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We prove this for deg $K_S<0$ here, and it is proved for $S$ a K3 surface in \\cite{MT}.\n  For K3 surfaces we calculate the invariants in terms of modular forms which generalise and prove conjectures of Vafa and Witten."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1702.08488","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-02-27T19:44:07Z","cross_cats_sorted":["hep-th","math.DG"],"title_canon_sha256":"c6840fe05f3986a6783a76851d2d94298fd718b32a444f58386a5f997489c5b5","abstract_canon_sha256":"fb59cb62071a998541c93bee1c8d52f4989442048a066a2f6e1032da1deedff4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:04:43.520603Z","signature_b64":"H+g7/glvEOK1mrNOHhhFDasANi9eNKQnRUXxCpZY/f3ARUZWOiHv4+/GFTBOzZ/1MiRW8Rrh7IQ5NBXK64lxDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9a5c75fc78a844a2ba1dbef64ab29f466186306b39106a98b994580f98bbc60","last_reissued_at":"2026-07-05T05:04:43.520179Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:04:43.520179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vafa-Witten invariants for projective surfaces II: semistable case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.DG"],"primary_cat":"math.AG","authors_text":"Richard P. 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