Higher-Form Currents with Nonzero Ghost Number

• Higher-form currents with nonzero ghost number are operator-valued differential forms defined via descent equations and graded by both form degree and ghost number.
• They extend conventional symmetry concepts in gauge, topological, and string theories, impacting anomaly matching and operator algebra closure.
• The integration of BRST and BV cohomological methods facilitates explicit construction and analysis of these ghostly operators in quantum field theories.

Higher-form currents with nonzero ghost number are operator-valued differential forms or generalized currents that possess both a form degree and a nonzero ghost number, as measured in the BRST, BV, or associated cohomological grading. Such currents extend the familiar higher-form symmetry constructs by incorporating the algebraic ghost structure fundamental to gauge-fixed and topological field theories. Their existence, structure, and physical implications have been established across superstring theory, gauge theory, string algebra models, and in the Batalin–Vilkovisky formalism, with explicit descent equations linking conventional and “ghostly” currents.

1. Definition and General Properties

A higher-form current is typically a (p + 1)-form operator-valued field $J^{(p+1)}$ obeying a conservation law of the schematic form:

$d^\dagger J^{(p+1)} = 0$

or, Hodge dualized,

$d\,\star J^{(p+1)} = 0$

In gauge-fixed and topological quantum theories, operator-valued forms may carry an additional grading: the ghost number $q$. Currents with nonzero ghost number $q \neq 0$—here termed ghostly higher-form currents (Editor's term)—generate symmetry transformations and Ward identities associated not to pointlike (0-form) charges but to extended objects, with the symmetry parameter itself being a differential form of degree $p$ and ghost number $q$.

In the Batalin–Vilkovisky (BV) setting, the algebra of local operators is graded by both form degree and ghost number, and descent equations organize conventional higher-form currents $J^{(p),0}$ and their ghostly descendants $J^{(p'),q'}$ into towers:

$d J^{(i)} = Q J^{(i+1)}$

where $Q$ is the BRST/BV differential and each $J^{(i)}$ has shifted form-degree and ghost number (Borsten et al., 19 Sep 2025).

2. Descent Equations and Symmetry Hierarchies

Chains of descent equations are central to the construction of higher-form currents with nonzero ghost number. Beginning with a ghost number zero conserved current, the BV/BRST differential $Q$ and the exterior derivative $d$ are employed recursively:

• General descent structure:

$$d J^{(0)} = Q J^{(1)}, \quad d J^{(1)} = Q J^{(2)}, \ldots$$

Each current $J^{(i)}$ is a local form of degree $p_i$ and ghost number $q_i$.

• Maxwell theory example (Borsten et al., 19 Sep 2025):

$$J^{(0)}_e = -B \quad (g.n. 0), \quad d J^{(0)}_e = Q A^+, \quad d A^+ = Q c^+, \quad d c^+ = 0$$

producing a tower:

$$J^{(0)}_e = -B \; (0), \quad J^{(1)}_e = A^+ \; (-1), \quad J^{(2)}_e = c^+ \; (-2)$$

This hierarchically extends higher-form symmetries to the ghost sector, with physical consequences for operator algebra closure, topological charge quantization, and anomaly matching.

3. Ghost Number in BRST and BV Cohomology

Ghost number is fundamental in BRST and BV quantized frameworks. Physical states, vertex operators, and symmetry currents are classified by their ghost number. The no-ghost theorem for superstring theory rigorously establishes that only states at adjacent ghost numbers $(A + 1/2)$ and $(A + 3/2)$ contribute to the absolute BRST cohomology, corresponding to copies of the light-cone spectrum (Dedushenko, 2012).

• The filtration degree is defined as:

$N_f = N^{gh} - N^{lc}$

• BRST cohomology is isomorphic to:

$$H^*((H_A, Q_B)) \cong \ker L_0 \cap (H_{A,\,A+1/2} \oplus H_{A,\,A+3/2})$$

Higher-form currents (e.g., conserved charges constructed from BRST-closed operators) must respect ghost number assignments to ensure physicality and unitarity.

4. Physical Examples: String Theory and Field Theory Realizations

String Vertex Operators

Closed string vertex operators with various ghost numbers are consistently constructed using the Faddeev–Popov (FP) procedure in tandem with the BRST descent equations (Kishimoto et al., 9 Feb 2024):

• The unintegrated vertex:

$$V^{(2)}(z,\bar{z}) = i\,c(z)\,\tilde{c}(\bar{z}) V(z,\bar{z}), \quad \text{ghost number 2}$$

• Multiplied by $$(i/(4\pi))[\partial c(z) - \bar{\partial} \tilde{c}(\bar{z})]$$ to obtain a ghost number 3 vertex
• Satisfy descent equations:

$$\delta_B \omega_2^0 = -d \omega_1^1, \quad \delta_B \omega_1^1 = d \omega_0^2, \quad \delta_B \omega_0^2 = 0$$

Gauge Theory and Higher Group Symmetry

Composite forms and ghost number grading organize symmetries in effective field theory (Brauner, 2020):

• In superfluid mixtures, composite currents arise via wedge products of duals of topological currents:

$\star(d\phi_1 \wedge d\phi_2)$

Indicate a higher-form symmetry of nonzero form degree—interpreted as nonzero ghost number.

• In axion electrodynamics, such composite terms encode mixed symmetry structures, resulting in higher-group symmetry.

5. Algebraic Structures and Topological Charges

In the context of topological string theory and conformal field theory, higher-form currents with nonzero ghost number emerge as composite operators and deformations of central operators (e.g., the $b$ ghost in pure spinor formalism (Jusinskas, 2013)). BRST-exact deformations and composite ghost-like fields carry nonzero ghost number, and their algebraic properties ensure the closure and nilpotency required for consistency of the operator algebra:

• The $b$ ghost deformation:

$$b' = b + [Q_0, B], \quad \text{with } B \text{ primary, ghost number } -2$$

• Composite $c$ ghost-like operator:

$b(z)c(y) \sim \frac{1}{z-y}$

Operator product expansions and the explicit conservation (or non-conservation) of ghost number in multi-component models are essential for maintaining the extended symmetry algebra.

6. Significance and Physical Implications

Higher-form currents with nonzero ghost number extend the catalogue of possible symmetry generators beyond those acting on physical states. In the BV formalism, their topological charges remain invariant under deformations of supporting submanifolds provided the non-closure is $Q$-exact—meaning integrated charges are well-defined modulo trivial (cohomologically null) variations.

Physically, such ghostly symmetries are crucial for:

• Matching anomalies in topological and gauge theories
• Organizing extended operator algebras, especially in presence of dualities or confinement phenomena in non-Abelian theories (Borsten et al., 19 Sep 2025)
• Constructing, via descent, towers of symmetry operators that act on Wilson or ’t Hooft loop operators, potentially controlling sectors of the quantum Hilbert space outside the unitary subspace

The overall structure of higher-form currents with nonzero ghost number is thus:

• Defined via descent equations and ghost number grading in the operator algebra
• Realized in physical models through ghost-enriched vertex operators, composite currents, and symmetry generators
• Interlinked with anomaly structure, duality transformations, and extended operator algebras across quantum field theory, superstring theory, and topological models

7. Summary Table: Descent Chains and Ghostly Symmetries

Theory Context	Conventional Current $J^{(0)}$	Ghostly Descendant $J^{(1)}$	Remarks
Maxwell theory	$-B$ (ghost 0)	$A^+$ (ghost –1)	Forms tower by $dJ^{(i)} = QJ^{(i+1)}$
Scalar field	$\star d\phi$ (0)	$\phi^+$ (–1)	Top degree symmetry
Abelian p-form gauge	$(d-p-1)$-form (0)	Ghost field antifield	Chain continues until degree exhausted
Yang-Mills centre	Čech 1-cocycle (0)	Descendant in cochain	Discrete symmetry, ghost grading

Ghostly higher-form currents systematically extend conventional global symmetry concepts, forming a hierarchy of operators classified by both form degree and ghost number, anchored in descent equations familiar in anomaly and topological field theory analyses.