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Boundary Chiral Algebras and Holomorphic Twists

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Abstract

We study the holomorphic twist of 3d \(\mathcal{N}=2\) gauge theories in the presence of boundaries, and the algebraic structure of bulk and boundary local operators. In the holomorphic twist, both bulk and boundary local operators form chiral algebras (a.k.a. vertex algebras). The bulk algebra is commutative, endowed with a shifted Poisson bracket and a “higher” stress tensor; while the boundary algebra is a module for the bulk, may not be commutative, and may or may not have a stress tensor. We explicitly construct bulk and boundary algebras for free theories and Landau–Ginzburg models. We construct boundary algebras for gauge theories with matter and/or Chern–Simons couplings, leaving a full description of bulk algebras to future work. We briefly discuss the presence of higher A-infinity like structures.

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Notes

  1. We use the terms “chiral algebra” and “vertex algebra” interchangeably. A priori neither term implies the existence of a stress tensor or other additional structures. In much of the literature, the term “vertex operator algebra” is used to refer to a chiral/vertex algebra with a stress tensor; we will avoid the term “vertex operator algebra” entirely to try to alleviate confusion. We will explain precisely which additional structures are present in the twist of 3d \(\mathcal{N}=2\) theories below.

  2. See Sect. 2.1 for the precise definition of spin J. Also, throughout this paper, we use R-charge rather than the more traditional fermion number as a cohomological grading—whence the appearance of \(e^{i\pi R}\) rather than \(e^{i\pi F}\) in (1.2). In the index, this is just a matter of convention; but at the level of the algebra \(\mathcal{V}\), it is convenient to use R-charge (when unbroken) as a cohomological grading because it gives a natural \({\mathbb {Z}}\) or \({\mathbb {Q}}\) refinement of fermion number.

  3. We expect that bulk chiral algebras in gauge theories could be constructed via a state-operator correspondence, analogous (on one hand) to the Braverman-Finkelberg-Nakajima construction [45, 46] in 3d \(\mathcal{N}=4\) theories, and (on the other hand) to the state-operator correspondence we eventually use to capture monopole operators on Dirichlet boundary conditions. However, we do not pursue bulk algebras further here.

  4. With some important caveats concerning the boundary degrees of freedom.

  5. We do not restrict to ghost-number zero, as is done in traditional BRST quantization of gauge theories. Restricting to ghost-number zero would amount to taking strict, rather than derived, invariants.

  6. A more precise statement is made in Sect. 3.7.

  7. Here the correct \(L_\infty \) structure on bulk operators is the natural one on the Hochschild cochains of the algebra of functions on the target, or equivalently the transferred \(L_\infty \) structure on the Hochschild cohomology.

  8. In the case of 2d topologically twisted theories, a sufficient condition for \(\beta _{\textrm{der}}\) to be an isomorphism is that \(\mathcal{B}\) generates the category of boundary conditions. Analogous conditions for the 3d holomorphic twist are not yet well understood.

  9. More precisely: on a 3-manifold with a transverse holomorphic foliation structure, the line operators should be supported on integral flows of the transverse vector field \(\partial /\partial t\).

  10. Such half-BPS boundary conditions defined by vacua appeared in classic work of Hori-Iqbal-Vafa on the 2d A-model [88]. They are sometimes called “thimble branes.” They were described in the general setting of massive 2d theories by [89, 90]. In 3d \(\mathcal{N}=4\) theories, boundary conditions defined by vacua were discussed in detail in [91, Sec 4] and [92, 93]. In 3d \(\mathcal{N}=2\) theories, many examples of \(\mathcal{N}=(0,2)\) boundary conditions defined by vacua have appeared in e.g. [27, 28, 31, 85].

  11. One may also consider manifolds with corners, as in [27, 100, 106,107,108].

  12. In the BV-BRST formalism, one would further add the untwisted BRST operator \(Q_{\textrm{BRST}}\) to the SUSY supercharge \(\overline{Q}_+\) to construct \(Q=Q_{\textrm{BRST}}+\overline{Q}_+\). We introduce the BV-BRST formalism later in Sect. 3.2.

  13. When \(U(1)_J\) charges are non-integral, extra topological conditions are required in order to define the holomorphic twist globally, on 3-manifolds with THF structure. Any 3-manifold with a THF structure has a canonical SO(2) principal bundle, defined as the orthogonal complement of the real codimension 2 foliation. This SO(2) gets identified with \(U(1)_J\). If the \(U(1)_J\) charges of fields/operators can all be chosen to be rational, belonging to \(\frac{1}{d}{\mathbb {Z}}\) for some d, then we must require the structure group of the canonical bundle to be equipped with a reduction to its d-fold cover. That is, we need a transverse d-spin structure. This will not be important in the current paper, since we are working in flat space. (In fact, even irrational charges are acceptable for us.)

  14. The definition (2.12) differs from that in [33] by a sign: \(\{\mathcal{O}_1{}_\lambda \mathcal{O}_2\}_{\text {Yagi-Oh}}\big |_{\lambda =0} = (-1)^{|\mathcal{O}_1|}\{\!\{ \mathcal{O}_1,\mathcal{O}_2\}\!\}\).

  15. There is one (somewhat trivial) exception to this argument. It may be that all holomorphic derivatives \(\partial \mathcal{O}\) are zero in cohomology. In this case the bulk chiral algebra \(\mathcal{V}\) is fully topological, and reduces to an ordinary graded-commutative algebra. Then one can just define the stress tensor in \(\mathcal{V}\) to be zero. In the full physical theory, one would expect \(T_{zz}\) to be Q-exact, though this is not strictly necessary.

  16. Technically, we are assuming that the identity on the boundary is not Q-exact. Otherwise, the entire chiral algebra \(\mathcal{V}_\partial \) would be zero—indicating a spontaneous breaking of (0,2) SUSY.

  17. In this table, we are actually giving the charges of the local operators called \(\phi \), \(\psi _+\), etc. This is standard and unspoken physics convention. Since the local operators are functionals on the space of fields, the fields technically have opposite charges.

  18. Explicitly, given any element \(\tau \in {\mathfrak {g}}\), we have \(\langle \mu (\phi ,\psi ),\tau \rangle = \psi \tau \phi \), where the RHS involves the action of \(\tau \) on \(\phi \in V\), and a contraction with \(\psi \in V^\vee \). More schematically, we have \(\mu (\phi ,\psi ) = \frac{\partial }{\partial \tau } (\psi \tau \phi ) \sim \phi \psi \).

  19. It is useful to think of the Chern–Simons level ‘k’ as a normalization of the Cartan-Killing form. For classical groups, the Cartan-Killing form at \(k=1\) correspond to the trace in the fundamental representation.

  20. We use the slash notation “\(\big |\)” as shorthand for evaluation at \(t=0\).

  21. These sorts of complex-codimension jumping loci are distinct from wall crossing, and featured in recent generalizations of the elliptic genus [132, 133].

  22. Here we mean “fields” in the standard sense of chiral algebras as well. Physically, they are really operators.

  23. See Footnote 14 (page 16) for a note on sign conventions. In the conventions of [33], which agree with more standard mathematical conventions for a -1 shifted Poisson algebra, one would have \(\{\!\{\phi ,\psi \}\!\} = 1\) and \(\{\!\{\psi ,\phi \}\!\} = -1\).

  24. We should stress that even the free chiral theory may admit higher operations. For example, our calculations in Sect. 5 may be seen as evidence for a non-trivial higher operation involving three operators colliding along the topological direction: two fermions and a polynomial in the bosons of at least cubic degree.

  25. Being more careful: the standard supersymmetric index is usually defined following the original [136] as \(\text {Tr}(-1)^Fq^J...\) rather than \(\text {Tr}e^{i\pi R}q^J...\). The two expressions are equivalent, up to a redefinition of q. Namely, physical fermion number is related to untwisted spin as \(F=2J_0\) (mod 2); thus, keeping in mind that \(J=R/2-J_0\), we see that shifting \(q\rightarrow e^{-2\pi i}q\) relates the character (4.11) to the standard index.

  26. We thank N. Paquette for discussions related to this idea [137].

  27. The term “flip” originates in [72], where a similar mechanism was used to relate half-BPS boundary conditions for 4d hypermultiplet.

  28. A constant is allowed because the integral of a constant superfield vanishes. This is equally true in the original physical 2d \(\mathcal{N}=(0,2)\) theory. However, preserving \(U(1)_R\) symmetry forces \(\sum _\alpha J_\alpha E^\alpha =0\).

  29. In principle, the the bulk algebra of a gauge theory could be obtained by taking the derived center of the boundary algebra on a Neumann boundary condition; or by generalizing some of the computations involving cohomology of moduli spaces of G-bundles from Sect. 7. It would be interesting to develop this in future work.

  30. It is conceivable, though, that the derived, algebraic geometric formulation of our final answer will apply to more general auxiliary chiral algebras.

  31. Physically, the derivatives in (6.6) should actually be covariant \(D_z\) derivatives. However, in the twisted formalism, there is no distinction between covariant and non-covariant derivatives. (The connection in the twisted formalism only has \(A_{{\bar{z}}}\) and \(A_t\) components.) A practical way to understand this is that, when describing the abstract structure of the algebra, it does not matter whether covariant or non-covariant derivatives are used. The connection never appears; nor does the \(F_{zz}\) curvature component, since it automatically vanishes.

  32. Quick comment on \({\mathfrak {g}}\) vs. \({\mathfrak {g}}^*\): In physics, the \({\textsf {c}}\) ghost is a \({\mathfrak {g}}\)-valued field. The operators formed from the \({\textsf {c}}\) ghost are functionals of this field, and are therefore elements of \({\mathfrak {g}}^*\). We can see this explicitly when “expanding \({\textsf {c}}\) into components”: in a formula \({\textsf {c}}= {\textsf {c}}^aT_a\), the generators \(T_a\) are elements of \({\mathfrak {g}}\) and the coefficients \({\textsf {c}}^a\), which are the actual local operators, are elements of \({\mathfrak {g}}^*\). This agrees with the standard mathematical formulation of the Chevalley–Eilenberg cochain complex, in terms of \({\mathfrak {g}}^*\).

  33. Closely related computations of the Q-cohomology of the Hilbert space of 3d \(\mathcal{N}=2\) gauge theories on Riemann surfaces were performed in [105]. It explained carefully therein that the Hilbert space is Dolbeault cohomology valued in a particular sheaf, depending on the fermionic matter of the theory, and twisted by a power of the canonical bundle due to Chern–Simons terms. In pure gauge theory on a disc with Neumann b.c. and a Chern–Simons level \(-{\textsf {h}}\), the overall twist cancels out, so we are left with the cohomology of the structure sheaf as in (6.17).

  34. A possible concern, which we will not explore further here, is that the presence of gauge fields weakens the argument we gave to exclude instanton corrections involving configurations of bulk fields which are constant in the direction normal to the boundary. In the presence of gauge fields, the bulk fields would only be covariantly constant, and the divergence of the instanton action should be revisited.

  35. Corresponding to these charge assignments, the bulk bosonic chirals (X, Y, Z) have R-charges (0, 1, 1). This ensures that the superpotential \(W=XYZ\) has R-charge 2, as required.

  36. We are using algebraic/sheaf notation for cohomology on the RHS. Analytically, the RHS would be written as \(H^{(0,\bullet )}_{{\bar{\partial }}}(\text {Bun}_G(\Sigma ), K^{\frac{1}{2}})\), i.e. it is the cohomology of the \({\bar{\partial }}\) operator acting on (0, p) forms valued in \(K^{\frac{1}{2}}\).

  37. If G has trivial center, then \(\text {Bun}_G\) is connected and \(H^2(\text {Bun}_G(\Sigma ),{\mathbb {Z}})={\mathbb {Z}}\), so one can simply say that \(c_1(\mathcal{L})\) generates. In general, \(c_1(\mathcal{L})\) is a multiple of the Kähler form, normalized to restrict to a generator of \(H^2\) on each connected component.

  38. The double quotient \(G(\mathcal{O})\backslash G(\mathcal{K})/G(\mathcal{O}) = G(\mathcal{O})\backslash \text {Gr}_G\), also known as the space of Hecke modifications, has appeared in the physics of bulk ’t Hooft lines in 4d Yang–Mills [4, 18], and in the BFN construction of 3d \(\mathcal{N}=4\) Coulomb branches [45, 46]. The one-sided quotient is appearing in the construction of boundary monopole operators here for a similar reason.

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Acknowledgements

We thank Natalie Paquette and Sam Raskin for useful discussions. This research was supported in part by a grant from the Krembil Foundation. K.C. and D.G. are supported by the NSERC Discovery Grant program and 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. The work of T.D. was supported by NSF CAREER Grant DMS-1753077. T.D.’s work was performed in part at the Aspen Center for Physics, which is supported by NSF grant PHY-1607611, as well as during visits to the Perimeter Institute. The authors have no financial or proprietary interests in any material discussed in this article.

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

    Kevin Costello & Davide Gaiotto

  2. Department of Mathematics and Center for Quantum Mathematics and Physics (QMAP), University of California, Davis, CA, 95616, USA

    Tudor Dimofte

  3. School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, UK

    Tudor Dimofte

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Costello, K., Dimofte, T. & Gaiotto, D. Boundary Chiral Algebras and Holomorphic Twists. Commun. Math. Phys. 399, 1203–1290 (2023). https://doi.org/10.1007/s00220-022-04599-0

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