Duan’s φ-Mapping Theory in Thermodynamics

• Duan’s φ-mapping theory is a topological-current formalism that converts the zero sets of two-component fields into localized, conserved currents representing topological defects.
• It classifies black hole thermodynamic branches, stability, and bifurcation structures by incorporating local indices like the Hopf index and Brouwer degree into winding numbers.
• The theory adapts to various systems via off-shell potentials and temperature-based criticality models, revealing universal yet dimension-dependent topological classes.

Duan’s $\phi$-mapping theory is a topological-current formalism in which the zero set of a two-component field $\phi^a$ is converted into a conserved current supported on localized defects. In recent black-hole thermodynamics, the framework is used to regard black holes, horizon sectors, and thermodynamic critical points as topological defects in an auxiliary parameter space constructed from an off-shell thermodynamic potential or from temperature-based criticality data. The local data at each zero—the Hopf index $\beta_i$, Brouwer degree $\eta_i$, and winding number $w_i=\beta_i\eta_i$—combine into an integer-valued global topological number, which is then used to classify thermodynamic branches, stability, and bifurcation structure (Du et al., 2023, Du et al., 2023, Hazarika et al., 2024).

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

The common mathematical core in these applications begins with a two-component field $\phi^a$ and its normalized unit vector

$n^a=\frac{\phi^a}{\|\phi\|}, \qquad a=1,2.$

Duan’s topological current is then written as

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

with $\mu,\nu,\rho=0,1,2$. The current is conserved,

$\phi^a$0

and, by the $\phi^a$1-mapping identity, it can be rewritten as

$\phi^a$2

so it is nonzero only at the zero points of $\phi^a$3 (Du et al., 2023, Sadeghi et al., 2023, Du et al., 2023).

The local charge content of the theory is expressed by the decomposition

$\phi^a$4

where $\phi^a$5 are isolated zeros, $\phi^a$6 is the Hopf index, and

$\phi^a$7

is the Brouwer degree. Their product

$\phi^a$8

is the winding number of the $\phi^a$9-th zero. The integrated topological number is therefore

$\beta_i$0

Across the literature summarized here, the integrated charge is denoted by $\beta_i$1, $\beta_i$2, $\beta_i$3, or $\beta_i$4, but its role is the same: it is the signed sum of local winding numbers (Du et al., 2023, Ali et al., 2023, Yerra et al., 2022).

A recurrent point is that the Jacobian is structurally decisive. For a simple isolated zero, $\beta_i$5; when $\beta_i$6, the defect can bifurcate, and the topology of the thermodynamic solution set can change (Du et al., 2023, Fairoos, 2023).

2. Thermodynamic embeddings of the theory

In black-hole thermodynamics, the most common embedding uses a generalized off-shell free energy. A standard choice is

$\beta_i$7

where $\beta_i$8 is an independent Euclidean-time parameter and the on-shell condition is recovered when $\beta_i$9 (Sadeghi et al., 2023, Fairoos et al., 2023). The associated field is typically taken as

$\eta_i$0

or equivalently with $\eta_i$1 and $\eta_i$2. The angular component is chosen so that it diverges at $\eta_i$3, forces the vector field outward at the boundary, and localizes zero points at

$\eta_i$4

The physical zeros are therefore obtained from

$\eta_i$5

and the first equation yields the on-shell branch $\eta_i$6 (Du et al., 2023, Sadeghi et al., 2023, Du et al., 2023).

Several papers use application-specific variants of the off-shell potential. For rotating BTZ black holes,

$\eta_i$7

while for charged BTZ black holes

$\eta_i$8

for de-Sitter spacetimes, separate generalized free energies are introduced for the event horizon and cosmological horizon,

$\eta_i$9

These variants alter the explicit zero loci but not the topological mechanism: the zeros still encode equilibrium black-hole solutions (Du et al., 2023, Du et al., 2023).

A second construction, used for thermodynamic criticality rather than branch classification, starts from the temperature. In six-dimensional charged Gauss–Bonnet AdS black holes and in Born-Infeld AdS black holes, the auxiliary scalar

$w_i=\beta_i\eta_i$0

or

$w_i=\beta_i\eta_i$1

is introduced, and one defines

$w_i=\beta_i\eta_i$2

or

$w_i=\beta_i\eta_i$3

In this formulation, thermodynamic critical points are the zero points of $w_i=\beta_i\eta_i$4 (Yerra et al., 2022, Ali et al., 2023).

A third variant provides a single vector field for two distinct phase transitions. The proposed common field is

$w_i=\beta_i\eta_i$5

or, in horizon-radius form,

$w_i=\beta_i\eta_i$6

Because

$w_i=\beta_i\eta_i$7

its zeros occur either at $w_i=\beta_i\eta_i$8 or at $w_i=\beta_i\eta_i$9, which are identified respectively with the Hawking-Page and Davies points (Hazarika et al., 2024).

3. Topological charge, stability, and classification

The global topological number is not a count of zero points; it is the signed sum of their winding numbers. This point is explicit in the BTZ analysis, where two zero points can still give $\phi$0 if their winding numbers are $\phi$1 and $\phi$2 (Du et al., 2023). In multiple black-hole applications, the local sign is also assigned a thermodynamic meaning: $\phi$3 corresponds to a locally stable branch, while $\phi$4 corresponds to a locally unstable branch (Fairoos et al., 2023, Fairoos, 2023).

The following representative results illustrate how the same formalism yields different global classes in different systems.

System	Zero-point structure	Global number
BTZ, $\phi$5	one zero, $\phi$6	$\phi$7
BTZ, rotating or charged	two zeros, $\phi$8, $\phi$9	$\phi^a$0
Schwarzschild-dS event horizon	one zero, $\phi^a$1	$\phi^a$2
Schwarzschild-dS cosmological horizon	one zero, $\phi^a$3	$\phi^a$4
Bardeen families studied	one zero, positive winding	$\phi^a$5
4D dRGT massive gravity	neutral: two zeros; charged: three zeros	$\phi^a$6; $\phi^a$7

For BTZ black holes, the uncharged nonrotating case has only one zero point and hence $\phi^a$8, while rotating or charged cases always have two zero points with opposite winding numbers and therefore $\phi^a$9 (Du et al., 2023). For four-dimensional de-Sitter black holes, the event horizon and cosmological horizon are topologically distinct thermodynamic systems: the Schwarzschild-dS event horizon has $n^a=\frac{\phi^a}{\|\phi\|}, \qquad a=1,2.$0, the Schwarzschild-dS cosmological horizon has $n^a=\frac{\phi^a}{\|\phi\|}, \qquad a=1,2.$1, and rotating or charged cases have $n^a=\frac{\phi^a}{\|\phi\|}, \qquad a=1,2.$2 because two defects cancel (Du et al., 2023).

The Bardeen family behaves differently. Regular Bardeen AdS black holes, Bardeen AdS black holes with quintessence, Bardeen black holes in massive gravity, and Bardeen black holes in 4D Einstein-Gauss-Bonnet gravity all exhibit one zero point with positive winding number and therefore the same topological classification,

$n^a=\frac{\phi^a}{\|\phi\|}, \qquad a=1,2.$3

In the examples plotted there, the number of on-shell black holes is one for a fixed $n^a=\frac{\phi^a}{\|\phi\|}, \qquad a=1,2.$4, $n^a=\frac{\phi^a}{\|\phi\|}, \qquad a=1,2.$5 decreases monotonically with $n^a=\frac{\phi^a}{\|\phi\|}, \qquad a=1,2.$6, and no phase transition appears (Sadeghi et al., 2023).

4. Dependence on dimension, horizon sector, and theory deformation

A major use of Duan’s $n^a=\frac{\phi^a}{\|\phi\|}, \qquad a=1,2.$7-mapping theory in this literature is to test whether topological classification is universal under changes of dimension, horizon type, or gravitational sector. The answer is mixed rather than uniform.

The de-Sitter analysis supports a three-class scheme with

$n^a=\frac{\phi^a}{\|\phi\|}, \qquad a=1,2.$8

and further shows that horizon type matters: the same spacetime can support different topological classes depending on whether the event horizon or the cosmological horizon is treated as the thermodynamic system (Du et al., 2023). By contrast, the BTZ study finds only two topological classes: the nonrotating uncharged BTZ black hole has $n^a=\frac{\phi^a}{\|\phi\|}, \qquad a=1,2.$9, while rotating or charged BTZ black holes have $$j^\mu=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$0. This was presented explicitly as a difference from the earlier higher-dimensional pattern and therefore as evidence that the topological classification is dimension-dependent (Du et al., 2023).

The response to theory deformations is also nonuniform. In the Bardeen sector, quintessence, massive gravity terms, and Gauss-Bonnet corrections do not change the topological number: all four families studied remain in the $$j^\mu=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$1 class (Sadeghi et al., 2023). In four-dimensional dRGT massive gravity, the charged black hole has the same topological structure as the AdS-RN black hole, and the paper argues that adding massive-gravity interaction terms does not alter the topological number in four dimensions. In higher-dimensional dRGT, however, the topological number becomes parameter-dependent; in particular, for neutral five-dimensional black holes the sign of $$j^\mu=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$2 determines whether the class is $$j^\mu=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$3 or $$j^\mu=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$4 (Fairoos et al., 2023).

Gauss-Bonnet corrections and Born-Infeld corrections are treated differently in the cited literature. For six-dimensional charged Gauss-Bonnet AdS black holes, the total topological charge of critical points remains

$$j^\mu=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$5

through all charge regimes considered, and the paper emphasizes that Gauss-Bonnet gravity does not change the topological class of the system’s critical points relative to the Einstein–Maxwell case (Yerra et al., 2022). In contrast, the Born-Infeld AdS analysis finds parameter regimes in which the topological nature changes, including transitions between an AdS-Schwarzschild-like class and an AdS-Reissner–Nordström-like class (Ali et al., 2023).

Einstein-Gauss-Bonnet branch topology gives a further dimension-sensitive pattern. In the defect-dynamics analysis, five-dimensional neutral EGB black holes behave topologically like RN-AdS black holes with $$j^\mu=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$6, while neutral EGB black holes in $$j^\mu=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$7 behave like neutral AdS black holes with $$j^\mu=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$8 (Fairoos, 2023).

5. Bifurcation theory and phase-transition interpretation

When the Jacobian does not vanish, a zero is regular and its trajectory can be followed. In the EGB defect-dynamics treatment, the defect velocity is written as

$$j^\mu=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\,\partial_\nu n^a\,\partial_\rho n^b,$$9

A bifurcation point occurs when

$\mu,\nu,\rho=0,1,2$0

and then defect motion requires Duan’s bifurcation theory, with local velocities determined by quadratic or cubic equations in $\mu,\nu,\rho=0,1,2$1 depending on the degeneracy (Fairoos, 2023).

This machinery is used to reinterpret phase transitions as defect dynamics. In five-dimensional neutral EGB gravity, a stable small black-hole defect with $\mu,\nu,\rho=0,1,2$2 collides with a static exotic defect at one bifurcation point, exchanges winding number, becomes unstable with $\mu,\nu,\rho=0,1,2$3, and later collides with a second exotic defect, exchanging winding number again and emerging as a stable large black hole with $\mu,\nu,\rho=0,1,2$4. The paper identifies this double interchange with the first-order small/large black-hole transition through an intermediate unstable branch (Fairoos, 2023). In neutral EGB black holes for $\mu,\nu,\rho=0,1,2$5, only one exotic defect appears, and the topology describes a single winding-number exchange from an unstable small black hole to a stable large black hole, with conserved total charge $\mu,\nu,\rho=0,1,2$6 (Fairoos, 2023).

Born-Infeld AdS black holes provide a closely related but more elaborate critical-point picture. There the zero points of $\mu,\nu,\rho=0,1,2$7 are thermodynamic critical points, and the charges of individual critical points are interpreted as vortices and anti-vortices. For $\mu,\nu,\rho=0,1,2$8, two critical points appear with

$\mu,\nu,\rho=0,1,2$9

so the total remains $\phi^a$00; the paper describes this as vortex/anti-vortex pair creation. At the isolated critical point, reached at the special pressure

$\phi^a$01

the topological charge is

$\phi^a$02

and it is interpreted as the annihilation point of one conventional and one unconventional critical point. A second threshold,

$\phi^a$03

marks a topological phase transition: for $\phi^a$04, the total charge is $\phi^a$05, whereas for $\phi^a$06 only one conventional critical point remains and the total charge becomes $\phi^a$07 (Ali et al., 2023).

The unified Hawking-Page/Davies construction gives a particularly compact phase-transition application. In Schwarzschild AdS, RN-AdS in the grand canonical ensemble, and Kerr AdS in the grand canonical ensemble, the same vector field detects two zeros: the Davies point carries topological charge $\phi^a$08 and the Hawking-Page point carries topological charge $\phi^a$09 (Hazarika et al., 2024).

6. Interpretive issues, limitations, and broader scope

A persistent misconception is that the global topological number is determined by the number of zero points alone. The formalism does not support that reading. BTZ rotating and charged cases, as well as rotating or charged de-Sitter horizons, explicitly exhibit two zero points with opposite winding numbers and therefore vanishing total charge (Du et al., 2023, Du et al., 2023). The invariant is the signed sum, not the defect count.

A second interpretive issue concerns the meaning of “conventional” and “novel” critical points. In the six-dimensional charged Gauss-Bonnet analysis, the older rule that a conventional critical point with $\phi^a$10 should be associated with a first-order phase transition is shown to fail in some parameter regimes. The proposed replacement is dynamical rather than equilibrium-based: novel critical points with $\phi^a$11 are interpreted as phase creation points, and conventional critical points with $\phi^a$12 as phase annihilation points (Yerra et al., 2022). This interpretation aligns more closely with later Born-Infeld analyses, where pair creation, pair annihilation, and isolated degenerate merger points are central (Ali et al., 2023).

The literature also indicates that claims of universality must be qualified. Some studies strengthen a three-class picture for Einstein–Maxwell systems and related de-Sitter cases (Du et al., 2023), while BTZ provides an explicit two-class exception (Du et al., 2023). Some deformations, such as the quintessence, massive-gravity, and Gauss-Bonnet terms in the Bardeen examples, leave the global class unchanged (Sadeghi et al., 2023); others, such as Born-Infeld nonlinear electrodynamics, can change the class and even produce a possible breakdown of the simpler topology–phase-transition correspondence in certain parameter ranges (Ali et al., 2023).

The scope of the framework is not limited to bulk black-hole thermodynamics. In the Born-Infeld study, the same $\phi^a$13-mapping method is transferred to the dual CFT. The isolated critical point on the CFT side again has zero topological charge, and the clean topological transition is organized by the per-degree-of-freedom control parameter

$\phi^a$14

with the threshold

$\phi^a$15

For $\phi^a$16, the total charge is $\phi^a$17; for $\phi^a$18, it is $\phi^a$19. The bulk and dual CFT therefore share the same topological classification, although the CFT control parameter is $\phi^a$20, not merely $\phi^a$21 (Ali et al., 2023).

Taken together, these developments suggest that Duan’s $\phi^a$22-mapping theory functions, in this research program, as a transferable defect-analysis framework. The choice of thermodynamic scalar or vector field is application-dependent, but the invariant content remains the same: zeros define the relevant thermodynamic objects, the Jacobian fixes their local orientation, and the global classification is encoded by the integer-valued sum of winding numbers.