{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:KCZYJGD2D5WLCHFNDZDP442Z6O","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3785ba76fbaba5c487990a5f8d8f8c00b3c455f9f9356f51211a102fb35ae67b","cross_cats_sorted":["math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-03-21T18:31:24Z","title_canon_sha256":"86d261b0f8086c0c517a33b692b6d70a22ec81104b2377a195dbb9407412f6e4"},"schema_version":"1.0","source":{"id":"1603.06530","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.06530","created_at":"2026-05-18T01:18:48Z"},{"alias_kind":"arxiv_version","alias_value":"1603.06530v1","created_at":"2026-05-18T01:18:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06530","created_at":"2026-05-18T01:18:48Z"},{"alias_kind":"pith_short_12","alias_value":"KCZYJGD2D5WL","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"KCZYJGD2D5WLCHFN","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"KCZYJGD2","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:082799a4457ba5a9ba8bb0ca99a49c2bbe4f3b57ff3f4f28846fbd83c6f55aea","target":"graph","created_at":"2026-05-18T01:18:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Inspired by the LG/CY correspondence, we study the local index theory of the Schr\\\"odinger operator associated to a singularity defined on ${\\mathbb C}^n$ by a quasi-homogeneous polynomial $f$. Under some mild assumption on $f$, we show that the small time heat kernel expansion of the corresponding Schr\\\"odinger operator exists and is a series of fractional powers of time $t$. Then we prove a local index formula which expresses the Milnor number of $f$ by a Gaussian type integral. Furthermore, the heat kernel expansion provides spectral invariants of $f$. Especially, we define torsion type inv","authors_text":"Hao Fang, Huijun Fan","cross_cats":["math.DG","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-03-21T18:31:24Z","title":"Torsion type invariants of singularities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06530","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:23663fbfe5c2545c37e8f7d407ee9bb8f5e6df2b7f8a1548813827e6f9affd34","target":"record","created_at":"2026-05-18T01:18:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3785ba76fbaba5c487990a5f8d8f8c00b3c455f9f9356f51211a102fb35ae67b","cross_cats_sorted":["math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-03-21T18:31:24Z","title_canon_sha256":"86d261b0f8086c0c517a33b692b6d70a22ec81104b2377a195dbb9407412f6e4"},"schema_version":"1.0","source":{"id":"1603.06530","kind":"arxiv","version":1}},"canonical_sha256":"50b384987a1f6cb11cad1e46fe7359f3ab6bb5f098e85a2ec8156579a6dd22ce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50b384987a1f6cb11cad1e46fe7359f3ab6bb5f098e85a2ec8156579a6dd22ce","first_computed_at":"2026-05-18T01:18:48.351481Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:18:48.351481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BqrL80p7iYll8c78AKuwseazUQxaepy+NpL2vZPBp9p3mAdLwN0h1GI2R+uVhqktIadWqj+FakM2Q8OZFe61DA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:18:48.351956Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.06530","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:23663fbfe5c2545c37e8f7d407ee9bb8f5e6df2b7f8a1548813827e6f9affd34","sha256:082799a4457ba5a9ba8bb0ca99a49c2bbe4f3b57ff3f4f28846fbd83c6f55aea"],"state_sha256":"2eb2edc33180e52a7b12f46cbb501510db305d5d261e5a089b024aa91ae64f93"}