Duan’s φ-Mapping Topological Current Method

• Duan’s Topological Current Method is a distributional framework that maps the zeros of an order-parameter field to quantized topological defects in black hole systems.
• It employs a δ-function decomposition and Jacobian vector analysis to determine local winding numbers and track defect trajectories in both static and dynamic settings.
• The method distinguishes between standard and exotic configurations—such as photon spheres with negative versus positive charges—shedding light on phase transitions and stability.

to=arxiv_search 彩神争霸有.json {"query":"Duan phi-mapping topological current black hole thermodynamics photon sphere", "max_results": 10, "sort_by": "relevance"} to=arxiv_search 彩彩票娱乐.json {"query":"(Wei, 2020) OR (Fairoos, 2023) OR (Fairoos et al., 2023) OR (Du et al., 2023) OR (Du et al., 2023) OR (Sadeghi et al., 2023) OR (Ali et al., 2023) OR (Du et al., 2023) OR (Sadeghi et al., 2023) OR (Rehan et al., 25 Mar 2026)", "max_results": 20, "sort_by": "relevance"} Duan’s Topological Current Method, also known as Duan’s $\phi$-mapping theory, is a distributional framework in which topological defects are identified with the zero set of an order-parameter field $\phi$. In contemporary black-hole research, the method is used to treat photon spheres, light rings, thermodynamic branches, and critical points as defects in an auxiliary parameter space, so that each isolated zero carries a quantized local charge and the sum of charges defines a global topological class (Wei, 2020, Fairoos, 2023, Fairoos et al., 2023).

1. Formal definition of the $\phi$-mapping current

In the two-component formulation most often used in black-hole applications, one introduces a field

$\phi(x)=(\phi^1,\phi^2)$

on a $(2+1)$-dimensional manifold with coordinates $x^\mu=(t,x^1,x^2)$, together with the normalized field

$$n^a=\frac{\phi^a}{\|\phi\|},\qquad \|\phi\|=\sqrt{\phi^a\phi^a}.$$

The associated topological current is

$$j^\mu=\frac{1}{2\pi}\,\epsilon^{\mu\nu\rho}\,\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$

and it satisfies the conservation law

$\partial_\mu j^\mu=0.$

A superpotential representation,

$$V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}n^a\partial_\rho n^b,$$

gives the same current through $\phi$0 (Wei, 2020).

The characteristic feature of Duan’s construction is its $\phi$1-function decomposition. Writing the current in terms of the Jacobian vector,

$\phi$2

one obtains

$\phi$3

The current therefore has support only at the zeros of $\phi$4. In this sense, the method provides a clean, distributional way to count and classify topological defects as the zeros of a vector field (Wei, 2020).

A standard charge on a two-dimensional domain $\phi$5 is defined by

$\phi$6

For isolated, regular zeros, the inner structure of the current becomes

$\phi$7

so that

$\phi$8

Here $\phi$9 is the Hopf index and $\phi$0 is the Brouwer degree (Wei, 2020). In the broader $\phi$1-component formulation, the same structure is written as

$\phi$2

with the corresponding Jacobian vector defined by antisymmetric derivatives of the $\phi$3 components of $\phi$4 (Fairoos et al., 2023).

2. Local indices, winding numbers, and bifurcation structure

The local topological charge of an isolated zero is the product

$\phi$5

For simple zeros in many black-hole applications, $\phi$6, so the sign of the Jacobian alone determines the local winding number. Equivalent contour formulas are also widely used: $\phi$7 with $\phi$8 (Wei, 2020).

The method is not restricted to static counting. In Duan’s kinematics, regular zeros define defect trajectories with velocity

$\phi$9

or, in spatial form,

$\phi(x)=(\phi^1,\phi^2)$0

This is the basis for treating black-hole branches as moving defects and certain non-propagating branches as static defects in thermodynamic parameter space (Fairoos, 2023).

When the Jacobian degenerates, Duan’s bifurcation theory predicts branch creation, annihilation, splitting, merging, or velocity exchange. In one frequently used criterion, a bifurcation point occurs when

$\phi(x)=(\phi^1,\phi^2)$1

Near such points, the local defect velocities are determined by a quadratic or cubic expansion, for example

$\phi(x)=(\phi^1,\phi^2)$2

or

$\phi(x)=(\phi^1,\phi^2)$3

In the photon-sphere setting, an equivalent local statement is that generation or annihilation occurs when $\phi(x)=(\phi^1,\phi^2)$4 vanishes at a critical point while another component remains nonzero; the conserved current enforces that the created or annihilated pair has opposite charges (Wei, 2020, Fairoos, 2023).

3. Construction of the order parameter in black-hole applications

The method is application-dependent through the choice of $\phi(x)=(\phi^1,\phi^2)$5. In photon-sphere problems for static, spherically symmetric spacetimes with

$\phi(x)=(\phi^1,\phi^2)$6

one introduces

$\phi(x)=(\phi^1,\phi^2)$7

and defines

$\phi(x)=(\phi^1,\phi^2)$8

The zeros of this field coincide with photon spheres, because the circular null-geodesic condition

$\phi(x)=(\phi^1,\phi^2)$9

is precisely the radial zero condition for the constructed vector field (Wei, 2020).

In thermodynamic topology, the auxiliary space is usually chosen as $(2+1)$0 or $(2+1)$1. One common construction defines

$(2+1)$2

where $(2+1)$3 serves as the order parameter and $(2+1)$4 is an auxiliary angular parameter (Fairoos, 2023). A closely related off-shell free-energy formulation uses

$(2+1)$5

so that zeros of $(2+1)$6 coincide with on-shell black-hole branches (Fairoos et al., 2023). A temperature-based variant introduces

$(2+1)$7

after eliminating one thermodynamic variable through a criticality condition (Alipour et al., 2023).

Across these constructions, the auxiliary angular component is chosen so that it vanishes at $(2+1)$8 or $(2+1)$9 and diverges at the boundaries. This enforces isolated zeros and supplies orientation data for the winding-number calculation. In the thermodynamic analyses of Einstein-Gauss-Bonnet black holes, the auxiliary angle $x^\mu=(t,x^1,x^2)$0 helps construct a two-component $x^\mu=(t,x^1,x^2)$1-field, and the physical conclusions do not depend on $x^\mu=(t,x^1,x^2)$2 beyond ensuring proper topological mapping (Fairoos, 2023).

4. Photon spheres, light rings, and topological charge

The photon-sphere application provides one of the clearest realizations of Duan’s method. For static, spherically symmetric black holes, the topological current is supported only at the zero points of the vector field that determines the photon-sphere location, so each photon sphere is assigned a topological charge. Considering the full exterior region, the total topological charge always equals $x^\mu=(t,x^1,x^2)$3 for asymptotically flat, asymptotically AdS, and asymptotically dS black holes (Wei, 2020).

The local sign of the Brouwer degree correlates with dynamical stability. In this construction, standard photon spheres, defined by $x^\mu=(t,x^1,x^2)$4, carry $x^\mu=(t,x^1,x^2)$5, whereas exotic photon spheres, defined by $x^\mu=(t,x^1,x^2)$6, carry $x^\mu=(t,x^1,x^2)$7 (Wei, 2020). The invariant total charge therefore enforces the existence of at least one standard photon sphere outside the horizon.

The method also distinguishes black holes from naked singularities. In the same setup, naked singularities have vanishing total topological charge,

$x^\mu=(t,x^1,x^2)$8

which places them in a different topological class from black holes (Wei, 2020). In the Reissner–Nordström limit discussed there, over-charging removes the horizons, the inner exotic photon sphere emerges, the total charge becomes $x^\mu=(t,x^1,x^2)$9, and the two photon spheres annihilate at

$$n^a=\frac{\phi^a}{\|\phi\|},\qquad \|\phi\|=\sqrt{\phi^a\phi^a}.$$0

in agreement with Duan’s annihilation criterion (Wei, 2020).

A later hyperscaling-violating analysis retained the same black-hole class $$n^a=\frac{\phi^a}{\|\phi\|},\qquad \|\phi\|=\sqrt{\phi^a\phi^a}.$$1 for photon spheres and $$n^a=\frac{\phi^a}{\|\phi\|},\qquad \|\phi\|=\sqrt{\phi^a\phi^a}.$$2 for naked singularities, but also proposed a new topological class for naked singularities, $$n^a=\frac{\phi^a}{\|\phi\|},\qquad \|\phi\|=\sqrt{\phi^a\phi^a}.$$3, with respect to $$n^a=\frac{\phi^a}{\|\phi\|},\qquad \|\phi\|=\sqrt{\phi^a\phi^a}.$$4. It further reported that for $$n^a=\frac{\phi^a}{\|\phi\|},\qquad \|\phi\|=\sqrt{\phi^a\phi^a}.$$5 the system either shows a naked singularity form with total topological charge $$n^a=\frac{\phi^a}{\|\phi\|},\qquad \|\phi\|=\sqrt{\phi^a\phi^a}.$$6 or has no solution, so that black-hole solutions occur only in $$n^a=\frac{\phi^a}{\|\phi\|},\qquad \|\phi\|=\sqrt{\phi^a\phi^a}.$$7 (Sadeghi et al., 2023).

5. Thermodynamic defects and topological classes of black holes

In thermodynamic applications, zeros of $$n^a=\frac{\phi^a}{\|\phi\|},\qquad \|\phi\|=\sqrt{\phi^a\phi^a}.$$8 are interpreted as thermodynamic defects. Depending on the construction, they correspond to black-hole branches, critical points, or off-shell extrema. In the Einstein-Gauss-Bonnet analysis, moving defects with nonzero velocity are black holes, while static defects with zero velocity are exotic defects, and a first-order small/large black-hole transition is interpreted as an interchange of winding numbers between a moving defect and a static exotic defect at bifurcation points (Fairoos, 2023).

Representative topological classes reported in the recent literature are summarized below.

System	Reported topological class	Source
Static, spherically symmetric black-hole photon spheres	$$n^a=\frac{\phi^a}{\|\phi\|},\qquad \|\phi\|=\sqrt{\phi^a\phi^a}.$$9; naked singularities have $$j^\mu=\frac{1}{2\pi}\,\epsilon^{\mu\nu\rho}\,\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$0	(Wei, 2020)
$$j^\mu=\frac{1}{2\pi}\,\epsilon^{\mu\nu\rho}\,\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$1 EGB-AdS / neutral EGB-AdS in $$j^\mu=\frac{1}{2\pi}\,\epsilon^{\mu\nu\rho}\,\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$2	$$j^\mu=\frac{1}{2\pi}\,\epsilon^{\mu\nu\rho}\,\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$3 / $$j^\mu=\frac{1}{2\pi}\,\epsilon^{\mu\nu\rho}\,\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$4	(Fairoos, 2023)
4D dRGT massive gravity, neutral / charged	$$j^\mu=\frac{1}{2\pi}\,\epsilon^{\mu\nu\rho}\,\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$5 / $$j^\mu=\frac{1}{2\pi}\,\epsilon^{\mu\nu\rho}\,\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$6	(Fairoos et al., 2023)
BTZ, $$j^\mu=\frac{1}{2\pi}\,\epsilon^{\mu\nu\rho}\,\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$7 / rotating or charged	$$j^\mu=\frac{1}{2\pi}\,\epsilon^{\mu\nu\rho}\,\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$8 / $$j^\mu=\frac{1}{2\pi}\,\epsilon^{\mu\nu\rho}\,\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$9	(Du et al., 2023)
Bardeen families studied there	$\partial_\mu j^\mu=0.$0	(Sadeghi et al., 2023)
Kerr–Sen AdS / GMGHS AdS / asymptotically flat Kerr–Sen	$\partial_\mu j^\mu=0.$1 / $\partial_\mu j^\mu=0.$2 / $\partial_\mu j^\mu=0.$3	(Rehan et al., 25 Mar 2026)

These examples show that the same formal current can classify very different phase structures. The $\partial_\mu j^\mu=0.$4 Einstein-Gauss-Bonnet case has $\partial_\mu j^\mu=0.$5 and exhibits a first-order small/large transition similar to Reissner–Nordström black holes in AdS space, whereas neutral Einstein-Gauss-Bonnet black holes in $\partial_\mu j^\mu=0.$6 have $\partial_\mu j^\mu=0.$7 and show a transition between unstable small and large stable black-hole phases (Fairoos, 2023). In dRGT massive gravity, the uncharged four-dimensional black hole has topological number $\partial_\mu j^\mu=0.$8, while the charged four-dimensional case has the same topological structure as the AdS–RN black hole, and in five dimensions the sign of $\partial_\mu j^\mu=0.$9 changes the class (Fairoos et al., 2023).

The method also reveals dimension- and horizon-dependence. BTZ black holes furnish only two topological classes: $$V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}n^a\partial_\rho n^b,$$0 for $$V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}n^a\partial_\rho n^b,$$1, and $$V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}n^a\partial_\rho n^b,$$2 for rotating or charged cases (Du et al., 2023). In de Sitter spacetime, the event horizon and the cosmological horizon can belong to different classes: for Schwarzschild–dS, the event horizon gives $$V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}n^a\partial_\rho n^b,$$3 while the cosmological horizon gives $$V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}n^a\partial_\rho n^b,$$4, whereas rotating or charged cases yield $$V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}n^a\partial_\rho n^b,$$5 in both sectors (Du et al., 2023). Conversely, in the Bardeen systems surveyed there, the topological classification stays fixed at $$V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}n^a\partial_\rho n^b,$$6, and the addition of massive-gravity, quintessence, or 4D Einstein–Gauss–Bonnet terms does not change the topological number (Sadeghi et al., 2023).

6. Alternative formulations, implications, and limitations

Several works have shown that Duan’s current admits equivalent analytic reformulations. For Einstein–power–Yang–Mills AdS black holes, the order parameter can be taken as

$$V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}n^a\partial_\rho n^b,$$7

after analytic continuation of the horizon coordinate into the complex plane, and the total topological number is then obtained through the argument principle,

$$V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}n^a\partial_\rho n^b,$$8

In that setting, the winding number from complex analysis coincides with the charge computed from Duan’s $$V^{\mu\nu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}n^a\partial_\rho n^b,$$9-mapping current (Du et al., 2023). A related complex residue method was later applied to Kerr–Sen AdS black holes, where residues of a characterized complex function directly encode the winding numbers and reproduce the same total charge found from the vector-field analysis (Rehan et al., 25 Mar 2026).

The method has also been used to analyze parametric changes in thermodynamic topology. In Born–Infeld–AdS black holes, the topological nature has a possible breakdown in certain parametric ranges, and the unconventional and the conventional phase critical points are identified as the creation and annihilation points of vortex–anti-vortex pairs (Ali et al., 2023). The same work distinguishes an AdS–Schwarzschild class with net topological charge $\phi$00 from an AdS–Reissner–Nordström class with net topological charge $\phi$01, and extends the classification to the dual CFT by using the charge per degree of freedom (Ali et al., 2023).

A later dynamical application classified thermodynamic critical points by $\phi$02 or $\phi$03 and then computed quasinormal modes of massless scalar perturbations near those points. It found that both the oscillation frequency and damping rate increase with the black hole radius at the critical temperature, and that the $\phi$04 case exhibits very similar dynamical characteristics across Reissner–Nordström–AdS, Born–Infeld–AdS, and quantum anomalous black holes (Chu et al., 22 Oct 2025). This suggests a nontrivial connection between thermodynamic topology and dynamics.

The method is nevertheless constrained by its construction. Existing thermodynamic studies typically assume a smooth $\phi$05-field derived from a well-defined off-shell thermodynamic potential, a canonical ensemble at fixed temperature, and fixed $\phi$06 when charge is present (Fairoos, 2023). Some analyses also require explicit pressure bounds for exotic defects to exist (Fairoos, 2023). Moreover, topological class and phase structure are not in one-to-one correspondence. One study of three black-hole models found that a five-dimensional Yang–Mills black hole in massive gravity can have different topological classes while exhibiting the same phase structure, and it explicitly noted that identifying conventional or novel critical points via $\phi$07 does not provide a sufficient condition for first-order transitions when there are multiple critical points (Alipour et al., 2023). This suggests that Duan’s method is best understood as a global classification scheme that complements, rather than replaces, standard free-energy and stability analyses.