AKSZ-type Topological Quantum Field Theories and Rational Homotopy Theory

AKSZ-type Topological Quantum Field Theories and Rational Homotopy Theory

Hyungrok Kim111Department of Physics, Stony Brook University
Abstract

We reformulate and motivate AKSZ-type topological field theories in pedestrian terms, explaining how they arise as the most general Schwartz-type topological actions subject to a simple constraint, and how they generalize Chern–Simons theory and other well known topological field theories, in that they are gauge theories of flat connections of higher gauge groups (infinity-Lie algebras).

Their Euler–Lagrange equations define quasifree graded-commutative differential algebras, or equivalently -algebras, the equivalent of the Lie algebra of the gauge group; we explain how integrating out auxiliary fields in physics corresponds to taking the Sullivan minimal model of this algebra, and how the correspondence between fields and gauge transformations realizes Koszul duality.

Using this dictionary, we can import topological invariants and notions (e.g. the rational LS-category) to apply to this class of theories.

1 Introduction

It is by now well accepted that particular classes of physical theories correspond to particular classes of geometries: e.g. a classical particle can live on a Riemannian manifold; more complicated physical models (e.g. including external electromagnetic fields) correspond to additional geometric structure on this manifold; a quantum particle sees the Laplacian spectrum instead; a classical or quantum string sees a yet different structure; certain superconformal field theories can be thought of as sigma models on the affine schemes defined by their chiral rings; vertex- or edge-weighted graphs correspond to Ising models with inhomogeneous magnetic field or temperature; 1-factorable graphs to dimer models; and so on.

In this work we consider a class of physical theories, the AKSZ-type TQFT, that represent spaces up to a very weak notion of equivalence (real homotopy equivalence), and equipped with a generalization of an invariant binary form. The study of these topological structures is called rational homotopy theory, initiated by Sullivan. AKSZ-type theories are a class of topological field theories with differential-form fields (including scalars and Yang–Mills fields), originally proposed in [aksz], motivated by considerations of BRST quantization and mirror symmetry; later deep relations with higher category theory and homotopy theory were found (see e.g. references in [frs]).

In this work, we explain how various constructions in rational homotopy theory, including Koszul duality and Sullivan resolution, correspond to natural operations on these physical theories, and conversely how homotopic invariants define interesting invariants of these physical theories.

The structure of gauge symmetries in this class of theories is richer than the typical case: while (in the minimal case) they form a Lie supergroup as usual, in addition they have higher-order operations, corresponding to higher-order brackets in an -algebra.

There are two central dichotomies that we wish to highlight in the class of topological field theories in question: simply connected vs. multiply connected and minimal vs. nonminimal. Simply connectedness means, in topology, that the degree-1 and degree-0 homotopy groups be trivial; the corresponding feature in physics is that fields of differential-form degrees 0 and 1 — that is, the scalars and the Yang–Mills fields — be absent. In topology, the failure of simply-connectedness is manifest in e.g. the failure of ratinoal homotopy theory to work well for such spaces;222In this subject, one typically requires several simplifying assumptions on these low-degree homotopy groups, viz. that the space in question be simple or nilpotent [fht]. in physics, this manifests as an essential non-Abelianness of the theories in question: higher-degree gauge fields of degree higher than that of the differential are, in a certain sense, Abelian, due to mere degree reasons.

The other dichotomy that we wish to highlight is that of minimality. The terminology is borrowed from rational homotopy theory, where one speaks of minimal or non-minimal Sullival algebras; essentially, non-minimal theories are those containing a “mass” (i.e. quadratic) term in their potentials, while in non-minimal theories these terms are absent. The presence of such terms renders some of the fields in the action to be auxiliary; their function is similar to that of the auxiliary fields occurring in the Mathai–Quillen formalism for cohomological (Witten-type) topological field theories in an action of the form

(see [lm, Chapter 6] for a pedagogical discussion). Being auxiliary, these fields can be integrated out and, in the simply connected case, leaves another AKSZ-type theory without the mass terms. Their elimination corresponds to the minimization of a non-minimal Sullivan algebra into a minimal one in topology, which essentially amounts to eliminating, in a cellular complex, filled balls, which do not affect the homotopy type of a space. Complications arise if both multiply-connectedness and non-minimality are present simultaneously: in that case, integrating out the auxiliary fields in general leaves the action in a non-AKSZ form.

1.1 Conventions

All coefficient fields will be that of the real numbers, unless otherwise specified: we regard algebra generators as fields, and these are real-valued in field theory in the physics sense. From a topological standpoint, rational homotopy theory, as its name implies, is usually done over the field of rational numbers; its discreteness mirrors that of the homotopy classes of topological spaces: that is, the space of homotopy classes of topological spaces is discrete, and does not have real moduli; and, if we forget torsion (and assume technical conditions about connectedness and simply-connectedness) these correspond to certain algebras over rationals. Working over the reals instead of the rationals, therefore, has the effect of continuously “interpolating” between different homotopy types, a concept bizarre to a topologist but more familiar to physicists.

We also neglect all super-phenomena.333Except in the trivial sense that we can quotient the -grading into , i.e. that odd-degree generators anticommute.. In physics terms, this means all fields are bosonic; in mathematical terms, this means we work with Lie algebras rather than Lie superalgebras, manifolds rather than supermanifolds, etc., and that things are -graded rather than -graded. Almost everything here can be super-ized in an obvious way, however, by throwing in additional minus signs appropriately.

1.2 Acknowledgements

I would like to thank Martin Roček for his highly helpful discussions, unflagging support, and for initial exploratory ideas on which this paper is based.

I would also like to thank Ingmar Saberi for his invaluable comments and suggestions, as well as Dennis Sullivan for helpful discussions.

2 Review of AKSZ-type theories

As much of the literature [aksz, frs, ikeda, roytenberg] [severa00, Letter 8] of this class of Schwartz-type TQFTs is couched or motivated in terms of either categorical homotopy theory (“-Chern–Weil theory”) or Batalin–Vilkovisky quantization, we find it worthwhile to write down, in plain language, the construction of this class of thoeries as merely Schwartz–type TQFTs.

The procedure boils down to these steps.

  1. Pick your favourite worldvolume dimension .

  2. Put in pairs of differential-form-valued fields, such that their degrees sum to . For example, if a 2-form field exists, then a corresponding -form field exists also. (A 0-form is exceptional in that it will in general live on a target manifold, making the theory a topological -model.) Let the fields be ; using this pairing we can raise and lower this index:

  3. Write down the topological “kinetic term” .

  4. Write down the most general possible “potential term” in terms of wedge products of differerential forms. There will be a finite number of possible terms unless a 0-form exists (in which case one will have an arbitrary smooth function of the scalar form). That is,

    where is a polynomial of wedge products of differential forms whose terms are homogeneous in degree , at least if scalars are absent. Here is an overall constant that is irrelevant in the classical theory.

  5. The most general potential involves a number of structure constants. One must solve for generalized Jacobi identities satisfied by these, which are required for invariance under small gauge transformations.

  6. The theory naturally comes with a set of gauge symmetries, which generalize that of Abelian -form fields and Yang–Mills fields. Every field (other than possibly scalars) will have an associated gauge symmetry.444For scalars, see the discussion in Example 8.

  7. The structure thus produced can be regarded as a flat connection for a flat gerbe or principal bundle (for an -algebra); one can introduce nontrivial boundary conditions instead, producing topologically nontrivial gerbes or bundles.555Note that this is a very different notion than nontrivial flat connections on a (possibly flat) topologically trivial bundles. That is, one only requires that the fields be defined patchwise with respect to an open cover, and allow gauge transformatinos as transition maps, with appropriate cocycle conditions (cf. the discussion in [frs]).666This step requires finite, exponentiated gauge transformations; therefore one must choose the gauge -group corresponding to the gauge algebra, e.g. between and for the algebra .

  8. In the quantum theory, the overall coefficient of this action may be quantized due to the requirement that the action change by under large gauge transformations.

It will be illustrative to look at a few examples.

Example 1 (Particle on symplectic manifold).

In one dimension, we can only have 0-forms (scalars). Let the fields be , to be raised and lowered by constants . Thus, the general action is

(1)

No potential terms are possible. Using integration by parts, we can arrange that . Thus is (locally) a nondegenerate antisymmetric matrix.

The scalars live on a manifold . Locally, the coefficients are constants; this means that is globally a symplectic matrix, by Darboux’s theorem. That is, abstractly regarding as defining a curve

the action is

where is a locally defined antiderivative of the symplectic form:

That is, for a worldvolume that is a closed curve that is the boundary of a surface , the action is

(2)

Now, quantization requires that be well defined regardless of the choice of the surface that bounds the closed worldline . If we normalize (without loss of generality) the de Rham cohomology class of to lie in integral cohomology (modulo torsion),

(3)

that is, require that for every closed surface , then in order for the quantum theory to be well defined.

The equations of motion of this theory are simply

That is, classically, this describes a particle that remains constant on a symplectic manifold. Explicitly, consider the case where is the cotangent bundle over a manifold . Then the action is

(in this case an antiderivative of the symplectic form exists globally) and it can be recognized as the infinite-mass limit of the standard action for a first-quantized particle

where is a Riemannian metic on .

Example 2 (String on Poisson manifold).

When the worldvolume is two-dimensional, we have a scalar and a one-form . The action is

(4)

where the function defining the potential is necessarily antisymmetric. Once again, the scalar lives on a manifold ; then is a covector at the point , and defines an antisymmetric (2,0)-tensor on . That is, the data defines a sigma model

on the cotangent bundle of a manifold . By rescaling and (dimensional analysis), the constant can be absorbed into .

The equations of motion are

(5)

For there to be any nontrivial solutions to the equations of motion, the equations of motion must be compatible with the nilpotence of the exterior derivative (i.e. the Bianchi identity); that is,

(6)

where denotes normalized total antisymmetrization. The bracketed expression can be recognized as the Schouten–Nijnhuis bracket ; that is, defines a Poisson structure on .

Example 3 (Membrane on a Lie algebra, a.k.a. Chern–Simons).

On a three-dimensional worldvolume, we can have a scalar/2-form pair or a 1-form/1-form pair. For simplicity let us treat only the 1-form/1-form case. (The general case is known in the literature as the Courant -model [roytenberg, roytenbergthesis, ikeda].)

Let the 1-form fields be . Then the action is

where is a symmetric matrix and are totally antisymmetric structure constants. The equations of motion are

(7)

The consistency condition reduces to the Jacobi identity for , which therefore defines a Lie algebra. This case is therefore the Chern–Simons theory on a Lie algebra equipped with an invariant nondegenerate bilinear form (not necessarily semisimple). It is well known to exhibit level quantization if this Lie algebra is non-Abelian.

Example 4 (Four dimensions).

On a four-dimensional worldvolume, again suppressing a scalar field, we have a 1-form and a 2-form , with the general action

(8)

The coefficients , and satisfy obvious (anti-)symmetries

(9)

The equations of motion are

(10)
(11)

In order for the Bianchi identities , to hold, the coupling constants must satisfy certain relations.

  • Taking the expression and leaving only the terms proportional to , we obtain

    (12)

    That is, the Jacobi identity holds up to a term proportional to , which can be thought of as the coefficient for a trilinear operation. (Later, we will see that this can be interpreted as a “homotopic Jacobi identity”, where is interpreted as a differential.) Taking the coefficient of in gives the same result.

  • Taking the expression and leaving only the terms proportional to , we obtain

    (13)

    Taking the coefficient of in gives the same result.

  • Taking the coefficient of in gives

    (14)
Example 5 (Six dimensions).

Six is the lowest dimension in which a nontrivial AKSZ-type theory without scalars or 1-forms exists. Avoiding scalars and 1-forms, the most general action contains a 2-form and a 3-form , with the action

(15)

The equations of motion are

(16)

In order for the Bianchi identities to hold, we must have

(17)

So, the fields can be partitioned into two noninteracting groups so that in each group at least one of the coefficients and vanish. When vanishes the theory is quadratic and is trivial. Thus, we can set without loss of generality.

Example 6 ( models).

Consider an -dimensional worldvolume with just a -form and an -form field , with trivial potential:

The equations of motion are

(18)

This is seen to be a generalized Abelian BF model.

If , then we may write a potential term

This is the non-Abelian BF model.

3 Review of mathematical preliminaries

We define certain mathematical terms that we will need for our exposition. All of our ground fields will be real, and all of our associative algebras will be unital; all of our graded algebras will be degreewise finite, and similarly the graded algebroids will be of finite type. The symbol as well as the term natural number will include zero. The mathematically sophisticated reader should feel free to skip this section.

3.1 Algebras with differential

The material in this section goes by the name of rational homotopy theory, and is developed in more detail in e.g. [hess, fht, bt, fot].

Definition 1.

A graded-commutive differential graded algebra, or cdg-algebra for short, is a vector space , with nonnegative-integer grading

with an associative product that respects the grading:

for homogeneous , equipped with a differential

such that

and

The algebra of differential forms forms a cdg-algebra.

A cdg-algebra is called semifree iff it is free as a graded-commutative algebra. A semifree cdg-algebra is called Sullivan iff the generators can be ordered such that the derivative of each generator only depends on the ones preceding it (and excluding itself). A Sullivan cdg-algebra is called minimal Sullivan iff this ordering can be chosen to be nondecreasing with respect to the degree.

This definition, while arbitrary-looking, can be derived from the general abstract framework known as model-category theory, which, given certain input data, generates a definition of “nice things” (bifibrant objects) and a canonical way of constructing approximations to things ((co-)fibrant resolutions) in terms of these nice things, so that homotopy theory can work properly. For the originally motivating example of topological spaces, the “nice” objects in question are cell complexes, constructed by adding “cells” iteratively; for cdg-algebras the “nice” objects are Sullivan algebras, and they can be viewed as analogues of cell complexes, obtained by adding generators iteratively.

Now, every semifree cdg-algebra that lacks degree-0 and degree-1 generators is Sullivan; and if in addition every derivative lacks products of length one (contains only terms of length two or longer), then it is minimal Sullivan. Therefore the difference between semifreeness and Sullivanness is a degree-1 affair.777In general, rational homotopy theory functions best when ignoring degree 1, that is, for simply connected things. This fact is, of course, well known in topology: only the degree-1 homotopy group (fundamental group) can be non-Abelian, for example.

By model-category theory, for every cdg-algebra , there exists a Sullivan algebra and a cdg-algebra homomorphism that induces isomorphisms on their cohomology algebras (“is a quasi-isomorphism”) and is surjective. But this is in general not unique. For every cdg-algebra, there exists a unique minimal Sullivan algebra and a unique cdg-algebra homomorphism that is a quasi-isomorphism; but this is not in general surjective.

The raison d’être of this algebraic framework is to classify topological spaces up to a rough sort of equivalence (“rational homotopy equivalence”), one that induces an equivalence on all homotopy groups tensored with a characteristic-0 field in question (for us, the real numbers). On every manifold , the algebra of differential forms is a real cdg-algebra. More generally, on every topological space in which a simplicial structure can be given, cdg-algebras of differential forms can be defined. The central statement of rational homotopy theory is that

(real) minimal Sullivan algebras are bijection with equivalence classes under real homotopy equivalence.

That is, a minimal Sullivan algebra uniquely and canonically encodes the data of this equivalence class.

3.2 Generalizations of Lie algebras

An -algebra is a homotopy-theoretic generalization of a graded Lie superalgebra. That is, it comes equipped with a differential, and the super-Jacobi identities hold up to exact terms. That is, the (binary) Lie bracket satisfies the super-Jacobi identity up to the differential of a certain ternary bracket; this ternary bracket satisfies a generalized Jacobi identity up to the differential of a 4-ary bracket, and so on. In fact, the differential fits into this scheme as the unary bracket, and its super-Jacobi identity is just the familiar Leibniz rule for derivations. The brackets are all strictly and totally graded-anticommutative.

While clear in concept, written out, the definition becomes quite complicated.

Definition 2.

An -algebra consists of an -graded vector space equipped with a series of -linear operators for each positive integer , each of degree , that is,

that is

  • totally graded-anticommutative, that is, switching any two elements of degrees and produces a sign ;

  • such that the homotopy Jacobi identity holds:

    where spans over the so-called shuffle permutations, which essentially means one does not overcount terms that are same except for some trivial permutation of the arguments of the brackets; and is the usual sign of a permutation, with an additional minus sign for each exchange of pair of odd elements.

This definition has the following special cases:

  • A differential graded Lie algebra is a graded Lie superalgebra equipped with a differential of degree that satisfies the graded Leibniz rule with respect to the Lie bracket; it is straightforward to check that this is exactly the same as an -algebra, where all brackets of arity higher than three vanish.

  • If all brackets other than the binary ones vanish (including the unary one), then an -algebra reduces to a graded Lie superalgebra. If, additionally, all elements are in degree , then this is equivalent to a Lie algebra.

  • If all brackets other than the unary one vanish, then an -algebra reduces to a cochain complex.

The above definition, while conceptually simple, is totally unmanageable. Thankfully, there exists an alternate, equivalent, and much simpler definition:

An -algebra is the same thing as a semifree cdg-algebra.

This equivalence goes by the name of Koszul duality. Concretely, given an -algebra with homogeneous basis , we construct a semifree cdg-algebra as follows. For each -basis element , we put in a cdg-algebra generator888We use raised indices because the Chevalley–Eilenberg algebra is (as an algebra) freely generated by the dual space of the -algebra. The canonical inner product between a vector space and its dual space appears in the differential, for example. We can choose not to dualize, but then we would be working with coalgebras, not algebras. of degree .999This annoying convention is needed to make the usual graded-commutativity rules work. These are equipped with a differential

where is the canonical pairing between the cdg-algebra generators and the -basis elements. This semifree cdg-algebra is called the Chevalley–Eilenberg algebra and denoted .

It is a tedious but straightforward exercise in combinatorics to check that the nilpotence of the Chevalley–Eilenberg differential is exactly equivalent to the homotopy Jacobi identities. Conversely, given a semifree cdg-algebra, one can decompose the differentials by number of terms to reconstruct the -algebra.

It is also worth noting that the case of Koszul duality for a Lie algebra is already familiar: it is the duality between the Lie algebra and the left-invariant differential forms on its associated Lie group. In particular, if the Lie group is compact, then (by an averaging argument) the cohomology of the Chevalley–Eilenberg algebra gives the topological cohomology of the Lie group.

All this is the infinitesimal part of a theory of -Lie groups, the precise formulation of which is best attempted after a long digression in higher category theory, and will not be attempted here.

3.3 Oidification

We shall also occasionally use the terminology -oid instead of . Essentially, this merely means that we consider a family of , parametrized by points of a smooth manifold (the moduli space).101010The definition of such a concept, here informally called the oidification, is also more formally known as horizontal categorification. The paradigmatic example is the relation between a vector space and a vector bundle. Similarly, a Lie algebroid is a moduli space of Lie algebras. A Lie algebra is given by a vector space with additional structure; similarly a Lie algebroid is given by a vector bundle with additional structure. Exactly what this structure is can be given in elementary differential-geometric terms (see e.g. [weinstein96] and references therein), but for us it is simpler to remark that this complication is already subsumed by that of Koszul duality: the concept of a Lie algebroid precisely coincides with that of a cdg-algebra, whose degree-0 part is of the form on a smooth manifold (the moduli space), and which is otherwise freely generated by only degree-1 generators. Relaxing the last condition gives us the concept of -algebroid, which is (the Koszul dual of) a cdg-algebra, freely generated except in degree 0, and whose degree-0 part is of the form . Once again, this boils down to a graded vector bundle on a manifold with a set of -ary operations on its sections satisfying certain complicated identities.

We will finally note that all this is to be regarded as the infinitesimal part of a theory of Lie (or ) groupoids. A Lie groupoid (again see [weinstein96]) is a parametrized family of Lie groups, except that, by virtue of the exponentiation, it consists of elements that “start” and “end” at two points on the moduli space; if one only remembers those elements that start and end at the same point, then one has a smooth family of Lie groups.

3.4 Invariant forms

Given the Chevalley–Eilenberg algebra of an -algebra , the Weil algebra is the cdg-algebra obtained by adjoining, to each generator , a new generator with degree , equipped with a new differential defined as

(19)

These may be summarized compactly as

(20)

(Structures of this kind, involving (partially-)commuting sets of nilpotent differentials, are analysed in detail in [ks].)

The Weil algebra comes equipped with the obvious quotient cdg-algebra homomorphism

(21)

obtained by killing the generators of the form . The raison d’être of the Weil algebra is that it has trivial cohomology; in fact, if is the Lie algebra of a compact Lie group , the above quotient homomorphism is an algebraic model of the (total space of the) classifying principal bundle .111111This observation, in the form of Lie algebra cohomology, is how the classical names of Chevalley, Eilenberg, and Weil became attached to these algebras.

In the Weil algebra , consider an element of degree that

  • consists entirely of (sums of products of) generators of the form , and

  • is closed (i.e. ).

In the Lie algebra case, this definition reduces to the classical notion of an invariant polynomial of degree (here is necessarily even); e.g. the Killing form is encoded as the degree-4 element .

4 Dictionary between Rational Homotopy Theory and Topological Quantum Field Theory

Using the heaviweight mathematical technology which we have sketched in the previous section, we now proceed to interpret the structure of AKSZ-type topologial field theories in terms of algebraic and topological structures, explaining and elucidating various features of this class of theories.

4.1 Fields as homomorphisms/continuous maps

Given a manifold and an -algebra with homogeneous basis , consider a cdg-algebroid morphism

Because the domain is a semifree cdg-algebra, this map is specified by the images of generators

That is, we obtain a series of differential forms of degree , that must satisfy identities of the form

(22)

corresponding to the preservation of . This can be thought of as flatness conditions for a field strength

In fact, this assertion can be suitably formalized; this datum defines a flat connection in the trivial -valued principal bundle on .121212 On nontrivial -valued principal bundles — sometimes called gerbes — of course potentials cannot be globally defined, and must be defined patchwise.

On the other hand, consider a morphism

Let the images of be and the images of be . Then the equivalent identities become

(23)

corresponding to the preservation of . Now the field strength can be nonzero, but it must nevertheless satisfy the Bianchi identity , corresponding to the nilpotence of in .

So we obtain the following dictionary:

cdg-Algebra Physics
operator covariant derivative
-differential exterior derivative
-differential nonderiv. part of covariant deriv.
nilpotence of Bianchi identity
cdg-algebra morphism potential with field strength
cdg-algebra morphism potential

In particular, if one specializes to the case where is a Lie algebra, one obtains the usual notions for Yang–Mills theory.

4.2 AKSZ-type Lagrangians as symplectic structures

If one examines the equations of motion for the AKSZ-type topological quantum field theories, one notices that the equations of motion are of the form (22). That is, their structure can be encoded by an -algebra , and can be regarded as a theory of flat connections for a -bundle.

We may ask the converse question: which cdg-algebras correspond to encodings of AKSZ-type theories? This is easily answered. First, the kinetic term of the AKSZ-type Lagrangian

defines a bilinear form on the -algebra . Furthermore, by integration by parts, can be assumed without loss of generality to be graded-antisymmetric with respect to the -algebra grading (and not the -grading).

Such forms can be encoded as closed elements in the Weil algebra

that consists of elements of the form , as explained in section 3.4; they necessarily have degree , where is the dimension of the worldvolume. -algebras equipped with the choice of such a structure has been called symplectic -algebras or -manifold in the literature [frs, severa05]: if one considers the equivalent -algebroid notion, then the -algebroid corresponds to a symplectic manifold, and the -algebroid to a Poisson manifold. The -algebra is a quadratic Lie algebra, that is, a Lie algebra equipped with an invariant symmetric binary form. The -algebroid is called a Courant algebroid in the literature [roytenberg, ikeda].

What about the potential in the AKSZ-type Lagrangian? It can be straigthforwardly verified, once one unwinds the definition, that the existence of the potential amounts to the existence of an antiderivative of the symplectic element , such that maps to under the canonical forgetful cdg-algebra homomorphism . An antiderivative of always exists, since the Weil algebra has trivial cohomology by construction.

That is, we have the following correspondence [severa00, severa05, frs]:

Geometry Physics
symplectic manifold 1-dimensional AKSZ-type theory
Poisson manifold 2-dimensional AKSZ-type theory
Courant algebroid 3-dimensional AKSZ-type theory
quadratic Lie algebra 3D AKSZ-type theory without scalars

4.3 Gauge transformations as Koszul duality

Given that AKSZ-type theories are a theory of flat connections over gerbes with -algebra fibre, it should therefore be the case that the gauge transformations are also valued in -algebras (at least for topologically trivial gerbes), the same way that a gauge transformation for a -bundle for a Lie group are smooth -valued functions (if the principal bundle is topologically trivial and/or is Abelian).

We explain how this is in fact the case. The relation between gauge transformations and gauge fields is the same one as that between -algebras and semifree cdg-algebras, or Koszul duality, as detailed in the following examples.

In general, in an -algebra, the Jacobi identity only holds up to homotopy (“on-shell”). However, the semifree cdg-algebra is minimal Sullivan, that is, if the derivatives of each generator does not have linear terms (only quadratic or higher), then the differential (unary bracket) in the -algebra is zero, and the Jacobi identity for the binary bracket holds exactly. In this case, the Jacobi identity holds exactly (“off-shell”), and the algebra of gauge transformations forms a Lie superalgebra (or a Lie supergroup, when exponentiated).

Nevertheless, even in this case, the algebra of gauge transformations is more than a mere Lie superalgebra, because the higher-arity brackets do not vanish in general.131313The following two statements are equivalent: (1) The -ary bracket (i.e. differential) vanishes; (2) the differential in the associated cdg-algebra lacks th-order terms. In other words, the algebra of gauge transformations has additional structure beyond the Lie (super-)bracket.

Example 7 (3D, continued).

The equations of motion of Chern–Simons theory, namely , determine a cdg-algebra, which is . Its dual -algebra is the Lie algebra . Gauge transformations (for a trivial principal bundle) are parametrized by maps ; infinitesimally they are given by maps .141414Of course, in general, on a principal bundle a gauge transformation is given by a section of the associated bundle (where acts on itself by conjugation), and an infinitesimal gauge transformation is given by a section of the associated bundle . When is trivial this reduces to and ; this identification is canonical, independent of the choice of a global section of .

The Yang–Mills potentials transform under a gauge transformation

as

That is, the algebra of infinitesimal gauge symmetries is the -algebra dual to the cdg-algebra of the fields; it exponentiates into a Lie group . Note that the torsion/fundamental group of is not determined by the cdg-algebra, which only determines the (torsion-free) cohomology with real coefficients.

Example 8 (2D, continued).

We continue the analysis of the Poisson sigma model (4). This theory has the following infinitesimal gauge symmetry [ikeda, (2.10)]:

(24)
(25)

where the gauge parameter is a section of the vector bundle

The allowed values of the gauge parameter depends on ; that is the gauge symmetry is described by a Lie algebroid, rather than a Lie algebra.

Taking the commutators of the infinitesimal gauge symmetries,

(26)

we obtain

In other words, the structure constants of the gauge Lie algebra are given by , which mirror those in the equation of motion for , as required by Koszul duality.

Example 9 (4D, continued).

Consider the action (8), with . It admits the gauge transformation

(27)

Now, instead of writing out the components and , we take the point of view that defines an -algebra-valued form (a gerbe connection), where the -algebra in question is the one Koszul-dual to the cdg-algebra defined by the field equations. Let the basis elements of the -algebra be and .151515These are analogues of the Pauli matrices for or the Gell-Mann matrices for . Then the gauge transformations are of the form

(28)

The gauge algebra is then expressed in terms of the brackets , etc. Now, the gauge algebra can be derived instead directly from the equations of motion: the condition that a Yang–Mills field be flat translates to the semifree cdg-algebra defined by

(29)

where is totally symmetric and are structure constants for a Lie algebra. (This case is not Sullivan due to degree-1 generators.)

This encodes (is Koszul-dual to) the -algebra of the gauge transformations:

(30)
(31)

All other brackets vanish. The binary brackets are an infinitesimal version of the finite gauge-symmetry composition law

(32)
(33)
(34)

where and are commuting parameters and and are anticommuting. However, this forgets the ternary bracket. The gauge symmetry algebra is more than just a Lie superalgebra; it carries an additional structure, the nonvanishing ternary bracket. In other words, this is an -algebra.

The ternary Jacobi identity means that satisfies the usual Jacobi identity. The next nontrivial Jacobi identity is the quaternary one, which in our case is

(35)

(All others are trivial.) In indices,

(36)
Example 10 (6D continued).

Suppose that we have the semifree cdg-algebra defined by

(37)

This encodes the -algebra

(38)

with all other brackets vanishing. Because only the binary bracket is nonzero, this is a graded superalgebra. The bosonic subalgebra, spanned by , exponentiates into an Abelian Lie group.

This is the infinitesimal version of the (finite) supergroup composition law

(39)

where and are anticommuting parameters and and are commuting.

There being only binary products, the Jacobi identity is the same as that for a Lie superalgebra, and these are all trivial (apart from .). Note that the requirement that be totally symmetric does not come from Jacobi identities; it comes from the existence of a realization as an AKSZ-type TQFT.

4.4 Sullivan resolution

We have seen how an AKSZ-type theory encodes an -algebra equipped with a Chern–Simons element for a symplectic structure, that is, to semifree cdg-algebras, and, if the coefficients can be made to be rational, this in turn encodes a topological space (up to rational homotopy), i.e. a rational homotopy type. However, in the latter step, multiple cdg-algebras may correspond to the same rational homotopy type, even if we ignore algebra isomorphisms. However, there exists a canonical such algebra, the so-called Sullivan minimal model. We discuss the physics interpreation of this canoncalization.

The process of taking this canonical form has two parts: one easy, one hard.

  • The easier part is in eliminating pairs of fields, of the form and , that can be eliminated. (Topologically, this can be thought of as adding a filled ball to a cell complex, which is topologically trivial.) Elimination of such pairs correspond to integrating out auxiliary fields via substitution into the action.

  • The more complicated part consists of dealing with degree-1 generators. In topological literature one usually makes the simplifying assumption that they are absent. In physics they correspond to Yang–Mills fields, and Sullivanization corresponds to replacing non-Abelian Yang–Mills fields by higher-degree Abelian fields corresponding to Chern–Simons forms.

We discuss each in turn.

4.4.1 Integrating out Yang–Mills fields

The obstruction to a semifree cdg-algebra being Sullivan lies in degree 1, or equivalently Yang–Mills gauge fields. Let us suppose that, in fact, the geometrical realization of is simply connected.161616This restriction can be slightly relaxed to allow Abelian fundamental groups that act trivially on higher homotopy groups, the so-called simple spaces. Then Sullivan resolution produces a theory with the same set of non-auxiliary171717that is, exclusing thing like pairs of auxiliary fields found in non-minimal Sullival algebras, but including possibly composite fields (such as the Chern–Simons forms) Abelian higher gauge fields, but with no Yang–Mills fields.

Note, however, that the resulting (minimal) Sullivan model may not admit a symplectic structure. This does not mean that they cannot be TQFTs; they can usually be embedded into a bigger minimal Sullivan algebra that can be given a symplectic structure (by an algebraic analogue of taking the cotangent bundle of a manifold).

Example 11.

As a paradigmatic example, we consider Chern–Simons theory.

Consider the cdg-algebra given by

(40)

This is a semifree model for the 3-sphere, and is the Chevalley–Eilenberg algebra of .

As a 3-sphere, its minimal model is simply

(41)

The cdg-algebra homomorphism

is a (non-injective) quasi-isomorphism.

If we are willing to sacrifice minimality, then we can instead use the non-minimal Sullivan algebra

(42)
(43)
(44)

The cdg-algebra homomorphism

(45)
(46)

is a surjective quasi-ismomorphism.

The Sullivan resolution, therefore, describes a theory of Abelian 3-form gauge field . Being one-dimensional, it does not, of course, admit a symplectic structure. In a three-dimensional worldvolume, the field equation is trivial (all 3-forms are automatically closed), but one can easily consider higher-dimensional AKSZ-type theories with a 1-form coupled to other fields:

Then the field equation of the Abelian 3-form field thus defined becomes nontrivial.

More generally, let be any compact Lie group of rank , with Lie algebra . Let the degrees of its invariant polynomials be . Then is rational-homotopy-equivalent to the product of odd-dimensional spheres:181818This is due to the fact that the cohomology ring of such a group is a Hopf algebra, and thus generated by odd-degree elements (Hopf’s theorem), coupled with the formality (in the rational-homotopy sense) of such groups as spaces.

The corresponding minimal Sullivan algebra is therefore of the form

(47)
(48)

That is, Sullivanization amounts to integrating out some degrees of freedom, leaving behind the composite Abelian gauge fields corresponding to Chern–Simons forms (of invariant polynomials).

4.4.2 Integrating out auxiliary terms

The previous section discussed the obstruction to semifree cdg-algebras being Sullivan; what about Sullivan cdg-algebras being minimal Sullivan? In the absence of degree-1 elements (Yang–Mills fields), this amounts to the absence of linear terms in the derivatives, which integrates to the absence of quadratic terms in the potential of an AKSZ-type field.

Example 12 (2d, continued).

At any given point , a neighbourhood of the Poisson manifold can be decomposed [Weinstein, Theorem 2.1]as

where is a symplectic manifold, is a Poisson manifold with , and such that the Poisson structure on coincides with the product Poisson structure. Due to this, the fields corresponding to and to decouple, and we may consider the two cases separately.

Symplectic case

Consider the case where the Poisson manifold is symplectic, with the inverse

In that case, the equation of motion

actually represents as the auxiliary field

Thus, it can be substituted away in the action, which leaves

This is the action of the A-model in topological string theory [aksz, ikeda].

Locally rank zero case

Consider the case where the Poisson manifold is such that at the “origin” . Then the cotangent space is naturally imbued with the structure of a real Lie algebra , whose coefficients are the first-order derivatives of at . In this case, the Lie-Poisson structure on provides a linear approximation of near [Weinstein]. Near , the action becomes

where the neighbourhood of has been identified with . Now, represents a -valued 1-form on , which we will regard as a -connection. Then is a scalar field taking values in the coadjoint representation of . Then the theory reduces to a model, up to higher-order terms; the equations of motion for and reduce to

(49)

up to higher-order terms: is a flat connection and is covariantly constant, to leading order.

Example 13 (4d, continued).

We continue the analysis of the action (8). Diagonalizing , we can partition the allowed values of the index into two sets, one of which we will label as and the other as , such that

(50)

and such that is a nondegenerate symmetric quadratic form (but not necessarily positive-definite). We will use and its inverse to raise and lower indices freely.

Using the equations of motion (13), we can see that

and

Thus there are the following components:

(51)

Now, from the equation of motion

(52)

it follows that are auxiliary fields, which can be substituted away in the action in favour of and .

Now, we can integrate out the nondegenerate part of as follows. The part of the action (8) containing are

(53)

Substituting, we find

(54)

where the square involves contraction of the -index.

Formally, this is caused by the cyclic dependency in the expressions of the derivatives, which are violations of the Sullivan condition for cdg-algebras. After integrating out in terms of and , we have the dependency graph

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