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Higher abelian gauge theory in BV formalism

This paper constructs a Batalin-Vilkovisky (BV) structure on the zero modes of quadratic elliptic higher abelian gauge theories, demonstrating the independence of the associated BV integral from the choice of zero-mode Lagrangian and deriving explicit partition functions for various $p$ -form abelian gauge theories.

Original authors: Shuhan Jiang

Published 2026-09-16

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CC BY 4.0

Original authors: Shuhan Jiang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a universe where the fundamental forces of nature are not just particles bumping into one another, but are instead described by smooth, continuous fields that ripple through space. In the most familiar version of this idea, electromagnetism is a field that can be measured at every point, but it also has a hidden, deeper layer. Just as a map of a city might show the streets but miss the fact that some roads loop back on themselves or that certain areas are connected by bridges that don't appear on the surface, these fields have a "global" structure. They carry information about the shape of the universe itself, including loops and holes that cannot be seen by looking at a tiny patch of space. Physicists call these hidden layers "topological sectors," and they are crucial for understanding how the universe behaves on the largest scales. For decades, scientists have struggled to write down a single, perfect mathematical description that captures both the smooth, wiggly parts of these fields and their rigid, topological loops simultaneously.

In this new work, Shuhan Jiang tackles this challenge by focusing on a specific type of field theory known as a "higher abelian gauge theory." Think of these as generalized versions of the electromagnetic field, but instead of describing lines of force, they describe higher-dimensional sheets or volumes that can wrap around the universe. The paper builds a rigorous framework to calculate the "partition function" of these fields. In simple terms, a partition function is a master sum that tells us how likely the universe is to be in any particular configuration. It is the ultimate accounting tool for a physical system, weighing every possible state to predict its overall behavior. The difficulty arises because these fields have two distinct types of behavior: "zero modes," which are the large-scale, global patterns that don't wiggle much, and "non-zero modes," which are the tiny, rapid fluctuations. While physicists have long known how to handle the tiny fluctuations, the global patterns have been notoriously difficult to pin down, especially when the universe has a complex shape with holes and twists.

Jiang's paper constructs a precise mathematical structure, known as a Batalin–Vilkovisky or BV structure, specifically designed to handle these global zero modes. This structure acts like a sophisticated accounting ledger that keeps track of both the smooth, continuous parts of the field and the discrete, integer-based loops that wrap around the universe. The author shows that by organizing these fields into a specific type of algebraic group, one can define a clear rule for summing up all possible configurations. The key breakthrough is proving that the final result of this sum does not depend on the arbitrary choices made during the calculation. In physics, it is common to have to pick a specific "gauge" or perspective to do the math, and often the final answer seems to change depending on that choice. Jiang demonstrates that for these higher-dimensional fields, if one follows the correct rules, the final partition function remains the same regardless of the perspective chosen, provided the universe is a closed, smooth shape like a sphere or a torus.

The paper provides explicit formulas for the partition function in several important cases, including Maxwell theory (which describes standard electromagnetism), Chern–Simons theory (a topological theory used in condensed matter physics), and a hybrid of the two. For the standard electromagnetic case, the result is a combination of three distinct ingredients: a volume term that measures the size of the space of smooth fields, a finite factor that counts the number of topological twists or "torsion" in the universe, and a sum over all possible ways the field can wrap around the universe's holes. The author finds that the way these topological twists are counted is slightly different from what previous researchers had proposed. This difference arises because the new method carefully accounts for "antifields," which are mathematical tools used to ensure the theory remains consistent when symmetries are present. By including these tools, the calculation captures a more complete picture of the global structure.

One of the most significant findings is that the method works consistently even when the fields are coupled to the shape of the universe in complex ways. The paper shows that for certain theories, the contribution from the global zero modes can be separated cleanly from the contribution of the tiny fluctuations. This separation is vital because it allows physicists to study the topological properties of the universe without getting bogged down by the infinite complexity of the tiny ripples. The author also explores a scenario where the theory might become "resonant," meaning the smooth fields and the topological loops interact in a way that creates new, unexpected states. In these specific cases, the standard formula needs to be extended, suggesting that there is still more to discover about how these fields behave under extreme conditions.

Ultimately, this work provides a solid, unshakeable foundation for calculating the global behavior of higher-dimensional fields. It confirms that the universe's hidden topological structure can be quantified with precision, offering a new tool for understanding the deep connection between the shape of space and the forces that govern it. The results are not just abstract mathematics; they offer a concrete way to compute the "weight" of different universes in a theoretical landscape, ensuring that the laws of physics remain consistent no matter how we choose to look at them. By resolving the ambiguity in how these global patterns are counted, the paper clears the path for future studies into the duality of forces and the nature of quantum fields in curved, complex spaces.

Technical Summary: Higher Abelian Gauge Theory in BV Formalism

Problem Statement
The paper addresses the formulation of the zero-mode partition function for quadratic $p$ -form abelian gauge theories on a closed, oriented Riemannian manifold $(X, g)$ . A central challenge in the global description of such theories is the necessity of retaining not only the perturbative differential-form fields but also their integral and torsion sectors. Standard perturbative approaches often fail to capture the global topological data inherent in these sectors. The author seeks to construct a rigorous Batalin–Vilkovisky (BV) structure on the space of zero modes that incorporates these global features, specifically within the framework of Cheeger–Simons differential cohomology.

Methodology
The construction proceeds in two primary stages: the abstraction of the BV structure on abelian Lie groups and its application to specific elliptic gauge theories.

Abelian BV Groups: The author introduces an abstract notion of an "abelian BV group," defined as a collection of abelian Lie groups $H = \{H^k\}$ equipped with a triple $(\omega, \kappa, w)$ .
- $\omega$ is a $(-1)$ -shifted symplectic pairing on the total Lie algebra $\mathfrak{h}$ .
- $\kappa$ is a direct sum of inner products on $\mathfrak{h}$ compatible with $\omega$ .
- $w$ is a complex quadratic weight defined on the component group $\pi_0(H^0)$ .
  The theory distinguishes between "integral" and "integrable" structures based on the properties of the exponential lattice $\Gamma \subset \mathfrak{h}$ . A "Lagrangian" $L \subset H$ is defined as a family of closed subgroups with a Lagrangian Lie algebra and injective component maps. The author defines "admissible" and "maximal" Lagrangians to ensure the convergence of the associated integrals.
Application to Gauge Theories: The framework is applied to quadratic elliptic higher abelian gauge theories formulated in Cheeger–Simons differential cohomology $\check{H}^{p+1}(X)$ .
- The classical fields are modeled by the $(p+1)$ -st smooth Deligne complex.
- The BV complex $\mathcal{F}$ is constructed as the $(-1)$ -shifted mapping cone of the equation-of-motion morphism $K$ .
- The space of zero modes is identified with the hypercohomology $H^\bullet(X; \mathcal{F})$ .
- Using Hodge theory and the Riemannian metric, the author establishes a canonical $(-1)$ -shifted symplectic pairing and an inner product on the perturbative zero modes (harmonic representatives).
- The global structure is recovered via the long exact sequence relating the perturbative complex, the full BV complex, and integral cohomology, which endows $H^\bullet(X; \mathcal{F})$ with the structure of an abelian Lie group with a specific exponential lattice and component groups.

Key Contributions and Results
The paper provides an explicit construction of the zero-mode partition function $Z_H(L)$ for a maximal admissible Lagrangian $L$ . The main result is Theorem 2.1, which states that for a "nice" abelian BV group (satisfying specific unimodularity and positivity conditions), the zero-mode partition function is independent of the choice of maximal admissible Lagrangian $L$ .

The partition function is shown to factorize into three distinct components:
$Z_H(L) = \text{vol}(L_0) \cdot T(H) \cdot \Theta(L)$

$\text{vol}(L_0)$ : A graded volume of the connected zero modes (compact harmonic tori), derived from the determinant of the lattice metric.
$T(H)$ : A finite torsion factor arising from the torsion subgroups of the component groups for non-zero degrees.
$\Theta(L)$ : A theta sum over the degree-zero topological sectors (the free part of the component group), weighted by the real part of the quadratic action.

The paper explicitly derives these formulas for three specific theories:

Maxwell Theory ( $p$ -form): The zero-mode partition function involves a Gaussian sum over the free part of $H^{p+1}(X; \mathbb{Z})$ and a specific torsion factor. The author notes that while the form is similar to previous works (e.g., [MS25]), the torsion factor differs due to the inclusion of antifields and the maximality requirement on the Lagrangian.
Abelian Chern–Simons Theory: For odd $p$ and dimension $n=2p+1$ , the theory is integrable and "nice" only when the level $k = \pm 1$ . The partition function reduces to a sum over torsion sectors weighted by a quadratic refinement of the cup product.
Maxwell–Chern–Simons Theory: The paper analyzes the interplay between the Maxwell and Chern–Simons terms. It finds that if no "resonant" zero modes exist (i.e., the kernel of the operator $e^{-2}\star d + k$ is trivial), the zero-mode partition function coincides with that of the pure Chern–Simons theory. If resonant modes exist, the theory is not integrable in the defined sense, and the zero-mode partition function requires a density-valued extension (which is noted as beyond the scope of the paper).

Significance and Claims
The paper claims to provide a rigorous, global BV formulation for higher abelian gauge theories that consistently separates the finite-dimensional zero-mode contribution from the regularized determinants of non-zero modes. By packaging the global topological data (integral and torsion sectors) into the structure of an abelian BV group, the author demonstrates that the zero-mode partition function is a well-defined, Lagrangian-independent invariant under mild hypotheses.

The author explicitly states that these formulas are intended for use in a companion paper [JLZ] for a BV-theoretic analysis of abelian duality. The work does not propose new experimental applications or future physical implications beyond this mathematical framework and its immediate application to duality. The significance lies in the precise handling of the global topological sectors within the BV formalism, offering a corrected torsion factor for Maxwell theory and a unified treatment of various quadratic abelian gauge theories.

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