Topological Classification of Symmetry Breaking and Vacuum Degeneracy

Abstract

We argue that a general system of scalar fields and gauge fields manifesting vacuum degeneracy induces a principal groupoid bundle over spacetime and that the pattern of spontaneous symmetry breaking and the Higgs mechanism are encoded by the singular foliation canonically induced on the moduli space of scalar vacuum expectation values by the Lie groupoid structure. Recent mathematical results in the classification of singular foliations then provide a qualitative classification of the possible patterns of vacuum degeneracy.

Figures (5)

• Figure 1: Deformations of the vacuum expectation value for G-type and S-type vacuum deformations. Above: G-type deformations require an experimenter with access to the entire region $R$ in which the vacuum will be deformed since the expectation value of the Goldstone boson $\langle\phi\rangle$ differs from $v_0$ in the entirety of $R$. Below: to perform an S-type deformation, an experimenter only needs access to the boundary $\partial R$ of the region, since the expectation of the longitudinal mode of the massive vector boson $\langle A_\mathrm{long}\rangle$ differs from $v_0$ only near $\partial R$.
• Figure 2: If one allows S-type vacuum deformations across phase boundaries, the equivalence relation becomes coarse. In the above, we have two phases --- the zero-dimensional, codimension-two leaf (black dot), where gauge symmetry is completely unbroken, and the one-dimensional, codimension-one leaves (black lines), where half of the gauge symmetry is broken. Under the equivalence relation defined by arbitrary S-type deformations that may cross phase boundaries (e.g. blue curve), all points on the moduli space are equivalent. Under the equivalence relation defined by S-type deformations that do not cross phase boundaries, the equivalence classes correspond to leaves of the singular foliation.
• Figure 3: A singular foliation consists of a partitioning of a smooth manifold into dovetailing layers ("leaves"), much like a mille-feuille cake, except that the dimensions of the leaves can vary.
• Figure 4: The transverse model to a leaf captures the pattern of vacuum deformations near a leaf.
• Figure 5: Some possible two-dimensional transverse models around the leaf $\bullet$: S-deformations for the vector field $a(x\partial_y-y\partial_x)+b(x^2+y^2)(x\partial_x+y\partial_y)$ for $(a,b)=(1,0)$, $(a,b)=(1,1)$, and $(a,b)=(0,1)$. In the first case, S-type deformations can never reach the leaf at the origin. In the second and third cases, S-type deformations can reach the leaf with a phase transition at the end.