We consider the sphere free energy F(b; mI) in $$ \mathcal{N} $$ N = 6 ABJ(M) theory deformed by both three real masses mI and the squashing parameter b, which has been computed in terms of an N dimensional matrix model integral using supersymmetric localization. We show that setting $$ {m}_3=i\frac{b-{b}^{-1}}{2} $$ m 3 = i b − b − 1 2 relates F(b; mI) to the round sphere free energy, which implies infinite relations between mI and b derivatives of F(b; mI) evaluated at mI = 0 and b = 1. For $$ \mathcal{N} $$ N = 8 ABJ(M) theory, these relations fix all fourth order and some fifth order derivatives in terms of derivatives of m1, m2, which were previously computed to all orders in 1/N using the Fermi gas method. This allows us to compute $$ {\partial}_b^4F\left|{}_{b=1}\right. $$ ∂ b 4 F b = 1 and $$ {\partial}_b^5F\left|{}_{b=1}\right. $$ ∂ b 5 F b = 1 to all orders in 1/N, which we precisely match to a recent prediction to sub-leading order in 1/N from the holographically dual AdS4 bulk theory.

The partition function of a 3d N = 4 gauge theory with rank N can be computed using supersymmetric localization in terms of a matrix model, which often can be formulated as an ideal Fermi gas with a non-trivial one-particle Hamiltonian. We show how OPE coefficients of protected operators correspond in this formalism to averages of n-body operators in the Fermi gas, which can be computed to all orders in 1/N using the WKB expansion. We use this formalism to compute OPE coefficients in the U(N) k × U(N) −k ABJM theory as well as the U(N) theory with one adjoint and N f fundamental hypermultiplets, both of which have weakly coupled M-theory duals in the large N and finite k or N f regimes. For ABJM we reproduce known results, while for the N f theory we compute the all orders in 1/N dependence at finite N f for the coefficient c T of the stress tensor two-point function.

We compute all four-point functions involving the operators J0 and J1 in large-N Chern-Simons fermionic theories, in the regime where all external momenta lie along the z-axis. We find that our result for 〈J0J0J0J0〉 agrees with previous computations, and that the other correlators fall in line with expectations from bootstrap arguments.

No abstract

Recently introduced connections between quantum codes and Narain CFTs provide a simple ansatz to express a modular-invariant function $$ Z\left(\tau, \overline{\tau}\right) $$ Z τ τ ¯ in terms of a multivariate polynomial satisfying certain additional properties. These properties include algebraic identities, which ensure modular invariance of $$ Z\left(\tau, \overline{\tau}\right) $$ Z τ τ ¯ , and positivity and integrality of coefficients, which imply positivity and integrality of the 𝔲(1)n × 𝔲(1)n character expansion of $$ Z\left(\tau, \overline{\tau}\right) $$ Z τ τ ¯ . Such polynomials come naturally from codes, in the sense that each code of a certain type gives rise to the so-called enumerator polynomial, which automatically satisfies all necessary properties, while the resulting $$ Z\left(\tau, \overline{\tau}\right) $$ Z τ τ ¯ is the partition function of the code CFT — the Narain theory unambiguously constructed from the code. Yet there are also “fake” polynomials satisfying all necessary properties, that are not associated with any code. They lead to $$ Z\left(\tau, \overline{\tau}\right) $$ Z τ τ ¯ satisfying all modular bootstrap constraints (modular invariance and positivity and integrality of character expansion), but whether they are partition functions of any actual CFT is unclear. We consider the group of the six simplest fake polynomials and denounce the corresponding Z’s as fake: we show that none of them is the torus partition function of any Narain theory. Moreover, four of them are not partition functions of any unitary 2d CFT; our analysis for other two is inconclusive. Our findings point to an obvious limitation of the modular bootstrap approach: not every solution of the full set of torus modular bootstrap constraints is due to an actual CFT. In the paper we consider six simple examples, keeping in mind that thousands more can be constructed.