Abstract
We construct the theory of a chiral boson with anisotropic scaling, characterized by a dynamical exponent z that takes positive odd integer values. The action reduces to that of Floreanini and Jackiw in the isotropic case (z = 1). The standard free boson with Lifshitz scaling is recovered when both chiralities are nonlocally combined. Its canonical structure and symmetries are also analyzed. As in the isotropic case, the theory is also endowed with a current algebra. Noteworthy, the standard conformal symmetry is shown to be still present, but realized in a nonlocal way. The exact form of the partition function at finite temperature is obtained from the path integral, as well as from the trace over hat{u} (1) descendants. It is essentially given by the generating function of the number of partitions of an integer into z-th powers, being a well-known object in number theory. Thus, the asymptotic growth of the number of states at fixed energy, including subleading correc- tions, can be obtained from the appropriate extension of the renowned result of Hardy and Ramanujan.
Highlights
To a quadratic term that contains higher spatial derivatives in the action
We construct the theory of a chiral boson with anisotropic scaling, characterized by a dynamical exponent z that takes positive odd integer values
Gravitational duals in which the anisotropic scaling symmetry is realized through the isometries of the socalled Lifshitz spacetimes have been studied in e.g. [10,11,12,13, 25,26,27,28,29,30,31,32]
Summary
H± can always be gauged away by virtue of the gauge transformation in (2.2) with h± = f±, so that (2.4) precisely reduces to the field equation of the anisotropic chiral boson in (1.3), i.e.,. The magnitude of the phase velocity is generically different for each mode, reflecting the fact that the theory is not Lorentz invariant (unless z = 1). The action (2.1) fulfills P[Sz±] = −Sz∓, regardless the field X± transforms as a (pseudo-)scalar. Chirality is swapped by a time reversal transformation T , defined through t → −t, i.e., T [Sz±] = −Sz∓. The joint action of parity and time reversal becomes a symmetry of the anisotropic chiral boson action (2.1) because PT [Sz±] = Sz±
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