{"paper":{"title":"The Callias Index Formula Revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fritz Gesztesy, Marcus Waurick","submitted_at":"2015-06-16T20:37:11Z","abstract_excerpt":"We revisit the Callias index formula for Dirac-type operators $L$ in odd space dimension $n$, and prove that \\begin{align}\n  \\text{ind} \\, (L)\n  =\\bigg(\\frac{i}{8\\pi}\\bigg)^{\\frac{n-1}{2}}\\frac{1}{2(\\frac{n-1}{2})!}\n  \\lim_{\\Lambda \\to\\infty}\\frac{1}{\\Lambda }\\sum_{i_{1},\\dots,i_{n} = 1}^n \\varepsilon_{i_{1}\\dots i_{n}}\n  \\int_{\\Lambda S_{n-1}}\\text{tr}_{\\mathbb{C}^d}\\, (U(x)(\\partial_{i_{1}}U)(x)\\dots\n  (\\partial_{i_{n-1}}U)(x)) x_{i_{n}}\\, d^{n-1} \\sigma(x), \\, (*) \\end{align} where $U(x) = \\text{sgn} \\,(\\Phi(x))$ and $L$ in $L^{2}(\\mathbb{R}^{n})^{2^{\\widehat n}d}$ is of the form \\[ L= \\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.05144","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}