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August 5, 2011

AKSZ Sigma-Models

Posted by Urs Schreiber

This is a continuation of the series of posts on sigma-model quantum field theories. It had started as a series of comments in

∞-Dijkgraaf-Witten theory

and continued in

String Topology Operations as a Sigma-Model.

Here I indicate the original definition of the class of models called AKSZ sigma-models (see there for a hyperlinked version of the following text).

In a previous post on exposition of higher gauge theories as sigma-models I had discussed how ordinary Chern-Simons theory is a σ\sigma-model. Indeed this is also a special case of the class of AKSZ σ\sigma-models.

In a followup post I will explain that AKSZ sigma-models are characterized as precisely those ∞-Chern-Simons theories that are induced from invariant polynomials which are both binary and non-degenerate. (Which is incidentally precisely the case in which all diffeomorphisms of the worldvolume can be absorbed into gauge transformations.)

Recall that a sigma-model quantum field theory is, roughly, one

  • whose fields are maps ϕ:ΣX\phi : \Sigma \to X to some space XX;

  • whose action functional is, apart from a kinetic term, the transgression of some kind of cocycle on XX to the mapping space Map(Σ,X)\mathrm{Map}(\Sigma,X).

Here the terms “space”, “maps” and “cocycles” are to be made precise in a suitable context. One says that Σ\Sigma is the worldvolume , XX is the target space and the cocycle is the background gauge field .

For instance an ordinary charged particle (such as an electron) is described by a σ\sigma-model where Σ=(0,t)\Sigma = (0,t) \subset \mathbb{R} is the abstract worldline, where XX is a smooth (pseudo-)Riemannian manifold (for instance our spacetime) and where the background cocycle is a circle bundle with connection on XX (a degree-2 cocycle in ordinary differential cohomology of XX, representing a background electromagnetic field : up to a kinetic term the action functional is the holonomy of the connection over a given curve ϕ:ΣX\phi : \Sigma \to X.

The σ\sigma-models to be considered here are higher generalizations of this example, where the background gauge field is a cocycle of higher degree (a higher bundle with connection) and where the worldvolume is accordingly higher dimensional – and where XX is allowed to be not just a manifold but an approximation to a _higher orbifold (a smooth ∞-groupoid).

More precisely, here we take the category of spaces to be smooth dg-manifolds. One may imagine that we can equip this with an internal hom Maps(Σ,X)\mathrm{Maps}(\Sigma,X) given by \mathbb{Z}-graded objects. Given dg-manifolds Σ\Sigma and XX , their canonical degree-1 vector fields v Σv_\Sigma and v Xv_X act on the mapping space from the left and right. In this sense their linear combination v Σ+kv Xv_\Sigma + k \, v_X for some kk \in \mathbb{R} equips also Maps(Σ,X)\mathrm{Maps}(\Sigma,X) with the structure of a differential graded smooth manifold.

Moreover, we take the “cocycle” on XX to be a graded symplectic structure ω\omega, and assume that there is a kind of Riemannian structure on Σ\Sigma that allows to form the transgression

Σev ω:=p !ev ω \int_\Sigma \mathrm{ev}^* \omega := p_! \mathrm{ev}^* \omega

by pull-push through the canonical correspondence

Maps(Σ,X)pMaps(Σ,X)×ΣevX, \mathrm{Maps}(\Sigma,X) \stackrel{p}{\leftarrow} \mathrm{Maps}(\Sigma,X) \times \Sigma \stackrel{ev}{\to} X \,,

where on the right we have the evaluation map.

Assuming that one succeeds in making precise sense of all this one expects to find that Σev *ω\int_\Sigma \mathrm{ev}^* \omega is in turn a symplectic structure on the mapping space. This implies that the vector field v Σ+kv Xv_\Sigma + k\, v_X on mapping space has a Hamiltonian SC (Maps(Σ,X))\mathbf{S} \in C^\infty(\mathrm{Maps}(\Sigma,X)). The grade-0 components S AKSZS_{\mathrm{AKSZ}} of S\mathbf{S} then constitute a functional on the space of maps of graded manifolds ΣX\Sigma \to X. This is the AKSZ action functional defining the AKSZ σ\sigma-model with target space XX and background field/cocycle ω\omega.

In the original article this procedure is indicated only somewhat vaguely. The focus of attention there is a discussion, from this perspective, of the action functionals of the 2-dimensional σ\sigma-models called the A-model and the B-model .

In a review by Dmitry Roytenberg, a more detailed discussion of the general construction is given, including an explicit and general formula for S\mathbf{S} and hence for S AKSZS_{\mathrm{AKSZ}} . For {x a}\{x^a\} a coordinate chart on XX that formula is the following.

Definition For (X,ω)(X,\omega) a symplectic dg-manifold of grade nn, Σ\Sigma a smooth compact manifold of dimension (n+1)(n+1) and kk \in \mathbb{R}, the AKSZ action functional

S AKSZ,k:SmoothGrMfd(𝔗Σ,X) S_{\mathrm{AKSZ},k} : \mathrm{SmoothGrMfd}(\mathfrak{T}\Sigma, X) \to \mathbb{R}

(where 𝔗Σ\mathfrak{T}\Sigma is the shifted tangent bundle)


S AKSZ,k:ϕ Σ(12ω abϕ ad dRϕ b+kϕ *π), S_{\mathrm{AKSZ},k} : \phi \mapsto \int_\Sigma \left( \frac{1}{2}\omega_{ab} \phi^a \wedge d_{\mathrm{dR}}\phi^b + k \, \phi^* \pi \right) \,,

where π\pi is the Hamiltonian for v Xv_X with respect to ω\omega and where on the right we are interpreting fields as forms on Σ\Sigma.

This formula hence defines an infinite class of σ\sigma-models depending on the target space structure (X,ω)(X, \omega), and on the relative factor kk \in \mathbb{R}. It was noticed from the beginning that ordinary Chern-Simons theory is a special case of this for ω\omega of grade 2, as is the Poisson sigma-model for ω\omega of grade 1 (and hence, as shown there, also the A-model and the B-model). The general case for ω\omega of grade 2 has been called the Courant sigma-model .

One nice aspect of this construction is that it follows immediately that the full Hamiltonian S\mathbf{S} on mapping space satisfies {S,S}=0\{\mathbf{S}, \mathbf{S}\} = 0. Moreover, using the standard formula for the internal hom of chain complexes one finds that the cohomology of (Maps(Σ,X),v Σ+kv X)(\mathrm{Maps}(\Sigma,X), v_\Sigma + k v_X) in degree 0 is the space of functions on those fields that satisfy the Euler-Lagrange equations of S AKSZS_{\mathrm{AKSZ}}. Taken together this implies that S\mathbf{S} is a solution of the “master equation” of a BV-BRST complex for the quantum field theory defined by S AKSZS_{\mathrm{AKSZ}}. This is a crucial ingredient for the quantization of the model, and this is what the AKSZ construction is mostly used for in the literature.

Posted at August 5, 2011 1:02 AM UTC

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4 Comments & 1 Trackback

Re: AKSZ Sigma-Models

in re: binary and non-degenerate.

why binary? you have in mind symplectic as later?

additional generality?

(Which is incidentally precisely the case in which all diffeomorphisms of the worldvolume can be absorbed into gauge transformations.)

assuming something about the dim of the world volume?

Posted by: jim stasheff on August 5, 2011 2:51 PM | Permalink | Reply to this

Re: AKSZ Sigma-Models

why binary? you have in mind symplectic as later?

There are different answers to this, depending on what you mean to ask.

The simple answer is: that’s the case that happens to be called the “AKSZ model”. If it weren’t binary, it would be called something else! :-)

The authors behind this acronym, in turn, were motivated by understanding the 2d σ\sigma-models called the A-model and the B-model (famous as “the topological string”) . The main point of their article was that these are obtained by choosing a special BV-gauge fixing of a 2-dimensional “AKSZ model”.

But of course then one can ask if there is generally something special about the case of L L_\infty-algebroid Chern-Simons theory where the invariant polynomial is binary.

And, yes, it turns out that an L L_\infty-algebroid that carries a binary and non-degenerate invariant polynomial of grade nn is special in that it is a higher generalization of a symplectic manifold: a symplectic Lie n-algebroid.

For n=0n = 0 this is an ordinary symplectic manifold. For n=1n = 1 it is a Poisson Lie algebroid. For n=2n = 2 it is a Courant Lie 2-algebroid.

We can ask further: what is so special about this higher symplectic case from the point of view of L L_\infty-algebroid Chern-Simons theory? I think one answer is what I suggested: in this case the action of any (small) diffeomorphism on the space of fields of the \infty-Chern-Simons theory is equivalent to some (small) gauge transformation on the fields. This is well known for ordinary Chern-Simons theory, and the proof directly generalizes.

And this is independent of the dimension. It depends however on the fact that the equations of motion imply that the curvature forms all vanish. This is what the binary non-degeneracy of the invariant polynomial guarantees.

All this is dicussed a bit in more detail in this entry here

∞-Chern-Simons theory – covariant phase space.

Posted by: Urs Schreiber on August 5, 2011 8:49 PM | Permalink | Reply to this

Re: AKSZ Sigma-Models

Not even sure what you mean by binary! bilinear?

Posted by: jim stasheff on August 6, 2011 1:54 PM | Permalink | Reply to this

Re: AKSZ Sigma-Models

Not even sure what you mean by binary! bilinear?

I mean: taking two arguments. Such as the Killing form

,:𝔤𝔤 \langle -,-\rangle : \mathfrak{g}\otimes \mathfrak{g} \to \mathbb{R}

on a semisimply Lie algebra, but as opposed to for instance the next higher (“second Pontryagin”) form

,,,:𝔤𝔤𝔤𝔤. \langle -,-,-,-\rangle : \mathfrak{g}\otimes \mathfrak{g} \otimes \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R} \,.

I think this should be called binary . Of course, generally, an invariant polynomial is multi-linear, hence an nn-ary one is nn-linear and a binary one is bilinear.

If we were talking about invariant polynomials on an ordinary Lie algebra only, then “binary” is often stated as “of degree 2”. But that is very dangerous terminology once we start talking about the generalization from Lie algebras to L L_\infty-algebras, because that introduces a notion of degree all by itself. In order not to get in conflict with that, I speak of the “arity” of the polynomial explicitly.

Posted by: Urs Schreiber on August 6, 2011 2:41 PM | Permalink | Reply to this
Read the post AKSZ-Models in Higher Chern-Weil Theory
Weblog: The n-Category Café
Excerpt: The class of AKSZ-sigma models has a natural origin in abstract higher Chern-Weil theory.
Tracked: August 6, 2011 2:57 PM

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