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Counting BPS states via holomorphic anomaly equations. (English) Zbl 1046.81086
Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3355-3/hbk). Fields Inst. Commun. 38, 57-86 (2003).
Let $$S$$ be a surface obtained by blowing up 9 base points of 2 generic cubics in $$P^2$$. The author studies the Gromov-Witten invariants $$N_g(\beta)$$ with $$\beta\in H_2(S, Z)$$ of a rational elliptic surface $$S$$ using holomorphic anomaly equation (HA eq.). Let $$F,\sigma$$ in $$H^2(S,Z)$$ be the fiber class and the class of a section of the elliptic fibration: $$S \to P^1$$. From $$N_g(d,n):= \sum_{\beta.\sigma=d, \beta.F=n}N_g (\beta)$$, $$Z_{g;n}(q):=\sum_{d\geq 0}N_g(d,n)q^d= P_{g,n}(E_2(q), E_4(q), E_6 (q))q^{n/ 2}/ \eta(q)^{12n}$$ is given. $$E_2,E_4,E_6$$ are Eisenstein series, $$\eta (q)= q^{1/24}\prod_{m>0} (1-q^m)$$. He treats the HA eq.: $\partial Z_{g;n}/ \partial E_2=24^{-1}\sum_{g'+g'' =g} \sum_{s=1\sim n-1} S(n-s)Z_{g';s^*} Z_{g'';n-s}+ n (n+1)Z_{g-1;n}/24,$ with the initial data $$Z_{0;1}= q^{1/2} E_4(q)/ \eta (q)^{1 2}$$. Using the affine $$E_8$$ symmetry which arises as isomorphisms of rational elliptic surfaces, he determines $$N_g(\beta)$$ with $$(\beta,F)=n= 1,2,3, 4$$ and genus $$g=2^{-1}\{(\beta,\beta)-(\beta,F)+2\}\leq 10$$. The conjectured numbers $$n_g(\beta)$$ of BPS states with spin $$g$$ and charge $$\beta$$ are obtained from $$N_g(\beta)$$. A conjecture relating to the ambiguity of $$F_g(p,q):= \sum_{n \geq 1}Z_{g;n}p^n$$ $$(g\geq 2)$$ is also given.
For the entire collection see [Zbl 1022.00014].

##### MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 14J81 Relationships between surfaces, higher-dimensional varieties, and physics 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations 11F23 Relations with algebraic geometry and topology
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