Geometry and field theory in multi-fractional spacetime

We construct a theory of fields living on continuous geometries with fractional Hausdorff and spectral dimensions, focussing on a flat background analogous to Minkowski spacetime. After reviewing the properties of fractional spaces with fixed dimension, presented in a companion paper, we generalize to a multi-fractional scenario inspired by multi-fractal geometry, where the dimension changes with the scale. This is related to the renormalization group properties of fractional field theories, illustrated by the example of a scalar field. Depending on the symmetries of the Lagrangian, one can define two models. In one of them, the effective dimension flows from 2 in the ultraviolet (UV) and geometry constrains the infrared limit to be four-dimensional. At the UV critical value, the model is rendered power-counting renormalizable. However, this is not the most fundamental regime. Compelling arguments of fractal geometry require an extension of the fractional action measure to complex order. In doing so, we obtain a hierarchy of scales characterizing different geometric regimes. At very small scales, discrete symmetries emerge and the notion of a continuous spacetime begins to blur, until one reaches a fundamental scale and an ultra-microscopic fractal structure. This fine hierarchy of geometries has implications for non-commutative theories and discrete quantum gravity. In the latter case, the present model can be viewed as a top-down realization of a quantum-discrete to classical-continuum transition.Comment: 1+82 pages, 1 figure, 2 tables. v2-3: discussions clarified and improved (especially section 4.5), typos corrected, references added; v4: further typos correcte

Interneuron- and GABAA receptor-specific inhibitory synaptic plasticity in cerebellar purkinje cells

Inhibitory synaptic plasticity is important for shaping both neuronal excitability and network activity. Here we investigate the input and GABA(A) receptor subunit specificity of inhibitory synaptic plasticity by studying cerebellar interneuron-Purkinje cell (PC) synapses. Depolarizing PCs initiated a long-lasting increase in GABA-mediated synaptic currents. By stimulating individual interneurons, this plasticity was observed at somatodendritic basket cell synapses, but not at distal dendritic stellate cell synapses. Basket cell synapses predominantly express β2-subunit-containing GABA(A) receptors; deletion of the β2-subunit ablates this plasticity, demonstrating its reliance on GABA(A) receptor subunit composition. The increase in synaptic currents is dependent upon an increase in newly synthesized cell surface synaptic GABA(A) receptors and is abolished by preventing CaMKII phosphorylation of GABA(A) receptors. Our results reveal a novel GABA(A) receptor subunit- and input-specific form of inhibitory synaptic plasticity that regulates the temporal firing pattern of the principal output cells of the cerebellum