$\displaystyle(X_{1},X_{2},X_{3},X_{4},X_{5})\to(\zeta_{16}^{8}X_{1},\zeta_{16}^{4}X_{2},\zeta_{16}^{2}X_{3},\zeta_{16}X_{4},\zeta_{16}X_{5})~{},$
$\displaystyle\sigma_{2}\in{\mathbb{Z}}_{4}:$
$\displaystyle(X_{1},X_{2},X_{3},X_{4},X_{5})\to(\zeta_{8}^{4}X_{1},\zeta_{8}^{2}X_{2},\zeta_{8}^{2}X_{3},X_{4},X_{5})~{},$
$\displaystyle\sigma_{3}\in{\mathbb{Z}}_{8}:$
$\displaystyle(X_{1},X_{2},X_{3},X_{4},X_{5})\to(X_{1},\zeta_{8}^{6}X_{2},\zeta_{8}X_{3},\zeta_{8}X_{4},X_{5})~{}.$
(4.66)

The discrete torsion group is $G_{\text{dt}}={\mathbb{Z}}_{4}$, and, as in the
previous example, the subgroup of $F$ that fixes a surface is
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{4}\subset{\mathbb{Z}}_{4}\times{\mathbb{Z}}_{8}$
generated by $\sigma_{2}^{2}$ and $\sigma_{3}^{2}$. So, once again we expect
that for the discrete torsion subgroup
${\mathbb{Z}}_{2}\subset{\mathbb{Z}}_{4}$, which we would like to identify
with $B(X)$, the Hodge numbers do note jump. This expectation is borne out by
explicit calculation:

$\displaystyle p$ $\displaystyle=0~{}:h^{1,2}=3~{},h^{1,1}=75~{},$
$\displaystyle p$ $\displaystyle=1~{}:h^{1,2}=3~{},h^{1,1}=27~{},$
$\displaystyle p$ $\displaystyle=2~{}:h^{1,2}=3~{},h^{1,1}=75~{},$
$\displaystyle p$ $\displaystyle=3~{}:h^{1,2}=3~{},h^{1,1}=27~{}.$ (4.67)

Our examples are consistent with the inclusion $B(X)\subseteq G_{\text{dt}}$,
and the jump in the Hodge numbers for $p\not\in B(X)$ is correlated with
resolution of surface singularities in the ambient toric variety. We also note
that for all of these examples we have known mirrors when $p=0$—these are
manifolds with $\pi_{1}(X^{\circ})\simeq B(X)$ described in [1]. When $p\neq
0$, it is straightforward to extend the construction presented in section 4.2
to find explicit mirror SCFTs.

##  5 Discussion

The discovery, exploration, and applications of mirror symmetry have enriched
a number of fields in physics and mathematics in the last thirty
years,292929Pedagogical reviews of some of the most important developments can
be found in [68, 50, 90, 43]. but many fundamental questions remain, and
finding the answers to these will lead to significant progress in our
understanding of duality in quantum field theory and string theory. Perhaps
the most difficult yet fundamental task that remains is to define precisely in
what category of quantum field theories mirror duality operates.303030IVM
thanks Ronen Plesser for emphasizing this point in many discussions over the
years. An instructive attempt at tackling this question in the context of
linear sigma models was made in [91], but further work is required even in
that setting. The progress in mirror symmetry at the level of topological
field theory—i.e. the homological mirror symmetry program—and the categorical
structures that emerge in that approach all suggest that we have much to learn
before we can give a fully satisfactory answer to this seemingly basic
question. Our work, based on the earlier efforts in [22, 23, 24, 25, 1],
suggests that both orbifolds and flat gerbes should be included in the answer,
and we hope that some of the methodology and ideas developed herein will be
useful in establishing a suitable framework.

Our study began with (1.2), which is well-motivated by the intuition that
mirror symmetry should exchange even and odd cohomology. We were able to show
that it holds as a consequence of open mirror symmetry and the splitting of
Atiyah-Hirzebruch sequences for the torsion K-groups, but with the assumption
that $A(X)$ has only elements of odd order. Since we also have examples where
$A(X)$ has even-order elements, but the sequences nevertheless split, it is
clear that the result holds beyond what we have been able to prove. Is it true
that for any CY threefold

$\displaystyle\\{K^{0}(X)\\}_{\text{tor}}=A(X)\oplus B(X)^{\ast}~{}?$ (5.1)

Another natural follow-up to our study would be to consider the relationship
between discrete torsion at Gepner points (and more generally Landau-Ginzburg
orbifold loci) and $B(X)$ more closely and systematically. Some efforts in
this direction have been made many years ago. For example, there is a
classification of all Landau-Ginzburg orbifolds with discrete torsion [24],
but the results obtained therein are limited to the computation of dimensions
of (a,c) and (c,c) rings. Those findings are already intriguing: for instance,
there are many entries with relatively small dimensions, and even one example
without marginal (2,2) deformations! It is quite likely that some of these are
related to Calabi-Yau manifolds with small Hodge numbers described in [65],
and the relation probably involves identification of $B(X)$ with a subgroup of
the discrete torsion. It should be useful to study this class of examples
along the lines developed in this work by focusing on the relationship between
the solvable CFT locus and the resolved geometry. Such an effort will lead to
a better understanding of aspects of the problem that appear obscure (at least
to us): for instance, how universal is the relationship between resolution of
fixed curves in the Calabi-Yau geometry and the jumping in the Hodge numbers
in the presence of discrete torsion? First steps in this direction involve re-
examining the classification of [24] with an eye to geometric interpretation,
and also a closer study of the examples in [65, 92] to determine the $A(X)$
and $B(X)$ torsion groups. It will also be useful to study self-mirror
examples, such $X_{2}$ and $X_{3}$ described in section 2.1.

We also envision a more detailed study of mirror symmetry in the presence of
discrete torsion at special orbifold points and related flat gerbes over a
smooth geometry. It will be useful to construct additional examples of mirror
pairs with flat gerbes and to elucidate the relationship between the
geometries on both sides of the mirror. In the context of Gepner models we
have shown how to construct mirror pairs with the inclusion of discrete
torsion. Can this be extended to other mirror constructions, such as that of
Berglund-Hübsch for Landau-Ginzburg orbifolds [93]?

There are also more general and perhaps more speculative questions concerning
the SCFT associated to a smooth CY $X$ with non-zero $A(X)$ or $B(X)$:

  1. 1.

Can we identify a flat gerbe on $X$ with a choice of discrete torsion for an
orbifold at some point in the moduli space?

  2. 2.

If $X$ has multiple orbifold points in the moduli space, what are the
relations between the discrete torsions at various points?

  3. 3.

Is there always a “parent” geometry $\overline{X}$ with
$A(\overline{X})=B(\overline{X})=0$, such that on a suitable locus in the
moduli space of ${\cal C}[\overline{X}]$ we can obtain ${\cal
C}[X,\beta_{\text{t}}]$ by an orbifold of ${\cal C}[\overline{X}]$?

  4. 4.

We have seen that given a mirror pair $X$, $X^{\circ}$, there is in general a
finite set of SCFTs ${\cal C}[Y^{\circ},\beta^{\circ}_{\text{t}}]$ that are
mirror to the discrete family ${\cal C}[X,\beta_{\text{t}}]$,
$\beta_{\text{t}}\in B(X)$, with ${\cal C}[X,0]\simeq{\cal C}[X^{\circ}]={\cal
C}[Y^{\circ},0]$. Which ${\cal C}[Y^{\circ},\beta^{\circ}_{\text{t}}]$ have a
geometric description? Can we describe the discrete family ${\cal
C}[Y^{\circ},\beta^{\circ}_{\text{t}}]$ directly, without appealing to the
mirror?

  5. 5.

There are additional topological invariants encoded in the cup product on
$\\{H(X,{\mathbb{Z}})\\}_{\text{tor}}$. Do these have an interpretation in the
SCFT ${\cal C}[X]$ and its mirror?

  6. 6.

Can we give an intrinsic SCFT definition of the groups $A(X)$ and $B(X)$
without reference to a geometry?313131We thank Paul Aspinwall for this
question.

Coming back to our discussion of $A(X)$ and its relation to a quantum symmetry
of the SCFT, we can also imagine extending the analysis to the situation where
$\pi_{1}(X)$ is non-abelian. It will be interesting to understand the
categorical symmetry structure in such a situation and to relate it to the
geometry and topology of $X$, as well as to interpret it on the mirror
$X^{\circ}$. This is one of the directions where these old questions can
benefit from the relatively new categorical symmetry methods, and at the same
time provide concrete examples to develop further intuition for these
remarkable structures. Our work also motivates a closer study of the
relationship between discrete torsion and gauged linear sigma models: are
there useful ways to lift this phase to the UV description? A related point
concerns the suggestions—see, e.g. [25, 78, 21], that certain flat gerbes can
be interpreted as a discrete $\theta$-angle in the gauged linear sigma model.
It would be useful to study these notions systematically, as they will
certainly have important ramifications for explicit mirror constructions.

We concentrated almost exclusively on the worldsheet SCFT, but the structures
we discussed have important implications for the effective spacetime theory.
At the most basic level, we expect that $A(X)$ should be interpreted as a
spacetime gauge symmetry, while turning on a non-trivial gerbe
$\beta_{\text{t}}\subset B(X)$ should affect the geometry of ${\cal
M}_{\text{ac}}(X)$—the moduli space of vector multiplets in a IIA
compactification on $X$. What, precisely, are the consequences of these
features for the effective theory? There should also be consequences for the
spectrum of BPS and non-BPS D-brane states, and the study of the interplay
between twisted K-theory and the Freed-Witten anomaly in these concrete
examples will surely have important general lessons and perhaps shed light on
open problems concerning the spectrum of branes in the presence of
topologically non-trivial fluxes.

The most speculative further directions concern generalizations to higher
dimensions. What are the consequences of these discrete topological features
for $\text{G}_{2}$ mirror symmetry or for mirror symmetry of CY $4$-folds?
While the former has a reasonable spacetime interpretation as a
$3$-dimensional string compactification, the worldsheet physics is more
difficult, since it involves a theory without extended supersymmetry. For the
latter, although the worldsheet theory is still a (2,2) SCFT, the spacetime
interpretation is subtle: it is a “compactification” to 1+1 dimensions with a
non-trivial tadpole for the Ramond-Ramond flux.

## References

  * [1] V. Batyrev and M. Kreuzer, “Integral cohomology and mirror symmetry for Calabi-Yau 3-folds,” arXiv:math/0505432.
  * [2] B. R. Greene, D. R. Morrison, and M. R. Plesser, “Mirror manifolds in higher dimension,” Commun.Math.Phys. 173 (1995) 559–598, arXiv:hep-th/9402119 [hep-th].
  * [3] P. S. Aspinwall and D. R. Morrison, “String theory on $K3$ surfaces,” in Mirror symmetry, II, vol. 1 of AMS/IP Stud. Adv. Math., pp. 703–716.  Amer. Math. Soc., Providence, RI, 1997.
  * [4] P. S. Aspinwall, “An N=2 dual pair and a phase transition,” Nucl. Phys. B 460 (1996) 57–76, arXiv:hep-th/9510142.
  * [5] V. Braun, M. Cvetic, R. Donagi, and M. Poretschkin, “Type II String Theory on Calabi-Yau Manifolds with Torsion and Non-Abelian Discrete Gauge Symmetries,” JHEP 07 (2017) 129, arXiv:1702.08071 [hep-th].
  * [6] R. Minasian and G. W. Moore, “K theory and Ramond-Ramond charge,” JHEP 11 (1997) 002, arXiv:hep-th/9710230.
  * [7] E. Witten, “D-branes and K-theory,” JHEP 12 (1998) 019, arXiv:hep-th/9810188.
  * [8] I. Brunner and J. Distler, “Torsion D-branes in nongeometrical phases,” Adv. Theor. Math. Phys. 5 (2002) 265–309, arXiv:hep-th/0102018.
  * [9] I. Brunner, J. Distler, and R. Mahajan, “Return of the torsion D-branes,” Adv. Theor. Math. Phys. 5 (2002) 311–352, arXiv:hep-th/0106262.
  * [10] I. Brunner, N. Carqueville, and D. Plencner, “A quick guide to defect orbifolds,” Proc. Symp. Pure Math. 88 (2014) 231–242, arXiv:1310.0062 [hep-th].
  * [11] L. Bhardwaj and Y. Tachikawa, “On finite symmetries and their gauging in two dimensions,” JHEP 03 (2018) 189, arXiv:1704.02330 [hep-th].
  * [12] K. Hashimoto and A. Kanazawa, “Calabi–Yau Three-folds of Type K (I): Classification,” Int. Math. Res. Not. 2017 no. 21, (2017) 6654–6693, arXiv:1409.7601 [math.AG].
  * [13] K. Hashimoto and A. Kanazawa, “Calabi-Yau threefolds of type K (II): mirror symmetry,” Commun. Num. Theor. Phys. 10 (2016) 157–192, arXiv:1511.08778 [math.AG].
  * [14] S. Ferrara, J. A. Harvey, A. Strominger, and C. Vafa, “Second quantized mirror symmetry,” Phys.Lett. B361 (1995) 59–65, arXiv:hep-th/9505162 [hep-th].
  * [15] P. S. Aspinwall and R. Y. Donagi, “The Heterotic string, the tangent bundle, and derived categories,” Adv.Theor.Math.Phys. 2 (1998) 1041–1074, arXiv:hep-th/9806094 [hep-th].
  * [16] J. de Boer, R. Dijkgraaf, K. Hori, A. Keurentjes, J. Morgan, D. R. Morrison, and S. Sethi, “Triples, fluxes, and strings,” Adv. Theor. Math. Phys. 4 (2002) 995–1186, arXiv:hep-th/0103170.
  * [17] Y. Tachikawa, “Frozen singularities in M and F theory,” JHEP 06 (2016) 128, arXiv:1508.06679 [hep-th].
  * [18] L. Bhardwaj, D. R. Morrison, Y. Tachikawa, and A. Tomasiello, “The frozen phase of F-theory,” JHEP 08 (2018) 138, arXiv:1805.09070 [hep-th].
  * [19] P. Cheng, I. V. Melnikov, and R. Minasian, “Flat equivariant gerbes: holonomies and dualities,” JHEP 04 (2023) 074, arXiv:2207.06885 [hep-th].
  * [20] S. Katz, A. Klemm, T. Schimannek, and E. Sharpe, “Topological Strings on Non-commutative Resolutions,” Commun. Math. Phys. 405 no. 3, (2024) 62, arXiv:2212.08655 [hep-th].
  * [21] S. Katz and T. Schimannek, “New non-commutative resolutions of determinantal Calabi-Yau threefolds from hybrid GLSM,” arXiv:2307.00047 [hep-th].
  * [22] C. Vafa and E. Witten, “On orbifolds with discrete torsion,” J.Geom.Phys. 15 (1995) 189–214, arXiv:hep-th/9409188 [hep-th].
  * [23] P. S. Aspinwall and D. R. Morrison, “Chiral rings do not suffice: N=(2,2) theories with nonzero fundamental group,” Phys. Lett. B334 (1994) 79–86, arXiv:hep-th/9406032 [hep-th].
  * [24] M. Kreuzer and H. Skarke, “Landau-Ginzburg orbifolds with discrete torsion,” Mod. Phys. Lett. A 10 (1995) 1073–1086, arXiv:hep-th/9412033.
  * [25] P. S. Aspinwall, D. R. Morrison, and M. Gross, “Stable singularities in string theory,” Commun. Math. Phys. 178 (1996) 115–134, arXiv:hep-th/9503208.
  * [26] M. Kreuzer and H. Skarke, “Orbifolds with discrete torsion and mirror symmetry,” Phys. Lett. B 357 (1995) 81–88, arXiv:hep-th/9505120.
  * [27] E. R. Sharpe, “Discrete torsion,” Phys. Rev. D 68 (2003) 126003, arXiv:hep-th/0008154.
  * [28] E. Sharpe, “Discrete torsion and shift orbifolds,” Nucl. Phys. B 664 (2003) 21–44, arXiv:hep-th/0302152.
  * [29] B. R. Greene and M. R. Plesser, “Duality in Calabi-Yau moduli space,” Nucl. Phys. B338 (1990) 15–37.
  * [30] D. D. Joyce, Compact manifolds with special holonomy.  Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000\.
  * [31] A. L. Besse, Einstein Manifolds.  Springer-Verlag, Berlin, Heidelberg, New York, 1987.
  * [32] A. Beauville, “Variétés kähleriennes dont la première classe de chern est nulle,” Journal of Differential Geometry 18 (1983) 755–782. https://api.semanticscholar.org/CorpusID:120533794.
  * [33] B. Wilking, “On fundamental groups of manifolds of nonnegative curvature,” Differential Geom. Appl. 13 (2000) 192–165.
  * [34] L. Charlap, Bieberbach groups and flat manifolds.  Universitext. Springer-Verlag, New York, 1 ed., 1986.
  * [35] L. Auslander and M. Kuranishi, “On the holonomy group of locally euclidean spaces,” Annals of Mathematics 65 no. 3, (1957) 411–415. http://www.jstor.org/stable/1970053.
  * [36] M. Fischer, M. Ratz, J. Torrado, and P. K. S. Vaudrevange, “Classification of symmetric toroidal orbifolds,” JHEP 01 (2013) 084, arXiv:1209.3906 [hep-th].
  * [37] W. Ballmann, Lectures on Kähler manifolds.  ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, 2006.
  * [38] A. Hatcher, Algebraic topology.  Cambridge University Press, 2002.
  * [39] S. Eilenberg and S. MacLane, “Relations between homology and homotopy groups of spaces,” Annals of Mathematics 46 no. 3, (1945) 480–509. http://www.jstor.org/stable/1969165.
  * [40] C. T. C. Wall, “Classification problems in differential topology. V,” Invent. Math. 1 no. 4, (1966) 355–374.
  * [41] P. S. Aspinwall, B. R. Greene, and D. R. Morrison, “Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory,” Nucl. Phys. B416 (1994) 414–480, arXiv:hep-th/9309097.
  * [42] J. Evslin, “What does(n’t) K-theory classify?,” arXiv:hep-th/0610328.
  * [43] P. S. Aspinwall, T. Bridgeland, A. Craw, M. R. Douglas, A. Kapustin, G. W. Moore, M. Gross, G. Segal, B. Szendroi, and P. M. H. Wilson, Dirichlet branes and mirror symmetry, vol. 4 of Clay Mathematics Monographs.  AMS, Providence, RI, 2009.  http://people.maths.ox.ac.uk/cmi/library/monographs/cmim04c.pdf.
  * [44] J. F. Davis and P. Kirk, Lecture notes in algebraic topology, vol. 35 of Graduate Studies in Mathematics.  Amer. Math. Soc., 2001.
  * [45] D.-E. Diaconescu, G. W. Moore, and E. Witten, “E(8) gauge theory, and a derivation of K theory from M theory,” Adv. Theor. Math. Phys. 6 (2003) 1031–1134, arXiv:hep-th/0005090.
  * [46] A. Hatcher, Vector bundles and K-theory.  online, 2.2 ed., May, 2017.
  * [47] D. Epstein and N. Steenrod, Cohomology operations, vol. AM-50 of Annals of mathematics studies.  Princeton University Press, 1962.
  * [48] G. W. Moore and E. Witten, “Selfduality, Ramond-Ramond fields, and K theory,” JHEP 05 (2000) 032, arXiv:hep-th/9912279.
  * [49] P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. R. Morrison, and E. Witten, eds., Quantum fields and strings: A course for mathematicians. Vol. 1, 2.  Amer. Math. Soc., 1999.
  * [50] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror symmetry, vol. 1 of Clay Mathematics Monographs.  American Mathematical Society, Providence, RI, 2003.  With a preface by Vafa.
  * [51] D. S. Freed, G. W. Moore, and G. Segal, “Heisenberg Groups and Noncommutative Fluxes,” Annals Phys. 322 (2007) 236–285, arXiv:hep-th/0605200.
  * [52] N. Seiberg and E. Witten, “Spin Structures in String Theory,” Nucl.Phys. B276 (1986) 272.
  * [53] P. H. Ginsparg, “Applied conformal field theory,” arXiv:hep-th/9108028 [hep-th].
  * [54] P. Cheng, I. V. Melnikov, and R. Minasian, “Flat F-theory and friends,” arXiv:2306.00865 [hep-th].
  * [55] L. J. Dixon, J. A. Harvey, C. Vafa, and E. Witten, “Strings on Orbifolds. 2,” Nucl. Phys. B274 (1986) 285–314.
  * [56] L. J. Dixon, “Some World Sheet Properties of Superstring Compactifications, on Orbifolds and Otherwise,”. Lectures given at the 1987 ICTP Summer Workshop in High Energy Phsyics and Cosmology, Trieste, Italy, Jun 29 - Aug 7, 1987.
  * [57] C. Vafa, “Modular Invariance and Discrete Torsion on Orbifolds,” Nucl. Phys. B 273 (1986) 592–606.
  * [58] I. Brunner, N. Carqueville, and D. Plencner, “Discrete torsion defects,” Commun. Math. Phys. 337 no. 1, (2015) 429–453, arXiv:1404.7497 [hep-th].
  * [59] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized Global Symmetries,” JHEP 02 (2015) 172, arXiv:1412.5148 [hep-th].
  * [60] C. Vafa, “Quantum symmetries of string vacua,” Mod.Phys.Lett. A4 (1989) 1615.
  * [61] J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string.  Cambridge University Press, 2007\.
  * [62] P. Candelas, G. T. Horowitz, A. Strominger, and E. Witten, “Vacuum Configurations for Superstrings,” Nucl. Phys. B258 (1985) 46–74.
  * [63] V. Braun, “On Free Quotients of Complete Intersection Calabi-Yau Manifolds,” JHEP 04 (2011) 005, arXiv:1003.3235 [hep-th].
  * [64] J. Gray and J. Wang, “Free quotients of favorable Calabi-Yau manifolds,” JHEP 07 (2022) 116, arXiv:2112.12683 [hep-th].
  * [65] P. Candelas, A. Constantin, and C. Mishra, “Calabi-Yau Threefolds With Small Hodge Numbers,” Fortsch. Phys. 66 (2018) 1800029, arXiv:1602.06303 [hep-th].
  * [66] A. Braun, A. Lukas, and C. Sun, “Discrete Symmetries of Calabi–Yau Hypersurfaces in Toric Four-Folds,” Commun. Math. Phys. 360 no. 3, (2018) 935–984, arXiv:1704.07812 [hep-th].
  * [67] V. V. Batyrev, “Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties,” J. Alg. Geom. 3 (1994) 493–545, arXiv:alg-geom/9310003.
  * [68] D. A. Cox and S. Katz, Mirror symmetry and algebraic geometry.  Amer. Math. Soc., 1999.  Providence, USA: AMS (2000) 469 p.
  * [69] E. Witten, “Phases of N = 2 theories in two dimensions,” Nucl. Phys. B403 (1993) 159–222, arXiv:hep-th/9301042.
  * [70] D. R. Morrison and M. Ronen Plesser, “Summing the instantons: quantum cohomology and mirror symmetry in toric varieties,” Nucl. Phys. B440 (1995) 279–354, arXiv:hep-th/9412236.
  * [71] I. V. Melnikov, “An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry,” Lect. Notes Phys. 951 (2019) .
  * [72] D. Cox, J. Little, and H. Schenck, Toric varieties, vol. 124 of Graduate Studies in Mathematics.  AMS, 2011.
  * [73] P. S. Aspinwall, B. R. Greene, and D. R. Morrison, “The monomial divisor mirror map,” Internat. Math. Res. Notices no. 12, (1993) 319–337, arXiv:alg-geom/9309007.
  * [74] G. Adams and I. V. Melnikov, “Marginal deformations of Calabi-Yau hypersurface hybrids with (2,2) supersymmetry,” arXiv:2305.05971 [hep-th].
  * [75] W. Stein et al., Sage Mathematics Software (Version 10.2).  The Sage Development Team, YYYY.  http://www.sagemath.org.
  * [76] A. Ashmore, L. Calmon, Y.-H. He, and B. A. Ovrut, “Calabi-Yau Metrics, Energy Functionals and Machine-Learning,” International Journal of Data Science in the Mathematical Sciences 1 no. 1, (2023) 49–61, arXiv:2112.10872 [hep-th].
  * [77] A. Ashmore, Y.-H. He, E. Heyes, and B. A. Ovrut, “Numerical spectra of the Laplacian for line bundles on Calabi-Yau hypersurfaces,” JHEP 07 (2023) 164, arXiv:2305.08901 [hep-th].
  * [78] K. Hori, “Duality In Two-Dimensional (2,2) Supersymmetric Non-Abelian Gauge Theories,” JHEP 10 (2013) 121, arXiv:1104.2853 [hep-th].
  * [79] D. Gepner, “Exactly solvable string compactifications on manifolds of SU(N) holonomy,” Phys.Lett. B199 (1987) 380–388.
  * [80] D. Gepner, “Lectures on N=2 String Theory,”. Lectures at Spring School on Superstrings, Trieste, Italy, Apr 3-14, 1989.
  * [81] C. Lutken and G. G. Ross, “Taxonomy of heterotic superconformal field theories,” Phys.Lett. B213 (1988) 152.
  * [82] E. J. Martinec, “Algebraic Geometry and Effective Lagrangians,” Phys. Lett. B217 (1989) 431.
  * [83] C. Vafa and N. P. Warner, “Catastrophes and the Classification of Conformal Theories,” Phys. Lett. B218 (1989) 51.
  * [84] W. Lerche, C. Vafa, and N. P. Warner, “Chiral rings in N=2 superconformal theories,” Nucl. Phys. B324 (1989) 427.
  * [85] K. A. Intriligator and C. Vafa, “Landau-Ginzburg orbifolds,” Nucl.Phys. B339 (1990) 95–120.
  * [86] Y. Tachikawa, “On gauging finite subgroups,” SciPost Phys. 8 no. 1, (2020) 015, arXiv:1712.09542 [hep-th].
  * [87] C. Cordova and G. Rizi, “Non-Invertible Symmetry in Calabi-Yau Conformal Field Theories,” arXiv:2312.17308 [hep-th].
  * [88] A. Adem, J. Leida, and Y. Ruan, Orbifolds and stringy topology, vol. 171 of Cambridge Tracts in Mathematics.  Cambridge University Press, Cambridge, 2007.  https://doi.org/10.1017/CBO9780511543081.
  * [89] A. Karch, D. Tong, and C. Turner, “A Web of 2d Dualities: ${\bf Z}_{2}$ Gauge Fields and Arf Invariants,” SciPost Phys. 7 (2019) 007, arXiv:1902.05550 [hep-th].
  * [90] P. S. Aspinwall, “D-branes on Calabi-Yau manifolds,” arXiv:hep-th/0403166.
  * [91] P. S. Aspinwall and M. R. Plesser, “General Mirror Pairs for Gauged Linear Sigma Models,” JHEP 11 (2015) 029, arXiv:1507.00301 [hep-th].
  * [92] A. Constantin, J. Gray, and A. Lukas, “Hodge Numbers for All CICY Quotients,” JHEP 01 (2017) 001, arXiv:1607.01830 [hep-th].
  * [93] P. Berglund and T. Hubsch, “A Generalized construction of mirror manifolds,” Nucl. Phys. B393 (1993) 377–391, arXiv:hep-th/9201014 [hep-th]. [AMS/IP Stud. Adv. Math.9,327(1998)].


# Black Holes Decohere Quantum Superpositions

Daine L. Danielson  Corresponding author<EMAIL_ADDRESS>Gautam
Satishchandran<EMAIL_ADDRESS>Robert M. Wald<EMAIL_ADDRESS>Enrico Fermi Institute and Department of Physics,
The University of Chicago, Chicago, Illinois 60637, USA

###### Abstract

We show that if a massive body is put in a quantum superposition of spatially
separated states, the mere presence of a black hole in the vicinity of the
body will eventually destroy the coherence of the superposition. This occurs
because, in effect, the gravitational field of the body radiates soft
gravitons into the black hole, allowing the black hole to acquire “which path”
information about the superposition. A similar effect occurs for quantum
superpositions of electrically charged bodies. We provide estimates of the
decoherence time for such quantum superpositions. We believe that the fact
that a black hole will eventually decohere any quantum superposition may be of
fundamental significance for our understanding of the nature of black holes in
a quantum theory of gravity.

Third prize essay written for the Gravity Research Foundation 2022 Essays on
Gravitation.

Black holes have long been known to be destroyers of quantum coherence. If one
member of an entangled pair of particles falls into a black hole, all that
will remain is the particle that stayed outside the black hole, which will be
in a mixed state. Much more generally, if matter of any kind falls into a
black hole, it will, in effect, eventually emerge as Hawking radiation and be
in a highly mixed state. While it may be debated as to whether the quantum
coherence is lost forever in this process (see, e.g., [1, 2]), there is a
general consensus that the state outside the black hole is highly mixed at
least up to the “Page time” in black hole evaporation.

The purpose of this essay is to show that black holes are even more insidious
destroyers of quantum coherence than has been previously known. If one puts
any quantum matter in a spatial superposition, the mere presence of a black
hole in the vicinity of the matter will eventually destroy the coherence of
this superposition. This happens because the long-range (i.e., electromagnetic
and gravitational) fields associated with the quantum matter affect the
quantum state of these fields on the black hole horizon. In effect, the black
hole thereby acquires “which path” information about the quantum
superposition. As we shall show, this inflicts a fundamental rate of
decoherence even on stationary superpositions outside its event horizon. This
is sufficient to decohere any quantum superposition over a sufficiently long
period of time.

To understand how this works, it is useful to first consider a quantum
superposition in flat spacetime and see how decoherence can be avoided,
following the analysis given in [3]. For simplicity and definiteness, we
consider an electrically charged body and the decoherence effects of the
electromagnetic field, but an exactly similar analysis will apply for a
massive body in the gravitational case. Below, we will refer to the charged
body as a “particle” although it need not be an elementary particle, e.g., it
could be an atom or a nanoparticle. Suppose an experimenter, Alice, sends a
particle of charge $q$ with spin initially in the positive $x$-direction
through a Stern-Gerlach apparatus oriented in the $z$-direction, so that the
state of her particle after the process is in a superposition state of the
form

$\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow;\mathcal{A}_{1}}+\ket{\downarrow;\mathcal{A}_{2}}\big{)}\,.$
(1)

Here $\ket{\mathcal{A}_{1}}$ and $\ket{\mathcal{A}_{2}}$ are spatially
separated wavepackets with separation $d$, with $\ket{\uparrow}$ and
$\ket{\downarrow}$ being eigenstates of the $z$-spin. We wish to know whether
the coherence of this superposition is preserved at a later time. In order to
make this into a well defined experimental/observational question, Alice can
put the particle through a reversing Stern-Gerlach apparatus at some later
time and measure the $x$-spin. If the coherence of the superposition Eq. 1 has
been maintained, the spin will always be found to be in the positive
$x$-direction, whereas if any coherence has been lost the spin will sometimes
be found to be in the negative $x$-direction.

We assume that there are no external influences whatsoever on Alice’s
particle. It might then seem obvious that coherence must be maintained.
However, this is not necessarily the case because, since the particle is
charged, an electromagnetic field is present and it is part of the system.
Heuristically, the state of the total system after passage through the initial
Stern-Gerlach apparatus is actually of the form

$\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow;\mathcal{A}_{1}}\otimes\ket{\psi_{1}}+\ket{\downarrow;\mathcal{A}_{2}}\otimes\ket{\psi_{2}}\big{)}$
(2)

where $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ formally correspond to the states
of the electromagnetic field for the charge-current sources determined by
$\ket{\mathcal{A}_{1}}$ and $\ket{\mathcal{A}_{2}}$, respectively. Since
$\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ clearly are distinguishable
electromagnetic fields, it might seem that Alice’s particle is already
decohered at the outset. However this decoherence is a “false decoherence” in
the sense of [4]. If Alice recombines her particle slowly enough so as to
avoid radiating, she will be able to fully restore the coherence of her
particle.

In order to give a precise description of the true decoherence of Alice’s
particle associated with the electromagnetic field, it is necessary to
separate the electromagnetic field into a “Coulomb part” (which is not an
independent degree of freedom and should cause only a false decoherence of
Alice’s particle) and a “radiation part” (which corresponds to the true
degrees of freedom of the electromagnetic field that should be responsible for
a true decoherence, observable by Alice). In general, this distinction is not
possible to make in a meaningful way at any finite time. However, the
situation improves considerably if we go to asymptotically late times. At
asymptotically late times, the electromagnetic field naturally decomposes into
a radiation field that propagates to null infinity and a Coulomb field that
follows Alice’s particle to timelike infinity. The asymptotic Coulomb field is
completely determined by the asymptotic state of Alice’s particle and does not
represent an independent degree of freedom. Thus, at asymptotically late
times, the state of the total system is of the form

$\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow;\mathcal{A}_{1}}_{i^{+}}\otimes\ket{\Psi_{1}}_{\mathscr{I}^{+}}+\ket{\downarrow;\mathcal{A}_{2}}_{i^{+}}\otimes\ket{\Psi_{2}}_{\mathscr{I}^{+}}\big{)}.$
(3)

Here $\ket{\uparrow;\mathcal{A}_{1}}_{i^{+}}$ and
$\ket{\downarrow;\mathcal{A}_{2}}_{i^{+}}$ represent the asymptotically late
time states of the components of Alice’s particle and
$\ket{\Psi_{1}}_{\mathscr{I}^{+}}$ and $\ket{\Psi_{2}}_{\mathscr{I}^{+}}$
represent the quantum states of the electromagnetic radiation at future null
infinity $\mathscr{I}^{+}$. If Alice has recombined her particle at some
finite time, then $\ket{\mathcal{A}_{1}}=\ket{\mathcal{A}_{2}}$. Thus, the
decoherence of Alice’s superposition will be determined by the orthogonality
of the radiation states

${\mathscr{D}}=1-\left\lvert\braket{\Psi_{1}}{\Psi_{2}}_{\mathscr{I}^{+}}\right\rvert.$
(4)

In the absence of any external influences, Alice can ensure that the coherence
of her particle is maintained (i.e., ${\mathscr{D}}\approx 0$) if she
recombines her particle in such a way that negligible entangling radiation is
emitted. As estimated in [5], this will be possible if the recombination is
done over a time span $T$ such that

$T\gg\frac{qd}{\sqrt{\epsilon_{0}c^{3}\hbar}}.$ (5)

In other words, if Eq. 5 holds, Alice can ensure that
$\ket{\Psi_{1}}_{\mathscr{I}^{+}}\approx\ket{\Psi_{2}}_{\mathscr{I}^{+}}\approx\ket{0}_{\mathscr{I}^{+}}$,
so ${\mathscr{D}}\approx 0$. Thus, in Minkowski spacetime, Alice can, in
principle, maintain the quantum coherence of her spatial superposition by
recombining the components of the superposition slowly enough.

We now consider how this situation changes if there is a black hole in the
vicinity of Alice. First, Alice must ensure that her lab doesn’t fall into the
black hole. One way of doing this would be for Alice to orbit the black hole.
However, this may result in some unwanted emission of radiation. Therefore it
would be better to equip Alice with a rocket engine that keeps her lab
stationary. She must then also apply some force to her particle (e.g., via a
uniform electric field) to keep it stationary. There also may be other effects
in her lab due to the spacetime curvature associated with the black hole.
However, Alice can take the effects of the gravitational field of the black
hole on her lab into account in such a way that they will not will not result
in the decoherence of her particle. Therefore, we shall ignore these affects.
However, as we shall now explain, the black hole itself will acquire “which
path” information about Alice’s particle, which will result in decoherence.

With regard to the decoherence of Alice’s particle, the key difference arising
when a black hole is present is that electromagnetic radiation can now
propagate through the black hole horizon as well as to null infinity. Thus,
when a black hole is present, the asymptotically late time state of Alice’s
particle and the electromagnetic field is now

$\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow;\mathcal{A}_{1}}_{i^{+}}\otimes\ket{\Psi_{1}}_{\mathscr{I}^{+}}\ket{\Phi_{1}}_{\mathscr{H}^{+}}+\ket{\downarrow;\mathcal{A}_{2}}_{i^{+}}\otimes\ket{\Psi_{2}}_{\mathscr{I}^{+}}\ket{\Phi_{2}}_{\mathscr{H}^{+}}\big{)}$
(6)

where $\ket{\Psi_{1}}_{\mathscr{I}^{+}}$ and
$\ket{\Psi_{2}}_{\mathscr{I}^{+}}$ are as before and
$\ket{\Phi_{1}}_{\mathscr{H}^{+}}$ and $\ket{\Phi_{2}}_{\mathscr{H}^{+}}$ are
the corresponding states of the electromagnetic field on the event horizon,
$\mathscr{H}^{+}$, of the black hole. The decoherence of Alice’s particle in
the presence of a black hole is now given by

${\mathscr{D}}=1-\left\lvert\braket{\Psi_{1}}{\Psi_{2}}_{\mathscr{I}^{+}}\braket{\Phi_{1}}{\Phi_{2}}_{\mathscr{H}^{+}}\right\rvert.$
(7)

As in Minkowski spacetime, if Alice recombines her particle adiabatically, she
can ensure that there is negligible radiation to infinity, so
$\ket{\Psi_{1}}_{\mathscr{I}^{+}}\approx\ket{\Psi_{2}}_{\mathscr{I}^{+}}\approx\ket{0}_{\mathscr{I}^{+}}$,
in which case any decoherence will be entirely due to radiation propagating
into the black hole

${\mathscr{D}}_{\rm
BH}=1-\left\lvert\braket{\Phi_{1}}{\Phi_{2}}_{\mathscr{H}^{+}}\right\rvert.$
(8)

It might be thought that, by performing her recombination adiabatically, Alice
also can ensure that no radiation enters the black hole. However this is not
the case.

To see this, we first consider a classical point charge outside of a
Schwarzschild black hole. The explicit solution for a static point charge
outside of a Schwarzschild black hole has long been known [6, 7, 8]. On the
horizon, the electric field of a static point charge is purely radial, i.e.,
the only nonvanishing component of the electric field on the horizon is
$E_{r}=cF_{ab}\ell^{a}n^{b}$, where $n^{a}$ denotes the affinely parametrized
null normal to the horizon and $\ell^{a}$ is the unique past-directed radial
null vector satisfying $\ell^{a}n_{a}=1$. Electromagnetic radiation on the
horizon is described by the pullback, $E_{A}$, of the electric field
$E_{a}=cF_{ab}n^{b}$ to the horizon, where capital Latin indices denote
angular components on the horizon. Since $E_{A}=0$ for a static point charge,
there is no radiation through the horizon, as would be expected. However,
suppose we now quasi-statically move the point charge to a new location. After
the charge has reached its new location, the electric field will again be
radial, but $E_{r}$ on the horizon will be different from what it was
initially. However, it follows from Maxwell’s equations at the horizon that

$\mathcal{D}^{A}E_{A}=-\partial_{V}E_{r}$ (9)

where $\mathcal{D}_{A}$ denotes the covariant derivative on the $2$-sphere
cross-sections of the horizon, angular indices are raised and lowered with the
metric, $q_{AB}$, on the cross-section, and $V$ is an affine parameter such
that $n^{a}=(\partial/\partial V)^{a}$. Therefore, we must have $E_{A}\neq 0$
on the horizon as the charge is being moved and, indeed, $\int E_{A}dV$ is
constrained by initial and final values of $E_{r}$, independently of how the
charge is moved between its initial and final positions. Thus, there is
necessarily some radiation that crosses the horizon of the black hole due to
the displacement of the charge. We can make the total energy flux of this
radiation through the horizon arbitrarily small by moving the charge very
slowly, but, as we will now show, we cannot make the “total photon flux” of
this radiation small by moving the charge quasi-statically.

In order to analyze quantum aspects of the radiation, we need to give a
precise specification of the quantum state of electromagnetic radiation on the
horizon of a black hole. For an unperturbed Schwarzschild black hole formed by
gravitational collapse, the state of the electromagnetic field on the horizon
of the black hole is described by the Unruh vacuum. However, we will be
concerned here only with low frequency phenomena ($\omega\ll 1$), in which
case the Unruh and Hartle-Hawking vacua near the horizon are essentially
indistinguishable. For the electromagnetic field in a gauge where
$A_{a}n^{a}=0$ on the horizon, the “free data” of the electromagnetic field on
the horizon is the pull-back, $A_{A}$, of the vector potential. In the Fock
space associated with the Hartle-Hawking vacuum, a “particle” corresponds to a
solution that is purely positive frequency with respect to affine parameter on
the horizon. The inner product on the one-particle Hilbert space is given by
[9]

$\braket{A_{1,B}}{A_{2,C}}_{\mathscr{H}^{+}}\mathrel{\equiv}\frac{2\epsilon_{0}c}{\hbar}\int_{\mathbb{S}^{2}}r_{\text{s}}^{2}d\Omega\int_{0}^{\infty}\frac{\omega
d\omega}{2\pi}~{}q^{BC}\overline{{\hat{A}_{1,B}}(\omega,x^{A})}{\hat{A}_{2,C}}(\omega,x^{A})$
(10)

where $r_{\text{s}}$ is the Schwarzschild radius of the black hole and
$\hat{A}_{A}$ is the Fourier transform of $A_{A}$ with respect to affine
parameter $V$. Eq. 10 corresponds to a Klein-Gordon type of inner product on
the positive frequency part of $A_{A}$. Now suppose that the black hole is
perturbed by a classical charge-current source of the quantum electromagnetic
field. The quantum state of the electromagnetic field will then be described
by the coherent state (relative to the unperturbed vacuum) associated with the
classical retarded solution. The expected number of “horizon photons” in this
electromagnetic state at the horizon is given by

$\langle N\rangle=\|A_{A}\|^{2}_{\mathscr{H}^{+}}$ (11)

where $A_{A}$ is the classical retarded solution and the norm of $A_{A}$ is
defined by the inner product Eq. 10.

Let us apply this result to the electromagnetic field of a point charge that
starts at a point $x$ outside the black hole, is moved to another point
$x^{\prime}$ outside of the black hole and remains at $x^{\prime}$ forever. We
have already seen in this case that $\int E_{A}dV\neq 0$. Since
$E_{A}=-c\partial_{V}A_{A}$, this means that $A_{A}$ does not return to its
initial value at the end of the process. This is closely analogous to the
memory effect that occurs at null infinity [10, 11]. The fact that $A_{A}$
does not return to its initial value implies that its Fourier transform
diverges as $1/\omega$ as $\omega\to 0$. It then follows immediately from Eq.
10 that $\|A_{A}\|^{2}_{\mathscr{H}^{+}}=\infty$. Thus, if one moves a point
charge from $x$ to $x^{\prime}$ and leaves the particle at $x^{\prime}$
forever, no matter how quickly or slowly the charge is moved, an infinite
number of “soft horizon photons” will be radiated into the black hole. This is
closely analogous to the infrared divergences at null infinity that arise in
scattering theory in quantum electrodynamics [12]. Note that the infinite
number of “soft photons” carry negligible energy, and by moving the charge
quasi-statically, the total electromagnetic energy radiated into the black
hole can be made to be arbitrarily small.

The case of more relevance for us is one in which the point charge is moved
from $x$ to $x^{\prime}$, is held at $x^{\prime}$ for a long proper time $T$,
and then is returned to $x$. In that case, $A_{A}$ returns to its initial
value at late times, so there is no infrared divergence in the sense that
$\langle N\rangle$ is finite. Nevertheless, the following estimates show that
$\langle N\rangle$ is very large when $T$ is very large. The radial electric
field of a point charge located a distance $b$ from the black hole is roughly
$E_{r}\sim q/\epsilon_{0}b^{2}$ [6, 7, 8]. The change in the radial electric
field when the charge is moved from $x$ to $x^{\prime}$ is therefore roughly
$\Delta E_{r}\sim qd/\epsilon_{0}b^{3}$, where $d$ is the distance between $x$
and $x^{\prime}$ and we have assumed that $d\ll b$. Taking account of the fact
that the $2$-spheres on the horizon are of radius $2GM/c^{2}$, it then follows
from Eq. 9 that the change in the vector potential, $A_{A}$, on the horizon
when the particle is moved from $x$ to $x^{\prime}$ is

$\Delta A_{A}\sim\frac{G^{2}M^{2}}{c^{5}}\frac{qd}{\epsilon_{0}b^{3}}.$ (12)

Eventually, when the particle is moved back to $x$, the change in $A_{A}$ will
be equal and opposite to Eq. 12. But if the charge is held at point
$x^{\prime}$ for a very long time $T$, the contribution of $A_{A}$ to the norm
defined by Eq. 10 will be dominated by the low frequency contribution arising
from the time interval over which Eq. 12 holds. We obtain

$\langle
N\rangle=\|A_{A}\|_{\mathscr{H}^{+}}^{2}\sim\frac{G^{4}M^{4}q^{2}d^{2}}{\hbar
c^{9}\epsilon_{0}b^{6}}\ln V$ (13)

where $V$ is the affine time on the horizon corresponding to the proper time
$T$ along the particle trajectory. However, the relation between affine time,
$V$, and Killing time, $v$, on the horizon of a black hole is given by
$V=\exp(\kappa v/c)$, where $\kappa=c^{4}/4GM$ is the surface gravity of the
black hole. Furthermore, the Killing time is related to the proper time of the
particle by the redshift factor. We shall assume that Alice’s lab is not
extremely close to the black hole and neglect the departure of the redshift
factor from $1$. We then obtain

$\langle
N\rangle=\|A_{A}\|_{\mathscr{H}^{+}}^{2}\sim\frac{G^{3}M^{3}q^{2}d^{2}}{\hbar
c^{5}\epsilon_{0}b^{6}}T.$ (14)

Thus, the number of “soft photons” radiated into the black hole in the above
process grows linearly with the time, $T$, that the point charge spends at
point $x^{\prime}$.

We now have all of the ingredients needed to analyze Alice’s coherence
experiment, under the assumption that Alice splits and recombines her particle
slowly enough that negligible radiation is emitted to infinity. Although our
results hold much more generally, it is easiest to consider the case where,
after passing through the Stern-Gerlach apparatus, the first component of
Alice’s particle remains at position $x$ and the second component of her
particle moves to position $x^{\prime}$. After these components stay at $x$
and $x^{\prime}$, respectively, for a time $T$, they are recombined in such a
way that the first component continues to remain at $x$ and the second
component moves from $x^{\prime}$ to $x$. In that case, no radiation is
emitted by the first component, so in Eq. 6, we have
$\ket{\Phi_{1}}_{\mathscr{H}^{+}}=\ket{0}_{\mathscr{H}^{+}}$. However, our
above analysis applies to the second component, which moves from $x$ to
$x^{\prime}$, stays at $x^{\prime}$ for a time $T$, and then returns to $x$.
Thus, $\ket{\Phi_{2}}_{\mathscr{H}^{+}}$ will be a state with expected number
of photons given by Eq. 14. If $\langle N\rangle\gtrsim 1$, then
$\ket{\Phi_{2}}_{\mathscr{H}^{+}}$ will be nearly orthogonal to
$\ket{\Phi_{1}}_{\mathscr{H}^{+}}=\ket{0}_{\mathscr{H}^{+}}$. This means
that—due entirely to the presence of a black hole—Alice’s particle will
decohere in a time

$\displaystyle T_{D}$ $\displaystyle\sim\frac{\epsilon_{0}\hbar
c^{6}b^{6}}{G^{3}M^{3}q^{2}d^{2}}$ (15) $\displaystyle\sim
10^{43}~{}\text{years}~{}\bigg{(}\frac{b}{\text{a.u.}}\bigg{)}^{6}\cdot\bigg{(}\frac{M_{\odot}}{M}\bigg{)}^{3}\cdot\bigg{(}\frac{e}{q}\bigg{)}^{2}\cdot\bigg{(}\frac{\rm{m}}{d}\bigg{)}^{2}.$
(16)

Thus, if our Sun were a black hole and if one separated an electron into two
components one meter apart in a laboratory experiment on Earth, it would not
be possible to maintain the coherence of the electron for more than $10^{43}$
years. On the other hand, if this experiment were done at $b=6M$, then
$T_{D}\sim 5\,{\rm minutes}$.

A closely parallel analysis can be given for the case of a gravitating
particle. In the gravitational case, the electric part of the Weyl tensor
$E_{ab}=c^{2}C_{acbd}n^{c}n^{d}$ plays a role closely analogous to the role
played by the electric field $E_{a}$ in the electromagnetic case. For a static
point mass outside a Schwarzschild black hole the only non-vanishing component
of the electric part of the Weyl tensor on the horizon is
$E_{rr}=c^{2}C_{acbd}\ell^{a}n^{c}\ell^{b}n^{d}$. Gravitational radiation on
the horizon is described by the pullback, $E_{AB}$, of $E_{ab}$, which
vanishes for a static point mass. However, the process of moving the particle
quasi-statically to a new location will involve a change in $E_{rr}$. The
(once-contracted) Bianchi identity on the horizon yields

$\mathcal{D}^{A}E_{AB}=-\partial_{V}E_{rB},\quad\quad\mathcal{D}^{A}E_{rA}=-\partial_{V}E_{rr}$
(17)

which implies

$\mathcal{D}^{A}\mathcal{D}^{B}E_{AB}=\partial_{V}^{2}E_{rr}$ (18)

in close analogy with Eq. 9. Thus, if a point mass is moved quasi-statically,
there necessarily will be radiation through the horizon. To determine the
number of gravitons emitted we treat the quantum gravitational field at the
level of linearized perturbation theory about the black hole background. For a
metric perturbation $h_{ab}$ in a gauge where $h_{ab}n^{a}=0=q^{AB}h_{AB}$ on
the horizon, the “free data” on the horizon is $h_{AB}$. As in the
electromagnetic case, a “particle” in the Fock space associated to the Hartle-
Hawking vacuum is a solution with positive frequency with respect to affine
parameter $V$. The inner product on the one-particle Hilbert space is given by
a direct analog of Eq. 10 with $A_{A}$ replaced by $h_{AB}$. Finally, $E_{AB}$
is given in terms of $h_{AB}$ by
$E_{AB}=-\tfrac{1}{2}c^{2}\partial^{2}_{V}h_{AB}$.

The analysis of the decoherence of a quantum superposition of a body of mass
$m$ in the presence of a black hole now proceeds in exact parallel with the
electromagnetic case. The only significant difference is that, for the same
reason as in the analysis of [5], it is now the effective mass quadrupole
$md^{2}$ of the superposition that enters, rather than the effective
electrostatic dipole $qd$ that entered the electromagnetic analysis. We find
that a black hole will decohere a quantum superposition of a massive body in a
time

$\displaystyle T_{D}^{\text{GR}}$ $\displaystyle\sim\frac{\hbar
c^{10}b^{10}}{G^{6}M^{5}m^{2}d^{4}}$ (19) $\displaystyle\sim 10\
\mu\text{s}~{}\bigg{(}\frac{b}{\text{a.u.}}\bigg{)}^{10}\cdot\bigg{(}\frac{\text{M}_{\odot}}{M}\bigg{)}^{5}\cdot\bigg{(}\frac{\text{M}_{\text{Earth}}}{m}\bigg{)}^{2}\cdot\bigg{(}\frac{\text{R}_{\mathrm{Earth}}}{d}\bigg{)}^{4}.$
(20)

Thus, if the Sun were a black hole and the Earth occupied a quantum state with
its center of mass spatially superposed by a separation on the order of its
own radius, this superposition would decohere due to the presence of the black
hole in about $10$ microseconds. Of course, it would not be easy to put the
Earth into such a quantum superposition.

In summary, we have found that black holes, in effect, gather information
about quantum superpositions of spatially separated components by means of the
long range fields sourced by the matter comprising these components.
Eventually, a black hole will decohere any quantum superposition. Although
this may not be of practical importance for any presently contemplated
experiments, it may be of fundamental significance for our understanding of
the nature of black holes in a quantum theory of gravity.

## Acknowledgements

D.L.D. acknowledges his support as a Fannie and John Hertz Foundation Fellow
holding the Barbara Ann Canavan Fellowship, and as an Eckhardt Graduate
Scholar in the Physical Sciences Division at the University of Chicago. This
research was supported in part by NSF grant 21-05878 to the University of
Chicago.

## References

  * [1] W.G. Unruh and R.M. Wald, _Information Loss_ , _Rept. Prog. Phys._ 80 (2017) 092002 [arXiv:1703.02140].
  * [2] D. Marolf, _The Black Hole information problem: past, present, and future_ , _Rept. Prog. Phys._ 80 (2017) 092001 [arXiv:1703.02143].
  * [3] D.L. Danielson, G. Satishchandran and R.M. Wald, _Gravitationally mediated entanglement: Newtonian field versus gravitons_ , _Phys. Rev. D_ 105 (2022) 086001 [arXiv:2112.10798].
  * [4] W.G. Unruh, _False loss of coherence_ , arXiv:1110.2199.
  * [5] A. Belenchia, R.M. Wald, F. Giacomini, E. Castro-Ruiz, v. Brukner and M. Aspelmeyer, _Quantum Superposition of Massive Objects and the Quantization of Gravity_ , _Phys. Rev. D_ 98 (2018) 126009 [arXiv:1807.07015].
  * [6] E.T. Copson and E.T. Whittaker, _On electrostatics in a gravitational field_ , _Proceedings of the Royal Society of London. Series A_ 118 (1928) 184.
  * [7] J.M. Cohen and R.M. Wald, _Point charge in the vicinity of a schwarzschild black hole_ , _Journal of Mathematical Physics_ 12 (1971) 1845.
  * [8] B. Linet, _Electrostatics and magnetostatics in the schwarzschild metric_ , _Journal of Physics A, General Physics_ 9 (1976) 1081.
  * [9] B.S. Kay and R.M. Wald, _Theorems on the Uniqueness and Thermal Properties of Stationary, Nonsingular, Quasifree States on Space-Times with a Bifurcate Killing Horizon_ , _Phys. Rept._ 207 (1991) 49.
  * [10] L. Bieri and D. Garfinkle, _An electromagnetic analogue of gravitational wave memory_ , _Class. Quant. Grav._ 30 (2013) 195009 [arXiv:1307.5098].
  * [11] G. Satishchandran and R.M. Wald, _Asymptotic behavior of massless fields and the memory effect_ , _Phys. Rev. D_ 99 (2019) 084007 [arXiv:1901.05942].
  * [12] K. Prabhu, G. Satishchandran and R.M. Wald, _Infrared Finite Scattering Theory in Quantum Field Theory and Quantum Gravity_ , arXiv:2203.14334.