Duality and mock modularity
We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional $\mathcal{N} =4$ super Yang-Mills theory on $\mathbb{CP}^{2}$ for the gauge group $SO(3)$ from the path integral of the effective theory on the Coulomb branch. The holomorphic kernel of th...
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| Format: | Article |
| Language: | English |
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SciPost
2020-11-01
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| Series: | SciPost Physics |
| Online Access: | https://scipost.org/SciPostPhys.9.5.072 |
| _version_ | 1818263131654193152 |
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| author | Atish Dabholkar, Pavel Putrov, Edward Witten |
| author_facet | Atish Dabholkar, Pavel Putrov, Edward Witten |
| author_sort | Atish Dabholkar, Pavel Putrov, Edward Witten |
| collection | DOAJ |
| description | We derive a holomorphic anomaly equation for the Vafa-Witten partition
function for twisted four-dimensional $\mathcal{N} =4$ super Yang-Mills theory
on $\mathbb{CP}^{2}$ for the gauge group $SO(3)$ from the path integral of the
effective theory on the Coulomb branch. The holomorphic kernel of this
equation, which receives contributions only from the instantons, is not modular
but `mock modular'. The partition function has correct modular properties
expected from $S$-duality only after including the anomalous nonholomorphic
boundary contributions from anti-instantons. Using M-theory duality, we relate
this phenomenon to the holomorphic anomaly of the elliptic genus of a
two-dimensional noncompact sigma model and compute it independently in two
dimensions. The anomaly both in four and in two dimensions can be traced to a
topological term in the effective action of six-dimensional (2,0) theory on the
tensor branch. We consider generalizations to other manifolds and other gauge
groups to show that mock modularity is generic and essential for exhibiting
duality when the relevant field space is noncompact. |
| first_indexed | 2024-12-12T19:14:09Z |
| format | Article |
| id | doaj.art-470a1f1bfcb84637936e81b786d8fef6 |
| institution | Directory Open Access Journal |
| issn | 2542-4653 |
| language | English |
| last_indexed | 2024-12-12T19:14:09Z |
| publishDate | 2020-11-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics |
| spelling | doaj.art-470a1f1bfcb84637936e81b786d8fef62022-12-22T00:14:46ZengSciPostSciPost Physics2542-46532020-11-019507210.21468/SciPostPhys.9.5.072Duality and mock modularityAtish Dabholkar, Pavel Putrov, Edward WittenWe derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional $\mathcal{N} =4$ super Yang-Mills theory on $\mathbb{CP}^{2}$ for the gauge group $SO(3)$ from the path integral of the effective theory on the Coulomb branch. The holomorphic kernel of this equation, which receives contributions only from the instantons, is not modular but `mock modular'. The partition function has correct modular properties expected from $S$-duality only after including the anomalous nonholomorphic boundary contributions from anti-instantons. Using M-theory duality, we relate this phenomenon to the holomorphic anomaly of the elliptic genus of a two-dimensional noncompact sigma model and compute it independently in two dimensions. The anomaly both in four and in two dimensions can be traced to a topological term in the effective action of six-dimensional (2,0) theory on the tensor branch. We consider generalizations to other manifolds and other gauge groups to show that mock modularity is generic and essential for exhibiting duality when the relevant field space is noncompact.https://scipost.org/SciPostPhys.9.5.072 |
| spellingShingle | Atish Dabholkar, Pavel Putrov, Edward Witten Duality and mock modularity SciPost Physics |
| title | Duality and mock modularity |
| title_full | Duality and mock modularity |
| title_fullStr | Duality and mock modularity |
| title_full_unstemmed | Duality and mock modularity |
| title_short | Duality and mock modularity |
| title_sort | duality and mock modularity |
| url | https://scipost.org/SciPostPhys.9.5.072 |
| work_keys_str_mv | AT atishdabholkarpavelputrovedwardwitten dualityandmockmodularity |