Generalized mirror transformation for open Gromov–Witten invariants

Let NN and kk be the parameters of the projective hypersurface, let PfP_f be the set of partitions σf=(f1,,fl(σf))\sigma_f=(f_1,\ldots,f_{l(\sigma_f)}) of ff, and let

S(σf):=i=1f1(mul(i,σf))!.S(\sigma_f):=\prod_{i=1}^{f}\frac{1}{(\operatorname{mul}(i,\sigma_f))!}.

Here wdiskN,k(Oha)2d1w_{disk}^{N,k}({\cal O}_{h^a})_{2d-1} is the B-model disk amplitude, Ohadisk,2d1\langle{\cal O}_{h^a}\rangle_{disk,2d-1} and the multipoint correlators are open Gromov–Witten invariants, and w(OhN3(kN)rOh0)0,rw({\cal O}_{h^{N-3-(k-N)r}}{\cal O}_{h^0})_{0,r} is the quantity specified by the referenced formula. Generalized mirror transformation.

wdiskN,k(Oha)2d1=Ohadisk,2d1+f=1d1σfPfS(σf)Ohaj=1l(σf)Oh1+(kN)fjdisk,2d2f1j=1l(σf)w(OhN3(kN)fjOh0)0,fjk.w_{disk}^{N,k}({\cal O}_{h^a})_{2d-1}=\langle{\cal O}_{h^a}\rangle_{disk,2d-1}+\sum_{f=1}^{d-1}\sum_{\sigma_f\in P_f}S(\sigma_f)\left\langle{\cal O}_{h^a}\prod_{j=1}^{l(\sigma_f)}{\cal O}_{h^{1+(k-N)f_j}}\right\rangle_{disk,2d-2f-1}\prod_{j=1}^{l(\sigma_f)}\frac{w({\cal O}_{h^{N-3-(k-N)f_j}}{\cal O}_{h^0})_{0,f_j}}{k}.

This is proposed as the open analogue of the closed generalized mirror transformation, translating B-model disk amplitudes into open Gromov–Witten invariants; the supplied parser status gives no evidence of resolution.

Sources & referencesView supporting material

Primary source

Masao Jinzenji and Masahide Shimizu, “Open Virtual Structure Constants and Mirror Computation of Open Gromov-Witten Invariants of Projective Hypersurfaces”, arXiv:1108.4766 (2011).

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