Using a recently developed bootstrapping method, we compute the phase diagram of scalar field theory on the fuzzy disc with quartic even potential. We find three distinct phases with second and third order phase transitions between them. In particular, we find that the second order phase transition happens approximately at a fixed ratio of the two coupling constants defining the potential. We compute this ratio analytically in the limit of large coupling constants. Our results qualitatively agree with previously obtained numerical results.

We examine the relation between the gauge groups of $$\mathrm {SU}(n)$$ SU ( n ) - and $$\mathrm {PU}(n)$$ PU ( n ) -bundles over $$S^{2i}$$ S 2 i , with $$2\le i\le n$$ 2 ≤ i ≤ n , particularly when n is a prime. As special cases, for $$\mathrm {PU}(5)$$ PU ( 5 ) -bundles over $$S^4$$ S 4 , we show that there is a rational or p-local equivalence $$\mathcal {G}_{2,k}\simeq _{(p)}\mathcal {G}_{2,l}$$ G 2 , k ≃ ( p ) G 2 , l for any prime p if, and only if, $$(120,k)=(120,l)$$ ( 120 , k ) = ( 120 , l ) , while for $$\mathrm {PU}(3)$$ PU ( 3 ) -bundles over $$S^6$$ S 6 there is an integral equivalence $$\mathcal {G}_{3,k}\simeq \mathcal {G}_{3,l}$$ G 3 , k ≃ G 3 , l if, and only if, $$(120,k)=(120,l)$$ ( 120 , k ) = ( 120 , l ) .

The gauge group of a principal G-bundle P over a space X is the group of G-equivariant homeomorphisms of P that cover the identity on X. We consider the gauge groups of bundles over $$S^4$$ S 4 with $${{\textrm{Spin}}}^{{\textrm{c}}}(n)$$ Spin c ( n ) , the complex spin group, as structure group and show how the study of their homotopy types reduces to that of $${{\textrm{Spin}}}(n)$$ Spin ( n ) -gauge groups over $$S^4$$ S 4 . We then advance on what is known by providing a partial classification for $${{\textrm{Spin}}}(7)$$ Spin ( 7 ) - and $${{\textrm{Spin}}}(8)$$ Spin ( 8 ) -gauge groups over $$S^4$$ S 4 .