sigma-model -- exposition of quantum sigma-models


Quantum field theory


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory




This is a sub-entry of sigma-model. See there for background and context.


Exposition of quantum sigma-models

Above we have discussed some standard classical sigma-models and higher gauge theories as sigma-models, also mostly classically. Here we talk about the quantization of these models (or some of them) to QFTs: quantum σ\sigma-models .

Particle on the line

for the moment see the discussion at

Topological string on a smooth manifold and string topology operations

Chas and Sullivan famously noticed, that the homology groups of the free loop space LXL X of a compact oriented smooth manifold XX are equipped with an interesting paring operation

H (LX)H (LX)H dimX(LX) H_\bullet(L X) \otimes H_\bullet(L X) \to H_{\bullet - dim X}(L X)

that generalizes the Goldman bracket on H 0(LX)Fπ 1(X)H_0(L X) \simeq F \pi_1(X). Since this operation is induced from concatenating loops, they called it the string product . Its study has come to be known as string topology, now a branch of differential topology.

It was soon realized that there indeed ought to be a relation to string physics: there ought to be a 2-dimensional quantum field theory associated with XX, as follows:

for Σ\Sigma a 2-dimensional surface with incoming and outgoing boundary components inΣinΣout outΣ\partial_{in} \Sigma \stackrel{in}{\to} \Sigma \stackrel{out}{\leftarrow} \partial_{out} \Sigma, the “space of states” of the theory ought to be given by the homology groups H (X inΣ)H_\bullet(X^{\partial_{in} \Sigma}) and H (X outΣ)H_\bullet(X^{\partial_{out} \Sigma}), and the path integral as a pull-push transform along Σ\Sigma ought to be given by push-forward and dual fiber integration

(X out) *(X in) !:H (X inΣ)H dimX(X outΣ) (X^{out})_* \circ (X^{in})^! : H_\bullet(X^{\partial_{in} \Sigma}) \to H_{\bullet- dim X}(X^{\partial_{out} \Sigma})

induced by the mapping space span

X Σ X in X out X inΣ X outΣ. \array{ && X^{\Sigma} \\ & {}^{\mathllap{X^{in}}}\swarrow && \searrow^{\mathrlap{X^{out}}} \\ X^{\partial_{in} \Sigma} &&&& X^{\partial_{out} \Sigma} } \,.

For ΣS 1S 1\Sigma \simeq S^1 \vee S^1 the 3-holed sphere with two incoming and one outgoing circle, this would describe an operation on string states induced by the merging of two closed strings to a single one

H (X S 1S 1)H (LX)×H (LX)H (LX) H_\bullet(X^{S^1 \coprod S^1}) \simeq H_\bullet(L X) \times H_\bullet(L X) \to H_\bullet(L X)

and this ought to be Chas-Sullivan string product operation.

That this is indeed the case was finally demonstrated by Ralph Cohen and Veronique Godin. (See string topology for all references.) While the idea is rather simple, the concrete realization, especially when taking open strings into account, is fairly technical (see for instance this MO discussion).

But there should be more to it: one expects that these operations on homology groups are just a shadow of a refined construction on chain complexes (for instance singular chains): while Godin’s construction gives an HQFT – a quantum field theory that depends only on the homology of the moduli spaces of the relevant cobordisms – one expect that this is the homology of a genuine extended TQFT (which in this dimension is widely but somewhat unfortunately known under the term “TCFT”). Remarks on how that might be obtained have been made in print by Costello and Lurie.

In the context of our discussion of σ\sigma-models, we would want to refine this even one further step and ask: is the string-topology TCFT of a manifold (given that it exists) formally a σ\sigma-model with target space that manifold, and using some suitable background gauge field?

Given that the string topology TCFT itself has not been fully identified yet, we cannot expect a complete answer to this at the moment, but we will discuss some crucial ingredients that are available.

Notably we can first ignore the dynamics of the system, just consider the kinematics and ask the simple question: which quantum σ\sigma-models on XX have (∞,n)-vector spaces of states whose decategorification are graded homology groups H (X Σ)H_\bullet(X^{\partial \Sigma}) of mapping spaces of XX?

The answer to this question is more more transparent after we formulate the question in more generality: as observed by Cohen and Godin in their A Polarized View of String Topology , we may assume without restriction that the homology groups here are with respect to the generalized homology with coefficients in any commutative ∞-ring KK (as long as this is an ”\infty-field” and as long as XX is KK-oriented):

H (X inΣ):=H (X inΣ,K). H_\bullet(X^{\partial_{in} \Sigma}) := H_\bullet(X^{\partial_{in}} \Sigma, K) \,.

Most every statement about ordinary commutative rings has its analog for commutative ∞-rings, and so we can just follow our nose:

An (∞,1)-vector space over KK is an KK-module spectrum and we have an (∞,1)-category KKMod of such \infty-vector spaces. Notice that these are a categorification of the ordinary notion of vector space only in the ”rr“-direction of the lattice of (r,n)-categories. A genuine (∞,n)-vector space over KK – as appears in the description of general nn-dimensional σ\sigma-models – is instead an object of (((KMod)Mod))Mod(\cdots ((K Mod) Mod) \cdots ) Mod . In particular the (,2)(\infty,2)-category of (,2)(\infty,2)-vector spaces over KK is something like (KMod)Mod(K Mod) Mod.

This means that an (∞,1)-vector bundle with flat ∞-connection over some manifold XX is equivalently encoded by an (∞,1)-functor

α:ΠXKMod \alpha : \Pi X \to K Mod

out of the fundamental ∞-groupoid of XX. This assigns to each point of XX a KK-module – the fiber of the (∞,1)-vector bundle thus encoded – to each path in XX an equivalence between the fibers over its endpoints, and so on: this is the higher parallel transport of a flat \infty-connection. We can also think of this as an ∞-representation of ΠX\Pi X on KK-modules, also called a representation up to homotopy . For instance if XX is the classifying space X=BGX = B G of a discrete ∞-group, then flat (∞,1)-vector bundles on XX are precisely ∞-representations of GG.

There is an evident full sub-(∞,1)-category

KLineKMod K Line \hookrightarrow K Mod

of 1-dimensional (,1)(\infty,1)-vector spaces: KK-lines – KK-modules that are equivalent to KK itself regarded as a KK-module.

An (,1)(\infty,1)-vector bundle :Π(X)KMod\nabla : \Pi(X) \to K Mod that factors through this inclusion is a KK-line \infty-bundle .

One finds, as for the case of ordinary 1-vector spaces, that

KLineBGL 1(K)BAut(K) K Line \simeq B GL_1(K) \simeq B Aut(K)

is the delooping of the automorphism ∞-group of KK. This means that KK-line \infty-bundles are equivalently GL 1(K)GL_1(K)-principal ∞-bundles. We can think of the inclusion

ρ:BGL 1(K)KLineKMod \rho : B GL_1(K) \stackrel{\simeq}{\to} K Line \hookrightarrow K Mod

as being the canonical linear ∞-representation of GL 1(K)GL_1(K); and for g:ΠXBGL 1(K)g : \Pi X \to B GL_1(K) a GL 1(K)GL_1(K)-principal ∞-bundle of the (,1)(\infty,1)-vector bundle

ΠXgBGL 1(K)ρKMod \Pi X \stackrel{g}{\to} B GL_1(K) \stackrel{\rho}{\to} K Mod

as the corresponding associated ∞-bundle.

Therefore it makes sense to consider σ\sigma-models with target space XX and background gauge field given by an KK-line \infty-bundle.

For instance for K=KUK = K U the K-theory spectrum, there is a canonical morphism B 2U(1)BGL 1(KU)B^2 U(1) \to B GL_1(KU) and hence to every circle 2-bundle α:ΠXB 2U(1)\alpha : \Pi X \to B^2 U(1) is associated the corresponding KUK U-line \infty-bundle

ΠXαB 2U(1)BGL 1(KU)ρKUMod. \Pi X \stackrel{\alpha}{\to} B^2 U(1) \stackrel{}{\to} B GL_1(K U) \stackrel{\rho}{\to} K U Mod \,.

Or for K=tmfK = tmf the tmf spectrum, there is a canonical morphism B 3U(1)BGL 1(tmf)B^3 U(1) \to B GL_1(tmf) and hence to every circle 3-bundle α:ΠXB 3U(1)\alpha : \Pi X \to B^3 U(1) is associated the corresponding tmftmf-line \infty-bundle

ΠXαB 3U(1)BGL 1(tmf)ρtmfMod. \Pi X \stackrel{\alpha}{\to} B^3 U(1) \stackrel{}{\to} B GL_1(tmf) \stackrel{\rho}{\to} tmf Mod \,.

This was amplified by Ando, Blumberg, Gepner, Hopkins, and Rezk (see the reference at (∞,1)-vector bundle), who notice much of the theory of K-(co)homology – including notably its Thom spectrum theory and its twisted cohomology – is neatly captured by simple statements about such AA-line (,1)(\infty,1)-bundles. For instance the notion of orientation in generalized cohomology simply boils down to the notion of trivialization of such KK-line \infty-bundles:

a vector bundle EXE \to X is KK-oriented precisely if the corresponding Thom space-bundle – which is a sphere spectrum-line \infty-bundle V:Π(X)SLineV: \Pi(X) \to S Line is such that the canonically associated KK-line bundle is trivializable:

(VisKorientable)((Π(X)VSLineKLine)const K). (V is K-orientable) \Leftrightarrow ( (\Pi(X) \stackrel{V}{\to} S Line \to K Line) \simeq const_K ) \,.

For our discussion here this means that Cohen-Godin’s finding that the string topology HQFT exists for KK such that XX is KK-orientable meaks that the KK-line \infty-bundle background field that we are to consider in this context are to be trivializable.

Recall that if we interpret an (,2)(\infty,2)-vector bundle as a background gauge field for a σ\sigma-model, then for Σ\Sigma any 2-dimensional cobordism the corresponding (∞,1)-vector space of states assigned to, say, the incoming boundary inΣ\partial_{in} \Sigma is defined to be the \infty-vector space of sections of the transgression of this (,2)(\infty,2)-vector bundle to an (∞,1)-vector bundle the mapping space X inΣX^{\partial_{in} \Sigma}. The transgression of a trivial bundle is again the trivial bundle. And the \infty-vector space of (co)sections is, in the discrete case, as we had discussed above, the (∞,1)-colimit

Γ inΣ:=lim (ΠX inΣSLineKMod). \Gamma_{\partial_{in} \Sigma} := \lim_\to(\Pi X^{\partial_{in} \Sigma} \to S Line \to K Mod ) \,.

This one can compute. By triviality of the bundle, Ando-Blumberg-Gepner-Hopkins-Rezk observe that this is the KK-homology spectrum of X inΣX^{\partial_{in} \Sigma}

(Σ X inΣ)KKMod. \cdots \simeq (\Sigma^\infty X^{\partial_{in} \Sigma}) \wedge K \in K Mod \,.

(Here the main point is that for the bundle not being trivial the result encodes the corresponding twisted cohomology , but for our purposes at the moment we want the oriented/trivializable case.)

This is hence the \infty-vector space of states over inΣ\partial_{in} \Sigma assigned by a σ\sigma-model with background gauge field a KK-line \infty-bundle over a KK-oriented target space.

Notice that for K=HK = H \mathbb{Z} the Eilenberg-MacLane spectrum for the integers, we have an equivalence

HModCh H \mathbb{Z} Mod \simeq Ch_\bullet

and in fact

HAlgdgAlg H \mathbb{Z} Alg \simeq dgAlg_{\mathbb{Z}}

betwee the (∞,1)-category of HH \mathbb{Z}-module spectra/HH \mathbb{Z}-algebra spectra (see there for details on this equivalence) and the (,1)(\infty,1)-category presented by the model structure on chain complexes/on dg-algebras. Under this equivalence the above module spectrum-space of states over the circle ought to be identified with the ordinary integral homology chain complex

(Σ X S 1)KC (X S 1). (\Sigma^\infty X^{S^1}) \wedge K \sim C_\bullet(X^{S^1}) \,.

The “decategorification” of this \infty-vector space of states is precisely the tower of homology groups of XX:

π (Γ Σ)=H (X inΣ,K). \pi_\bullet(\Gamma_{\partial \Sigma}) = H_\bullet(X^{\partial_{in} \Sigma}, K) \,.

The topological AA-model string and Gromov-Witten theory

The quantum string-sigma model whose target space is a symplectic manifold (X,ω)(X, \omega) – often called the A-model – is given by the

of XX. For the moment see there for more details.

Quantum abelian Chern-Simons

For the moment see

Created on August 3, 2011 16:12:56 by Urs Schreiber (