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Cobordism Invariants from BPS q-Series

Abstract

Many BPS partition functions depend on a choice of additional structure: fluxes, Spin or \(\hbox {Spin}^c\) structures, etc. In a context where the BPS-generating series depends on a choice of \(\hbox {Spin}^c\) structure, we show how different limits with respect to the expansion variable q and different ways of summing over \(\hbox {Spin}^c\) structures produce different invariants of homology cobordisms out of the BPS q-series.

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Notes

  1. 1.

    The fact that \({\hat{Z}}_b (M_3,q)\) is labeled by \(\hbox {Spin}^c\) structures was carefully explained in [15] and recently justified further from a different perspective [16], by realizing \({\hat{Z}}_b (M_3,q)\) as a Rozansky–Witten theory with the target space based on the affine Grassmannian.

  2. 2.

    Another choice of conventions used in the literature is \(\mu (M_3, s) := \frac{\sigma (M_4)}{8}\) mod 2.

  3. 3.

    Note that modulo 2 the correction terms of [20] reduce to a more classical invariant (considered e.g. in [24]):

    $$\begin{aligned} d(M_3,b)\; = \; \frac{c_1({\tilde{b}})^2-\sigma (M_4)}{4} \mod 2 \end{aligned}$$

    where \(M_4\) is a 4-manifold with boundary \(\partial M_4=M_3\) and a \(\hbox {Spin}^c\) structure \({\tilde{b}}\) that extends \(b = {\tilde{b}}|_{M_3}\).

  4. 4.

    We use the normalization of the WRT invariant such that \(\mathrm {WRT}(S^3,k)=1\), which is standard in the mathematical literature. Note that the partition function of \(SU(2)_k\) Chern–Simons theory, \(Z_{SU(2)_k}\), is naturally normalized such that \(Z_{SU(2)_k}(S^2\times S^1)=1\) instead, but has \(Z_{SU(2)_k}(S^3)=\sqrt{\frac{2}{k}}\sin \frac{\pi }{k}\).

  5. 5.

    for simplicity, stated here for a 3-manifold with \(b_1 (M_3) = 0\); generalization to \(b_1 (M_3) > 0\) is not much more complicated [14, 25], but will need here.

  6. 6.

    We follow the normalization convention of [26]. The normalization of [28] differs from the one in [26] when \(b_1(M_3)\ne 0\). This, however, does not affect the analysis done in this section, because we will assume that \(M_3\) is a rational homology sphere.

  7. 7.

    The generalization to graphs with loops was considered in [25], where the relation between ordinary WRT invariants and the BPS q-series invariants \({\hat{Z}}_b (q)\) was studied. It would be interesting to extend the analysis of [25] to Spin-refined WRT invariants discussed here.

  8. 8.

    In general, the pairing is only defined on the torsion subgroup.

  9. 9.

    There is a typo in the last term of Theorem 2.13 in [31].

  10. 10.

    It is important to pay attention to orientation conventions used in the literature. For example, since negative definite plumbings were favored in [14], the lens space L(p, 1) was naturally defined as a \(-p\) surgery on the unknot. The same convention was used in [31]. On the other hand, the opposite choice of orientation is used in [20], where L(p, 1) is a \(+p\) surgery on the unknot. Here, we follow this latter choice.

  11. 11.

    In order to fully account for BPS states in 3d \({{\mathcal {N}}}=2\) theory with 2d (0, 2) boundary condition or in relation to vertex algebras [22], one also needs to restore factors of \((q;q)_{\infty }\) that correspond to the “center-of-mass” chiral multiplet in 3d \({{\mathcal {N}}}=2\) theory and are also usually omitted. The normalization of \({\hat{Z}}_b (M_3,q)\) that relates to topological invariants of \(M_3\), such as WRT invariants, never includes such factors, though, and so they will not play any role here.

  12. 12.

    Recall, S consists of all vertices \(I \in \text {Vert} (\Gamma )\) that satisfy

    $$\begin{aligned} \sum _{I \in S} Q_{IJ} \; \equiv \; Q_{JJ} \quad \text {mod}~2 \end{aligned}$$
    (3.50)

    for any \(J \in \text {Vert} (\Gamma )\). This is nothing but the familiar condition (2.11) expressed in terms of the adjacency matrix of the plumbing graph. The corresponding Kirby diagram is obtained by replacing every vertex of the graph \(\Gamma \) by a copy of the unknot.

  13. 13.

    The invariants \({\hat{Z}}_b (M_3,q)\) provide a non-perturbative definition of \(SL(2,{{\mathbb {C}}})\) Chern-Simons theory (that behaves well under cutting and gluing) since the perturbative expansion of the latter is reproduced in the limit \(\hbar \rightarrow 0\) (with \(q = e^{\hbar }\)). And, \({{\mathcal {H}}}(T^2)\) can be thought of as the Hilbert space in this theory [15], produced by quantizing the classical phase space \({{\mathcal {M}}}_{\text {flat}} (T^2, SL(2,{{\mathbb {C}}})) \cong \frac{{{\mathbb {C}}}^* \times {{\mathbb {C}}}^*}{{{\mathbb {Z}}}_2}\).

  14. 14.

    This equivalence can be shown by 3d Kirby moves.

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Acknowledgements

We are especially grateful to Francesca Ferrari, Ciprian Manolescu, and Yi Ni for their help and insightful comments. It is also a pleasure to thank Rob Kirby and Paul Melvin for stimulating discussions and inspiration at the triple-header birthday conference “Topology in Dimensions 3, 3.5 and 4” in Berkeley (June, 2018). We also thank Cumrun Vafa for encouragement and comments. The work of S.G. is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632, and by the National Science Foundation under Grant No. NSF DMS 1664227. The research of S.P. is supported by Kwanjeong Educational Foundation.

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Gukov, S., Park, S. & Putrov, P. Cobordism Invariants from BPS q-Series. Ann. Henri Poincaré (2021). https://doi.org/10.1007/s00023-021-01089-2

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