Toda Darboux transformations and vacuum expectation values
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911993524387840 |
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| author | Wang, Chengwei Chen, Mengyao Cheng, Jipeng |
| author_facet | Wang, Chengwei Chen, Mengyao Cheng, Jipeng |
| contents | Determinant formulas for vacuum expectation values $\langle s+k+n-m,-s|e^{H(\mathbf{t})}β_m^{*}\cdotsβ_1^{*}β_n\cdotsβ_1g|k\rangle $ are given by using Toda Darboux transformations. Firstly notice that 2--Toda hierarchy can be viewed as the 2--component bosonizations of fermionic KP hierarchy, then two elementary Toda Darboux transformation operators $T_{+}(q)=Λ(q)\cdotΔ\cdot q^{-1}$ and $T_{-}(r)=Λ^{-1}(r)^{-1}\cdotΔ^{-1}\cdot r$ are constructed from the changes of Toda (adjoint) wave functions by using 2--component boson--fermion correspondence. Based on this, the above vacuum expectation values now can be realized as the successive applications of Toda Darboux transformations. So the corresponding determinant formulas can be derived from the determinant representations of Toda Darboux transformations. Finally by similar methods, we also give the determinant formulas for $\langle n-m|e^{\mathcal{H}(\mathbf{x})}β_m^{*}\cdotsβ_1^{*}β_n\cdotsβ_1g|k\rangle $ related with KP tau functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toda Darboux transformations and vacuum expectation values Wang, Chengwei Chen, Mengyao Cheng, Jipeng Exactly Solvable and Integrable Systems Mathematical Physics 35Q53, 37K10, 37K40 Determinant formulas for vacuum expectation values $\langle s+k+n-m,-s|e^{H(\mathbf{t})}β_m^{*}\cdotsβ_1^{*}β_n\cdotsβ_1g|k\rangle $ are given by using Toda Darboux transformations. Firstly notice that 2--Toda hierarchy can be viewed as the 2--component bosonizations of fermionic KP hierarchy, then two elementary Toda Darboux transformation operators $T_{+}(q)=Λ(q)\cdotΔ\cdot q^{-1}$ and $T_{-}(r)=Λ^{-1}(r)^{-1}\cdotΔ^{-1}\cdot r$ are constructed from the changes of Toda (adjoint) wave functions by using 2--component boson--fermion correspondence. Based on this, the above vacuum expectation values now can be realized as the successive applications of Toda Darboux transformations. So the corresponding determinant formulas can be derived from the determinant representations of Toda Darboux transformations. Finally by similar methods, we also give the determinant formulas for $\langle n-m|e^{\mathcal{H}(\mathbf{x})}β_m^{*}\cdotsβ_1^{*}β_n\cdotsβ_1g|k\rangle $ related with KP tau functions. |
| title | Toda Darboux transformations and vacuum expectation values |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics 35Q53, 37K10, 37K40 |
| url | https://arxiv.org/abs/2408.09457 |