## Algebra & Number Theory

### Root systems and the quantum cohomology of ADE resolutions

#### Abstract

We compute the $ℂ∗$-equivariant quantum cohomology ring of $Y$, the minimal resolution of the DuVal singularity $ℂ2∕G$ where $G$ is a finite subgroup of $SU(2)$. The quantum product is expressed in terms of an ADE root system canonically associated to $G$. We generalize the resulting Frobenius manifold to nonsimply laced root systems to obtain an $n$ parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Gromov–Witten potential of $[ℂ2∕G]$.

#### Article information

Source
Algebra Number Theory, Volume 2, Number 4 (2008), 369-390.

Dates
Revised: 9 May 2008
Accepted: 9 May 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513797266

Digital Object Identifier
doi:10.2140/ant.2008.2.369

Mathematical Reviews number (MathSciNet)
MR2411404

Zentralblatt MATH identifier
1159.14028

Keywords

#### Citation

Bryan, Jim; Gholampour, Amin. Root systems and the quantum cohomology of ADE resolutions. Algebra Number Theory 2 (2008), no. 4, 369--390. doi:10.2140/ant.2008.2.369. https://projecteuclid.org/euclid.ant/1513797266

#### References

• K. Behrend and B. Fantechi, “The intrinsic normal cone”, Invent. Math. 128:1 (1997), 45–88.
• A. Bertram, “Another way to enumerate rational curves with torus actions”, Invent. Math. 142:3 (2000), 487–512.
• N. Bourbaki, Groupes et algèbres de Lie, Chapitres IV–VI, Actualités scientifiques et industrielles 1337, Hermann, Paris, 1968.
• J. Bryan and A. Gholampour, “The quantum McKay correspondence for polyhedral singularities”, preprint, 2008.
• J. Bryan and T. Graber, “The crepant resolution conjecture”, in Algebraic Geometry (Seattle, 2005), 2008. To appear.
• J. Bryan and Y. Jiang, “The Crepant Resolution Conjecture for the orbifold $\mathbf{C^2/Z_4}$”. In preparation.
• J. Bryan, S. Katz, and N. C. Leung, “Multiple covers and the integrality conjecture for rational curves in Calabi–Yau threefolds”, J. Algebraic Geom. 10:3 (2001), 549–568.
• J. Bryan, T. Graber, and R. Pandharipande, “The orbifold quantum cohomology of ${\mathbb C}\sp 2/Z\sb 3$ and Hurwitz–Hodge integrals”, J. Algebraic Geom. 17:1 (2008), 1–28.
• T. Coates, A. Corti, H. Iritani, and H.-H. Tseng, “The Crepant Resolution Conjecture for type A surface singularities”, preprint, 2007.
• G. Gonzalez-Sprinberg and J.-L. Verdier, “Construction géométrique de la correspondance de McKay”, Ann. Sci. École Norm. Sup. $(4)$ 16:3 (1983), 409–449.
• S. Katz and D. R. Morrison, “Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups”, J. Algebraic Geom. 1:3 (1992), 449–530.
• G. Lusztig, Introduction to quantum groups, Progress in Mathematics 110, Birkhäuser, Boston, 1993.
• D. Maulik, “Gromov–Witten theory of A-resolutions”, preprint, 2008, \codarefarXiv:0802.2681
• J. McKay, “Graphs, singularities, and finite groups”, pp. 183–186 in The Santa Cruz Conference on Finite Groups (Santa Cruz, 1979), edited by B. Cooperstein and G. Mason, Proc. Sympos. Pure Math. 37, Amer. Math. Soc., Providence, R.I., 1980.
• M. Reid, “La correspondance de McKay”, Astérisque 276 (2002), 53–72. http://www.ams.org/mathscinet-getitem?mr=2003h:14026MR 2003h:14026
• T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics 9, Birkhäuser, Boston, 1998.