Atiyah–Bott formula

Short description: On the cohomology ring of the moduli stack of principal bundles

In algebraic geometry, the Atiyah–Bott formula says the cohomology ring

${\mathrm{H}}^{*}({\mathrm{Bun}}_G(X),{\mathbb{\mathbb{Q}}}_l)$

of the moduli stack of principal bundles is a free graded-commutative algebra on certain homogeneous generators. The original work of Michael Atiyah and Raoul Bott concerned the integral cohomology ring of ${\mathrm{Bun}}_G(X)$.

See also

• Borel's theorem, which says that the cohomology ring of a classifying stack is a polynomial ring.

References

• Atiyah, Michael F.; Bott, Raoul (1983). "The Yang-Mills equations over Riemann surfaces". Philosophical Transactions of the Royal Society of London. Ser. A 308 (1505): 523–615. doi:10.1098/rsta.1983.0017. Bibcode: 1983RSPTA.308..523A. 
• Gaitsgory, Dennis; Lurie, Jacob (2019), Weil's Conjecture for Function Fields, Annals of Mathematics Studies, 199, Princeton, NJ: Princeton University Press, http://www.math.harvard.edu/~lurie/papers/tamagawa.pdf