Exact grand partition functions, modularity, and quantization conditions for the orientifold ABJ Fermi gas at general Chern-Simons level $k$

The orientifold projection of the ABJ Fermi gas rewrites the grand canonical partition functions of the $\mathcal{N}=5$ $O(n)\times USp(n')$ Chern-Simons-matter theories as spectral determinants $\Xi_\pm(\mu,k,M)=\det(1+e^\mu\rho_\pm)$ of parity-projected density matrices. Closed forms in terms of a single Jacobi theta function, and therefore exact quantization conditions from its zeros, exist only at Chern-Simons levels $k=1,2,4$; at $k=8$ the grand partition functions are sums of theta blocks whose coefficient functions are invariant under the modular generators $S$ and $T^k$ but only $4\pi i$-periodic in the chemical potential, and for $k=3,6,12$ and general $k$ the all-genus orientifold worldsheet-instanton series has no closed resummation. The problem is to complete this exact-solution program at general positive integer $k$: find closed-form expressions for the grand partition functions and for the resummation function $W(\mu)$, identify the modular subgroup of $SL(2,\mathbb{Z})$ acting on them for general $(k,M)$, derive exact quantization conditions for the sum-of-theta-block cases, and prove the chiral-projection functional relations beyond $k=2,4$ (notably $k=8$), ideally by a uniform structural derivation, with the first-principles real-topological-string and two-dimensional-conformal-field-theory interpretations as flagged structural strands. Later work resolved the theory-identification conjectures (Honda; Moriyama-Nosaka) and the $k=4$ functional relations (Kubo); the remaining core is open.

Background

The ABJM theory and its unequal-rank deformation ABJ are three-dimensional superconformal Chern-Simons-matter theories with gauge group $U(N+M)_k\times U(N)_{-k}$, where the subscripts are quantized integer Chern-Simons levels: the Chern-Simons gauge action is first order in time derivatives, so the level $k$ plays the role of an inverse coupling. ABJM describes $N$ M2-branes, two-dimensional membranes, probing a curved background of M-theory (the eleven-dimensional extension of string theory). These theories admit an orientifold: one quotients by a $\mathbb{Z}_2$ symmetry that combines spacetime parity with an internal reflection of the transverse coordinates, breaking the supersymmetry from $\mathcal{N}=6$ to $\mathcal{N}=5$ and changing the gauge group to an orthogonal-symplectic pair $O(2N+2M)_{2k}\times USp(2N)_{-k}$, with $O$ the orthogonal group and $USp$ the unitary symplectic group. Physically these orientifold theories describe M2-branes moving in the presence of an orientifold plane, the fixed locus of the orientifold involution.

The partition function of such a theory on the three-sphere is exactly computable by supersymmetric localization, which reduces it to a finite-dimensional matrix integral. Reorganizing the canonical partition functions $Z(N)$ into a grand canonical generating function $\Xi(\mu)=\sum_N e^{\mu N}Z(N)$, with chemical potential $\mu$, turns the matrix model into the statistical mechanics of a quantum-mechanical ideal Fermi gas: $\Xi(\mu)=\det(1+e^\mu\rho)$, a Fredholm determinant (spectral determinant) of a positive single-particle density operator $\rho$ acting on wavefunctions on a half-line. Exact knowledge of $\Xi(\mu)$ is equivalent to exact knowledge of the spectrum of $\rho$, and the zeros of $\Xi(\mu)$ in the complex $\mu$-plane encode exact quantization conditions, that is, discrete Bohr-Sommerfeld-type equations for the single-particle energy levels.

Okuyama (2016) implemented the orientifold projection directly on the ABJ Fermi gas. The $\mathbb{Z}_2$ action acts on the single-particle coordinate as $R:x\mapsto -x$; the even and odd projected density matrices are $\rho_\pm=\rho(1\pm R)/2$, the projected grand partition functions are $\Xi_\pm(\mu,k,M)=\det(1+e^\mu\rho_\pm)$, and the full orientifold grand partition function is $\Xi=\Xi_+\Xi_-$. The unprojected $\Xi$ has a periodic-sum structure, $\Xi(\mu)=\sum_{n\in\mathbb{Z}}e^{J(\mu+2\pi i n)}$, where $J(\mu)$ is the grand potential. The grand potential contains membrane-instanton terms (M2-brane contributions suppressed as $e^{-\mu}$) and worldsheet-instanton terms (suppressed as $e^{-4\mu/k}$ and organized by genus like a string-loop expansion). Because of this, the periodic sum collapses to a single Jacobi theta function $\vartheta_3$ — a standard quasi-periodic entire function with explicitly known zeros — exactly when $e^{-4\mu/k}$ is itself $2\pi i$-periodic, which happens only for $k=1,2,4$. In the cases $k=2$ and $k=4$ worked out explicitly in the paper, Okuyama obtained closed forms $\Xi_\pm\propto e^{j_\pm}\vartheta_3(\ldots)$ and, following the technique introduced by Grassi, Hatsuda and Marino for the unprojected ABJ(M) theories, simple exact quantization conditions from the zeros of the theta function.

At $k=8$ the periodic sum splits into blocks labeled by $\mathbb{Z}_k$ charge and $\Xi_\pm$ becomes a sum of two theta-function blocks whose coefficient functions $G_{o,e}(\mu)$ (a single function $G(\mu)$ and its inverse in the paper's formulae) are invariant under the standard generators $S:\tau\mapsto -1/\tau$ and $T:\tau\mapsto\tau+1$ of the modular group $SL(2,\mathbb{Z})$ of transformations of the torus modulus (more precisely under $S$ and $T^k$), while being only $4\pi i$-periodic in $\mu$ rather than $2\pi i$-periodic. For $k=6$, and generically for $k\notin\{1,2,4,8\}$ such as $k=3,12$, the worldsheet-instanton part of $J$ receives contributions at every genus in the orientifold theory, unlike the unprojected case, and the available resummation is a function $W(\mu)$ defined only through its series expansion, with no known closed form. The paper also records two identification statements, equalities (2.48) and (2.49): $\Xi_-(\mu,k,0)$ and $\Xi_+(\mu,k,1)$ coincide with the grand partition functions of the $\mathcal{N}=5$ $O(2N+1)_{2k}\times USp(2N)_{-k}$ and $O(2N)_{2k}\times USp(2N)_{-k}$ Chern-Simons theories, the first observed by Honda and the second conjectured by Okuyama from matching the canonical partition functions computed by Moriyama and Suyama. Okuyama also proved, for $k=2,4$ by exploiting the closed forms, a set of functional relations between $\Xi_+$ and $\Xi_-$ of the same quantum Wronskian type conjectured by Grassi, Hatsuda and Marino for the unprojected theories — Wronskian-type relations between two linearly independent solutions of an auxiliary difference equation, reflecting the two possible eigenvalues of the orientifold parity operator.

The closing section of the paper lists as open problems: closed forms for general integer $k$, the closed form of $W(\mu)$, the precise modular subgroup for general $(k,M)$, quantization conditions for the theta-block cases, proofs of the functional relations beyond $k=2,4$, a first-principles interpretation of the orientifold worldsheet-instanton coefficients (for example through the real topological string, the extension of topological string theory to worldsheets carrying an orientation-reversing involution, here on the local $\mathbb{P}^1\times\mathbb{P}^1$ Calabi-Yau geometry), and the precise relation to torus partition functions of chiral fermions in two-dimensional conformal field theory.

Problem Statement

Complete the exact solution of the orientifold ABJ Fermi gas at general positive integer Chern-Simons level $k$: determine the exact grand partition functions $\Xi_\pm(\mu,k,M)=\det(1+e^\mu\rho_\pm)$ of the parity-projected density matrices and their analytic structure for arbitrary positive integer $k$ and non-negative integer rank parameter $M$. The requested results, which jointly support this single objective, are: items 1–4 are the required deliverables, which together constitute the resolution of the problem; item 5 is a supporting interpretive strand that is desirable but is not required for the problem to count as solved. Each of items 1–4 asks for the general-$(k,M)$ result, and the specific values named inside an item (such as $k=3,6,12$ and $k=8$) are the instances on which that general result must be demonstrated and checked, not restricted targets in their own right. Each of items 1–4 has independent standing and may be reported as progress on its own, but the objective is resolved only when all four are settled.

1. Closed forms. For $k\notin\{1,2,4,8\}$, notably $k=3,6,12$, find closed-form expressions for the exact grand partition functions $\Xi_\pm(\mu,k,M)$ and for the all-genus worldsheet-instanton resummation function $W(\mu)$ (or an equivalent modular or special-function object), in the sense in which $k=2,4$ reduce to a single Jacobi theta function and $k=8$ to a finite sum of theta blocks.

2. Modularity. Identify the precise subgroup $\Gamma\subseteq SL(2,\mathbb{Z})$ under which the exact grand partition functions, including the block coefficient functions $G_{o,e}(\mu)$, transform for general $(k,M)$, and characterize the role of modularity and of the observed reduction of $2\pi i$-periodicity in $\mu$ to $4\pi i$-periodicity in the projected theory.

3. Spectrum. Derive exact quantization conditions and spectral-determinant characterizations for the cases where $\Xi_\pm$ are sums of theta-function blocks, determining the set of $k$ for which this finite theta-block decomposition holds (it holds at $k=8$) and the associated conditions, analogous to the conditions obtained from the zeros of single theta functions at $k=2,4$.

4. Functional relations. Prove the conjectured chiral-projection (quantum-Wronskian-type) relations between $\Xi_+$ and $\Xi_-$ for $k$ beyond $2$ and $4$, notably $k=8$, or give a uniform structural derivation valid for all $k$.

5. Structural interpretation. Explain the orientifold worldsheet-instanton coefficients from first principles, for example via a real topological string on local $\mathbb{P}^1\times\mathbb{P}^1$, and make precise the observed resemblance between the exact grand partition functions and torus partition functions of chiral fermions in two-dimensional conformal field theory.

The already-resolved sub-questions — the identification of $\Xi_\pm$ with the grand partition functions of the $\mathcal{N}=5$ $O(n)\times USp(n')$ theories, and the $k=4$ functional relations — are not part of the target; the problem is the remaining core above.

References

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3. Masazumi Honda, "Exact relations between M2-brane theories with and without Orientifolds", JHEP 06 (2016) 123. DOI: 10.1007/JHEP06(2016)123. arXiv:1512.04335. https://arxiv.org/abs/1512.04335
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