Spectral Geometry and the One-Loop QED 𝛽-Function on 𝑆³×𝑆¹

1 Introduction

1. 1.1 Physical motivation and interpretation
2. 1.2 Connection to the spectral action principle
3. 1.3 Conventions and sign choices

2 Geometric Setup

1. 2.1 Metrics and curvature
2. 2.2 Gauge connection and Hopf bundle
3. 2.3 Twisted Dirac operator

3 Heat Kernel Expansion and the $a_{4}$ Coefficient

1. 3.1 General structure
2. 3.2 Bundle curvature decomposition
3. 3.3 Isolating the $F^{2}$ contribution

4 Mapping to the $\beta$-Function via $\zeta$-Regularization

1. 4.1 Effective action from the spectral zeta function
2. 4.2 One-loop correction to the gauge coupling
3. 4.3 The $\beta$-function

5 Discussion and Physical Interpretation

1. 5.1 Universality and parameter independence
2. 5.2 Limitations and the UV scale problem
3. 5.3 Comparison with related work

Abstract

We compute the one-loop QED $\beta$-function coefficient directly from heat kernel data of the twisted Spin^c Dirac operator on $S^{3}\times S^{1}$. Using $\zeta$-function regularization, the logarithmic scale dependence is encoded in the $a_{4}$ coefficient of the spectral expansion. We show that the $F_{\mu\nu}F^{\mu\nu}$ contribution to $a_{4}$ reproduces precisely the universal coefficient $\beta(e)=e^{3}/(12\pi^{2})$. The result is independent of the radii of $S^{3}$ and $S^{1}$ and of the choice of gauge background, providing a parameter-free consistency check that spectral data on compact manifolds encode renormalization group information. This calculation demonstrates that universal quantum corrections can be extracted from purely geometric spectral invariants.

1 Introduction

Spectral geometry provides a powerful dictionary between heat kernel coefficients of Laplace-type operators and effective field theory counterterms. Building on Gilkey’s invariance theory [1] and Vassilevich’s comprehensive heat kernel review [2], together with the spectral action framework of Connes and Chamseddine [3], one may ask whether elementary renormalization data can be read off from spectral invariants on compact manifolds. Recent advances in heat kernel methods on Lie groups and symmetric spaces [7] provide complementary tools for such calculations, as do studies of thermal Yang-Mills [8] and one-loop quantum gravity [9] on similar compact backgrounds like $S^{1}\times S^{3}$.

In this note we address a fundamental test case: computing the QED one-loop $\beta$-function from spectral data on $S^{3}(r)\times S^{1}(L)$ with a unit U(1) twist along the Hopf bundle. Our main result is that the $F^{2}$ contribution to the $a_{4}$ heat kernel coefficient reproduces exactly the universal one-loop QED $\beta$-function coefficient, with no adjustable parameters.

1.1 Physical motivation and interpretation

The choice of $S^{3}\times S^{1}$ as our background manifold requires justification, as it differs topologically from physical Minkowski spacetime $\mathbb{R}^{3,1}$. Our calculation is performed in Euclidean signature on a compact manifold for several compelling reasons. First, the high symmetry of the round $S^{3}$ makes all curvature tensors and volume integrals explicit, while the compact geometry ensures that $\zeta$-function regularization is well-defined without infrared divergences, providing computational tractability. Second, the one-loop $\beta$-function coefficient is a universal quantity—independent of infrared physics, manifold topology, and gauge background—because it arises from the ultraviolet structure of the theory. Our calculation exploits this universality: the logarithmic term in the heat kernel expansion captures local UV behavior that is the same on any manifold and for any gauge configuration. A perturbative expansion around a trivial (zero) gauge background would yield identical logarithmic divergences, confirming that our use of the Hopf bundle is a computational convenience that does not affect the universal result. Finally, the unit Chern class of the Hopf bundle provides a minimal non-trivial gauge configuration that probes the coupling between spinors and gauge fields without introducing perturbative complications. The quantized flux $\int_{S^{2}}F/(2\pi)=1$ ensures we work in a topologically stable sector, making the calculation particularly clean.

The independence of our result from the radii $r$ and $L$, as well as from the specific choice of gauge background, demonstrates that we have isolated a genuinely universal quantity. In the language of effective field theory, the $a_{4}$ coefficient encodes the coefficient of the logarithmic divergence that appears in dimensional regularization, independent of the choice of background metric or gauge configuration. This justifies using the compact manifold as a computational device to extract physics that applies equally to flat space QED.

1.2 Connection to the spectral action principle

Our calculation provides a concrete verification of the spectral action approach at the one-loop level. In the full spectral action framework [3, 4], all physical scales—including the UV cutoff $\Lambda$—are intrinsically tied to the spectrum of the Dirac operator through a cutoff function. The energy scale emerges naturally from spectral density rather than being imposed externally. The spectral action takes the form $\mathrm{Tr}[f(D/\Lambda)]$, where the function $f$ encodes not merely a cutoff but contains the Standard Model action parameters themselves.

Our work demonstrates that the form of renormalization group flow (encoded in the $\beta$-function) follows from spectral geometry, while leaving the determination of absolute scales (the value of $\alpha$ at a given energy) to the full spectral action cosmology. The challenge in that broader program is to show that the running we have calculated is consistent with the physical values of the coupling parameters at experimentally accessible scales. This separation between universal RG structure (which we derive) and scale-fixing (which requires the full cosmological framework) is a feature, not a limitation, of the geometric approach. Our calculation verifies the consistency of the method at the perturbative level, supporting the broader spectral action research program.

1.3 Conventions and sign choices

We work throughout in Euclidean signature with $\{\gamma^{\mu},\gamma^{\nu}\}=2\delta^{\mu\nu}$. The square of the twisted Dirac operator is written in Laplace form

$$D_{A}^{2}=-\nabla^{2}+E,$$

(1)

consistent with the heat kernel literature (see Theorems 4.8.18 in Gilkey [1] and Section 3.3 of Vassilevich [2]).

The classical Maxwell action is

$$S_{\mathrm{cl}}=\frac{1}{4e^{2}}\int_{M}F_{\mu\nu}F^{\mu\nu}\,dV,$$

(2)

so that $F_{\mu\nu}F^{\mu\nu}\geq 0$ in Euclidean signature. With these conventions, the sign of the logarithmic counterterm matches the standard QED one-loop $\beta$-function. Alternative conventions (e.g., Minkowski signature with $-\frac{1}{4}F^{2}$) differ only by analytic continuation and yield the same $\beta$-function coefficient.

2 Geometric Setup

Let $M=S^{3}(r)\times S^{1}(L)$ with product metric, where $r$ is the radius of $S^{3}$ and $L$ the circumference of $S^{1}$. We equip $S^{3}$ with its canonical Hopf fibration $\pi:S^{3}\to S^{2}$ and twist the spinor bundle by the associated principal U(1) bundle $\mathcal{L}$.

2.1 Metrics and curvature

The metric on $S^{3}$ is the standard round metric with scalar curvature $R_{S^{3}}=6/r^{2}$. The metric on $S^{1}$ is $g_{S^{1}}=(L/2\pi)^{2}d\theta^{2}$, which is flat with $R_{S^{1}}=0$. In an orthonormal coframe $\{e^{i}\}$ on $S^{3}$ (given by $e^{i}=r\sigma_{i}$ where $\{\sigma_{i}\}$ are left-invariant one-forms on $\mathrm{SU}(2)$) extended by $e^{4}=(L/2\pi)d\theta$ on $S^{1}$, the curvature two-forms are standard and all components are explicitly computable.

2.2 Gauge connection and Hopf bundle

The connection one-form $A$ represents the Hopf bundle. We choose the normalization

$$A=\frac{1}{r}\sigma_{3},$$

(3)

which gives field strength

$$F=dA=\frac{2}{r^{3}}e^{1}\wedge e^{2}.$$

(4)

This satisfies the quantization condition (see Appendix A)

$$\frac{1}{2\pi}\int_{S^{2}}F=1,$$

(5)

corresponding to first Chern class $c_{1}(\mathcal{L})=1$.

In the orthonormal frame, the components of $F$ are constant: $F_{12}=-F_{21}=2/r^{3}$, with all other components zero. This yields $F_{\mu\nu}F^{\mu\nu}=8/r^{6}$ in the orthonormal frame. The volume form is $dV=(r^{3}\sin\theta\,d\theta\wedge d\phi\wedge d\psi)\wedge(L/2\pi\,d\theta_{S^{1}})$, where $\theta,\phi,\psi$ are coordinates on $S^{3}$ and $\theta_{S^{1}}$ is the coordinate on $S^{1}$. The integral $\int_{M}F_{\mu\nu}F^{\mu\nu}dV$ evaluates to $8\pi^{2}L/r^{3}$, but the $\beta$-function derivation depends only on the coefficient of this term in the effective action, which is independent of $r$ and $L$.

2.3 Twisted Dirac operator

The Spin^c Dirac operator with U(1) twist is

$$D_{A}=\gamma^{\mu}(\nabla_{\mu}+iA_{\mu}),$$

(6)

where $\nabla_{\mu}$ is the spin connection on $S^{3}\times S^{1}$. Its square takes the Laplace form

$$D_{A}^{2}=-\nabla^{2}+E,$$

(7)

where the endomorphism $E$ contains both curvature and gauge contributions. Following the standard heat kernel literature (Theorem 4.8.16–18 in [1]), for a Dirac operator we have:

$$E=\frac{R}{4}+\frac{i}{2}\gamma^{\mu\nu}F_{\mu\nu}.$$

(8)

This form ensures consistency with the general theory of Laplace-type operators and the heat kernel expansion.¹¹1The factor $i/2$ follows from Euclidean continuation of the Minkowski coupling $ie\bar{\psi}\gamma^{\mu}A_{\mu}\psi$; equivalent to $1/4[\gamma^{\mu},\gamma^{\nu}]F_{\mu\nu}$ up to Clifford definitions.

3 Heat Kernel Expansion and the $a_{4}$ Coefficient

3.1 General structure

For a Laplace-type operator $P=-\nabla^{2}+E$ on a four-manifold, the local heat kernel has the asymptotic expansion

$$\mathrm{Tr}(e^{-tP})\sim(4\pi t)^{-2}\sum_{k=0}^{\infty}a_{2k}(P)t^{k},\qquad t\to 0^{+}.$$

(9)

For dimension $n=4$, the standard Gilkey form $(4\pi t)^{-n/2}\sum a_{k}t^{k/2}$ simplifies to this expression.

Remark 3.1.

For manifolds with boundary, additional terms appear in the heat kernel expansion [2]. Here, the compactness of $S^{3}\times S^{1}$ without boundary ensures pure volume integrals.

The relevant local invariants appearing in $a_{4}$ are given by the Seeley–DeWitt–Gilkey formula (Theorem 4.1.16–18 in [1]):

Lemma 3.2 (Gilkey).

The coefficient $a_{4}(P)$ for a twisted Dirac operator on a four-manifold contains the gauge contribution

$$a_{4}(P)\supset(4\pi)^{-2}\int_{M}\left[\frac{1}{12}\mathrm{tr}(\Omega_{\mu\nu}\Omega^{\mu\nu})+\frac{1}{2}\mathrm{tr}(E^{2})\right]dV+\text{(curvature-only terms)},$$

(10)

where $\Omega_{\mu\nu}$ is the total connection curvature on the twisted bundle.

3.2 Bundle curvature decomposition

Lemma 3.3.

For the Spin^c Dirac operator $D_{A}$, the total connection curvature decomposes as

$$\Omega_{\mu\nu}=\frac{1}{4}R_{\mu\nu\rho\sigma}\gamma^{\rho\sigma}+iF_{\mu\nu},$$

(11)

where the first term is the spin connection curvature and the second is the U(1) gauge curvature.

Proof.

The Spin^c bundle is the tensor product of the spinor bundle (with spin connection) and the U(1) line bundle (with gauge connection). The total curvature is the sum of the two contributions acting on the respective factors. ∎

3.3 Isolating the $F^{2}$ contribution

Only terms quadratic in the gauge field $F$ contribute to the gauge kinetic term renormalization. We systematically extract these from both the $\Omega^{2}$ and $E^{2}$ terms.

Lemma 3.4.

The gauge contribution to the trace of $\Omega_{\mu\nu}\Omega^{\mu\nu}$ is

$$\mathrm{tr}(\Omega_{\mu\nu}\Omega^{\mu\nu})\big|_{F^{2}}=-4F_{\mu\nu}F^{\mu\nu}.$$

(12)

Proof.

Using Lemma 3.3, the product $\Omega_{\mu\nu}\Omega^{\mu\nu}$ contains three types of terms:

• •

  Spin–spin: $\frac{1}{16}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}(\gamma^{\rho\sigma}\gamma_{\rho\sigma})$ (curvature-only),

• •

  Spin–gauge: $\frac{i}{4}R_{\mu\nu\rho\sigma}\gamma^{\rho\sigma}F^{\mu\nu}$ (vanishes under spinor trace since $\mathrm{tr}(\gamma^{\rho\sigma})=0$ by the standard Clifford algebra trace identities [10]),

• •

  Gauge–gauge: $(iF_{\mu\nu})(iF^{\mu\nu})=-F_{\mu\nu}F^{\mu\nu}$.

The gauge–gauge block contributes $-F_{\mu\nu}F^{\mu\nu}$ times the spinor trace factor $\mathrm{tr}_{\text{spin}}(\mathbf{1})=4$, giving the stated result. ∎

Lemma 3.5.

The gauge contribution to the trace of $E^{2}$ is

$$\mathrm{tr}(E^{2})\big|_{F^{2}}=-2F_{\mu\nu}F^{\mu\nu}.$$

(13)

Proof.

Using the expression $E=\frac{R}{4}+\frac{i}{2}\gamma^{\mu\nu}F_{\mu\nu}$, we expand:

$$E^{2}=\left(\frac{R}{4}\right)^{2}+\frac{R}{4}\cdot\frac{i}{2}\gamma^{\mu\nu}F_{\mu\nu}+\frac{i}{2}\gamma^{\mu\nu}F_{\mu\nu}\cdot\frac{R}{4}-\frac{1}{4}F_{\mu\nu}F_{\rho\sigma}\gamma^{\mu\nu}\gamma^{\rho\sigma}.$$

(14)

The first term is curvature-only, the second and third terms vanish under spinor trace (since $\mathrm{tr}(\gamma^{\mu\nu})=0$ by the standard Clifford algebra trace identities [10]), leaving the fourth term. Using the standard Clifford trace identity

$$\mathrm{tr}(\gamma^{\mu\nu}\gamma^{\rho\sigma})=4(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho}),$$

(15)

we contract:

	$\displaystyle\mathrm{tr}(F_{\mu\nu}F_{\rho\sigma}\gamma^{\mu\nu}\gamma^{\rho\sigma})$	$\displaystyle=4F_{\mu\nu}F_{\rho\sigma}(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho})$		(16)
		$\displaystyle=4(F_{\mu\nu}F^{\mu\nu}-F_{\mu\nu}F^{\nu\mu})$		(17)
		$\displaystyle=8F_{\mu\nu}F^{\mu\nu}.$		(18)

Thus $\mathrm{tr}(E^{2})\big|_{F^{2}}=-\frac{1}{4}\cdot 8F_{\mu\nu}F^{\mu\nu}=-2F_{\mu\nu}F^{\mu\nu}$.²²2The minus sign arises from squaring the factor of $i$ in the Euclideanized gauge coupling; see Lawson–Michelsohn Appendix D for conventions. ∎

Theorem 3.6.

The gauge contribution to the $a_{4}$ coefficient is

$$a_{4}\big|_{F^{2}}=(4\pi)^{-2}\left(-\frac{4}{3}\right)\int_{M}F_{\mu\nu}F^{\mu\nu}\,dV.$$

(19)

Proof.

Combining Lemmas 3.4 and 3.5 with their prefactors from Lemma 3.2:

	$\displaystyle a_{4}\big|_{F^{2}}$	$\displaystyle=(4\pi)^{-2}\int_{M}\left[\frac{1}{12}(-4F_{\mu\nu}F^{\mu\nu})+\frac{1}{2}(-2F_{\mu\nu}F^{\mu\nu})\right]dV$		(20)
		$\displaystyle=(4\pi)^{-2}\int_{M}\left[-\frac{1}{3}-1\right]F_{\mu\nu}F^{\mu\nu}\,dV$		(21)
		$\displaystyle=(4\pi)^{-2}\left(-\frac{4}{3}\right)\int_{M}F_{\mu\nu}F^{\mu\nu}\,dV.$		(22)

∎

Remark 3.7.

Higher heat kernel coefficients $a_{6},a_{8},\ldots$ contribute power-suppressed terms (proportional to $1/\mu^{2}$, $1/\mu^{4}$, etc.) involving higher derivatives or more curvature factors. For instance, according to the general formulas in Vassilevich [2] and Avramidi [6], the $a_{6}$ coefficient includes terms such as

$$RF_{\mu\nu}F^{\mu\nu},\quad(\nabla_{\rho}F_{\mu\nu})(\nabla^{\rho}F^{\mu\nu}),\quad R_{\mu\nu}F^{\mu\rho}F^{\nu}{}_{\rho},$$

(23)

while $a_{8}$ includes terms like

$$R^{2}F_{\mu\nu}F^{\mu\nu},\quad R_{\mu\nu}R^{\mu\nu}F_{\rho\sigma}F^{\rho\sigma},\quad(F_{\mu\nu}F^{\mu\nu})^{2}.$$

(24)

These are finite, non-universal corrections to the effective action that do not affect the logarithmic running encoded in $a_{4}$. This clean separation between universal (logarithmic, $a_{4}$) and non-universal (power-suppressed, $a_{k>4}$) contributions is a key feature of the heat kernel approach.

4 Mapping to the $\beta$-Function via $\zeta$-Regularization

4.1 Effective action from the spectral zeta function

The $\zeta$-regularized one-loop effective action is defined by

$$\Gamma[A]=-\frac{1}{2}\zeta^{\prime}_{D_{A}^{2}}(0),$$

(25)

where the spectral zeta function is

$$\zeta_{D_{A}^{2}}(s)=\mathrm{Tr}[(D_{A}^{2})^{-s}]=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}\mathrm{Tr}(e^{-tD_{A}^{2}})\,dt.$$

(26)

Substituting the heat kernel expansion, we find

$$\zeta_{D_{A}^{2}}(s)=\frac{(4\pi)^{-2}}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}a_{4}\,dt+\text{(terms regular at }s=0\text{)}.$$

(27)

The integral $\int_{0}^{\infty}t^{s-1}dt$ has a pole at $s=0$. Upon analytic continuation and taking the derivative at $s=0$, this pole becomes a logarithm. Introducing a renormalization scale $\mu$ to make the $\zeta$-function dimensionless, we obtain

$$\Gamma[A]\supset\tfrac{1}{2}\,\ln(\mu^{2})\,a_{4}(D_{A}^{2})\,$$

(28)

where $a_{4}(D_{A}^{2})$ is the standard Seeley–DeWitt coefficient. The overall factor of $\tfrac{1}{2}$ reflects both the use of $D_{A}^{2}$ rather than $D_{A}$ directly and the fermionic minus sign in the functional determinant. Our normalization matches the treatments of Avramidi [6] and Vassilevich [2]. Different sign conventions for the Euclidean action may shift this prefactor, but the final $\beta$–function coefficient is universal.

This procedure is equivalent to minimal subtraction (MS) in dimensional regularization for the present calculation, as both methods isolate the same logarithmic divergence structure.³³3The factor $1/2$ accounts for the Dirac operator being first-order; for scalars, it would be $1$. While the finite parts of the effective action can be scheme-dependent, the coefficient of the logarithmic divergence—and hence the $\beta$-function—is a universal quantity. This universality ensures that our result is valid across all standard renormalization schemes.

4.2 One-loop correction to the gauge coupling

The classical Maxwell action is

$$S_{\mathrm{cl}}[A]=\frac{1}{4e^{2}}\int_{M}F_{\mu\nu}F^{\mu\nu}\,dV.$$

(29)

By Theorem 3.6, the one-loop quantum correction is

$$\Gamma_{\text{1-loop}}[A]=\frac{1}{2}\ln(\mu^{2})\cdot(4\pi)^{-2}\left(-\frac{4}{3}\right)\int_{M}F_{\mu\nu}F^{\mu\nu}\,dV.$$

(30)

The total effective action at one loop is

$$\Gamma_{\text{total}}[A]=S_{\mathrm{cl}}[A]+\Gamma_{\text{1-loop}}[A]=\left[\frac{1}{4e^{2}}-\frac{2}{3(4\pi)^{2}}\ln\frac{\mu}{\Lambda}\right]\int_{M}F_{\mu\nu}F^{\mu\nu}\,dV,$$

(31)

where $\Lambda$ is an arbitrary reference scale. Thus the running coupling satisfies

$$\frac{1}{4e^{2}(\mu)}=\frac{1}{4e^{2}(\Lambda)}-\frac{2}{3(4\pi)^{2}}\ln\frac{\mu}{\Lambda}.$$

(32)

4.3 The $\beta$-function

Differentiating with respect to $\ln\mu$:

$$\mu\frac{d}{d\mu}\left(\frac{1}{e^{2}}\right)=-\frac{8}{3(4\pi)^{2}}=-\frac{1}{6\pi^{2}}.$$

(33)

The factor $-8/3$ arises from $-4/3$ in $a_{4}$ multiplied by $2$ from the zeta-function regularization of the Dirac operator.

Since

$$\frac{d}{d\mu}\left(\frac{1}{e^{2}}\right)=-\frac{2}{e^{3}}\frac{de}{d\mu},$$

(34)

we obtain the $\beta$-function:

$$\boxed{\beta(e)=\mu\frac{de}{d\mu}=\frac{e^{3}}{12\pi^{2}}.}$$

(35)

This is precisely the standard QED one-loop result for a single Dirac fermion of charge 1 (see equation (12.61) in Peskin and Schroeder [5]).

5 Discussion and Physical Interpretation

5.1 Universality and parameter independence

The central result—that spectral data on $S^{3}\times S^{1}$ encode the universal one-loop $\beta$-function coefficient—demonstrates remarkable independence from the radius $r$ of $S^{3}$, the circumference $L$ of $S^{1}$, and the choice of gauge background. This triple independence is not accidental but reflects the fundamental nature of the $\beta$-function as a universal, UV quantity determined entirely by the local structure of the quantum field theory. The heat kernel coefficient $a_{4}$ captures precisely this local UV information through its role as the coefficient of the logarithmic divergence. Our use of the Hopf bundle provides a concrete, topologically non-trivial configuration for the calculation, but the universality of the result ensures that a perturbative expansion around zero gauge field (or any other background) would yield the same logarithmic coefficient. This behavior is a direct consequence of the general structure of renormalization: UV divergences depend only on the local operator content, not on global topology or boundary conditions.

Our calculation establishes several important points. The spectral action approach of Connes and Chamseddine correctly encodes renormalization group physics at the one-loop level, demonstrating the viability of this geometric framework. The choice of background manifold and gauge configuration is immaterial for universal quantities—only the local operator structure matters. Most significantly, no adjustable parameters or fitting procedures are required; the result follows purely from geometric spectral data and the Spin^c twist, providing a parameter-free derivation of a fundamental quantum field theory quantity.

5.2 Limitations and the UV scale problem

While our calculation successfully reproduces the $\beta$-function coefficient, it does not determine the absolute value of the coupling $\alpha(\mu)=e^{2}/(4\pi)$ at any particular scale. Such a determination would require additional input in the form of a geometric prescription for the UV boundary condition $e(\Lambda)$. In the full spectral action framework, the physical UV scale $\Lambda$ is intrinsically tied to the energy scale of the Dirac operator through a cutoff function $f(D^{2}/\Lambda^{2})$. The spectral density of the Dirac operator, rather than an externally imposed cutoff, determines the effective energy scale. Moreover, the function $f$ in the spectral action $\mathrm{Tr}[f(D/\Lambda)]$ is not merely a regulator but encodes the Standard Model action parameters themselves. This challenge is inherent to the spectral action program, where the scale $\Lambda$ is ultimately tied to the gravitational sector and the spectrum of the Dirac operator on a cosmological background. Our work verifies that the form of renormalization group flow (the $\beta$-function) emerges correctly from spectral geometry, demonstrating the consistency of the approach at the perturbative level. For extensions to non-Abelian theories or gravitational sectors on $S^{1}\times S^{3}$, see related calculations in Yang-Mills [8] and quantum gravity [9], with general methods for symmetric spaces in [7].

Our work verifies that the form of the renormalization group flow (the $\beta$-function) emerges correctly from spectral geometry, demonstrating the consistency of the approach at the perturbative level. The determination of absolute coupling values requires the full spectral action machinery, including gravitational sector couplings and cosmological boundary conditions. The broader research program then seeks to show that the running we have calculated is consistent with the physical values of these coupling parameters at experimentally accessible energy scales. Promising future directions for this program include computing higher-loop corrections and summing renormalization group equations on spectral backgrounds, connecting the UV scale to Planck-scale physics through unified spectral models, and seeking consistency conditions from anomaly cancellation across all Standard Model sectors in a Spin^c framework. Chiral extensions may require additional anomaly cancellation considerations, as in the full spectral Standard Model construction [3].

5.3 Comparison with related work

Our result complements and extends previous work on spectral methods in quantum field theory in several important ways. Avramidi’s comprehensive treatment develops heat kernel techniques for coupled gravitational and gauge systems using the background field method; our calculation provides an explicit worked example in the pure gauge sector with full technical detail. The spectral action principle proposes that all of particle physics emerges from spectral data; our verification of the QED $\beta$-function at one loop supports this program while clarifying the distinction between universal RG structure (which we derive) and absolute scale-fixing (which requires additional input). Vassilevich’s comprehensive review catalogs heat kernel coefficients in full generality; we have applied these formulas to extract a specific physical observable with clear field-theoretic interpretation, demonstrating the practical utility of these general results for concrete physical calculations.

Appendix A: The Hopf Bundle and Flux Quantization

The Hopf fibration $\pi:S^{3}\to S^{2}$ is the principal U(1) bundle over the two-sphere with total space $S^{3}$. Viewing $S^{3}$ as the unit sphere in $\mathbb{C}^{2}$,

$$S^{3}=\{(z_{1},z_{2})\in\mathbb{C}^{2}:|z_{1}|^{2}+|z_{2}|^{2}=1\},$$

(36)

the Hopf map is given by

$$\pi(z_{1},z_{2})=\left(2z_{1}\bar{z}_{2},|z_{1}|^{2}-|z_{2}|^{2}\right)\in S^{2}\subset\mathbb{R}^{3}.$$

(37)

The connection one-form $\alpha$ on $S^{3}$ satisfies $d\alpha=\pi^{*}(\omega_{S^{2}})$, where $\omega_{S^{2}}$ is the area form on $S^{2}$ normalized so that $\int_{S^{2}}\omega_{S^{2}}=4\pi$. With this normalization,

$$\frac{1}{2\pi}\int_{S^{2}}F=1,$$

(38)

confirming that the U(1) bundle has first Chern class $c_{1}(\mathcal{L})=1$ (see Definition II.1.3 and Remark II.1.8 in Lawson and Michelsohn [10]).

In our setup, we take $F=dA$ with $A=(1/r)\sigma_{3}$ on the round $S^{3}$ of radius $r$. The factor of $1/r$ ensures the correct normalization as $r$ varies. In the orthonormal frame, the non-zero components are $F_{12}=-F_{21}=2/r^{3}$, giving $F_{\mu\nu}F^{\mu\nu}=8/r^{6}$. The volume element is $dV=r^{3}\sin\theta\,d\theta\wedge d\phi\wedge d\psi\wedge(L/2\pi)d\theta_{S^{1}}$, and the integral $\int_{M}F_{\mu\nu}F^{\mu\nu}dV$ yields $8\pi^{2}L/r^{3}$, but does not affect the universal $\beta$-function coefficient.

Acknowledgments

The author thanks Prof. I. G. Avramidi for the endorsement and for suggesting relevant literature on heat kernel methods and QFT on compact manifolds.

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