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Twisted formalism for 3d \({\mathcal {N}}=4\) theories

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Abstract

We describe the topological A and B twists of 3d \({\mathcal {N}}=4\) theories of hypermultiplets gauged by \({\mathcal {N}}=4\) vector multiplets as certain deformations of the holomorphic–topological (HT) twist of those theories, utilizing the twisted superfields of Aganagic–Costello–Vafa–McNamara describing HT-twisted 3d \({\mathcal {N}}=2\) theories. We rederive many known results from this perspective, including state spaces on Riemann surfaces, deformations induced by flavor symmetries, the boundary VOAs of Costello–Gaiotto, and the category of line operators as proposed by Costello–Dimofte–Gaiotto–Hilburn–Yoo. Along the way, we show how the secondary product of local operators in the holomorphic–topological twist is related to the secondary product in the fully topological twist.

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Notes

  1. Somewhat more precisely, if we write \(Q_{A} = Q_{HT} + \delta _{A}\) then it is immediate that \(\delta _A\) anti-commutes with \(Q_{HT}\) and is nilpotent. Thus, \(\delta _A\) descends to a nilpotent symmetry of the HT-twisted theory. The A twist, i.e.,  twisting with \(Q_A\), can then be realized by a spectral sequence whose first page corresponds to taking HT twist, whose second page corresponds to taking \(\delta _A\) cohomology of the HT-twisted theory, and so on. There are similar considerations for the B-twist. In the present paper, we do not quite take this approach but use it as inspiration. Instead of taking the HT twist, we consider a yet different theory (roughly obtained by formulating the theory in the Batalin–Vilkovisky formalism and integrating out \(Q_{HT}\)-exact fields) that is equivalent to the HT-twisted theory at the level of cohomology, i.e.,  they are quasi-isomorphic. The deformations describe here are obtained by passing \(\delta _A, \delta _B\) through this quasi-isomorphism; importantly, we deform the theory at chain level, i.e.,  we consider the total complex, rather than a spectral sequence.

  2. See also the work of Elliott–Gwilliam–Williams [21] for a similar discussion of the other twists of 4d \({\mathcal {N}}=4\).

  3. The reason why we do not need to deform the boundary conditions after passing to the holomorphic–topological twist is that the deformations are \(Q_{HT}\)-exact. For example, the deformed (0, 4) Neumann boundary condition of an (cf. Equation (E.9) of [35]) takes the form

    $$\begin{aligned} {\partial }_t {\overline{Y}}+ \zeta {\partial }_{{\overline{z}}} X = 0 \qquad -{\partial }_t {\overline{X}}+ \zeta {\partial }_{{\overline{z}}} Y = 0 \end{aligned}$$

    where \(\zeta \) is the deformation parameter, i.e., \(\zeta = 0\) corresponds to the undeformed boundary condition. Indeed, \({\partial }_{{\overline{z}}}X, {\partial }_{{\overline{z}}}Y\) are \(Q_{HT}\)-exact.

  4. As in [26], the state spaces described in this paper are not quite the same as those in [37], although they come from different choices of polarizations of the same phase space. In particular, the polarizations chosen in this paper has bounded cohomological degrees with finite-dimensional graded components, at the cost of lacking a Hermitian inner product, cf.  [26, Sec 2.5.3] For the A-twisted theories, it is the same as the recent paper [38] describing BPS state spaces in A-twisted theories across various dimensions, although the authors of that paper use a slightly different twisting homomorphism than the one used in this paper – they use the \(U(1)_M\) flavor symmetry described below to make the field we call X a scalar on \(\Sigma \) rather than a spinor.

  5. In the topological A twist, vector multiplets localize to holomorphic bundles (or, more generally, monopole configurations) while twisted vector multiplets localize to bundles equipped with flat connections; these roles are exchanged in the B twist. Thus, the A-twisted (resp. B-twisted) theory can be deformed by background holomorphic \(G_H\) (resp. \(G_C\)) bundles and \(G_C\) (resp. \(G_H\)) bundles with flat connection.

  6. We work in conventions such that parity alone determines the graded commutativity of observables. Indeed, the R-charge, and hence the cohomological grading, in the above class of \({\mathcal {N}}=2\) theories may be non-integral.

  7. The paper [5] uses the machinery of the homotopy RG flow of [64] to make mathematically precise statements in perturbative theory. In this paper, we will almost entirely ignore these details and work at infinitely long length scales \(L \rightarrow \infty \) to simplify the discussion.

  8. We thank J. Hilburn for bringing this to our attention.

  9. It may happen that the torus \(U(1)_M\) is part of the gauge group, hence not a flavor symmetry. Nonetheless, so long as the hypermultiplet representation is split \(V \oplus V^*\) then this \(U(1)_M\) action commutes with the gauge group and hence can be used to redefine the \(U(1)_R\) charge.

  10. A similar phenomenon happens for monopole operators in the A twist, where one should include a Coulomb branch flavor symmetry in the cohomological grading. We will briefly comment on the need to do this in Sect. 4.1.2.

  11. We restrict to constant vector fields to simplify the discussion. A similar analysis to what follows can be applied to more general holomorphic vector fields V, showing that they too act trivially in Q cohomology.

  12. It is important to note that this need not be the case: in principle, there may be anomalies that cause the classical Q-exactness of the stress tensor to fail. Even the quantum existence of the stress tensor \(\textbf{T}\) can fail; see, e.g.,  [65] for a related discussion in the context of 2d \({\mathcal {N}}=(0,2)\) theories. Although we expect it not to happen in the examples discussed in the present paper, there may be further non-perturbative obstructions, cf.  [66] for the corresponding statements in 2d \({\mathcal {N}}=(0,2)\) theories. We leave the verification of the non-perturbative Q-exactness of \(\textbf{T}\) in these and related examples to future work.

  13. We thank B. Williams for explaining the following argument.

  14. Somewhat more generally, the deformation \(\delta \) need not be an exact symmetry of the HT twist but still satisfies the Maurer–Cartan equation \(\{Q, \delta \} + \delta ^2 = 0\). Moreover, the elements \(Q_t, Q_{\overline{z}}\) trivializing \({\partial }_t, {\partial }_{\overline{z}}\) in the Q twist and the \({\mathbb {Q}}\) twist could be different. Neither of these situations arise in the examples considered in this paper, so we restrict to this simplified case. We expect that it is possible to generalize these arguments to more general cases.

  15. In the \(HT^A\) twist, \(\varepsilon \) is a scalar of cohomological degree 1. In the \(HT^B\) twist, it has spin 1 (transforms as \(\mathop {}\!\textrm{d}z\)) and cohomological degree \(-1\).

  16. More precisely, \(\textbf{X}, \textbf{Y}\) are coordinate functions on \(\text {Sect}(\Sigma _{{\overline{\partial }}}, T^*{{\mathbb {C}}}\otimes K^{1/2}_{\Sigma })\), and \(-\mathbf {\Psi }_{\textbf{Y}}, \mathbf {\Psi }_{\textbf{X}}\) are linear functions on the fiber of the odd tangent bundle, i.e.,  1-forms over \( \text {Sect}(\Sigma _{{\overline{\partial }}}, T^*{{\mathbb {C}}}\otimes K^{1/2}_{\Sigma })\).

  17. The homology theory one should work with is quite subtle, especially once we include gauge fields in Sect. 4. As described in [38, Sec 6] for general A-twisted \(\sigma \)-models, and in particular Example 6.5 of loc. cit., the correct choice for the present case is Borel–Moore homology. Indeed, this homology theory was originally proposed by Nakajima [71] and later used by Braverman–Finkelberg–Nakajima [40] to realize a mathematically precise definition of Coulomb branches of the 3d \({\mathcal {N}}=4\) theories described in this paper. In contrast to more traditional homology, Borel–Moore homology allows cycles to be non-compact and is dual to compactly supported cohomology. See, e.g.,  [72, Ch 2] for details about Borel–Moore homology in the context of geometric representation theory.

  18. We could alternatively choose a slightly different twisting homomorphism where \(\textbf{X}\) is a scalar and \(\textbf{Y}\) is a section of \(K_\Sigma \). This corresponds to working with the twisted spin \(\widetilde{J}_A = J_A - \tfrac{1}{2} M\).

  19. The cohomological degree 0 fields are the 1-form part of \({\widehat{\textbf{A}}}\) and the 2-form part of \({\widehat{\mathbf {\Phi }}}\). The latter can be identified with the auxiliary field in the adjoint chiral multiplet inside the \({\mathcal {N}}=4\) multiplet, which is equal to \({\overline{\mu }}_{{\mathbb {C}}}\) on-shell. Thus, noting that \(({\widehat{\textbf{A}}})|_{1\text {-form}} = ({\widehat{A}}_t -i {\widehat{\sigma }}) \mathop {}\!\textrm{d}t + {\widehat{A}}_{\overline{z}}\mathop {}\!\textrm{d}{\overline{z}}\), the first equation reads \(F_{{\overline{z}}t} - i D_{\overline{z}}\sigma \sim {\overline{\mu }}_{{\mathbb {C}}}\), which is exactly the \(\mathop {}\!\textrm{d}{\overline{z}}\mathop {}\!\textrm{d}t\) part of the Bogomolny equations.

  20. Strictly speaking, these are the brackets in the perturbative algebra of local operators in the HT-twist of a free \({\mathcal {N}}=4\) vector multiplet. Nonetheless, we expect that these brackets can be used in the perturbative algebra of local operators in the HT-twisted interacting theory described above, cf.  [6, Sec 3.4].

  21. This is always true for \(G = {{\mathbb {C}}}^\times \) as \(h = 0\) and \(T_V = n^2\) for a representation of charge n. For \(G = SL(N)\), which has \(h = N\), with K fundamental hypermultiplets, which each have \(T_{\textrm{fund}} = 1\), this constraint says that \(K \ge 2N\). If the representation is not sufficiently large, we could instead choose boundary chiral multiplets to cancel the anomaly.

  22. We note that it is important here to redefine the naive cohomological grading by the Coulomb branch flavor symmetry to arrive at this answer. This choice of R-symmetry is precisely a choice of dimension theory on the above space of triples, cf. [76, Section 1.3]. This allows for the definition the necessary semi-infinite cohomology used in defining the Coulomb branch.

  23. The result for the Hilbert space in Rozansky–Witten theory in terms of the sheaf cohomology of exterior powers of the tangent bundle derived in [17], cf. Equation (3.37), need not hold in gauge theory, where one really has to contend with the Higgs branch as stack rather than simply the smooth hyperkähler or holomorphic symplectic manifold used in Rozansky–Witten theory.

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Acknowledgements

We would like to thank Tudor Dimofte for his support during the preparation of this paper and his suggestion for investigating this problem. We would also like thank Kevin Costello, Thomas Creutzig, Justin Hilburn, Brian Williams, and Keyou Zeng for useful conversations during the development of this project. N.G. acknowledges support from the University of Washington and previous support from T. Dimofte’s NSF CAREER grant DMS 1753077.

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  1. Department of Physics, University of Washington, Seattle, WA, 98195, USA

    Niklas Garner

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  1. Niklas Garner
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Correspondence to Niklas Garner.

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Garner, N. Twisted formalism for 3d \({\mathcal {N}}=4\) theories. Lett Math Phys 114, 16 (2024). https://doi.org/10.1007/s11005-023-01758-9

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