On Sp(n)-Instantons and the Fourier-Mukai Transform of Complex Lagrangians
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| Format: | Preprint |
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2024
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| author | Madnick, Jesse Windes, Emily Autumn |
| author_facet | Madnick, Jesse Windes, Emily Autumn |
| contents | The real Fourier-Mukai (RFM) transform relates calibrated graphs to so-called "deformed instantons" on Hermitian line bundles. We show that under the RFM transform, complex Lagrangian graphs in $R^{2n} \times T^{2n}$ correspond to Sp($n$)-instantons over $R^{2n} \times (T^{2n})^*$. In other words, the deformed Sp($n$)-instanton equation coincides with the usual Sp($n$)-instanton equation.
Motivated by this observation, we study Sp($n$)-instantons on hyperkahler manifolds $X^{4n}$, with an emphasis on conical singularities. First, when $X = C(M)$ is a hyperkahler cone, we relate Sp($n$)-instantons on $X$ to tri-contact instantons on the 3-Sasakian link $M$ and consider various dimensional reductions. Second, when $X$ is an asymptotically conical (AC) hyperkahler manifold of rate $ν\leq -\frac{2}{3}(2n+1)$, we prove a Lewis-type theorem to the following effect: If the set of AC Sp($n$)-instantons is non-empty, then every AC Hermitian Yang-Mills connection over $X$ with sufficiently fast decay at infinity is an Sp($n$)-instanton. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_06412 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Sp(n)-Instantons and the Fourier-Mukai Transform of Complex Lagrangians Madnick, Jesse Windes, Emily Autumn Differential Geometry The real Fourier-Mukai (RFM) transform relates calibrated graphs to so-called "deformed instantons" on Hermitian line bundles. We show that under the RFM transform, complex Lagrangian graphs in $R^{2n} \times T^{2n}$ correspond to Sp($n$)-instantons over $R^{2n} \times (T^{2n})^*$. In other words, the deformed Sp($n$)-instanton equation coincides with the usual Sp($n$)-instanton equation. Motivated by this observation, we study Sp($n$)-instantons on hyperkahler manifolds $X^{4n}$, with an emphasis on conical singularities. First, when $X = C(M)$ is a hyperkahler cone, we relate Sp($n$)-instantons on $X$ to tri-contact instantons on the 3-Sasakian link $M$ and consider various dimensional reductions. Second, when $X$ is an asymptotically conical (AC) hyperkahler manifold of rate $ν\leq -\frac{2}{3}(2n+1)$, we prove a Lewis-type theorem to the following effect: If the set of AC Sp($n$)-instantons is non-empty, then every AC Hermitian Yang-Mills connection over $X$ with sufficiently fast decay at infinity is an Sp($n$)-instanton. |
| title | On Sp(n)-Instantons and the Fourier-Mukai Transform of Complex Lagrangians |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2407.06412 |