Abstract
The one-loop dilatation operator in the holomorphic 3-scalar sector of the dynamical fishnet theory is studied. Due to the non-unitary nature of the underlying field theory this operator, dubbed in [1] the eclectic spin chain Hamiltonian, is non-diagonalisable. The corresponding spectrum of Jordan blocks leads to logarithms in the two-point functions, which is characteristic of logarithmic conformal field theories. It was conjectured in [2] that for certain filling conditions and generic couplings the spectrum of the eclectic model is equivalent to the spectrum of a simpler model, the hypereclectic spin chain. We provide further evidence for this conjecture, and introduce a generating function which fully characterises the Jordan block spectrum of the simplified model. This function is found by purely combinatorial means and is simply related to the q-binomial coefficient.
Highlights
The corresponding spectrum of Jordan blocks leads to logarithms in the two-point functions, which is characteristic of logarithmic conformal field theories
We introduced a generating function ZL,M,K(q) that we conjectured to fully enumerate the Jordan block spectrum of the hypereclectic model introduced in [2], for any sector of particles labelled by L, M, K
It takes a form reminiscent of a partition function, where one traces a certain kind of number operator over the state space. It may be expressed as a sum over products of q-binomial coefficients, which elegantly reduces to a single q-binomial for the case of one wall, i.e. K = 1
Summary
We consider local single-trace operators in the holomorphic 3-scalar sector of the theory (1.1). In N = 4 SYM the one-loop dilatation operator in the analogous sector can be written as a sum over permutation operators and enjoys an su(3) symmetry [24]. In the strongly twisted theory (1.1) this symmetry is broken and the one-loop dilatation operator Hec : C3 ⊗L → C3 ⊗L is a sum over chiral permutation operators [2]. The chiral permutation operators Hi : C3 ⊗ C3 → C3 ⊗ C3 act as follows:. The Hamiltonian (2.2) scans a state for neighboring fields in chiral order |32 , |13 , or |21 , and swaps them to anti-chiral order |23 , |31 , and |12 respectively. H3 corresponds to the one-loop dilatation operator in the fishnet theory, where we consider K non-dynamical insertions φ3, which act as walls. It corresponds essentially to a chiral version of the XY-model [25]
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