Spinors on a curved noncommutative space: coupling to torsion and the Gross–Neveu model

Abstract

We analyse the spinor action on a curved noncommutative space, the so-called truncated Heisenberg algebra, and in particular, the nonminimal coupling of spinors to the torsion. We find that dimensional reduction of the Dirac action gives the noncommutative extension of the Gross-Neveu model, the model which is, as shown by Vignes-Tourneret, fully renormalisable. * majab@ipb.ac.rs, madore@th.u-psud.fr, lnenadovic@ipb.ac.rs * As we do not specify the representation here, we have only one product: the one which defines the algebra, (2.6).

One-loop structure of the U(1) gauge model on the truncated Heisenberg space

“…The result is not what we expected or hoped for. Namely, in related models with scalar and spinor matter it was possible to attribute renormalizability to the background geometry, that is, to an adequate inclusion of geometric quantities in the Lagrangian [21]. It is well known on the other hand that on commutative curved spaces scalar and spinor theories are renormalizable only if matter is nonminimally coupled to the background curvature and torsion [35], and this pattern is exactly followed in the Grosse-Wulkenhaar and VignesTourneret models.…”

“…It defines a noncommutative wedge product. The Hodge dual on the other hand cannot be defined in the general case as it depends on (the existence of) the trace: in our case it is almost unique [21]. Finally, one specifies the connection: the metric-compatible connection used in [20] defines a noncommutative space with curvature and torsion.…”

“…The idea is based on the result that the two-dimensional GW action can be viewed as an action defined on particular curved three-dimensional space after the Kaluza-Klein (KK) reduction [20]. The VT action, similarly, is the spinor action on the same space [21]. In both cases matter is nonminimally coupled to the background curvature and torsion.…”

Detecting scaling in phase transitions on the truncated Heisenberg algebra

“…Grosse-Wulkenhaar (GW) model [9][10][11][12][13] is one of rare NC models immune to UV/IR mixing [14][15][16]. It describes a self-interacting real scalar field on the NC Moyal space confined in the external harmonic oscillator potential.…”

“…Critical exponents are ν = 1.00(15), α = −0.41(6), β = 0.42(2) and γ = −0.99(7). ∆C = C − 0.84(6) = −0.67(14) • N α/ν and γ 1 is the exponent of the correction to the scaling behaviour of susceptibility ∆χ = χ − 1.13(2) • N γ/ν = 7.5(4) • N −2.00(6) . Different colors represent different matrix sizes up to N = 50.…”