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Henri Poincar\u00e9"],"published-print":{"date-parts":[[2024,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>K3 surfaces play a prominent role in string theory and algebraic geometry. The properties of their enumerative invariants have important consequences in black hole physics and in number theory. To a K3 surface, string theory associates an Elliptic genus, a certain partition function directly related to the theory of Jacobi modular forms. A multiplicative lift of the Elliptic genus produces another modular object, an Igusa cusp form, which is the generating function of BPS invariants of<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{K3} \\times E$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mtext>K3<\/mml:mtext><mml:mo>\u00d7<\/mml:mo><mml:mi>E<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In this note, we will discuss a refinement of this chain of ideas. The Elliptic genus can be generalized to the so-called Hodge-Elliptic genus which is then related to the counting of refined BPS states of<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{K3} \\times E$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mtext>K3<\/mml:mtext><mml:mo>\u00d7<\/mml:mo><mml:mi>E<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We show how such BPS invariants can be computed explicitly in terms of different versions of the Hodge-Elliptic genus, sometimes in closed form, and discuss some generalizations.<\/jats:p>","DOI":"10.1007\/s00023-023-01375-1","type":"journal-article","created":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T05:02:22Z","timestamp":1697000542000},"page":"2731-2779","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Hodge-Elliptic Genera, K3 Surfaces and Enumerative Geometry"],"prefix":"10.1007","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9940-3250","authenticated-orcid":false,"given":"Michele","family":"Cirafici","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,10,11]]},"reference":[{"key":"1375_CR1","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/JHEP01(2021)147","volume":"01","author":"S Alexandrov","year":"2021","unstructured":"Alexandrov, S., Nampuri, S.: Refinement and modularity of immortal dyons. 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