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ABJM S3 partition function as a Cartan matrix integral

Physical scope

Source-reported formal localization formula for the zero-mass, zero-FI ABJM model with gauge group U(N)×U(N) on the round Euclidean S3; no general N=2 result, continuum measure construction, or empirical claim is asserted.

Example

Fix the adopted zero-mass, zero-Fayet–Iliopoulos ABJM model on the round sphere of radius $r > 0$ , with $N \in Z_{\geq 1}$ and nonzero integer level $k$ . Let $s_{i}, {\hat{s}}_{i} \in R$ be the eigenvalues of the two constant Hermitian Cartan scalars, each of physical dimension $r^{- 1}$ , and put $λ_{i} = r s_{i}$ and ${\hat{λ}}_{i} = r {\hat{s}}_{i}$ . In the Kapustin–Willett–Yaakov Euclidean level convention, the source-reported formal localization expression is

Z_{A B J M} (S_{r}^{3}) = \frac{1}{(N!)^{2}} \int_{R^{2 N}} (\prod_{i = 1}^{N} d λ_{i} d {\hat{λ}}_{i}) e^{- i π k (\sum_{i} λ_{i}^{2} - \sum_{i} {\hat{λ}}_{i}^{2})} \frac{\prod_{1 \leq i < j \leq N} {(2 \sinh ⁣ (π (λ_{i} - λ_{j})))}^{2} {(2 \sinh ⁣ (π ({\hat{λ}}_{i} - {\hat{λ}}_{j})))}^{2}}{\prod_{i, j = 1}^{N} {(2 \cosh ⁣ (π (λ_{i} - {\hat{λ}}_{j})))}^{2}} .

Each Cartan variable is integrated over the real line. The integer level and the dimensionless eigenvalues fix every exponent and hyperbolic-function argument; the factor $(N!)^{- 2}$ is the inverse order of the two Weyl groups. Mariño's convention uses $μ_{i} = 2 π λ_{i}$ and $g_{s} = 2 π i / k_{M a r i n o}$ ; it gives the same displayed phase after setting $k_{M a r i n o} = - k$ . The displayed real-contour expression is oscillatory. It is the source's matrix-integral result under its formal localization and determinant prescriptions, not a construction of the continuum functional measure, a proof of the localization limit, or a formula for every three-dimensional $N = 2$ theory.

Example references

Kapustin, Willett, and Yaakov, §§3–4

Mariño, §§6.2–6.5

Facts & Assumptions

Given: The source-specific ABJM field content, levels, radius, zero mass/FI parameters, selected supercharge, real Cartan contour, and formal measure qualifications of the Postulate.

[A1]

Adopted ABJM localization model on the round three-sphere: the adopted model has global group $U (N) \times U (N)$ , levels $k, - k$ , four chiral multiplets in two conjugate bifundamental pairs, and $Δ = R = \frac{1}{2}$ . Its field-theory measure and localization argument are formal inputs, not a constructed continuum measure.

[F1]

Kapustin, Willett, and Yaakov, §§3.1–3.5 and §4, give the combined $Q_{g f} = Q + s$ prescription, with the full insertion $Q_{g f} Ψ_{g f}$ (including $Q Ψ_{g f}$ ) rather than its BRST part alone, and the gauge-fixed Cartan reduction, root/Vandermonde factor, one-loop determinants, and the ABJM real-line matrix-integral convention. The result is specific to their superconformal Chern–Simons–matter setup.

[F2]

Mariño, §6.3 and Eq. (6.67), gives the zeta-regularized chiral determinant for canonical dimension $Δ = \frac{1}{2}$ in a self-conjugate representation. Its §6.5 Eq. (6.81) combines four bifundamental chirals into the ABJM denominator and states the opposite Euclidean level-phase convention.

Proof

1.1givenA1F1construct

Let $h = {diag (x_{1}, \dots, x_{N}) : x_{i} \in R}$ and $\hat{h} = {diag ({\hat{x}}_{1}, \dots, {\hat{x}}_{N}) : {\hat{x}}_{j} \in R}$ . Define the coordinate functionals $e_{i} (diag x) = x_{i}$ and ${\hat{e}}_{j} (diag \hat{x}) = {\hat{x}}_{j}$ . In the complexified Lie algebra, for $i \neq j$ the matrix unit $E_{i j}$ satisfies $[diag x, E_{i j}] = (x_{i} - x_{j}) E_{i j}$ , so its root is $e_{i} - e_{j}$ ; choose the roots with $i < j$ as the positive system. The bifundamental representation $(N, \overline{N})$ is $C^{N} \otimes (C^{N})^{*}$ , with Lie-algebra action $X \mapsto H X - X \hat{H}$ for $(H, \hat{H}) \in h \oplus \hat{h}$ . On the matrix-unit basis vector $E_{i j}$ this action is multiplication by $x_{i} - {\hat{x}}_{j}$ , so its weight is $e_{i} - {\hat{e}}_{j}$ . The reversed bifundamental has weights $- e_{i} + {\hat{e}}_{j} = - (e_{i} - {\hat{e}}_{j})$ . At the constant Cartan fields in the Postulate, these weights give scalar eigenvalues $s_{i} - {\hat{s}}_{j}$ and their negatives. Finally, $λ_{i} = r s_{i}$ and ${\hat{λ}}_{i} = r {\hat{s}}_{i}$ are dimensionless because $r$ has units of length and the scalar eigenvalues have units $r^{- 1}$ .

1.2givenA1F1F2algebra

At the Postulate's locus, the real functional from the first gauge factor is $S_{C S, k} = \frac{k}{4 π} \int_{S_{r}^{3}} d^{3} x \sqrt{g} 2 {tr}_{N} (D σ) = - π k \sum_{i} (r s_{i})^{2}$ , since $vol (S_{r}^{3}) = 2 π^{2} r^{3}$ and $D = - σ / r$ ; the second factor contributes $+ π k \sum_{i} (r {\hat{s}}_{i})^{2}$ . The Euclidean weight is $e^{+ i S_{C S}}$ by the Postulate, so its classical factor is the displayed $e^{- i π k (\sum_{i} λ_{i}^{2} - \sum_{i} {\hat{λ}}_{i}^{2})}$ . Independently, Mariño's dimensionless variables satisfy $μ_{i} = 2 π λ_{i}$ and $g_{s} = 2 π i / k_{M a r i n o}$ , so $- \frac{1}{2 g_{s}} \sum_{i} (μ_{i}^{2} - {\hat{μ}}_{i}^{2}) = i π k_{M a r i n o} \sum_{i} (λ_{i}^{2} - {\hat{λ}}_{i}^{2})$ . Setting $k_{M a r i n o} = - k$ gives the same phase. The variables are integrated on the real Cartan lines; the phase is oscillatory.

2.1A1F2step 1.1algebra

Group a conjugate pair of chiral multiplets into the self-conjugate total representation $R \oplus R^{*}$ before applying [F2]. Its weights are $e_{i} - {\hat{e}}_{j}$ and their negatives. Equation (6.67), applied to this whole self-conjugate representation, gives $(2 \cosh (π (λ_{i} - {\hat{λ}}_{j})))^{- 1 / 2} (2 \cosh (- π (λ_{i} - {\hat{λ}}_{j})))^{- 1 / 2} = (2 \cosh (π (λ_{i} - {\hat{λ}}_{j})))^{- 1}$ for each weight pair. This calculation does not assign the real half-power to an isolated non-self-conjugate chiral determinant, which can carry a phase. ABJM has two conjugate pairs, giving the denominator power $- 2$ for every $(i, j)$ .

2.2F1step 1.1algebra

The adjoint determinant for the first $U (N)$ factor is the product over its roots. For each positive root $α = e_{i} - e_{j}$ , the two factors are $2 \sinh (π α (λ))$ and $2 \sinh (- π α (λ))$ , whose product is $- (2 \sinh (π (λ_{i} - λ_{j})))^{2}$ . There are $N (N - 1) / 2$ positive roots; the second $U (N)$ factor has the same number, so their two root-sign factors multiply to $(- 1)^{N (N - 1)} = 1$ . The remaining factors are exactly the two displayed squared-sinh products. The residual Weyl group is $S_{N} \times S_{N}$ of order $(N!)^{2}$ , giving the prefactor.

3.1givenA1F1F2step 1.2step 2.1step 2.2algebra∎

Multiplying the classical phase from step 1.2, the two vector determinants from step 2.2, the chiral determinant from step 2.1, the Cartan measures, and the Weyl quotient gives the displayed matrix integral. The adopted formal localization assumptions use the completed $Q_{g f}$ -closed gauge-fixing action in [A1] and [F1]. This reproduces the source-specific formula only under those assumptions; it does not prove that the continuum field-space measure or localization limit exists.

Depends on

Adopted ABJM localization model on the round three-sphere