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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 17th 2020
   • (edited Nov 17th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexa bare list of references, to be `!include`-ed into relevant entries (such as _[[Witten genus]]_, _[[M5-brane elliptic genus]]_ but also inside _[[elliptic cohomology -- references]]_) -- for ease of harmonizing lists of references <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/1">v1</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

   a bare list of references, to be !include-ed into relevant entries (such as Witten genus, M5-brane elliptic genus but also inside elliptic cohomology – references) – for ease of harmonizing lists of references

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 19th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Matthew Ando]], [[Eric Sharpe]], _Elliptic genera of Landau-Ginzburg models over nontrivial spaces_, Adv. Theor. Math. Phys. 16 (2012) 1087-1144 ([arXiv:0905.1285](https://arxiv.org/abs/0905.1285)) <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/7">v7</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

   added pointer to:

   • Matthew Ando, Eric Sharpe, Elliptic genera of Landau-Ginzburg models over nontrivial spaces, Adv. Theor. Math. Phys. 16 (2012) 1087-1144 (arXiv:0905.1285)

   diff, v7, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 29th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * M. Nouman Muteeb, _Domain walls and M2-branes partition functions: M-theory and ABJM Theory_ ([arXiv:2010.04233](https://arxiv.org/abs/2010.04233)) <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/8">v8</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

   added pointer to:

   • M. Nouman Muteeb, Domain walls and M2-branes partition functions: M-theory and ABJM Theory (arXiv:2010.04233)

   diff, v8, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 29th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexand this one: * [[Stefan Hohenegger]], [[Amer Iqbal]], Soo-Jong Rey, _M String, Monopole String and Modular Forms_, Phys. Rev. D 92, 066005 (2015) ([arXiv:1503.06983](https://arxiv.org/abs/1503.06983)) <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/8">v8</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

   and this one:

   • Stefan Hohenegger, Amer Iqbal, Soo-Jong Rey, M String, Monopole String and Modular Forms, Phys. Rev. D 92, 066005 (2015) (arXiv:1503.06983)

   diff, v8, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 24th 2021
   • (edited Mar 24th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * [[Yuji Tachikawa]], _Topological modular forms and the absence of a heterotic global anomaly_ ([arXiv:2103.12211](https://arxiv.org/abs/2103.12211)) <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/11">v11</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

   added pointer to today’s

   • Yuji Tachikawa, Topological modular forms and the absence of a heterotic global anomaly (arXiv:2103.12211)

   diff, v11, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 25th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded these early references: * [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomalies, characters and strings*, Nuclear Physics B Volume 287, 1987, Pages 317-361 (<a href="https://doi.org/10.1016/0550-3213(87)90108-8">doi:10.1016/0550-3213(87)90108-8</a>) * [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomalies and modular invariance in string theory*, Physics Letters B 177 (3-4), 317-323, 1986 (<a href="https://doi.org/10.1016/0370-2693(86)90760-4">doi:10.1016/0370-2693(86)90760-4</a>) * [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomaly cancelling terms from the elliptic genus*, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (<a href="https://doi.org/10.1016/0550-3213(88)90468-3">doi:10.1016/0550-3213(88)90468-3</a>) <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/12">v12</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

   added these early references:

   • A. N. Schellekens, Nicholas P. Warner, Anomalies, characters and strings, Nuclear Physics B Volume 287, 1987, Pages 317-361 (doi:10.1016/0550-3213(87)90108-8)

   • A. N. Schellekens, Nicholas P. Warner, Anomalies and modular invariance in string theory, Physics Letters B 177 (3-4), 317-323, 1986 (doi:10.1016/0370-2693(86)90760-4)

   • Wolfgang Lerche, Bengt Nilsson, A. N. Schellekens, Nicholas P. Warner, Anomaly cancelling terms from the elliptic genus, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (doi:10.1016/0550-3213(88)90468-3)

   diff, v12, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 26th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added/expanded ([here](https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references#InterpretationInTMFCohomology)) the section "Conjectural interpretation in tmf-cohomology", which now, after the previous pointer to Stolz & Teichner, continues with these more recent pointers: *** and the more explicit suggestion that, under this identification, the [[Chern-Dold character]] from [[tmf]] to [[modular forms]], sends a [[2d SCFT]] to its [[partition function]]/[[elliptic genus]]/supersymmetric index: * [[Davide Gaiotto]], [[Theo Johnson-Freyd]], Section 5 of: *Holomorphic SCFTs with small index*, Canadian Journal of Mathematics ([arXiv:1811.00589](https://arxiv.org/abs/1811.00589), [doi:10.4153/S0008414X2100002X](https://doi.org/10.4153/S0008414X2100002X)) This perspective is also picked up in [Gukov, Pei, Putrov & Vafa 18](#GukovPeiPutrovVafa18). Discussion of the [[2d SCFTs]] (namely [[supersymmetry|supersymmetric]] [[SU(2)]]-[[WZW-models]]) conjecturally corresponding, under this conjectural identification, to the elements of $\mathbb{Z}/24$ [$\simeq$](tmf#BoardmanHomomorphismInTmfIs6Connected) $ tmf^{-3}(\ast) = \pi_3(tmf) $ [$\simeq$](tmf#BoardmanHomomorphismInTmfIs6Connected) $\pi_3(\mathbb{S})$ (the [[third stable homotopy group of spheres]]): * [[Davide Gaiotto]], [[Theo Johnson-Freyd]], [[Edward Witten]], p. 17 of: *A Note On Some Minimally Supersymmetric Models In Two Dimensions*, ([arXiv:1902.10249](https://arxiv.org/abs/1902.10249)) in S. Novikov et al. *Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry*, Proc. Symposia Pure Math., 103(2), 2021 ([ISBN: 978-1-4704-5592-7](https://bookstore.ams.org/pspum-103-2)) * [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Mock modularity and a secondary elliptic genus* ([arXiv:1904.05788](https://arxiv.org/abs/1904.05788)) * [[Theo Johnson-Freyd]], *Topological Mathieu Moonshine* ([arXiv:2006.02922](https://arxiv.org/abs/2006.02922)) *** <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/13">v13</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

   I have added/expanded (here) the section “Conjectural interpretation in tmf-cohomology”, which now, after the previous pointer to Stolz & Teichner, continues with these more recent pointers:

   ---

   and the more explicit suggestion that, under this identification, the Chern-Dold character from tmf to modular forms, sends a 2d SCFT to its partition function/elliptic genus/supersymmetric index:

   • Davide Gaiotto, Theo Johnson-Freyd, Section 5 of: Holomorphic SCFTs with small index, Canadian Journal of Mathematics (arXiv:1811.00589, doi:10.4153/S0008414X2100002X)

   This perspective is also picked up in Gukov, Pei, Putrov & Vafa 18.

   Discussion of the 2d SCFTs (namely supersymmetric SU(2)-WZW-models) conjecturally corresponding, under this conjectural identification, to

   the elements of $\mathbb{Z}/24$ $\simeq$ $tmf^{-3}(\ast) = \pi_3(tmf)$ $\simeq$ $\pi_3(\mathbb{S})$ (the third stable homotopy group of spheres):

   • Davide Gaiotto, Theo Johnson-Freyd, Edward Witten, p. 17 of: A Note On Some Minimally Supersymmetric Models In Two Dimensions, (arXiv:1902.10249) in S. Novikov et al. Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry, Proc. Symposia Pure Math., 103(2), 2021 (ISBN: 978-1-4704-5592-7)

   • Davide Gaiotto, Theo Johnson-Freyd, Mock modularity and a secondary elliptic genus (arXiv:1904.05788)

   • Theo Johnson-Freyd, Topological Mathieu Moonshine (arXiv:2006.02922)

   ---

   diff, v13, current

8. • CommentRowNumber8.
   • CommentAuthornLab edit announcer
   • CommentTimeSep 19th 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexadded pointer to: * Ying-Hsuan Lin, [[Du Pei]], *Holomorphic CFTs and topological modular forms* &lbrack;[arXiv:2112.10724](https://arxiv.org/abs/2112.10724)&rbrack; Anonymous <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/17">v17</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

   added pointer to:

   • Ying-Hsuan Lin, Du Pei, Holomorphic CFTs and topological modular forms [arXiv:2112.10724]

   Anonymous

   diff, v17, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeOct 28th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * Jan Albert, Justin Kaidi, Ying-Hsuan Lin, *Topological modularity of Supermoonshine* $[$[arXiv:2210.14923](https://arxiv.org/abs/2210.14923)$]$ <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/18">v18</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

   added pointer to today’s

   • Jan Albert, Justin Kaidi, Ying-Hsuan Lin, Topological modularity of Supermoonshine $[$arXiv:2210.14923$]$

   diff, v18, current

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have polished up the publication data for Witten"s "Index of Dirac operator on loop space" and then added to it pointer to this lesser known article, where the origin of the idea is described in more detail: * [[Edward Witten]], p. 92-94 in: *Global anomalies in string theory*, in: W. Bardeen and A. White (eds.) *[Symposium on Anomalies, Geometry, Topology](https://inspirehep.net/conferences/965785?ui-citation-summary=true)*, World Scientific (1985) 61-99 &lbrack;[[WittenGlobalAnomaliesInStringTheory.pdf:file]], [spire:214913](https://inspirehep.net/literature/214913)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/19">v19</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

    I have polished up the publication data for Witten”s “Index of Dirac operator on loop space” and then added to it pointer to this lesser known article, where the origin of the idea is described in more detail:

    • Edward Witten, p. 92-94 in: Global anomalies in string theory, in: W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, World Scientific (1985) 61-99 [WittenGlobalAnomaliesInStringTheory.pdf:file, spire:214913]

    diff, v19, current

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to today's * Kimyeong Lee, Kaiwen Sun, Xin Wang, *Twisted Elliptic Genera* &lbrack;[arXiv:2212.07341](https://arxiv.org/abs/2212.07341)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/20">v20</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

    added pointer to today’s

    • Kimyeong Lee, Kaiwen Sun, Xin Wang, Twisted Elliptic Genera [arXiv:2212.07341]

    diff, v20, current

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded publication data for these two items: * Murad Alim, [[Babak Haghighat]], Michael Hecht, [[Albrecht Klemm]], Marco Rauch, Thomas Wotschke, *Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes*, Comm. Math. Phys. **339** (2015) 773–814 $[$[arXiv:1012.1608](https://arxiv.org/abs/1012.1608), [doi:10.1007/s00220-015-2436-3](https://doi.org/10.1007/s00220-015-2436-3)$]$ * [[Sergei Gukov]], [[Du Pei]], [[Pavel Putrov]], [[Cumrun Vafa]], *4-manifolds and topological modular forms*, J. High Energ. Phys. **2021** 84 (2021) $[$[arXiv:1811.07884](https://arxiv.org/abs/1811.07884), <a href="https://doi.org/10.1007/JHEP05(2021)084">doi:10.1007/JHEP05(2021)084</a>, [spire:1704312](https://inspirehep.net/literature/1704312)$]$ <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/21">v21</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

    added publication data for these two items:

    • Murad Alim, Babak Haghighat, Michael Hecht, Albrecht Klemm, Marco Rauch, Thomas Wotschke, Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes, Comm. Math. Phys. 339 (2015) 773–814 $[$arXiv:1012.1608, doi:10.1007/s00220-015-2436-3$]$

    • Sergei Gukov, Du Pei, Pavel Putrov, Cumrun Vafa, 4-manifolds and topological modular forms, J. High Energ. Phys. 2021 84 (2021) $[$arXiv:1811.07884, doi:10.1007/JHEP05(2021)084, spire:1704312$]$

    diff, v21, current

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeFeb 16th 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to today's: * [[Yuji Tachikawa]], [[Mayuko Yamashita]], [[Kazuya Yonekura]], *Remarks on mod-2 elliptic genus* $[$[arXiv:2302.07548](https://arxiv.org/abs/2302.07548)$]$ <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/22">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/22">v22</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

    added pointer to today’s:

    • Yuji Tachikawa, Mayuko Yamashita, Kazuya Yonekura, Remarks on mod-2 elliptic genus $[$arXiv:2302.07548$]$

    diff, v22, current

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 25th 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to today's * Kohki Kawabata, Shinichiro Yahagi, *Elliptic genera from classical error-correcting codes* $[$[arXiv:2308.12592](https://arxiv.org/abs/2308.12592)$]$ <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/25">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/25">v25</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

    added pointer to today’s

    • Kohki Kawabata, Shinichiro Yahagi, Elliptic genera from classical error-correcting codes $[$arXiv:2308.12592$]$

    diff, v25, current

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2024
    • (edited Mar 15th 2024)
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to today's: * [[Yuji Tachikawa]], Hao Y. Zhang, *On a $\mathbb{Z}_3$-valued discrete topological term in 10d heterotic string theories* \[<a href="https://arxiv.org/abs/2403.08861">arXiv:2403.08861</a>\] <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/27">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/27">v27</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

    added pointer to today’s:

    • Yuji Tachikawa, Hao Y. Zhang, On a $\mathbb{Z}_3$-valued discrete topological term in 10d heterotic string theories [arXiv:2403.08861]

    diff, v27, current

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 10th 2024
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to today's * [[Theo Johnson-Freyd]], [[Mayuko Yamashita]], *On the 576-fold periodicity of the spectrum SQFT: The proof of the lower bound via the Anderson duality pairing* \[<a href="https://arxiv.org/abs/2404.06333">arXiv:2404.06333</a>\] <a href="https://ncatlab.org/nlab/revision/diff/elliptic+genera+as+partition+functions+--+references/28">diff</a>, <a href="https://ncatlab.org/nlab/revision/elliptic+genera+as+partition+functions+--+references/28">v28</a>, <a href="https://ncatlab.org/nlab/show/elliptic+genera+as+partition+functions+--+references">current</a>

    added pointer to today’s

    • Theo Johnson-Freyd, Mayuko Yamashita, On the 576-fold periodicity of the spectrum SQFT: The proof of the lower bound via the Anderson duality pairing [arXiv:2404.06333]

    diff, v28, current