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# Superradiant Instability in 2D Black Hole Backgrounds: A Detailed Analysis

Chen Wenxiang

► To cite this version:

Chen Wenxiang. Superradiant Instability in 2D Black Hole Backgrounds: A Detailed Analysis. 2025. [arXiv:0409.5571]

HAL Id: hal-04985571

https://hal.science/hal-04985571v1

Preprint submitted on 11 Mar 2025

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# Superradiant Instability in 2D Black Hole Backgrounds: A Detailed Analysis

Chen Wenxiang*

## Abstract

In this paper, we explore the phenomenon of superradiant instability in two-dimensional black hole spacetimes. Using the method of scalar field perturbation, we derive the effective potential in the radial direction and demonstrate how a negative potential region can lead to superradiant behavior, analogous to the more commonly studied higher-dimensional black holes. By examining the dynamics of scalar fields in simple 2D black hole backgrounds, we reveal that certain configurations exhibit instability in a way reminiscent of the classical Kerr black hole superradiance. We also discuss the potential implications for quantum gravity, black hole thermodynamics, and holography.

## 1 Introduction

Traditionally, two-dimensional black holes, especially the BTZ black hole, are considered superradiance-stable in some studies. This is because in two-dimensional spacetimes, the radiation mechanisms and particle dynamics of black holes often exhibit strong symmetry and relatively simple dynamics, which may make superradiance phenomena difficult to arise or less likely to occur.

In two-dimensional spacetimes, lower-dimensional black holes, such as the BTZ black hole, typically possess higher symmetries, especially in anti-de Sitter (AdS) spacetimes. This leads to relatively simple and symmetric gravitational fields and radiation mechanisms. This symmetry might suppress the generation of superradiance to some extent. In lower-dimensional spacetimes, there is generally no complex gravitational wave propagation and radiation mechanism, as seen in three- or four-dimensional spacetimes. The propagation of gravitational waves in two-dimensional black holes is simplified, potentially lacking the capacity to produce high-energy superradiance effects.

Although two-dimensional black holes can still produce Hawking radiation, this radiation tends to be relatively smooth and stable. In lower dimensions, the possibility of particle interactions and collective effects is reduced, limiting the formation of superradiance.

In a two-dimensional gravitational field, due to the lower dimensionality of spacetime, the event horizon of the black hole and the relativistic effects may be milder. Consequently, the superradiance mechanism might not manifest as significantly as it does in four-dimensional spacetimes.

Traditionally, two-dimensional black holes (such as the BTZ black hole) are considered superradiance-stable. This is primarily due to the symmetry, simplified gravitational fields, and

<footnote>*Department of Astronomy, College of Physics and Materials Science, Guangzhou University. Email: wx-chen4277@qq.com</footnote>
radiation mechanisms in lower-dimensional spacetimes, which make superradiance effects less likely to occur. However, this does not mean that similar superradiance-like phenomena are entirely impossible under special conditions. It simply suggests that traditional models exhibit strong stability in this regard.

The study of black holes has long been a central part of theoretical physics, where they serve as a laboratory for testing ideas in general relativity, quantum field theory, and thermodynamics. One of the most interesting phenomena associated with black holes is superradiance, a process by which a wave can extract energy from a rotating black hole, causing the black hole to lose energy over time. This phenomenon has been studied extensively in higher-dimensional spacetimes, particularly for rotating black holes (Kerr black holes) [1–4]. However, there has been growing interest in studying black hole dynamics in lower-dimensional spacetimes, where the complexity of the theory can be reduced while still capturing essential features of black hole physics.

In this paper, we investigate superradiant instability in two-dimensional (2D) black holes. Despite the lack of true angular momentum in 2D spacetime, we show that a scalar field can still exhibit superradiant-like behavior due to the presence of an effective potential that allows for amplification of field modes. This phenomenon is studied within the context of 2D models, such as the Jackiw-Teitelboim (JT) gravity model and the Callan-Giddings-Harvey-Strominger (CGHS) model, which are known to provide useful analogs to higher-dimensional black holes in terms of their thermodynamics and dynamical properties. [5–8]

The main goal of this paper is to demonstrate how a negative potential in the effective radial equation can lead to superradiant instability in 2D black hole backgrounds. We will explore the scalar field equation in these backgrounds, show how it can be reduced to a one-dimensional scattering problem, and analyze the conditions under which superradiant instability occurs. By drawing parallels to the well-understood case of rotating black holes, we aim to provide insights into the broader implications of superradiance in lower-dimensional gravity theories.

## 2 2D Black Hole Background and Scalar Field Equation

In order to study the scalar field perturbations around a 2D black hole background, we begin by considering the metric for a static, spherically symmetric black hole in 2D. The general form of such a metric is given by the line element:

$$ds^2 = -f(r) dt^2 + \frac{dr^2}{f(r)}, \qquad (1)$$

where $f(r)$ is a function that determines the geometry of the black hole, and $r$ is the radial coordinate. For a 2D black hole, the function $f(r)$ typically has a zero at $r = r_H$, which defines the event horizon of the black hole. The metric thus describes a static black hole whose geometry is governed by the function $f(r)$, which is a key ingredient in understanding the perturbations of fields in this background.

The next step is to consider a massless scalar field $\Phi(t, r)$ propagating in this background. The equation of motion for the scalar field is obtained from the Klein-Gordon equation, which in a curved spacetime reads:

$$\frac{1}{\sqrt{-g}} \partial_{\mu} (\sqrt{-g} g^{\mu\nu} \partial_{\nu} \Phi) = 0, \qquad (2)$$

where $g$ is the determinant of the metric $g_{\mu\nu}$. This equation governs the dynamics of the scalar
field $\Phi(t, r)$ in a curved spacetime. For our specific 2D black hole background, the nonzero components of the inverse metric $g^{\mu\nu}$ are: [5–12]

$$g^{tt} = -\frac{1}{f(r)}, \quad g^{rr} = f(r). \qquad (3)$$

Substituting these components into the Klein-Gordon equation, we obtain the following equation for the scalar field $\Phi(t, r)$:

$$-\frac{1}{f(r)} \frac{\partial^2 \Phi}{\partial t^2} + f(r) \frac{\partial^2 \Phi}{\partial r^2} + f'(r) \frac{\partial \Phi}{\partial r} = 0. \qquad (4)$$

This is a second-order partial differential equation that governs the evolution of the scalar field perturbations in the background of a 2D black hole. The term $f'(r)$ represents the radial derivative of the metric function, which influences the behavior of the scalar field along the radial direction.

To explore the perturbative behavior of the scalar field, we will next introduce a separation of variables technique, where the scalar field $\Phi(t, r)$ is written as a product of temporal and radial functions:

$$\Phi(t, r) = e^{-i\omega t} \psi(r), \qquad (5)$$

where $\omega$ is the frequency of the perturbation, and $\psi(r)$ is the radial profile of the field. Substituting this ansatz into equation (4), we obtain the following equation for $\psi(r)$:

$$\frac{d^2\psi(r)}{dr^2} + \left(\omega^2 f(r) - \frac{f'(r)}{f(r)}\right) \psi(r) = 0. \qquad (6)$$

This is a Schrödinger-like equation for the radial part of the scalar field. It describes how the scalar field propagates in the curved spacetime of the 2D black hole, where the effective potential is determined by the metric function $f(r)$.

At this point, it is important to consider the boundary conditions of the problem. The scalar field must be regular at the horizon $r = r_H$, and the asymptotic behavior of the field at large $r$ must be considered to ensure that the solution behaves appropriately at spatial infinity.

These boundary conditions are essential in determining the allowed modes of the scalar field and analyzing its stability or instability in the black hole background. Depending on the choice of $f(r)$, these modes may correspond to quasinormal modes, which are characteristic oscillations of the black hole that carry information about the black hole’s properties, such as its temperature and mass.

In the next section, we will examine specific choices for the function $f(r)$ that describe different types of 2D black holes and analyze the corresponding scalar field perturbations.

## 3 Separation of Variables and Effective Potential

To simplify the analysis, we perform a separation of variables. We assume a solution of the form: [5–12]

$$\Phi(t, r) = e^{-i\omega t} \psi(r), \qquad (7)$$

where $\omega$ is the frequency of the perturbation. Substituting this into the Klein-Gordon equation, we obtain the following radial equation:
$$ \frac{d^2\psi}{dr^2} + [\omega^2 - V_{\text{eff}}(r)] \psi(r) = 0, \qquad (8) $$

where the effective potential $V_{\text{eff}}(r)$ takes the form:

$$ V_{\text{eff}}(r) = f(r) \left[ \frac{\ell(\ell + 1)}{r^2} \right] \quad \text{for angular momentum } \ell, \qquad (9) $$

where $\ell$ is the angular momentum quantum number. For a 2D black hole in a static configuration, the angular momentum is effectively zero, i.e., $\ell = 0$. This assumption greatly simplifies the analysis, reducing the effective potential to:

$$ V_{\text{eff}}(r) = 0 \quad \text{for } \ell = 0. \qquad (10) $$

In more general 2D black hole models, such as the CGHS and JT models, the effective potential $V_{\text{eff}}(r)$ may exhibit more complex behavior. Specifically, the potential can become negative or form a well, leading to interesting physical phenomena such as superradiance. These behaviors are most prominent in the regions where the effective potential dips below zero, which can be expressed as:

$$ V_{\text{eff}}(r) < 0 \quad \text{for certain ranges of } r. \qquad (11) $$

In these regions, the scalar field perturbations may undergo an amplification due to the interaction with the potential well. The result is a net gain in energy, resembling superradiance observed in higher-dimensional black holes. This phenomenon occurs when the scalar field is reflected back by the negative potential, gaining energy in the process.

To explore these phenomena further, we consider the behavior of the scalar field perturbations in these negative potential regions. Using the separation of variables approach and examining the boundary conditions at $r_H$ and at infinity, we can derive the precise conditions for the occurrence of superradiance. These conditions depend on the specific form of the metric function $f(r)$, the value of the frequency $\omega$, and the structure of the effective potential.

Moreover, we can study the quasinormal modes (QNM) of the system, which represent the characteristic oscillations of the black hole under perturbation. These modes are determined by solving the radial equation with appropriate boundary conditions, where the energy of the scalar field is either extracted or amplified by the black hole, depending on the relationship between the frequency $\omega$ and the potential structure.

The general solution to the radial equation can be expressed in terms of a WKB approximation for high-energy perturbations, or through a more detailed numerical approach for low-energy modes. These solutions provide insight into the stability of the black hole and the possible amplification of scalar field perturbations in the presence of an effective potential well.

As we move forward, we will explore specific forms for $f(r)$ in the context of the CGHS and JT black holes and analyze the implications for scalar field perturbations, focusing on superradiance and the resulting impact on black hole thermodynamics.

# 4 Reflection and Transmission Coefficients

The relations derived allow us to express B, C, and D in terms of A, as shown in Appendix A. Specifically, we define $B = R(k, g)A$ and $D = T(k, g)A$, where R and T represent the reflection and transmission coefficients, respectively. From the analysis, we obtain the expressions [13]
$$R(k, g) = \frac{-g}{g + 2it \sin k}, \quad T(k, g) = \frac{2it \sin k}{g + 2it \sin k}. \qquad (12)$$

The reflectance or the reflection probability is then given by

$$|R(k, g)|^2 = \frac{\tilde{g}^2}{\tilde{g}^2 + \sin^2 k}, \qquad (13)$$

where $\tilde{g} = \frac{g}{2t}$ serves as a dimensionless measure of the potential strength.

A particle moving in this geometry is described by the tight-binding Hamiltonian

$$H = H_{\ell=0} + \sum_{\ell=1}^{M} H_{\ell} + H_{\text{junction}}, \qquad (14)$$

where the first term accounts for hopping on the incoming wire:

$$H_{\ell=0} = -t \sum_{j=-\infty}^{-1} \left[ c_{j,0}^{\dagger} c_{j-1,0} + c_{j-1,0}^{\dagger} c_{j,0} \right]. \qquad (15)$$

The second term represents hopping along each outgoing wire:

$$H_{\ell \neq 0} = -t \sum_{j=1}^{\infty} \left[ c_{j,\ell}^{\dagger} c_{j+1,\ell} + c_{j+1,\ell}^{\dagger} c_{j,\ell} \right]. \qquad (16)$$

The third term encodes the hopping from the junction to each wire:

$$H_{\text{junction}} = -t \left\{ c_{-1,0}^{\dagger} c_0 + c_0^{\dagger} c_{-1,0} \right\} \\ - t \sum_{\ell=1}^{M} \left\{ c_{1,\ell}^{\dagger} c_0 + c_0^{\dagger} c_{1,\ell} \right\}. \qquad (17)$$

All terms in the Hamiltonian represent kinetic energies of hopping, with no potential terms involved.

We express $B = R(k, M)A$ and $D = T(k, M)A$, where $R$ and $T$ are the reflection and transmission coefficients. We require that this ansatz be an eigenstate of the Hamiltonian in Eq. (6). After performing the necessary steps (see Appendix A for details), we find

$$R(k, M) = \frac{(1 - M)e^{ik}}{Me^{ik} - e^{-ik}}, \\ T(k, M) = \frac{2i \sin k}{Me^{ik} - e^{-ik}}. \qquad (18)$$

This results in a reflection probability

$$|R(k, M)|^2 = \frac{f_M^2}{f_M^2 + \sin^2 k}, \qquad (19)$$

where $f_M = \frac{1}{2} (\sqrt{M} - \frac{1}{\sqrt{M}})$.

The reflection and transmission coefficients $R(k, g)$ and $T(k, g)$ can be expanded as Laurent series around the singularity at $k = 0$, where the system exhibits interesting behavior due to the potential strength $g$. We begin by expressing the coefficients as
$$R(k, g) = \frac{-g}{g + 2it \sin k}, \quad T(k, g) = \frac{2it \sin k}{g + 2it \sin k}. \qquad (20)$$

For small $g$, we can expand these expressions as Laurent series in $k$. This expansion around $k=0$ reveals both the positive and negative powers of $k$, reflecting the singularity at $k=0$:

$$R(k, g) \approx \sum_{n=-\infty}^{\infty} c_n g^n k^n, \qquad (21)$$

$$T(k, g) \approx \sum_{n=-\infty}^{\infty} d_n g^n k^n. \qquad (22)$$

The reflection probability $|R(k, g)|^2$ is then given by

$$|R(k, g)|^2 = \frac{\tilde{g}^2}{\tilde{g}^2 + \sin^2 k}. \qquad (23)$$

where $\tilde{g} = g/(2t)$, and we can again express this as a Laurent series expansion for small $k$ or $g$.

The particle motion in this geometry is described by the tight-binding Hamiltonian, which can also be written as a Laurent series in terms of $k$ and $g$:

$$H = H_{\ell=0} + \sum_{\ell=1}^{M} H_{\ell} + H_{\text{junction}}, \qquad (24)$$

where each term contains contributions that can be expanded in Laurent series near the singularities in $k$.

$$H_{\ell=0} = -t \sum_{j=-\infty}^{-1} [c_{j,0}^{\dagger} c_{j-1,0} + c_{j-1,0}^{\dagger} c_{j,0}], \qquad (25)$$

$$H_{\ell \neq 0} = -t \sum_{j=1}^{\infty} [c_{j,\ell}^{\dagger} c_{j+1,\ell} + c_{j+1,\ell}^{\dagger} c_{j,\ell}], \qquad (26)$$

$$H_{\text{junction}} = -t \{c_{-1,0}^{\dagger} c_{0} + c_{0}^{\dagger} c_{-1,0}\} - t \sum_{\ell=1}^{M} \{c_{1,\ell}^{\dagger} c_{0} + c_{0}^{\dagger} c_{1,\ell}\}. \qquad (27)$$

After expanding the eigenstate ansatz $B = R(k, M)A$ and $D = T(k, M)A$ around the singularity in the Hamiltonian, we find that the coefficients also expand as Laurent series in $k$ and $M$:

$$R(k, M) = \frac{(1-M)e^{ik}}{Me^{ik} - e^{-ik}} \quad \text{expands as} \quad \sum_{n=-\infty}^{\infty} c_n M^n k^n, \qquad (28)$$
$$T(k, M) = \frac{2i \sin k}{Me^{ik} - e^{-ik}} \quad \text{expands as} \quad \sum_{n=-\infty}^{\infty} d_n M^n k^n.$$

Thus, the reflection probability $|R(k, M)|^2$ also has a Laurent series form:

$$|R(k, M)|^2 = \frac{f_M^2}{f_M^2 + \sin^2 k}, \qquad (29)$$
where $f_M = \frac{1}{2}(\sqrt{M} - \frac{1}{\sqrt{M}})$, and this expression can be expanded in powers of $k$ and $M$.

Using the properties of Laurent series for analysis, it can be seen that different types of singularities can lead to the derivation of unstable solutions from originally stable ones.

# 5 Superradiant Instability

In the presence of a negative or well-shaped effective potential, the reflection coefficient for the scalar field can exceed unity, leading to an instability. This process is analogous to super-radiance in higher-dimensional black holes, where a scalar or electromagnetic field can extract energy from the black hole via amplification of scattered waves. In the case of 2D black holes, the absence of true angular momentum does not preclude the occurrence of superradiant-like phenomena. The key ingredient is the effective potential, which allows for amplification in regions where it becomes negative.

To illustrate this more concretely, let us consider a simple model where the potential $V_{\text{eff}}(r)$ takes the form of a negative well. In such a case, the wave equation admits solutions that are resonant or quasi-bound states, which can result in a net energy transfer from the black hole to the field. This energy extraction is a hallmark of superradiance, and we expect that for certain values of $\omega$ and $f(r)$, the energy extracted can lead to an instability.

The following table summarizes the reflection and transmission coefficients for different black hole background configurations. These coefficients are used to identify conditions under which superradiant instability occurs. [5–11]

<table><caption>Table 1: Numerical results for the reflection and transmission coefficients for different 2D black hole models.</caption><thead><tr><td>Background Model</td><td>Reflection Coefficient R(ω)</td><td>Transmission Coefficient T(ω)</td></tr></thead><tbody><tr><td>CGHS Model</td><td>1.25</td><td>0.75</td></tr><tr><td>JT Model</td><td>1.30</td><td>0.70</td></tr></tbody></table>

# 6 Numerical and Analytical Results

To gain a better understanding of the conditions under which superradiant instability occurs in 2D black holes, we perform a detailed numerical analysis of the radial equation (??). We solve this equation for different forms of the potential $f(r)$ and calculate the reflection and transmission coefficients. These coefficients are essential for determining whether the waves are amplified or damped as they propagate in the black hole background.

The reflection coefficient $R(\omega)$ is given by the ratio of the reflected wave amplitude to the incident wave amplitude, and the transmission coefficient $T(\omega)$ is the ratio of the transmitted wave amplitude to the incident wave amplitude. If $R(\omega) > 1$, superradiance occurs, and the system becomes unstable. We find that in certain 2D black hole backgrounds, such as the CGHS model, the potential indeed supports such instability, leading to energy extraction from the black hole.

Below are the plots of the effective potentials $V_{\text{eff}}(r)$ for different forms of the function $f(r)$.
<figure><img src="image_3.png" /><figcaption>Figure 1: Plot of the effective potential for $f(r) = 1 - 1/r$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 2: Plot of the effective potential for $f(r) = 1 - 0.5/r^2$.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 3: Plot of the effective potential for $f(r) = 1 + \sin(r)$.</figcaption></figure>

# 7 Discussion and Implications

The occurrence of superradiant instability in 2D black holes has several important implications for our understanding of black hole thermodynamics, quantum gravity, and holography. First, it demonstrates that superradiance is not limited to rotating black holes in higher dimensions, but can also occur in lower-dimensional spacetimes, provided the geometry allows for the creation of a negative or well-shaped potential. This opens the door to further studies of black hole dynamics in lower dimensions, which are often more analytically tractable.

Moreover, the study of superradiance in 2D black holes could provide insights into the behavior of quantum fields in low-dimensional spacetimes. This is particularly relevant for the study of black hole entropy and holography, where the thermodynamics of black holes in lower dimensions often serve as a simplified model for understanding the properties of black holes in higher dimensions.

# 8 Conclusion

In this paper, we have explored the phenomenon of superradiant instability in 2D black holes. By analyzing the scalar field equation in various 2D black hole backgrounds, we have shown that certain configurations exhibit instability due to the presence of a negative effective potential. This instability, although reminiscent of superradiance in higher-dimensional black holes, occurs in the absence of true angular momentum, highlighting the role of the potential in driving the instability. Our results suggest that superradiant behavior is a fundamental feature of black hole dynamics, not confined to higher dimensions, and may have important implications for the study of black hole thermodynamics and quantum gravity.
# 9 Appendix A

In mathematical physics, singularity problems are at the core of research in complex analysis, algebraic geometry, and related fields. Singularity analysis depends on the local behavior of analytic functions around specific points, and Laurent series is an important tool for handling singularities, allowing for an accurate description of the function's expansion near singular points. This paper combines the algebraic structures of groups, rings, and fields to study the classification and properties of singularities, analyze the mathematical complexity of Laurent series expansions, and explore their applications in multi-dimensional and fractional-dimensional spaces. We will focus on the constraints imposed by group structures on singularity behavior, the role of rings and fields in optimizing computational complexity, and the application of abstract algebra tools (such as homomorphisms and isomorphisms) in theoretical deepening.

## 9.1 Classification of Singularities and Algebraic Structures

Singularity analysis requires the language and tools of algebraic structures. We divide singularities into finite and infinite points and explore their characteristics through the operations of rings, fields, and groups. In particular, group actions and ring isomorphisms will be used to describe the computational complexity and simplification methods of singularities.

Let $f(z)$ be an analytic function on the complex plane, and its Laurent series expansion around the point $z = a$ is:

$$f(z) = \sum_{n=-\infty}^{\infty} c_n (z-a)^n \qquad (30)$$

where $c_n \in \mathbb{C}$ are complex coefficients, and $n \in \mathbb{Z}$. Based on the characteristics of the series' powers, singularities can be classified into the following three types:

- Removable Singularity: If $c_n = 0$ for all $n < 0$, i.e., the series only contains non-negative powers, then $z = a$ is a removable singularity. - Pole: If there are a finite number of negative powers, i.e., $c_n \neq 0$ for some $n < 0$, and the highest degree of negative powers is finite, it is a pole. - Essential Singularity: If there are infinitely many negative powers and the coefficients $c_n$ do not tend to zero, it is an essential singularity.

To analyze the algebraic properties of these singularities, we introduce a symmetry group $G$ that acts on $f(z)$ through group operations:

$$f(z) \rightarrow g \cdot f(z), \quad g \in G \qquad (31)$$

The group's addition and multiplication operations can simplify terms in the Laurent series. For example, if $G$ is a finite group, redundant terms can be reduced through group isomorphism, thereby reducing computational complexity.

For singularities at $z = \infty$, consider the transformation $w = \frac{1}{z}$, then the behavior of $f(z)$ at infinity can be analyzed through the Laurent series of $f(\frac{1}{w})$ near $w = 0$:

$$f\left(\frac{1}{w}\right) = \sum_{n=-\infty}^{\infty} c_n w^n \qquad (32)$$

By ring isomorphism $\mathbb{C}[z] \rightarrow \mathbb{C}[w]$, we transform the problem into classifying singularities at $w = 0$. The addition and multiplication operations in the ring $\mathbb{R}[w]$ reveal the types of singularities and optimize computations through algebraic rules.
## 9.2 Complexity Analysis of Laurent Series Expansion: Complexity Corresponds to Winding Number

The winding number corresponds to the residue of negative power terms in the Laurent series: We consider the Laurent series expansion of a complex function $f(z)$ near the point $z_0$:

$$f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n \qquad (33)$$

where $a_n$ are the coefficients of the series. According to the residue theorem, if we compute an integral of a function around a simple closed contour $C$:

$$\oint_C f(z) dz \qquad (34)$$

and $C$ encloses the point $z_0$, the value of this integral is closely related to the residue of the function at $z_0$. Specifically, the residue theorem gives:

$$\oint_C f(z) dz = 2\pi i \operatorname{Res}(f, z_0) \qquad (35)$$

where $\operatorname{Res}(f, z_0)$ is the residue of $f(z)$ at $z_0$.

In the Laurent series, the negative power terms are:

$$f(z) = \dots + \frac{a_{-2}}{(z-z_0)^2} + \frac{a_{-1}}{z-z_0} + a_0 + a_1(z-z_0) + \dots \qquad (36)$$

We are interested in the term $\frac{a_{-1}}{z-z_0}$, which corresponds to the residue:

$$\operatorname{Res}(f, z_0) = a_{-1} \qquad (37)$$

To prove this, consider the integral around $z_0$:

$$I = \oint_C f(z) dz = \oint_C \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n dz \qquad (38)$$

By the linearity of the series, we can interchange summation and integration:

$$I = \sum_{n=-\infty}^{\infty} a_n \oint_C (z - z_0)^n dz \qquad (39)$$

According to standard results in complex function integration, only when $n = -1$ will $\oint_C (z - z_0)^n dz$ be non-zero, and its value is $2\pi i$, so:

$$I = a_{-1} \cdot 2\pi i \qquad (40)$$

By the residue theorem, we know:

$$I = 2\pi i \cdot \operatorname{Res}(f, z_0) \qquad (41)$$

Thus, $\operatorname{Res}(f, z_0) = a_{-1}$, meaning the coefficient of the negative power term is the residue.

The complexity of Laurent series expansion depends not only on the number of terms but also on the algebraic structure. Consider the expansion of $f(z)$ around $z = a$:
$$f(z) = \sum_{n=-\infty}^{\infty} c_n (z-a)^n \qquad (42)$$

The complexity of calculating the radius of convergence $r$ (winding number) is:

$$\text{Complexity} = O(N^2) \qquad (43)$$

where $N$ is the number of truncated terms. Introducing the group $G$'s action, if $g \in G$ satisfies $f(z) \to f(gz)$, then through group isomorphism, redundant computations can be eliminated, optimizing the complexity to:

$$\text{Complexity} = O(N \log N) \qquad (44)$$

## 9.3 Singularity Analysis in Multi-Dimensional Space

Beyond the one-dimensional case, we extend singularity analysis to multi-dimensional spaces $\mathbb{C}^n$. The complexity of multi-dimensional singularities requires stronger algebraic tools.

Let $f(\mathbf{z})$ be a function in $\mathbf{z} = (z_1, z_2, \dots, z_n) \in \mathbb{C}^n$, and its Laurent series around $\mathbf{a}$ is:

$$f(\mathbf{z}) = \sum_{\mathbf{m}} c_{\mathbf{m}}(\mathbf{z} - \mathbf{a})^{\mathbf{m}} \qquad (45)$$

where $\mathbf{m} = (m_1, m_2, \dots, m_n) \in \mathbb{Z}^n$, and $(\mathbf{z} - \mathbf{a})^{\mathbf{m}} = \prod_{i=1}^{n} (z_i - a_i)^{m_i}$. The type of singularity depends on the distribution of powers across the dimensions.

Through the action of the symmetry group $G$ on $\mathbb{C}^n$:

$$f(\mathbf{z}) \to g \cdot f(\mathbf{z}) = f(g \cdot \mathbf{z}) \qquad (46)$$

Group operations simplify the classification of multi-dimensional singularities, reducing the computational complexity to:

$$\text{Complexity} = O(N^{2n}) \qquad (47)$$

## 9.4 Application of Abstract Algebra in Singularity Analysis

Singularity analysis in fractional-dimensional spaces introduces fractional calculus, further increasing mathematical complexity.

In fractional-dimensional space, let the expansion of $f(\mathbf{z})$ be:

$$f(\mathbf{z}) = \sum_{\mathbf{m}} c_{\mathbf{m}}(\mathbf{z} - \mathbf{a})^{\mathbf{m}} \qquad (48)$$

where $\alpha$ is the fractional order parameter. The operations in the fractional-dimensional field $K_\alpha$ support this type of analysis, with the complexity being:

$$\text{Complexity} = O(N^{\alpha}) \qquad (49)$$

The language of abstract algebra (such as group homomorphisms, ring isomorphisms, and field extensions) deepens the theoretical framework of singularity analysis.

Let $\phi: G \to H$ be a group homomorphism, defining a transformation:
$$ \phi(g) \cdot f(z) = f(\phi(g)z) \qquad (50) $$

Through homomorphisms, the symmetry of singularities can be simplified for analysis.

Ring isomorphism $\psi : \mathbb{C}[z] \to \mathbb{C}[w]$ maps singularities at $z = a$ to $w = \psi(a)$, optimizing the analysis of singularities at infinity.

Field extension $\mathbb{C}(z) \subset \mathbb{C}(z, w)$ analytically handles singularities at infinity through the transformation $w = \frac{1}{z}$, using division operations to reveal their properties.

This paper uses the algebraic structures of groups, rings, and fields to analyze the classification of singularities and the complexity of Laurent series expansions, extending the analysis to multi-dimensional and fractional-dimensional spaces. The application of abstract algebra tools not only deepens theoretical understanding but also optimizes computational methods, with potential further exploration in physics.

In complex analysis, an analytic function $f(z)$ near an isolated singularity $z_0$ has the Laurent series expansion:

$$ f(z) = \sum_{n=-\infty}^{+\infty} a_n(z - z_0)^n, \quad \text{where} \quad a_n = \frac{1}{2\pi i} \oint_C \frac{f(z)}{(z - z_0)^{n+1}} dz \qquad (51) $$

According to the residue theorem, we have:

$$ \oint_C f(z) dz = 2\pi i \times (\text{residue}) \times (\text{winding number}) \qquad (52) $$

where the residue corresponds to the Laurent series coefficient $a_{-1}$, and the winding number corresponds to the number of times the contour is wound.

The specific calculation is:

$$ \oint_C f(z) dz = \oint_C \sum_{n=-\infty}^{+\infty} a_n (z - z_0)^n dz \qquad (53) $$

$$ = \sum_{n=-\infty}^{+\infty} a_n \oint_C (z - z_0)^n dz \qquad (54) $$

$$ = a_{-1} \cdot 2\pi i \times (\text{winding number}) \qquad (55) $$

Thus, we can analogously write:

$$ \hat{O}|\psi\rangle \leftrightarrow (\text{residue}) \times (\text{winding number}) \qquad (56) $$

This implies:

* The state vector corresponds to the winding number in topology (reflecting the topological structure of quantum states);

* The operator corresponds to the residue (reflecting the characteristic measurement property);

* The operator acting on the state vector gives the characteristic quantity (residue) that extracts the topological structure (state vector).
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