\) is therefore injective on semi-invariants and maps \(\sum_j s_j\otimes f_j\) to \(\sum_j{\rm alt}(s_j)f_j\). The twist \({\rm sgn}^k\) comes from the permutation matrices needed to make the restriction \(S_n\)-equivariant.
Numerically, the reduced space reproduces Equation 1 for \(n=2\) (total degree \(\le12\)), \(n=3\) (degree \(\le10\)) and \(n=4\), \(k\le1\) (degree \(\le9\)). The scheme-theoretic space agrees with it only on the \(S_n\)-invariant part. For \(n=2\) the excess is \[ \mathcal H_{\rm sch}-\mathcal H_{\rm red}=s_{1,1}\,\frac{qt\,h_{k-1}(q,t)}{(1-q)(1-t)},\qquad h_{k-1}(q,t)=\sum_{j=0}^{k-1}t^jq^{k-1-j}, \] which consists of states supported on the diagonal, such as \(\det[I,B_1I](y_1-y_2)-\det[I,B_2I](x_1-x_2)\) at \(k=1\). These states are nilpotent on the scheme and vanish on the reduced scheme. We also confirmed this \(n=2\) statement with the matrix degrees of freedom kept explicitly. Imposing the constraints derivedly (by Koszul fermions over \(\Lambda\)) gives the Euler characteristic \(\chi(\mathcal{O}(k))Z_P(n,0)/\chi(\mathcal{O}(0))\). This again agrees with \(Z_P(n,k)\) only on the invariant part (and at \(k=0\)). Thus the original partition function \(\chi(\mathcal{O}(k))\) cannot distinguish the prescriptions, and their differences sit in the non-trivial \(S_n\) isotypic components. Only the reduced prescription produces \(P\otimes\mathcal{O}(k)\).
Flags. Ginzburg’s DG resolution (Ginzburg 2012) suggests a gauge-theoretic way to reduce: pass to a Borel or parabolic subgroup. For a composition \(\gamma=(\gamma_1,\dots,\gamma_m)\) of \(n\) we take block upper triangular \(B_1,B_2\in\mathfrak p_\gamma\) and \(I\in V\). The field \(J\) must vanish on the first \(m-1\) blocks, since that is the only way to have \(IJ\in\mathfrak p_\gamma\). We impose \([B_1,B_2]+IJ=0\) by Koszul fermions and treat the Levi factor \(\prod U(\gamma_i)\) by a Molien integral and the nilradical by ghosts. The resulting virtual count \(Z_{\rm virt}(\gamma,k)\) has the following properties.
For \(\gamma=(1^n)\) it reduces to the residue formula \[ \langle Z_P(n,k),h_1^n\rangle=\frac{1}{((1-q)(1-t))^n}\,\mathrm{CT}_z\Big[\prod_{i<j}\frac{(1-z_i/z_j)(1-qtz_i/z_j)}{(1-qz_i/z_j)(1-tz_i/z_j)}\,{\det}^k h_{kn}(z^{-1})\Big], \] which follows from (Bellamy et al. 2010; Ginzburg 2012). We checked it against the Atiyah–Bott sum as rational functions for \(n\le4\).
It equals \(\langle Z_P(n,k),h_\gamma\rangle\) exactly when \(\gamma=(1,\dots,1,m)\), which is the case where the virtual dimension \(n+\gamma_m+m-1\) equals \(2n\). We checked this for all compositions with \(n\le4\) and \(k\le1\); it is the virtual-dimension criterion of (Bonelli et al. 2021, 2024).
For \(k=0\) it agrees, for all compositions with \(n\le4\), with the formula obtained from (Minddal 2026) with the normalization of (Bonelli et al. 2021). For \(\gamma=(2,2)\), i.e. the nested Hilbert scheme \(N(1,4,2)=\mathrm{Hilb}^{(4,2)}\), which is normal with rational singularities (Ramkumar and Sammartano 2024), the deviation is \(-qt(q+t)(1+qt)/((1-q)^4(1-t)^4(1+q)^2(1+t)^2)\). The deviation therefore comes from excess dimension, not from non-reducedness.
The literature treats \(k=0\) (or, in (Bonelli et al. 2021), \(k=-1\)) only. Our computations suggest a closed form for all \(k\): \[ Z_{\rm virt}(\gamma,k)=\Big\langle Z_P(n,k),\;h_{\gamma_m}\prod_{i<m}\frac{h_{\gamma_i}[(1-qt)X]}{1-qt}\Big\rangle . \tag{2}\] Equivalently, at each fixed point \(\nu\) the sum over the tails of the flag equals \(\langle\widetilde H_\nu,h_{\gamma_m}\prod_{i<m}h_{\gamma_i}[(1-qt)X]/(1-qt)\rangle\), independently of \(k\). Since \(h_1[(1-qt)X]/(1-qt)=h_1\), the deviation vanishes exactly in the one-box-step case. We checked Equation 2 for all compositions with \(n\le4\) and \(k\le2\), and for seven compositions with \(n=5\) and \(k\le1\) (total degree \(\le8\)–\(10\)). Independently, the fixed-point values tabulated in (Bonelli et al. 2021) for \(N(1,4,2)\) coincide with \(\langle\widetilde H_\nu,h_2(h_2-qt\,e_2)\rangle\) for all five \(\nu\). We have not attempted a proof.
In summary, the Procesi bundle is reached by the first-order model only through a specific treatment of a non-complete-intersection constraint, either by reduction or by a flag resolution with one-box steps. The two natural alternatives, scheme-theoretic and derived imposition, differ from it in a controlled way that is invisible in \(\chi(\mathcal{O}(k))\). This extends the theme of the previous sections from level shifts to \(S_n\)-refined characters.
8 Conclusions
One rule with two exponents per field organizes the quantum shifts of first-order gauged matrix models. Part of the resulting dictionary confirms identifications already made by the original authors (Barns-Graham et al. 2018; Hu et al. 2024; Hellerman and Susskind 2001; Cappelli and Riccardi 2005). Its use is to make the remaining distinctions sharp and to resolve conflicting attributions. It separates three kinds of shift: Gauss-law (dictionary) shifts (\(s_w\)), zero-point and Calogero-type shifts (\(s_\varepsilon\)), and genuine spin\(^c\) twists (\(s_\varepsilon=s_w=\tfrac12\)). The comparison between trivial and non-trivial canonical bundles is essential: on \(\mathrm{Hilb}^n\) several inequivalent prescriptions cannot be told apart.
The same theme reappears for the Procesi bundle: the \(S_n\)-refined character depends on how a non-complete-intersection constraint is imposed, although the unrefined partition function does not. A natural next step is a phase space with both non-trivial \(K\) and a holomorphic constraint, together with non-linear Hamiltonians in the gauged setting, and a proof of Equation 2.
9 Remarks on JK residues and wall crossing
For rank-one ADHM we also verified the following.
The fixed-contour Molien integral selects the chamber \(\mathrm{sign}(\zeta)=\mathrm{sign}(k)\), consistently with \(\zeta=k'\).
The wall-crossing difference is \[ Z_+-Z_-=\chi(\mathcal{O}(k))-(qt)^{-(k+1)n}\chi(\mathcal{O}(k))^\vee . \] It vanishes at \(k=0\) and factorizes at \(k=1\) because \(\chi(\mathcal{O}(1))=\chi(\mathcal{O}(-1))\), which we observed for \(n\le5\).
In rank two there is no wall crossing for \(|k|<N\).
These follow from the general results of (Hori et al. 2015; Hwang et al. 2015). A case analysis of the background charge already appears in (Barns-Graham 2018).
10 Scope of the numerical checks
All checks except those marked otherwise are automated tests (429 in total). Power series in \(q,t\) are compared up to total degree 10–16, and numerical integrals are compared to relative accuracy \(10^{-4}\)–\(10^{-12}\). The ranges are:
rank-one ADHM: \(n\le4\), \(0\le k\le3\); JK residues for \(n\le5\);
lattices: \(n\le3\);
rank two: \(n\le2\);
Grassmannians: \(n\le2\), \(N\le4\);
Jordan quiver: \(n\le3\), \(f\le2\);
isospectral constraints (Section 7): \(n=2\) with \(k\le3\), \(n=3\) with \(k\le2\) and \(n=4\) with \(k\le1\), by linear algebra modulo a prime below \(2^{23}\) (total degree 9–12); for \(n=2\) also with explicit matrices, exactly over \(\mathbb Q\) (\(k\le2\), total degree \(\le6\));
flags: residue formula exactly as rational functions for \(n\le4\); virtual counts as series up to total degree 8–10 for all compositions with \(n\le4\), \(k\le2\), and for seven compositions with \(n=5\), \(k\le1\) (the latter outside the automated tests).
For \(k\ge2\), the Hilbert series of products of alternants is computed probabilistically, from ranks modulo a large prime.