## Nagoya Mathematical Journal

### Twisted orbifold Gromov–Witten invariants

Valentin Tonita

#### Abstract

Let $\mathcal {X}$ be a smooth proper Deligne–Mumford stack over $\mathbb{C}$. One can define twisted orbifold Gromov–Witten invariants of $\mathcal {X}$ by considering multiplicative invertible characteristic classes of various bundles on the moduli spaces of stable maps $\mathcal {X}_{g,n,d}$, cupping them with evaluation and cotangent line classes, and then integrating against the virtual fundamental class. These are more general than the twisted invariants introduced by Tseng. We express the generating series of the twisted invariants in terms of the generating series of the untwisted ones. We derive the corollaries which are used in a paper with Givental about the quantum $K$-theory of a complex compact manifold $X$.

#### Article information

Source
Nagoya Math. J., Volume 213 (2014), 141-187.

Dates
First available in Project Euclid: 17 December 2013

https://projecteuclid.org/euclid.nmj/1387313478

Digital Object Identifier
doi:10.1215/00277630-2393950

Mathematical Reviews number (MathSciNet)
MR3161407

Zentralblatt MATH identifier
1303.14065

#### Citation

Tonita, Valentin. Twisted orbifold Gromov–Witten invariants. Nagoya Math. J. 213 (2014), 141--187. doi:10.1215/00277630-2393950. https://projecteuclid.org/euclid.nmj/1387313478

#### References

• [1] D. Abramovich, “Lectures on Gromov–Witten invariants of orbifolds” in Enumerative Invariants in Algebraic Geometry and String Theory, Lecture Notes in Math. 1947, Springer, Berlin, 2008, 1–48.
• [2] D. Abramovich, T. Graber, M. Olsson, and H.-H. Tseng, On the global quotient structure of the space of twisted stable maps to a quotient stack, J. Algebraic Geom. 16 (2007), 731–751.
• [3] D. Abramovich, T. Graber, and A. Vistoli, “Algebraic orbifold quantum products” in Orbifolds in Mathematics and Physics (Madison, Wis., 2001), Contemp. Math. 310, Amer. Math. Soc., Providence, 2002, 1–24.
• [4] D. Abramovich, T. Graber, and A. Vistoli, Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math. 130 (2008), 1337–1398.
• [5] D. Abramovich and A. Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), 27–75.
• [6] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), 45–88.
• [7] W. Chen and Y. Ruan, “Orbifold Gromov–Witten theory” in Orbifolds in Mathematics and Physics (Madison, Wis., 2001), Contemp. Math. 310, Amer. Math. Soc., Providence, 2002, 25–85.
• [8] T. H. Coates, Riemann–Roch theorems in Gromov–Witten theory, Ph.D. dissertation, University of California, Berkeley, Berkeley, Calif., 2003.
• [9] T. Coates and A. Givental, Quantum Riemann–Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), 15–53.
• [10] A. B. Givental, Gromov–Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001), 551–568, 645.
• [11] A. B Givental, “Symplectic geometry of Frobenius structures” in Frobenius Manifolds, Aspects Math. E36, Friedr. Vieweg, Wiesbaden, 2004, 91–112.
• [12] A. Givental and V. Tonita, The Hirzebruch–Riemann–Roch theorem in true genus $0$ quantum $K$-theory, preprint, arXiv:1106.3136v1 [math.AG].
• [13] T. J. Jarvis and T. Kimura, “Orbifold quantum cohomology of the classifying space of a finite group” in Orbifolds in Mathematics and Physics (Madison Wis., 2001), Contemp. Math. 310, Amer. Math. Soc., Providence, 2002, 123–134.
• [14] A. Kabanov and T. Kimura, A change of coordinates on the large phase space of quantum cohomology, Comm. Math. Phys. 217 (2001), 107–126.
• [15] Y.-P. Lee, A formula for Euler characteristics of tautological line bundles on the Deligne–Mumford moduli spaces, Int. Math. Res. Not. IMRN 1997, 393–400.
• [16] D. McDuff and D. Salamon, $J$-Holomorphic Curves and Quantum Cohomology, Univ. Lecture Ser. 6, Amer. Math. Soc., Providence, 1994.
• [17] C. Teleman, The structure of 2D semi-simple field theories, Invent. Math. 188 (2012), 525–588.
• [18] B. Toën, Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford, $K$-Theory 18 (1999), 33–76.
• [19] H.-H. Tseng, Orbifold quantum Riemann–Roch, Lefschetz and Serre, Geom. Topol. 14 (2010), 1–81.