Nonstandard Ghost-Number Assignments

• Nonstandard ghost-number assignments are vital for constructing vertex operators beyond traditional ranges, refining BRST cohomology and symmetry analysis.
• In string theory, these assignments facilitate gauge-fixing and accurate amplitude computations by incorporating nonminimal variables in pure spinor frameworks.
• They also underpin modular representation theory by defining refined invariants that control the structure of stable module categories and ghost map lengths.

Nonstandard ghost-number assignments refer to the systematic construction and employment of vertex operators, module maps, or cohomology classes at ghost numbers outside the ranges traditionally dictated by physical state conditions or standard descent formalisms in both string theory and representation theory. These assignments are indispensable for understanding the structure of BRST cohomology, the algebra of module categories, and compute amplitudes—in particular wherever naive prescriptions fail or additional gauge-fixing and symmetry requirements arise.

1. Ghost Number Assignments: Definitions and Formalisms

In the BRST and pure spinor frameworks, the ghost number $g$ tracks the grading with respect to the canonical $b,c$ or pure spinor variables. For closed string backgrounds and superstring field theory, conventional assignments yield physical (massless) vertex operators at specific ghost numbers: typically, $g=1$ for unintegrated vertices and $g=2$ for physical states (e.g., linearized SUGRA fluctuations), as in $H^2(Q)$ for Type IIB pure spinor superstrings (Mikhailov, 2014).

However, both in worldsheet and algebraic settings, rigorous analysis uncovers the necessity and subtlety of considering vertex operators or morphisms at higher, lower, or negative ghost numbers. For example, in the non-minimal 11D pure spinor superparticle, negative and zero ghost number physical operators (e.g., $C_{ab}$, $U^{(0)}$) are essential for a covariant and consistent amplitude prescription (Guillen et al., 27 Aug 2025). In categorical representation theory, related "ghost number" invariants control the structure of homological projective classes and the failure of generating hypotheses (Christensen et al., 2013).

2. Nonstandard Ghost-Number Cohomology in String Theory

In the pure spinor formalism, the cohomology $H^g(Q)$ at ghost number $g$ appears in distinct roles:

• $g=1$: Generators of global symmetries (super-Poincaré currents).
• $g=2$: Linearized supergravity multiplets.
• $g=3$: Nonstandard cohomology classes, transforming like the physical multiplet itself, implying $H^3(Q) \cong H^2(Q)$ up to finite discrete states (Mikhailov, 2014).

Ghost-number-3 representatives can be constructed via tensoring lower ghost-number states; for instance, by acting with a momentum operator on a plane-wave $g=2$ state,

$V_3 = (\text{momentum operator}) \cdot V_2$

where $V_2$ is BRST-closed but not BRST-exact unless a Dirac-type overdetermined pre-image exists. Under physical (semi-relative) cohomology conditions—e.g., enforcing $(b_0 - \bar b_0)V = 0$—the nonstandard ghost-number classes are removed, restoring the expected vanishing of $H^g$ for $g>2$ (Mikhailov, 2014).

3. Ghost Number in Pure Spinor and Minimal/Non-Minimal Formalisms

The introduction of non-minimal variables is crucial in higher dimensions (notably 11D). In the minimal pure spinor superparticle, it is impossible to construct a Lorentz-invariant, BRST-closed ghost-number-zero vertex fulfilling the descent relation with the standard ghost-number-one vertex. Augmenting the formalism with extra pairs $(\bar\lambda_a, \bar w^a)$ and $(r_a, s^a)$, one constructs negative and zero ghost-number physical operators, such as $C_{ab}$. These negative operators, when paired with supergravity superfields, enable the explicit construction of the ghost-number-zero vertex operator $U^{(0)}$ (Guillen et al., 27 Aug 2025): $$U^{(0)} = W^a\,L_{h\,ab} + A^{bc}\,P_a\,h_{bca} - (\lambda\gamma^a)^S T_{ba}{}^c C_{ab} + \cdots$$ This $U^{(0)}$ satisfies generalized descent relations and underlies manifestly covariant calculations for tree and loop amplitudes.

4. Gauge-Fixing, Cohomological Descent, and Nonstandard Vertices

General prescriptions for closed string vertex operator insertions clarify the emergence of nonstandard ghost numbers through gauge-fixing of worldsheet diffeomorphism and conformal Killing symmetries—particularly on the disk and sphere. Faddeev–Popov procedures yield extra ghost insertions that directly shift the ghost number by two or more units (Kishimoto et al., 2024). For a weight-$(1,1)$ matter primary, the vertices

$$\omega_0^3 = -\frac{1}{4\pi}(\partial c - \bar\partial\tilde c)c\tilde c\,V$$

and related forms $\omega^g_n$ with varying $g$ and $n$ (number of forms) systematically exhaust the cohomological sequence.

Amplitudes involving such higher ghost-number operators are required, for example, for the correct computation of the disk dilaton tadpole, where the nonstandard $g=3$ dilaton vertex is essential to reproduce the $D+2$ anomaly coefficient in 26D (Kishimoto et al., 2024).

5. Nonstandard Ghost Number in Representation Theory

In triangulated categories such as $\operatorname{StMod}(kG)$ for modular representation theory, the ghost number of a group algebra is defined via the minimal length of composite ghost maps (those inducing zero in Tate cohomology). Nonstandard ghosts are directly connected to refined invariants: the simple ghost number and the strong ghost number, measuring, respectively, the vanishing on all simple modules or upon restriction to all subgroups (Christensen et al., 2013). These ghost numbers have concrete group-theoretic and cohomological implications, such as:

• For cyclic $p$-groups, $gn(kC_{p^r}) = \lfloor p^r/2 \rfloor$,
• For dihedral groups, explicit computations reveal ghost numbers depending on Sylow subgroups and their normality.

The relationships among these invariants—$gn(kG)\leq sgn(kG)\leq stgn(kG)$—reflect deeper structural properties of the stable module category and the subtleties of induction/restriction.

Invariant	Definition	Key Property
Ghost number ($gn$)	Length of composite ghost maps (trivial in Tate)	Upper bound for all ghost-type invariants
Simple ghost number	Vanishing after precomposition with simples	Agrees with $gn$ for cyclic $p$-groups or when Sylow $p$-subgroup normal
Strong ghost number	Vanishing upon restriction to all subgroups	Controls invariants over the whole subgroup lattice

6. Physical and Computational Implications

Nonstandard ghost-number assignments are not artifacts but are physically and mathematically necessary. In closed string theory, two-point amplitude prescriptions in the pure spinor formalism demand nonstandard total ghost numbers—conventional choices yield identically vanishing amplitudes due to pure spinor zero-mode counting. The only consistent prescription employs ghost-number-(2,•) and ghost-number-(3,•) vertices, which saturate the requisite $(3,3)$ zero modes in both left- and right-movers (Kashyap, 28 Dec 2025). The uniqueness of this structure is guaranteed by the vanishing of pure spinor cohomology for total ghost number $\geq 4$, a result previously established by Mikhailov.

In higher-dimensional pure spinor systems (e.g., 11D supergravity), the ability to construct negative and zero ghost number vertices via nonminimal variables is essential for consistent SUGRA amplitude formulations, manifest BRST invariance, and the construction of higher-point tree and loop amplitudes (Guillen et al., 27 Aug 2025).

7. Role in Cohomology, Amplitude Consistency, and Future Directions

Careful control of ghost-number assignments ensures the correct characterization of deformation spaces, the absence of cohomological obstructions to marginality, and the closure of supermultiplet structures under supersymmetry. Nonstandard ghost numbers thus enforce the consistency of string backgrounds, cohomological completeness, and the correct anomaly structure. The lessons from nonstandard ghost-number constructions generalize to other contexts, including pure spinor supermembrane formulations, higher-dimensional field theories, and the study of stable module categories in algebra.

These insights not only deepen the mathematical structure of BRST (and derived) categories but are indispensable in practical amplitude computations in contemporary string and supergravity theory (Mikhailov, 2014, Kishimoto et al., 2024, Guillen et al., 27 Aug 2025, Kashyap, 28 Dec 2025, Christensen et al., 2013).