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This gives interacting equations of motion for the modes of the string. Loop variable techniques are used to obtain gauge invariant equations. Since this method is not tied to flat space time or any particular background metric, it is manifestly background independent. The technique can be applied to both open and closed strings. Thus gauge invaria"},"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":"1508.03692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2015-08-15T03:22:13Z","cross_cats_sorted":[],"title_canon_sha256":"b3309ed3f0cf21c9366f10e1de796ee25302d5cc997780b5517e0499b4f70fd8","abstract_canon_sha256":"a5983a4f75d4acddf6e9c5a99c51c64a0dbad25a079e1ad000003edf680d1a6f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:53.167992Z","signature_b64":"6uhvQxtVWbNKVsg9unNu/yDQaGBraoEhjEHs1XdLNwi9C5uV9ylyyAtx9xgAfR26briNJkC88u6EmPxOj88OAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e48a8eac61401c5c6e439e4d0148e391e745e9b8f1761cee9e1cef8a3ab71a2","last_reissued_at":"2026-05-18T01:24:53.167299Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:53.167299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exact Renormalization Group and Loop Variables: A Background Independent Approach to String Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"hep-th","authors_text":"B. 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