Geometries of Quantum Field Theories

Abstract: This is an English translation of a book originally published in Japanese by Saiensu-sha in 2015. The book is an introduction to the 3d-3d correspondence, the relation between 3d $\mathcal{N}=2$ supersymmetric gauge theories and the geometry of 3-manifolds, which arises from the compactification of the 6d $\mathcal{N}=(2,0)$ theory. The presentation is bottom-up rather than top-down: instead of starting from the 6d theory, we begin by asking what a quantum field theory is, and let the geometry emerge on its own. Along the way we discuss renormalization and low-energy effective theories, gauging as an operation which glues field theories together, the resulting $Sp(2n, \mathbb{Z})$-action on 3d theories, 3d $\mathcal{N}=2$ supersymmetric theories and their dualities, the squashed three-sphere partition function and supersymmetric localization, duality domain walls of 4d $\mathcal{N}=2$ theories, quantum Teichmuller theory, complex Chern-Simons theory and the quantization of the moduli space of flat $SL(2, \mathbb{C})$-connections on 3-manifolds, with quantum dilogarithm functions as a recurring thread. The emphasis throughout is on geometric and algebraic structures which are invisible in a single field theory, and appear only in the theory space of quantum field theories. Appendices summarize supersymmetry in various dimensions, classical and quantum dilogarithm functions, and cluster algebras. Exercises with difficulty ratings are included in each chapter.

• The paper presents a structural program outlining clean definitions for data identifying QFTs, renormalization, and organized dualities.
• In three-dimensional $\mathcal{N}=2$ theories, gauging operations create a dual web, and the quantum geometry of moduli space distinguishes different IR fixed states.
• Complex-torus compactification of the six-dimensional $(2,0)$ theory aligns mapping-class transformations to $SL(2,\mathbb{Z})$ dualities and establishes TQFT principles, with the $S^3_b$ partition function quantifying nonperturbative equivalences in various duality scenarios

Conceptual objective and scope

“Geometries of Quantum Field Theories” presents a structural program for understanding quantum field theory through the geometry of theory space rather than through isolated Lagrangians. Its central claim is that a quantum field theory should not be identified naively with a particular Lagrangian and set of couplings. Renormalization-group flows, dualities, gauging operations, boundary conditions, and compactifications generate networks of descriptions whose organization can itself possess geometric structure. The book develops this perspective primarily for supersymmetric gauge theories, with emphasis on three-dimensional $\mathcal{N}=2$ theories and their relation to three-manifolds (2609.28210).

The presentation is deliberately bottom-up. Rather than beginning with the compactification of the six-dimensional $(2,0)$ theory and postulating a field-theory/geometry dictionary, the discussion begins with foundational questions: what data specify a QFT, which of those data are physical, and why distinct Lagrangians can encode identical infrared physics. Geometry is introduced only after the roles of renormalization, duality, gauging, and partition functions have been established. This organization is important because it treats the 3d–3d correspondence as one realization of a more general principle concerning the geometry of QFTs, rather than as an isolated correspondence.

The work is not a comprehensive account of supersymmetric field theory, nor does it attempt to update the subject through the date of the English edition. It is instead a conceptual synthesis of developments that were already available by approximately 2015, supplemented by corrections and editorial revision. The author explicitly identifies the absence of a rigorous general mathematical foundation for QFT as a background limitation. The analysis therefore proceeds through concrete protected structures—vacuum moduli spaces, partition functions, duality transformations, and topological sectors—rather than through a fully axiomatized definition of arbitrary QFTs.

From Lagrangians to theory space

The opening argument concerns the inadequacy of the textbook identification of a field theory with a Lagrangian. In the effective-field-theory framework, the relevant description at a scale $\mu$ consists not merely of a formal Lagrangian $L$, but of the running parameters $\vec c(\mu)$ and the scale at which observables are being considered. Wilsonian renormalization explains why low-energy observables are insensitive to most ultraviolet details, while simultaneously making the inverse problem—reconstructing a microscopic theory from observables—non-unique.

This non-uniqueness is not merely an experimental or computational difficulty. Gauge fields are redundant variables, and many fields are not themselves gauge-invariant observables. Fermionic and gauge degrees of freedom can therefore admit descriptions in terms of different variables. The book uses bosonization, the Aharonov–Bohm effect, and the Stückelberg mechanism to illustrate the distinction between fields and observables. In particular, gauge symmetry is treated correctly as a redundancy of description rather than as an ordinary physical symmetry. A formulation based only on local gauge potentials is consequently not canonical.

The text makes the stronger claim that even complete knowledge of all physical observables need not determine a unique Lagrangian. Several mechanisms produce this ambiguity:

• different parameter regions of one Lagrangian can describe the same physics;
• distinct ultraviolet Lagrangians can flow to the same infrared fixed point;
• gauge and nongauge descriptions can be equivalent;
• some QFTs may not possess a conventional Lagrangian description at all.

The final point is especially significant for the later use of the six-dimensional $(2,0)$ theory. The book treats that theory as a physically meaningful object supported by string-theoretic and compactification arguments, while plainly acknowledging that a conventional non-Abelian Lorentz-invariant Lagrangian is not known.

Duality is therefore not an accidental equivalence between specially chosen models. It is a structural relation in theory space induced by the fact that the fields used in a Lagrangian are not directly observable. The examples include 4d $\mathcal{N}=4$ $SL(2,\mathbb{Z})$ duality, 4d $\mathcal{N}=1$ Seiberg duality, and 3d mirror symmetry. Their logical roles differ. Montonen–Olive duality acts on the coupling of a fixed class of Lagrangians, Seiberg duality relates distinct ultraviolet gauge theories at an infrared fixed point, and the 3d $\mathcal{N}=2$ SQED–$(2,0)$0 duality relates a gauge theory to a theory with no gauge field.

For Seiberg duality, the electric $(2,0)$1 theory with $(2,0)$2 flavors and the magnetic $(2,0)$3 theory with mesons and superpotential $(2,0)$4 are asserted to flow to the same fixed point in the conformal window

$(2,0)$5

The implication is direct: the infrared physics does not determine even the rank of the ultraviolet gauge group. In the 3d $(2,0)$6 example, $(2,0)$7 SQED with one flavor and the $(2,0)$8 model are equivalent in the infrared, with the meson and monopole operators mapped to the three elementary chiral fields. This provides a particularly transparent instance in which a gauge description and a nongauge description represent the same fixed point.

Gauging as an operation on theory space

The book’s central organizational operation is gauging. A theory with global symmetry $(2,0)$9 can be coupled to a background gauge field for $\mu$0; gauging promotes that background field to a dynamical field and integrates over it. Conversely, ungauging replaces a dynamical gauge field by a background field. If two theories share a global symmetry, gauging the diagonal subgroup combines them into a new interacting theory. Ungauging can decompose the resulting theory into constituent sectors.

This operation generates more than isolated dual pairs. Starting from a duality $\mu$1, one can couple either theory to a third theory through a shared global symmetry and obtain a new duality. Iterating this process produces a duality web: a collection of duality frames related by gauging, ungauging, and changes of variables. The associated geometric interpretation is that different duality frames correspond to different decompositions of a common geometric object.

The argument is strongest in three dimensions because gauging a $\mu$2 symmetry produces a new topological $\mu$3 symmetry. For an Abelian gauge field,

$\mu$4

is conserved by the Bianchi identity, independently of the equations of motion. The dual photon $\mu$5 makes this symmetry manifest as a shift symmetry. Thus, gauging does not simply remove a global symmetry; it transforms the symmetry structure and creates a new channel through which further gauging can occur.

The coupling of the topological current to a background gauge field is an off-diagonal Chern–Simons term. Together with diagonal Chern–Simons terms, these operations generate an $\mu$6 action on 3d theories with a $\mu$7 global symmetry. The generators are

$\mu$8

They satisfy $\mu$9 and $L$0, where the minus sign in $L$1 is charge conjugation. With $L$2 Abelian symmetries, the action generalizes to $L$3.

This $L$4 action is not generally a symmetry of one 3d theory. It maps one theory to another by changing its couplings, background fields, and dynamical variables. The book later identifies it with electromagnetic duality acting on a 4d bulk theory, thereby explaining why the same group appears in both dimensions.

Three-dimensional $L$5 dynamics

The specialization to 3d $L$6 theories supplies the protected setting in which the preceding structural claims become computable. The theory has four supercharges and can be obtained by dimensional reduction of 4d $L$7 supersymmetry. Its vector multiplet contains a gauge field, gauginos, a real scalar $L$8, and an auxiliary field. The scalar $L$9 is a specifically three-dimensional degree of freedom and becomes a coordinate on the Coulomb branch.

The book emphasizes the distinction between complex and real masses. Complex masses arise from superpotential terms and preserve holomorphy. Real masses arise from background vector multiplets and are equivalent, after gauging, to expectation values of vector-multiplet scalars. Because real masses are real parameters, they can produce codimension-one walls across which the vacuum structure changes discontinuously. This distinction is essential for the chamber structure of 3d moduli spaces.

Chern–Simons terms are another genuinely three-dimensional ingredient. Their levels are quantized, and integrating out a massive Dirac fermion shifts the effective level by

$\vec c(\mu)$0

The resulting parity anomaly means that a theory with an odd number of appropriately charged fermions cannot generally preserve both gauge invariance and parity without a compensating half-integer counterterm. In the book, this anomaly is used not only as a consistency condition but also as a nontrivial diagnostic of proposed dualities.

The vacuum moduli space is organized into Coulomb, Higgs, mixed, and topological branches. Quantum corrections can alter the classical decomposition, lift branches, or connect branches that are classically distinct. The monopole operator provides the complex coordinate on the Coulomb branch by combining $\vec c(\mu)$1 with the dual photon: $\vec c(\mu)$2 It is a disorder operator: its definition imposes a singular magnetic-flux boundary condition rather than being given directly by a polynomial in elementary fields.

The SQED–$\vec c(\mu)$3 example illustrates the quantum geometry of moduli space. Classically, SQED with one flavor has a Higgs branch parametrized by the meson $\vec c(\mu)$4 and a cylindrical Coulomb branch. Quantum corrections split the Coulomb branch into two components parametrized by monopole operators $\vec c(\mu)$5 and $\vec c(\mu)$6, which meet the Higgs branch at a common singular point. These three branches match the three coordinate axes of the $\vec c(\mu)$7 model subject to

$\vec c(\mu)$8

The operator map is

$\vec c(\mu)$9

The implication is not merely that the two theories have equal spectra: their protected moduli-space geometries and global-symmetry actions are identified.

The book also states an important qualification. In 3d $(2,0)$0 theories, mirror symmetry should not generally be described as a simple exchange of Higgs and Coulomb branches. That slogan is more accurate for $(2,0)$1 theories. Quantum corrections can blur the distinction between branches, and a nongauge dual description may have only ordinary Higgs-type coordinates corresponding to both mesonic and monopole operators of the gauge theory.

Localization and the partition-function dictionary

The $(2,0)$2 partition function is introduced as a controlled projection of QFT data. It is not claimed to encode every observable, but it is sufficiently rich to test dualities and sufficiently protected to compute exactly. Supersymmetric localization reduces the infinite-dimensional path integral to a finite-dimensional integral over constant Coulomb-branch variables. The result is expressed through quantum dilogarithm functions.

For a 3d $(2,0)$3 theory, the localized partition function receives classical contributions from Chern–Simons and FI terms and one-loop contributions from vector and chiral multiplets. Schematically, the latter are products of double-sine or quantum-dilogarithm factors. The gauge coupling does not appear in the localized answer because the Yang–Mills kinetic term is $(2,0)$4-exact. This has a strong consequence: the partition function computed in a weakly coupled ultraviolet description can equal the partition function of the strongly coupled infrared fixed point.

The book uses this fact to translate field-theory dualities into exact mathematical identities. For the SQED–$(2,0)$5 duality, the partition functions are

$(2,0)$6

and

$(2,0)$7

Their equality is the pentagon identity of the quantum dilogarithm. The implication is methodological: a nonperturbative field-theory equivalence can be established through a finite-dimensional integral identity, provided the relevant protected observable is sufficiently discriminating.

The $(2,0)$8 partition function also exposes the relation between three-dimensional and two-dimensional theories. In the limit $(2,0)$9, the quantum dilogarithm becomes a classical dilogarithm, and the squashed sphere approaches an $\mathcal{N}=4$0 compactification relevant to a 2d $\mathcal{N}=4$1 theory. The leading asymptotics define an effective twisted superpotential $\mathcal{N}=4$2, whose critical-point equations determine the two-dimensional vacua. Kaluza–Klein modes appear as the infinite tower encoded in the logarithmic asymptotics of the hyperbolic sine. Thus the same partition-function identity has both a quantum 3d interpretation and a semiclassical 2d interpretation.

Domain walls, canonical transformations, and quantization

A major conceptual step is the reinterpretation of the $\mathcal{N}=4$3 partition function as a wave function. If a 3d theory has a $\mathcal{N}=4$4 global symmetry with scalar parameter $\mathcal{N}=4$5, then $\mathcal{N}=4$6 transforms under $\mathcal{N}=4$7 as a wave function under canonical transformations. Specifically, $\mathcal{N}=4$8 multiplies the wave function by a quadratic phase,

$\mathcal{N}=4$9

while $SL(2,\mathbb{Z})$0 acts by Fourier transformation,

$SL(2,\mathbb{Z})$1

For a general matrix

$SL(2,\mathbb{Z})$2

the transformation is an integral operator with quadratic kernel determined by a generating function $SL(2,\mathbb{Z})$3. The associated canonical variables satisfy the $SL(2,\mathbb{Z})$4 transformation law. In the semiclassical limit, the integral reduces to a Legendre transformation; at finite $SL(2,\mathbb{Z})$5, it is a quantum-mechanical transformation with effective Planck constant $SL(2,\mathbb{Z})$6.

This formalism gives a physical origin for the Hilbert space: the 3d theory is a boundary or domain-wall theory of a 4d $SL(2,\mathbb{Z})$7 theory. The 4d gauge fields appear as background fields from the 3d perspective, so the 3d theory acts as an operator between Hilbert spaces associated with the two sides of the wall. A theory with two flavor symmetries becomes an operator kernel,

$SL(2,\mathbb{Z})$8

Composition of domain walls corresponds to composition of operators, and reversing orientation corresponds to taking an inverse or dual morphism.

The resulting categorical language is not merely terminological. It explains why gauging, gluing, and duality transformations compose algebraically. It also identifies the 3d $SL(2,\mathbb{Z})$9 action with the duality group of the 4d bulk. The $\mathcal{N}=1$0 transformation is a jump in the four-dimensional theta angle across the wall; the $\mathcal{N}=1$1 transformation exchanges electric and magnetic variables. In the non-Abelian case, the $\mathcal{N}=1$2 theory supplies the domain wall associated with $\mathcal{N}=1$3-duality, including the exchange between $\mathcal{N}=1$4 and its Langlands-dual global form.

The six-dimensional origin and the 3d–3d correspondence

The geometric framework becomes explicit through compactification of the 6d $\mathcal{N}=1$5 theory. Compactification on a two-manifold $\mathcal{N}=1$6 produces 4d $\mathcal{N}=1$7 theories, while compactification on a three-manifold $\mathcal{N}=1$8 produces 3d $\mathcal{N}=1$9 theories after an appropriate partial topological twist. The same six-dimensional construction therefore relates 4d duality webs to 3d field theories and relates the decomposition of manifolds to the decomposition and gauging of QFTs.

For a torus, the complex structure parameter $\mathcal{N}=2$0 is identified with the complexified coupling of 4d $\mathcal{N}=2$1 theory. Its mapping class group is

$\mathcal{N}=2$2

which acts by fractional-linear transformations on $\mathcal{N}=2$3. The geometric invariance of the torus under changes of basis in its one-cycles becomes the field-theoretic $\mathcal{N}=2$4 duality. This is one of the book’s clearest examples of a nonperturbative gauge-theory statement emerging from a simple geometric statement.

For a general Riemann surface, the mapping class group is identified with the duality group of the associated 4d theory. For a three-manifold with boundary

$\mathcal{N}=2$5

the manifold is a cobordism between the two-dimensional surfaces. It therefore defines a map between the Hilbert spaces associated with $\mathcal{N}=2$6 and $\mathcal{N}=2$7. Gluing cobordisms corresponds to composing maps, while orientation reversal corresponds to dualization. These are precisely the Atiyah–Segal axioms of a TQFT.

The resulting three-dimensional topological theory is complex Chern–Simons theory. The identification is motivated by exchanging the order of compactification: compactifying the six-dimensional theory first on the three-manifold gives a 3d theory on the remaining spacetime, while compactifying first on the spatial $\mathcal{N}=2$8 produces a theory on the three-manifold whose canonical quantization yields a Hilbert space on $\mathcal{N}=2$9. The cobordism structure and the gluing rules then force the TQFT interpretation.

The geometric claim can be summarized as follows:

A duality frame corresponds to a decomposition of a manifold; gauging corresponds to gluing geometric pieces; field-theory parameters and vacua correspond to geometric data; and duality corresponds to the non-uniqueness of the decomposition.

The later chapters develop this claim using ideal tetrahedral decompositions, hyperbolic geometry, complex Chern–Simons theory, Teichmüller theory, knot and braid constructions, and cluster algebras. The quantum dilogarithm identity associated with 3d mirror symmetry becomes the algebraic shadow of a Pachner move between triangulations. Consequently, a local change in a triangulation induces an infrared duality between the associated 3d theories.

Limitations and open questions

The paper’s principal limitation is foundational rather than computational. The general space of QFTs is not defined mathematically, and the proposed “geometry of theory space” is developed through selected protected quantities and supersymmetric constructions. Equality of $(2,0)$00 partition functions, vacuum moduli spaces, or anomaly data is strong evidence for duality, but does not by itself establish equality of all observables.

Several arguments also depend on conjectural input. The existence and properties of the 6d $(2,0)$01 theory are not derived intrinsically within ordinary field theory. The correspondence between compactification data and lower-dimensional theories is supported by string theory, duality checks, and protected computations. Likewise, the 4d $(2,0)$02 $(2,0)$03 duality is described as extensively checked but not rigorously proven in the generality used.

The gauging construction requires further qualifications. Anomaly cancellation, global forms of gauge groups, discrete quotients, counterterms, and the renormalization of newly introduced gauge couplings can affect whether an infrared duality survives gauging. The text acknowledges that dualities valid only at an IR fixed point do not automatically remain valid after adding dynamical gauge fields. The subsequent 3d examples are supported by partition-function identities and geometric constructions, but the general closure of dualities under gauging remains conditional.

The localized partition function also has normalization subtleties. Gravitational Chern–Simons terms and framing anomalies can alter mass-independent phases and constants. The book often suppresses these factors because they do not affect the parameter-dependent identities under consideration. This is adequate for many duality checks but insufficient for a complete functorial or fully extended TQFT formulation.

Finally, the geometric correspondence is developed most concretely for supersymmetric theories with Lagrangian or effectively Lagrangian descriptions. Its applicability to nonsupersymmetric theories, nonrelativistic systems, and genuinely non-Lagrangian sectors is proposed as a structural possibility rather than established by the analysis.

Conclusion

The book’s principal contribution is a coherent framework for interpreting dualities, gauging, partition functions, domain walls, and compactification as manifestations of geometry in QFT theory space. Its central examples show how nonperturbative equivalences become geometric identities: mapping-class-group actions yield gauge-theory dualities, quantum-dilogarithm identities encode 3d mirror symmetry, and manifold decompositions encode duality frames. The 3d–3d correspondence is therefore presented not as an isolated dictionary but as a realization of a broader principle: the organization of quantum field theories, their observables, and their relations can carry geometric and categorical structure.