Modular symmetry of massive free fermions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913682030592000 |
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| author | Downing, Max Murthy, Sameer Watts, Gerard M. T. |
| author_facet | Downing, Max Murthy, Sameer Watts, Gerard M. T. |
| contents | We construct an infinite set of conserved tensor currents of rank $2n$, $n=1,2,\dots$, in the two-dimensional theory of free massive fermions, which are bilinear in the fermionic fields. The one-point functions of these currents on the torus depend on the modular parameter $τ$ and spin structure $(α,β)$. We show that, upon scaling the mass $m$ so as to keep the combination $m^2$Im($τ$) invariant, the one-point functions are non-holomorphic Jacobi forms of weights $(2n,0)$ or $(0,2n)$ and index 0, with respect to the modular parameter $τ$ and elliptic parameter $z=ατ+β$. In particular, we express the one-point functions as Kronecker-Eisenstein-type sums over the lattice $\mathbb{Z}τ+\mathbb{Z}$, which makes the modular symmetry manifest. We show that there is an action of three differential operators on these Jacobi forms which form an $\mathfrak{sl}_2(\mathbb{R})$ Lie algebra. Further we show that these Jacobi forms obey three differential equations arising from the representation theory of the Jacobi group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_01251 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Modular symmetry of massive free fermions Downing, Max Murthy, Sameer Watts, Gerard M. T. High Energy Physics - Theory We construct an infinite set of conserved tensor currents of rank $2n$, $n=1,2,\dots$, in the two-dimensional theory of free massive fermions, which are bilinear in the fermionic fields. The one-point functions of these currents on the torus depend on the modular parameter $τ$ and spin structure $(α,β)$. We show that, upon scaling the mass $m$ so as to keep the combination $m^2$Im($τ$) invariant, the one-point functions are non-holomorphic Jacobi forms of weights $(2n,0)$ or $(0,2n)$ and index 0, with respect to the modular parameter $τ$ and elliptic parameter $z=ατ+β$. In particular, we express the one-point functions as Kronecker-Eisenstein-type sums over the lattice $\mathbb{Z}τ+\mathbb{Z}$, which makes the modular symmetry manifest. We show that there is an action of three differential operators on these Jacobi forms which form an $\mathfrak{sl}_2(\mathbb{R})$ Lie algebra. Further we show that these Jacobi forms obey three differential equations arising from the representation theory of the Jacobi group. |
| title | Modular symmetry of massive free fermions |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2302.01251 |