English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The modular graph $C_{a,b,c,d}$ on the torus is a three loop planar graph inwhich two of the vertices have coordination number four, while the others havecoordination number two. We obtain an eigenvalue equation satisfied by$C_{a,b,c,d}$ for generic values of $a,b,c$ and $d$, where the source termsinvolve various modular graphs. This is obtained by varying the graph withrespect to the Beltrami differential on the toroidal worldsheet. Use of severalauxiliary graphs at various intermediate stages of the analysis is crucial inobtaining the equation. In fact, the eigenfunction is not simply $C_{a,b,c,d}$but involves subtracting from it specific sums of squares of non--holomorphicEisenstein series characterized by $a,b,c$ and $d$.

Eigenvalue equation for the modular graph $C_{a,b,c,d}$