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H 1 A string theorist view point on the genus-two Kawazumi-Zhang invariant

Auteurs : Pioline, Boris (Auteur de la Conférence)
CIRM (Editeur )

Résumé : The genus-two Kawazumi-Zhang (KZ) invariant is a real-analytic modular function on the Siegel upper half-plane of degree two, which plays an important role in arithmetic geometry. In String theory, it appears as part of the integrand in two-loop four-graviton scattering amplitudes. With hindsight from String theory, I will show that the KZ invariant can be obtained as a generalized Borcherds lift from a weak Jacobi form of index 1 and weight 2. This implies that the KZ invariant is an eigenmode of the quadratic and quartic Casimir operators, and gives access to the full asymptotic expansion in all possible degeneration limits. It also reveals a mock-type holomorphic Siegel modular form underlying the KZ invariant. String theory amplitudes involves modular integrals of the KZ invariant (times lattice partition functions) on the Siegel upper half-plane, which provide new examples of automorphic objects on orthogonal Grassmannians, beyond the usual Langlands-Eisenstein series.

Note: this talk is based on the preprint "A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces" available on arXiv:1504.04182. Following up on a question asked during the talk (which was answered very poorly), the author obtained shortly after a proof of the conjecture stated in this preprint and during the talk. The proof is available in the revised version arXiv:1504.04182v2.

Codes MSC :
81T30 - String and superstring theories; other extended objects (e.g., branes)
83E30 - String and superstring theories

 Informations sur la Vidéo Langue : Anglais Date de publication : 18/06/15 Date de captation : 27/05/15 Sous collection : Research talks Format : quicktime ; audio/x-aac arXiv category : High Energy Physics ; Dynamical Systems Domaine : Mathematical Physics ; Dynamical Systems & ODE Durée : 00:49:56 Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2015-05-27_Pioline.mp4 Informations sur la rencontre Nom de la rencontre : Automorphic forms: advances and applications / Formes automorphes: avancées et applicationsOrganisateurs de la rencontre : Bringmann, Kathrin ; Lovejoy, Jérémy ; Richter, OlavDates : 25/05/15 - 29/05/15 Année de la rencontre : 2015 URL Congrès : http://conferences.cirm-math.fr/1108.html Citation Data DOI : 10.24350/CIRM.V.18769203 Cite this video as: Pioline, Boris (2015). A string theorist view point on the genus-two Kawazumi-Zhang invariant. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18769203 URI : http://dx.doi.org/10.24350/CIRM.V.18769203

Bibliographie

1. [1] Pioline, B. (2015). A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces. - http://arxiv.org/abs/1504.04182v2

2. [2] D'Hoker, E., Green, M. B., Pioline, B., Russo, R. (2014). Matching the $\mathcal{D}^6 \mathcal{R}^4$ interaction at two-loops. - http://arxiv.org/abs/1405.6226

3. [3] Kawazumi, N. (2008). Johnson's homomorphisms and the Arakelov-Green function. < arXiv:0801.4218> - http://arxiv.org/abs/0801.4218

4. [4] Zhang, S.-W. (2010). Gross-Schoen cycles and dualising sheaves. Inventiones Mathematicae 179(1), 1-73 - http://dx.doi.org/10.1007/s00222-009-0209-3

5. [5] de Jong, R. (2013). Second variation of Zhang’s $\lambda$-invariant on the moduli space of curves. American Journal of Mathematics 135(1), 275-290 - http://dx.doi.org/10.1353/ajm.2013.0008

6. [6] Wentworth, R. (1991). The asymptotics of the Arakelov-Green's function and Faltings' delta invariant. Communications in Mathematical Physics 137(3), 427-459 - http://projecteuclid.org/euclid.cmp/1104202735

7. [7] de Jong, R. (2009). Admissible constants for genus 2 curves. Bulletin of the London Mathematical Society 42(3), 405-411 - http://dx.doi.org/10.1112/blms/bdp132

8. [8] D'Hoker, E., Green, M. B. (2013). Zhang-Kawazumi Invariants and Superstring Amplitudes. - http://arxiv.org/abs/1308.4597

9. [9] Kawai, T. (1995). $\mathcal{N}=2$ heterotic string threshold correction, $\mathcal{K}3$ surface and generalized Kac-Moody superalgebra. - http://arxiv.org/abs/hep-th/9512046

10. [10] Angelantonj, C., Florakis, I., Pioline, B. (2015). Threshold corrections, generalised prepotentials and Eichler integrals. - http://arxiv.org/abs/1502.00007

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