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Poisson vertex algebras in supersymmetric field theories

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Abstract

A large class of supersymmetric quantum field theories, including all theories with \({\mathcal {N}}= 2\) supersymmetry in three dimensions and theories with \({\mathcal {N}}= 2\) supersymmetry in four dimensions, possess topological–holomorphic sectors. We formulate Poisson vertex algebras in such topological–holomorphic sectors and discuss some examples. For a four-dimensional \({\mathcal {N}}= 2\) superconformal field theory, the associated Poisson vertex algebra is the classical limit of a vertex algebra generated by a subset of local operators of the theory.

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Notes

  1. We are grateful to Dylan Butson for explaining this point to us.

  2. In the path integral formulation, the action of a conserved charge X on a local operator \({\mathcal {O}}\) located at a point x is implemented by the integration of the Hodge dual of the associated conserved current over a simply-connected codimension-1 cycle surrounding x. If we take this cycle to be the difference of two time slices sandwiching x, we obtain the familiar formula \(X \cdot {\mathcal {O}}= [X, {\mathcal {O}}]\) as an operator acting on the Hilbert space of states. The cycle can also be infinitesimally small, so the operation \({\mathcal {O}}\mapsto X \cdot {\mathcal {O}}\) is local.

  3. Since \(H_{d+2}(\mathrm {Conf}_2(\mathbb {R}^d \times \mathbb {C}^\times )) \cong \mathbb {Z}\), to establish this relation we just need to evaluate a suitable cohomology class on these cycles. We can take this class to be \(\mathrm {d}(\theta _1 + \theta _2) \wedge \mathrm {d}^{-1}(\bigwedge _\mu \delta (x_{12}^\mu ) \mathrm {d}x_{12}^\mu )\), where \(\theta = \arg z\) and \(\delta (x)\) is the delta function. To compute the homology group, let \(U = \mathrm {Conf}_2(\mathbb {R}^d \times \mathbb {C}^\times )\) and \(V \simeq \mathbb {R}^{2d+3} \times S^1\) be a normal neighborhood of the diagonal \(\varDelta \) of \((\mathbb {R}^d \times \mathbb {C}^\times )^2\). Then, \(U \cup V = (\mathbb {R}^d \times \mathbb {C}^\times )^2 \simeq \mathbb {R}^{2d+2} \times S^1 \times S^1\) and \(U \cap V = V {\setminus } \varDelta \simeq \mathbb {R}^{d+2} \times S^1 \times S^{d+1}\). From the Mayer–Vietoris sequence

    $$\begin{aligned} \cdots \rightarrow H_{d+3}(U \cup V) \rightarrow H_{d+2}(U \cap V) \rightarrow H_{d+2}(U) \oplus H_{d+2}(V) \rightarrow H_{d+2}(U \cup V) \rightarrow \cdots \,, \end{aligned}$$
    (34)

    we get \(H_{d+2}(U) \cong H_{d+2}(S^1 \times S^{d+1}) \cong \mathbb {Z}\).

  4. Think of \({\mathcal {O}}_1^*\), \({\mathcal {O}}_2^*\) and \({\mathcal {O}}_3^*\) as the moon, the earth and the sun, respectively, and consider the heliocentric versus geocentric descriptions of their relative motion. The origin of \(\mathbb {C}^\times \) can be the center of our galaxy.

  5. The \({\mathcal {N}}= 2\) supersymmetry algebra in \(2+1\) dimensions can be obtained from the \({\mathcal {N}}= 1\) supersymmetry algebra in \(3+1\) dimensions by dimensional reduction. For the latter algebra, we follow the conventions of [31], except that we rescale the supercharges by a factor of \(1/\sqrt{2}\). We have chosen to perform the reduction along the \(x^2\)-direction and subsequently renamed \(x^3\) to \(x^2\).

  6. The fact that the Q-cohomology of local operators is a Poisson vertex algebra was mentioned in [2,3,4].

  7. Here, we cannot use the \(\mathbb {C}^\times \)-action because we need to place local operators away from \(w = 0\) in order to define these structures. The action by \(\alpha \in \mathbb {C}^\times \) transforms \({}^{t,Z}{\mathcal {O}}(w,{\bar{w}},z,{\bar{z}})\) to \({}^{\alpha t,Z}{\mathcal {O}}(\alpha w, \alpha ^{-1}{\bar{w}},z,{\bar{z}})\) (assuming, for simplicity, that \({\mathcal {O}}\) is a scalar operator). For \(\alpha w\) and \(\alpha ^{-1} {\bar{w}}\) to be complex conjugate to each other, we must have \(\alpha \in \mathrm {U}(1)\) or \(w = 0\).

  8. In [26], the construction of vertex algebras was extended to \({\mathcal {N}}= 2\) supersymmetric gauge theories which are not necessarily conformal. For a nonconformal theory, the associated vertex algebra is anomalous, but the anomaly can be canceled if an \({\mathcal {N}}= (0,2)\) supersymmetric surface defect carrying an appropriate vertex algebra is inserted at \(w = 0\). Although the combined vertex algebra is well defined, it may not have a classical limit since the vertex algebra on the surface defect may not have one. What goes wrong in the above argument is the assertion that the OPE between Q-cohomology classes is regular.

  9. In terms of the spherical coordinates \((\psi ,\theta ,\varphi ) \in [0,\pi ] \times [0,\pi ] \times [0,2\pi )\), defined by

    $$\begin{aligned} \begin{aligned} x^1_1 - x^1_2&= r \sin \psi \sin \theta \cos \varphi \,, \\ x^2_1 - x^2_2&= r \sin \psi \sin \theta \sin \varphi \,, \\ x^3_1 - x^3_2&= r \cos \psi \,, \\ x^4_1 - x^4_2&= r \sin \psi \cos \theta \,, \end{aligned} \end{aligned}$$
    (114)

    the 2-disk \(D^2_{x_2}\) is located at \(\varphi = 0\) and has the boundary \(S^1_{z_2}\) at \(\theta = 0\), \(\pi \). One can easily show that for an equivariantly closed form, the integral over \(S^3_{x_2}\) reduces to an integral over \(S^1_{z_2}\).

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Acknowledgements

We would like to thank Kevin Costello for invaluable advice and illuminating discussions, and Dylan Butson, Tudor Dimofte and Davide Gaiotto for helpful comments. The research of JO is supported in part by Kwanjeong Educational Foundation, by the Visiting Graduate Fellowship Program at the Perimeter Institute for Theoretical Physics and by the Berkeley Center of Theoretical Physics. The research of JY is supported by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.

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  1. Department of Physics, University of California, Berkeley, CA, 94720, USA

    Jihwan Oh

  2. Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada

    Jihwan Oh & Junya Yagi

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Oh, J., Yagi, J. Poisson vertex algebras in supersymmetric field theories. Lett Math Phys 110, 2245–2275 (2020). https://doi.org/10.1007/s11005-020-01290-0

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