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Complex actions in two-dimensional topology change

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Jorma Louko, Rafael D. Sorkin

General Relativity and Quantum Cosmology · Submitted 6 Nov 1995 · 72 references

Pith's verdict: The two-dimensional computation is clean and worth citing; the abstract's D≥3 suppression/enhancement claim runs ahead of what Section VI actually establishes.

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arXiv IDgr-qc/9511023 (version 3)
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CitationJorma Louko, Rafael D. Sorkin. "Complex actions in two-dimensional topology change." arXiv:gr-qc/9511023 (1995). https://pith.science/paper/3HAMA4FY
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The paper

The paper claims

By regularizing two families of degenerate two-dimensional metrics with a small positive imaginary part, the paper computes the action at a Lorentzian conical singularity and finds it complex: $+2πi$ at a topology-changing crotch, and $β -

The reading

The paper asks what the Einstein–Hilbert action $S = (1/2κ)∫ d^Dx √{-g} R$ is worth on the degenerate metrics that make topology change possible. In two dimensions every cobordism is built from two elementary pieces, the 'yarmulke' (a Misner-space region plus one point, like a cosmological initial singularity) and the 'trousers' (whose crotch is where the topology actually changes, and which is also the fundamental vertex of string theory). The authors write down explicit smooth Morse-theory metrics on $R^2$ that are flat everywhere except at one point, deform them by a small imaginary regulator chosen so that a scalar-field path integral converges, and then take the regulator to zero. The scalar curvature density becomes a delta-function at the singular point whose strength is complex: $β - 2πi$ for the yarmulke, where $β$ is the Misner rapidity, and $+2πi$ for the trousers, independent of every parameter. This matters because when such a two-dimensional geometry is multiplied by flat extra dimensions, the imaginary part of the action scales with the volume of those dimensions, so topology change through a crotch is exponentially suppressed by $e^{-A/4G}$ in four dimensions while creation or annihilation of a universe through a yarmulke is exponentially enhanced.

What carries the argument

The load-bearing device is the two-parameter metric family $ds^2 = (x^2 + y^2 + γ)(dx^2 + dy^2) - ζ(x dx + ǫ y dy)^2$, built from the Morse function $f = ½(x^2 + ǫy^2)$ with a flat auxiliary metric, which is analytic and flat away from the origin but degenerates there. The regulator $γ$ is taken positive for the Euclidean cases and purely imaginary ($γ = +iσ$) for the Lorentzian ones, because a real regulator destroys invertibility; the sign of $Im γ$ is fixed by scalar-field convergence. Integrating $½√{-g}R$ against a test function, integrating by parts, and using dominated convergence to drop the surviving integral gives the delta-function strengths directly. Two independent cross-checks carry the generality claim: a suitably complexified Gauss–Bonnet theorem, in which a closed two-manifold with Euler number $χ$ satisfies $½∫d^2x√{-g}R = -2πiχ$, and Lorentzian Regge calculus with complex defect angles. The Gauss–Bonnet route is illustrated by closing off a trousers with three yarmulkes to make an $S^2$, and by adding two points to a patch of $(1+1)$-dimensional anti-de Sitter space.

Extended reading notes

Core claim

The central claim is that the Ricci scalar density $½√{-g}R$ of a Lorentzian conical-type singularity is a delta-function with a complex coefficient, and that the sign of the imaginary part is fixed by the requirement that the regulated spacetime admit a convergent scalar-field path integral (equivalently $Im(g_{ab}) > 0$). For the yarmulke family the coefficient is $β - 2πi$; for the trousers family it is $+2πi$. The real part reproduces the ordinary defect-angle result known from Euclidean conical singularities, while the imaginary part is new and, in the trousers case, universal: all values of the metric parameter $ζ > 1$ describe the same geometry and give the same $2πi$. The authors then argue that these singular metrics are not stationary points of the action in $D ≥ 3$ dimensions — the conical contribution scales with the $(D-2)$-dimensional volume and is not stationary under variations of that volume — so topology change is classically forbidden in the metric formulation, and quantum-mechanically the imaginary part adds an exponential suppression for the trousers and an exponential enhancement for the yarmulke. A closing contrast is drawn with the vielbein-cum-connection formulation, in which the trousers apparently does solve the field equations.

Load-bearing premise

The claim that any adequate regulator gives the same complex delta-function strength rests on the convexity of the domain $Im(g_{ab}) > 0$ together with a complex Gauss–Bonnet theorem that the authors state they have not proven, and their own footnote gives an explicit complex metric on $S^2$ for which the naive version of that theorem fails.

Editorial extensions

If this is right

  • In the metric formulation of Einstein gravity in $D ≥ 3$, spacetimes containing a crotch singularity are not classical solutions, so topology change is forbidden in the classical limit.
  • In the gravitational path integral, trousers histories are suppressed by $\exp(-A/4G)$ in $3+1$ dimensions, where $A$ is the area of the two flat dimensions, making topology change through a crotch exponentially disfavoured.
  • Yarmulke histories are enhanced by the same exponential factor, which the authors read as possible evidence for enhanced creation or annihilation of a universe, to be balanced against the non-stationarity of the real part of the action.
  • The metric and vielbein-cum-connection formulations of gravity appear to be physically inequivalent, since the trousers solves the field equations in the latter but not the former.
  • Because the curvature density is a weight-one scalar density, the result is independent of the differentiable structure used to approach the singular point, unlike distributional Riemann or Ricci tensors, which the authors show are highly regularization-dependent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the imaginary part of the action is real and universal rather than an artifact of the chosen regulator, it should be recoverable from an independent discrete construction — a Regge or causal-set sum over the same cobordism — and the two numbers should agree in sign as well as magnitude.
  • The sign asymmetry between trousers (suppressed) and yarmulke (enhanced) suggests a one-loop consistency check: the same imaginary part should show up as the phase of a matter determinant on the singular background, so a heat-kernel computation of the effective action near the crotch is a concrete place to look for agreement.
  • The authors' own remark that the coefficient $8π/κ = 1/4G$ coincides with the one appearing in black-hole entropy hints that the imaginary action of a singular surface and horizon entropy may be two readings of the same boundary term; that is a conjecture the paper raises but does not pursue.
  • The proposal that 'generalized isometries' — homeomorphisms preserving causal order and the volume element — should replace diffeomorphisms as gauge equivalences for degenerate metrics is testable in principle: two of the paper's trousers metrics with different $ζ$ should be physically identified, and any path integral that fails to do so overcounts histories.
Original abstract

We investigate topology change in (1+1) dimensions by analyzing the scalar-curvature action $1/2 \int R dV$ at the points of metric-degeneration that (with minor exceptions) any nontrivial Lorentzian cobordism necessarily possesses. In two dimensions any cobordism can be built up as a combination of only two elementary types, the ``yarmulke'' and the ``trousers.'' For each of these elementary cobordisms, we consider a family of Morse-theory inspired Lorentzian metrics that vanish smoothly at a single point, resulting in a conical-type singularity there. In the yarmulke case, the distinguished point is analogous to a cosmological initial (or final) singularity, with the spacetime as a whole being obtained from one causal region of Misner space by adjoining a single point. In the trousers case, the distinguished point is a ``crotch singularity'' that signals a change in the spacetime topology (this being also the fundamental vertex of string theory, if one makes that interpretation). We regularize the metrics by adding a small imaginary part whose sign is fixed to be positive by the condition that it lead to a convergent scalar field path integral on the regularized spacetime. As the regulator is removed, the scalar density $1/2 \sqrt{-g} R$ approaches a delta-function whose strength is complex: for the yarmulke family the strength is $\beta -2\pi i$, where $\beta$ is the rapidity parameter of the associated Misner space; for the trousers family it is simply $+2\pi i$. This implies that in the path integral over spacetime metrics for Einstein gravity in three or more spacetime dimensions, topology change via a crotch singularity is exponentially suppressed, whereas appearance or disappearance of a universe via a yarmulke singularity is exponentially enhanced.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,133 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.49 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.92 · compares with prior results 0.11 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We investigate topology change in (1+1) dimensions by analyzing the scalar-curvature action $1/2 \int R dV$ at the points of metric-degeneration that (with minor exceptions) any nontrivial Lorentzian cobordism necessarily possesses. In two dimensions any cobordism can be built up as a combination of only two elementary types, the ``yarmulke'' and the ``trousers.'' For each of these elementary cobordisms, we consider a family of Morse-theory inspired Lorentzian metrics that vanish smoothly at a single point, resulting in a conical-type singularity there. In the yarmulke case, the distinguished point is analogous to a cosmological initial (or final) singularity, with the spacetime as a whole being obtained from one causal region of Misner space by adjoining a single point. In the trousers case, the distinguished point is a ``crotch singularity'' that signals a change in the spacetime topology (this being also the fundamental vertex of string theory, if one makes that interpretation). We regularize the metrics by adding a small imaginary part whose sign is fixed to be positive by the condition that it lead to a convergent scalar field path integral on the regularized spacetime. As the regulator is removed, the scalar density $1/2 \sqrt{-g} R$ approaches a delta-function whose strength is complex: for the yarmulke family the strength is $\beta -2\pi i$, where $\beta$ is the rapidity parameter of the associated Misner space; for the trousers family it is simply $+2\pi i$. This implies that in the path integral over spacetime metrics for Einstein gravity in three or more spacetime dimensions, topology change via a crotch singularity is exponentially suppressed, whereas appearance or disappearance of a universe via a yarmulke singularity is exponentially enhanced.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We investigate topology change in (1+1) dimensions by analyzing the scalar-curvature action $1/2 \int R dV$ at the points of metric-degeneration that (with minor exceptions) any nontrivial Lorentzian cobordism necessarily possesses. contribution 0.97 0.87 0.06 asserted 0.97
s1 In two dimensions any cobordism can be built up as a combination of only two elementary types, the ``yarmulke'' and the ``trousers.'' For each of these elementary cobordisms, we consider a family of Morse-theory inspired Lorentzian metrics that vanish smoothly at a single point, resulting in a conical-type singularity there. method 0.46 0.51 0.03 asserted 1.00
s2 In the yarmulke case, the distinguished point is analogous to a cosmological initial (or final) singularity, with the spacetime as a whole being obtained from one causal region of Misner space by adjoining a single point. interpretation 0.42 0.50 0.03 asserted 0.99
s3 In the trousers case, the distinguished point is a ``crotch singularity'' that signals a change in the spacetime topology (this being also the fundamental vertex of string theory, if one makes that interpretation). contribution 0.42 0.29 0.04 asserted 0.93
s4 We regularize the metrics by adding a small imaginary part whose sign is fixed to be positive by the condition that it lead to a convergent scalar field path integral on the regularized spacetime. method 0.99 0.91 0.02 asserted 0.99
s5 As the regulator is removed, the scalar density $1/2 \sqrt{-g} R$ approaches a delta-function whose strength is complex: for the yarmulke family the strength is $\beta -2\pi i$, where $\beta$ is the rapidity parameter of the associated Misner space; for the trousers family it is simply $+2\pi i$. result 1.00 0.92 0.22 asserted 1.00
s6 This implies that in the path integral over spacetime metrics for Einstein gravity in three or more spacetime dimensions, topology change via a crotch singularity is exponentially suppressed, whereas appearance or disappearance of a universe via a yarmulke singularity is exponentially enhanced. interpretation 0.88 0.72 0.84 asserted 0.95

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References
How later papers used it

74 later papers cite it

Evidence: substantial. 72 of the citing papers share no author with it, and 14 of those carry a confident label from Pith's citation classifier.

Pith read 19 of the 179 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Builds on it 1 of 5 papers with this label
How the literature uses it

72 arXiv papers cite this in 168 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 128 sentences without a confident label

Builds on it 4 papers
Uses its method or tool 5 papers
  • The idea was to follow [ref] and [ref] in using the complex version of the Gauss-Bonnet theorem to define the Einstein-Hilbert action on such spacetimes, and to treat higher-derivative corrections via a formal analytic continuation of the Euclidean power series analysis performed in [ref] .

    Gravitational thermodynamics without the conformal factor problem: Partition functions and Euclidean saddles from Lorentzian Path Integrals · What spacetimes contribute to the real Lorentz-signature gravitational path integral? · 0.90

  • The action of such singular geometries was defined following [ref] which, as we will review, can be thought of as taking the defining contour for the path integral to be given by a certain $iϵ$ prescription[An interesting result of this prescription is that the Lorentz-signature action $S$ of such configurations picks up an imaginary part whose sign is given by $2-n$.

    Euclidean and complex geometries from real-time computations of gravitational R\'enyi entropies · Introduction · 0.84

  • Then the metric ([ref]) will be [equation] Using the coordinate transformation [ref] , the metric is simplified to [equation] coincident with the flat Minkowski plane, covered twice by the transformation $(x,y)→(u,v)$.

    Wormholes and Teleporters · Opening wormholes, details of construction · 0.85

Compares with it 2 papers
Cites it as support 3 papers
Cites it as background 20 papers
  • The genesis of this criterion goes back to Louko & Sorkin [ref] , where they formulated a first version for scalar fields in $1+1$ dimensions.

    Uses of Complex Metrics in Cosmology · The LSKSW allowability criterion · 0.96

  • Without aiming to be exhaustive, we just mention that complex metrics have been considered in the study of the Hartle-Hawking no-boundary proposal for the wave function of the universe [ref] and of the topology-changing processes [ref] .

    Note on complex metrics, complex time and periodic universes · Introduction · 0.99

  • The Kontsevich-Segal-Witten criterion [ref] is a proposed constraint on the type of allowed complex metrics as saddle points of the gravitational path integral.

    Comments on the no boundary wavefunction and slow roll inflation · Constraints from the Kontsevich-Segal-Witten criterion · 0.91

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