Web Duality in QFT and Algebra

Updated 7 July 2026

Web Duality is a framework that generates dual descriptions from seed equivalences using operations like gauging, modular transformations, and topological counterterms across varying spacetime dimensions.
It systematically connects boson–boson, fermion–fermion, and boson–fermion dualities in non-supersymmetric quantum field theories, incorporating examples such as the 3D Chern–Simons–matter web and its 4D extension with Maxwell and θ-terms.
The concept also extends to algebraic settings, providing exact pairing between web invariants and immanants in Grassmannians and linking diagrammatic categories to quantum representation theory.

In the cited literature, web duality denotes a structured family of dual descriptions generated from one or more seed equivalences by repeated operations such as gauging global symmetries, adding topological counterterms, performing electric–magnetic transformations, or orbifolding discrete symmetries. In non-supersymmetric quantum field theory, this notion organizes boson–boson, fermion–fermion, and boson–fermion dualities across $1+1$, $2+1$, and $3+1$ spacetime dimensions; in particular, the four-dimensional construction of Murugan and Nastase extends the three-dimensional Chern–Simons–matter web to theories with Maxwell and $\theta$ -terms and is conjectured to be the final thread in a multi-dimensional duality web (Murugan et al., 2021). In a distinct mathematical usage, the same expression labels an exact pairing phenomenon between web invariants and web immanants for Grassmannians (Banaian et al., 21 Jul 2025).

1. Interdimensional emergence of duality webs

In $1+1$ dimensions, the web is built by gauging $\mathbb Z_2$ global symmetries while tracking background spin structures $\rho$ and Arf invariants. Starting from the continuum Jordan–Wigner bosonization of a single Majorana fermion, the resulting network connects Ising duality, Kramers–Wannier duality, bosonization of a Dirac fermion, T-duality, and both modular-invariant and non-modular-invariant $c=1$ conformal branches (Karch et al., 2019).

In $2+1$ dimensions, the web begins from the conjectural infrared equivalence between a free Dirac fermion and a Wilson–Fisher complex scalar coupled to a dynamical $U(1)$ gauge field with Chern–Simons and BF terms. The parity anomaly supplies the half-integer counterterms needed for gauge invariance, and repeated applications of time reversal, particle–vortex duality, and BF/CS manipulations generate boson–boson, boson–fermion, and fermion–fermion nodes of the web (Seiberg et al., 2016). A complementary formulation emphasizes that on a generic oriented 3-manifold the relevant background field must be a spin$2+1$0 connection, with global consistency expressed through the Atiyah–Patodi–Singer index theorem and the cancellation between half-level gauge and gravitational Chern–Simons terms (Ma, 2018).

The four-dimensional construction replaces Chern–Simons data by Maxwell and $2+1$1-terms, with complexified coupling

$2+1$2

Murugan and Nastase formulate a non-supersymmetric web in $2+1$3 dimensions containing boson–boson, fermion–fermion, and boson–fermion dualities, and they argue that dimensional reduction on $2+1$4 recovers the familiar three-dimensional $2+1$5 and $2+1$6 moves, with $2+1$7 in the reduced Chern–Simons term (Murugan et al., 2021).

2. Generating moves and modular structure

A duality web is not a single equivalence but a closure of a seed theory under a small set of operations. In the four-dimensional web, the three basic moves are explicitly identified as $2+1$8, $2+1$9, and bosonization: $3+1$0 is the electric–magnetic Legendre transform, $3+1$1 shifts $3+1$2 and contributes a local contact term, and the bosonization step is anchored in the Burgess–Quevedo map together with the four-dimensional particle–vortex “string” duality. Acting with these operations generates transformed couplings

$3+1$3

and the paper summarizes the construction as: gauge a global $3+1$4, introduce Lagrange multipliers to enforce flatness, integrate out fields in the opposite order, and read off the transformed dual Lagrangians (Murugan et al., 2021).

The three-dimensional lattice realization makes the same modular structure explicit in theory space. There,

$3+1$5

so that $3+1$6 adds one unit of Chern–Simons level and $3+1$7 promotes a background field to a dynamical one while introducing a BF coupling. Repeated use of $3+1$8, $3+1$9, and time reversal generates fermionization, Son’s duality, and boson–vortex duality (Son et al., 2018).

The same literature also records an important caveat. In continuum treatments, Maxwell regulators are needed to suppress UV gauge fluctuations, and then the exact $\theta$ 0 algebra generally fails microscopically unless one assumes a hierarchy such as $\theta$ 1. On the lattice, exact partition-function identities are available without Maxwell terms, but $\theta$ 2 remains an infrared assumption rather than a theorem (Son et al., 2018). This suggests that “web” language is most precise as an IR statement about equivalence classes of theories rather than a strict UV isomorphism.

3. The four-dimensional non-supersymmetric web

The bosonic seed in the four-dimensional construction is a charge- $\theta$ 3 complex scalar $\theta$ 4 coupled to a $\theta$ 5 gauge field $\theta$ 6: $\theta$ 7 At the Wilson–Fisher point one sends $\theta$ 8 and tunes $\theta$ 9, so that $1+1$0 flows to the non-trivial IR fixed point. The basic bosonization statement equates a massive Dirac fermion partition function to a scalar-plus-flux ensemble up to a contact $1+1$1 term: $1+1$2 The path-integral derivation proceeds through the four-dimensional particle–vortex “string” duality, introduces a two-form $1+1$3, and yields the IR operator map

$1+1$4

Most derivations are carried out in the infrared limit, with mass deformations used as regulators and Maxwell couplings taken small (Murugan et al., 2021).

From that seed, boson–boson duality follows by acting on $1+1$5 with

$1+1$6

The $1+1$7-move adds $1+1$8, while the $1+1$9-move exchanges electric and magnetic fields through a Legendre transform with a two-form Lagrange multiplier. Fermion–fermion duality is then obtained by promoting the background field to a dynamical one, adding BF couplings, and using the bosonization map to derive a four-dimensional analogue of Son’s composite-fermion duality. In the resulting schematic form,

$\mathbb Z_2$ 0

up to contact shifts. Combining the particle–vortex-string duality with the Burgess–Quevedo map also gives a direct boson–fermion equivalence in which a Wilson–Fisher scalar at coupling $\mathbb Z_2$ 1 is dual in the IR to a Dirac fermion at coupling $\mathbb Z_2$ 2, again up to regulators and contact $\mathbb Z_2$ 3-terms (Murugan et al., 2021).

Node	Definition	Moves
$\mathbb Z_2$ 4	Wilson–Fisher scalar $\mathbb Z_2$ 5	$\mathbb Z_2$ 6, $\mathbb Z_2$ 7
$\mathbb Z_2$ 8	Dirac fermion $\mathbb Z_2$ 9	bosonization to $\rho$ 0, then $\rho$ 1 to composite $\rho$ 2

This organization is the explicit four-dimensional duality web given in the paper, and its somewhat singular character is presented as the four-dimensional analogue of the better-known three-dimensional web (Murugan et al., 2021).

4. Anomalies, boundaries, and condensed-matter realizations

The parity anomaly is central to the $\rho$ 3-dimensional web. Integrating out a single massive Dirac fermion produces an Abelian Chern–Simons term with coefficient $\rho$ 4, and at finite temperature the same term is multiplied by $\rho$ 5. Because half-level Chern–Simons terms are not well-defined on arbitrary $\rho$ 6 bundles, the background field must be treated as a spin $\rho$ 7 connection, and the anomaly is cancelled only after including the appropriate gravitational Chern–Simons contribution (Ma, 2018). This is the global input that makes the web well-defined beyond local perturbation theory.

The four-dimensional uplift is explicitly linked to boundary physics. Dimensional reduction on $\rho$ 8 sends the $\rho$ 9-dimensional Maxwell+ $c=1$ 0 system to a $c=1$ 1-dimensional Maxwell–Chern–Simons theory, and the paper argues that a four-dimensional bulk $c=1$ 2 gauge field coupled to duality-twisted matter provides a regulator for three-dimensional boundary Dirac fermions or Wilson–Fisher scalars. In this picture, the $c=1$ 3D web descends to the $c=1$ 4D Chern–Simons–matter web and is proposed to clarify the appearance of Son’s composite Dirac liquid and bosonic topological orders on the surface of a four-dimensional bulk (Murugan et al., 2021).

A parallel $c=1$ 5-dimensional fermion–fermion duality sharpens the condensed-matter interpretation. There, a charged Dirac fermion $c=1$ 6 in a background electromagnetic field $c=1$ 7 is dual at low energies to a neutral Dirac fermion $c=1$ 8 coupled to a dynamical one-form $c=1$ 9 and a Kalb–Ramond two-form $2+1$0,

$2+1$1

In the massive regime, varying with respect to $2+1$2 enforces $2+1$3, reproducing the original $2+1$4-response; in the massless limit, the same construction is applied to $2+1$5D Dirac semimetals, where the dual description predicts string-like vortex excitations arising from $2+1$6 (Palumbo, 2019).

The three-dimensional web is also used directly in condensed matter. The bosonic and fermionic dualities are applied to the half-filled Landau level, where the Dirac composite fermion description yields a Fermi surface at $2+1$7 with particle–hole symmetry, and to topological-insulator surface states, whose anomaly inflow is naturally expressed through a $2+1$8-dimensional electric–magnetic viewpoint (Seiberg et al., 2016).

5. Critical, lattice, and non-invertible extensions

The web paradigm has been extended from gapped dual descriptions to quantum critical theories. In $2+1$9 dimensions, the seed $U(1)$ 0 fermion–boson duality is used to recast Gross–Neveu–Yukawa transitions of $U(1)$ 1 Dirac fermions in terms of $U(1)$ 2 complex bosons and partner gauge fields. The resulting web includes a single-flavour superconducting transition with emergent $U(1)$ 3 supersymmetry, multi-flavour chiral Ising, $U(1)$ 4, and Heisenberg Gross–Neveu–Yukawa nodes, as well as offspring dualities relating deconfined quantum criticality to gauged $U(1)$ 5 GNY theories. The same framework also produces large- $U(1)$ 6 monopole scaling dimensions such as $U(1)$ 7, $U(1)$ 8, and $U(1)$ 9 for $2+1$00, $2+1$01 (Witczak-Krempa, 2023).

Lattice constructions make the web microscopic. A cubic Euclidean lattice realization generalizes Abelian bosonization to the full $2+1$02-dimensional web, provides exact operator mappings such as

$2+1$03

and exhibits hidden symmetries, including the $2+1$04 self-duality with emergent $2+1$05. The same work also emphasizes that the usual modular manipulations carry implicit assumptions about gauge suppression and hierarchy of couplings (Son et al., 2018).

In $2+1$06 dimensions, the web idea has been pushed into multicritical phase diagrams with $2+1$07 symmetry. There, repeated discrete operations—gauging one or both $2+1$08 symmetries and stacking the nontrivial bosonic SPT—generate bosonic and fermionic webs with nine multicritical conformal field theories each, every node connecting four gapped phases. The continuum Kennedy–Tasaki map is identified as

$2+1$09

and it exchanges symmetry-broken and SPT phases while leaving the trivial phase invariant (Karch et al., 19 Feb 2025).

A further generalization replaces invertible dualities by intrinsically non-invertible ones. For $2+1$10 subsystem symmetries on the square lattice, subsystem Kramers–Wannier and subsystem Kennedy–Tasaki maps connect trivial, subsystem-symmetry-broken, and subsystem SPT phases. On open lattices the maps are unitary and invertible, while on a torus they become non-unitary and non-invertible in the original Hilbert space, with fusion rules governed by projectors onto fixed twist sectors (Maity et al., 24 Nov 2025). This suggests that “web duality” now includes both invertible and non-invertible algebraic structures.

6. Diagrammatic and algebraic meanings of web duality

Outside quantum field theory, the term acquires a more specialized algebraic meaning. For Grassmannian cluster algebras, Fraser, Lam, and Le observed a phenomenon named web duality in which web immanants coincide with web invariants. Banaian, Catania, Gaetz, Moore, Musiker, and Wright formulate this as a tableau-transposition statement: when $2+1$11 and $2+1$12, there should be a rotation-invariant basis $2+1$13 of $2+1$14 webs indexed by standard $2+1$15 Young tableaux $2+1$16, together with a dual basis $2+1$17 of $2+1$18 webs indexed by transposed tableaux, such that

$2+1$19

Their main theorem proves this for $2+1$20, $2+1$21 and $2+1$22, with $2+1$23 basis webs on each side and pairing

$2+1$24

The same paper combines this duality with a twisted higher boundary measurement map to recover and extend formulas for twists of cluster variables (Banaian et al., 21 Jul 2025).

A related, but not identical, use of web language appears in representation theory through quantum skew Howe duality. The $2+1$25-spider category is presented by planar webs modulo local relations, and the key theorem identifies the spider with the tensor-generated subcategory of $2+1$26. The paper explicitly summarizes the outcome by saying that the diagrammatic web category is dual to the idempotented quantum group $2+1$27 under skew Howe duality (Cautis et al., 2012). Extensions to types $2+1$28 similarly use web categories to realize quantum Howe dualities for orthogonal and symplectic coideal subalgebras (Sartori et al., 2017).

Taken together, these mathematical results indicate that “web duality” can denote either a duality between theories generated by modular and gauging operations, or a duality between web bases and algebraic representation data. The two usages are technically distinct, but both organize complicated equivalence classes through a small set of generators, local relations, and exact pairings.