nLab Dijkgraaf-Witten theory

Idea

Dijkgraaf-Witten theory in dimension $n$ is the topological sigma-model quantum field theory whose target space is the classifying space of a discrete group and whose background gauge field is a circle n-bundle with connection on $\mathbf{B}G$ (necessarily flat) which is equivalently a cocycle in the group cohomology of $G$ with coefficients in the circle group.

Viewed in a broader context and generalizing: Dijkgraaf-Witten theory is the ∞-Chern-Simons theory induced from a characteristic class $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ on a discrete ∞-groupoid $\mathbf{B}G := Disc B G$. If $G$ here is an ordinary discrete group this is traditional Dijkgraaf-Witten theory, if $G$ is a discrete 2-group and the background field is a circle 4-bundle, then this is called the Yetter model.

This are the first two steps in filtering of target spaces by homotopy type truncation of ∞-Chern-Simons theory with discrete target spaces.

Concise survey of the ingredients

We may think of this as describing the quantum mechanics of an $(n-1)$-brane with $n$-dimensional worldvolume $\Sigma$ propagating on $B G$ and being charged under a higher analog of the electromagnetic field:

a field configuration over $\Sigma$ (a $\Sigma$-shaped trajectory) is a morphism $\phi : \Sigma \to \mathbf{B}G$, hence equivalently a $G$-principal bundle on $\Sigma$. The configuration space of fields over $\Sigma$ is the groupoid of $G$-principal bundles over $\Sigma$.

The background gauge field is a morphism $\alpha : \mathbf{B}G \to \mathbf{B}^n U(1)$ – hence a characteristic class for $G$: a cocycle of degree $n$ in the group cohomology of $G$.

The value of the Lagrangian $L(\phi)$ on a field configuration $\phi$ is the characteristic class of this bundle with respect to the universal characteristic class of the given circle n-bundle:

$$L : (\Sigma \stackrel{\phi}{\to} \mathbf{B}G) \mapsto (\alpha(\phi) : \Sigma \stackrel{\phi}{\to} \mathbf{B}G \stackrel{\alpha}{\to} \mathbf{B}^n U(1)).$$

This is the classical field theory input of the model. The extended quantum field theory defined by this is supposed to be a rule that assigns space of states to lower dimensional pieces of $\Sigma$ and to $n$-dimensional $\Sigma$s a propagator.

The space of states assigned to a $\Sigma$ of dimension $n-k$ for $k \in \mathbb{N}$ is the k-groupoid of sections of the higher line bundle associated to the circle (n-k)-bundle $\tau_\Sigma \alpha$ obtained by transgression of $\alpha$ to the mapping space $\mathbf{H}(\Sigma, \mathbf{B}G)$.

The propagator on $\Sigma$ of dimension $n$ is given by the path integral computed with measure the groupoid cardinality of $\mathbf{B}G$ and integral kernel given by the action functional

$$\exp(i S(-)) : G Bund(\Sigma) \to U(1)$$

that sends a field $\phi$ to the evaluation of $\alpha(\phi)$ on the fundamental class of $\Sigma$

$$\exp(i S(\phi)) = \int_\Sigma \alpha(\phi) \,.$$

Gentle exposition

(…)

Details of DW-theory as an extended TQFT

The Dijkgraaf-Witten model is an example of (fully) extended topological quantum field theory. Namely, the above data not only assign an element in $U(1)$ to any closed $n$-dimensional manifold, but also a vector space to any closed $(n-1)$-dimensional manifold, a 2-vector space to any closed $(n-2)$ manifold, and so on, ending with an n-vector space assigned to the point. Also, manifolds with boundary corresponds to (higher) linear operators between these (higher) vector spaces. According to the cobordism hypothesis, the whole structure of the Dijkgraaf-Witten model as an fully extended TQFT is contained in the datum of the $n$-Vector space it assigns to the point. This is the space of sections of the flat $n$-vector bundle $\mathbf{B}G\to n Vect$ induced by the background field $\mathbf{B}G\to \mathbf{B}^n U(1)$.

Finite Group Cohomology

Since the target space of Dijkgraaf-Witten theory is the delooping groupoid $\mathbf{B}G$ of a group $G$ (internal to Set), any background field given by a morphism $\alpha : \mathbf{B}G \to A$ in ∞Grpd is a cocycle in the group cohomology of $G$, as described there.

Here we have a finite (or discrete) group $G$, and a discrete abelian group $A$, and we want to define $H^n(G;A)$. A way of doing this is to realize everything topologically: from $G$ we build the classifying space $\mathcal{B}G$, and from $A$ the Eilenberg-MacLane space $\mathcal{B}^n A=K(A,n)$. Then we consider the space of maps $hom(\mathcal{B}G,\mathcal{B}^n A)$ (these are our cocycles) and take its $\pi_0$.

This way we have a familiar description, in a certain sense (topological spaces, continuous maps, homotopies,..), of the set $H^n(G;A)$. The drawback is that the topological spaces involved here are “gigantic” (infinite dimensional CW-complexes), where we had started with a very “little” datum: a finite group. So one can wonder if there is a finite model for the above construction, and the homotopy hypothesis serves it on a silver plate. Namely, since $G$ is discrete, $\mathcal{B}G$ is a 1-type, and nothing but the geometric realization of the delooping groupoid $\mathbf{B}G$ (boldface $B$ here); similarly $\mathcal{B}^n A$ is the topological geometric realization of the $n$-groupoid $\mathbf{B}^n A$, and the space of cocycles is $hom(B G,B^n A)$. since $G$ is a finite group, $B G$ is a finite groupoid, and so $hom(B G,B^n A)$ is a finite set. This set is the finite model for $hom(\mathcal{B}G,\mathcal{B}^n A)$ we were looking for.

To be continued…

The $k$-vector spaces of states in codimension $k$

The k-vector space associated with a closed oriented $(n-k)$-dimensional manifold $X_{n-k}$ admits a simple description in terms of sections:

The background field $\alpha : \mathbf{B}G \to A$ is transgressed to the mapping space $[\Pi(X_{n-k}), \mathbf{B}G]$ by forming the internal hom

$$[\Pi(X_{n-k}), \mathbf{B}G] \stackrel{[\Pi(X_{n-k}), \alpha]}{\to} [\Pi(X_{n-k}), A] \stackrel{\tau_{\leq k}}{\to} \tau_{\leq k} [\Pi(X_{n-k}), A] \,,$$

where the last morphism is the projection on the k-truncation. This defines a cocycle on the space of fields $[\Pi(X_{n-k}), \mathbf{B}G]$ over $X_{n-k}$, which classifies some principal ∞-bundle on this space. Given a canonical representation of the spaces of phases $\tau_k [\Pi(X_{n-k}), A]$ on a k-vector space we obtain the corresponding associated bundle over the space of fields. The $(k-1)$-category assigned by the extended topological quantum field theory to the closed $X_{n-k}$ is the category of sections of this $k$-vector bundle.

Proposition

We have

$$\tau_k [\Pi(X_{n-k}), \mathbf{B}^n U(1)] \simeq \mathbf{B}^k U(1)$$

Proof

By general abstract reasoning (recalled at cohomology and fiber sequence, for instance) we have for the homotopy groups that

$$\pi_i[\Pi(X_{n-k}),\mathbf{B}^n U(1)] \simeq H^{n-i}(X_{n-k}, U(1)) \,.$$

Now use the universal coefficient theorem, which asserts that we have an exact sequence

$$0 \to Ext^1(H_{n-i-1}(X_{n-k},\mathbb{Z}),U(1)) \to H^{n-i}(X_{n-k},U(1)) \to Hom(H_{n-i}(X_{n-k},\mathbb{Z}),U(1)) \to 0 \,.$$

Since $U(1)$ is an injective $\mathbb{Z}$-module we have

$$Ext^1(-,U(1))=0 \,.$$

Together this means that we have an isomorphism

$$H^{n-i}(X_{n-k},U(1)) \simeq Hom_{Ab}(H_{n-i}(X_{n-k},\mathbb{Z}),U(1))$$

that identifies the cohomology group in question with the internal hom in Ab from the integral homology group of $X_{n-k}$ to $U(1)$.

For $i\lt k$, the right hand side is zero, and so

$$\pi_i[\Pi(X_{n-k}),\mathbf{B}^n U(1)]=0 \;\;\;\; for i\lt k \,.$$

For $i=k$, instead, $H_{n-i}(X_{n-k},\mathbb{Z})\simeq \mathbb{Z}$, since $X_{n-k}$ is a closed $(n-k)$-manifold and so

$$\pi_k[\Pi(X_{n-k}),\mathbf{B}^n U(1)]\simeq U(1) \,.$$

Remark

Another proof of the isomorphism $H^{n-k}(X_{n-k},U(1))\cong U(1)$ and of the identities $H^{n-i}(X_{n-k},U(1))=0$ for $i\lt k$ can be obtained as follows. Consider the short exact sequence of locally constant sheaves of abelian groups

$$0\to \mathbb{Z}\to \mathbb{R}\to U(1)\to 1.$$

This induces a long exact sequence in cohomology

$$\cdots \to H^{n-i-1}(X_{n-k},U(1))\to H^{n-i}(X_{n-k},\mathbb{Z})\to H^{n-i}(X_{n-k},\mathbb{R}) \to H^{n-i}(X_{n-k},U(1))\to H^{n-i+1}(X_{n-k},\mathbb{Z})\to \cdots$$

For $i\lt k$ we have $H^{n-i}(X_{n-k},U(1))=0$ by dimensional reasons, while for $i=k$ we find the exact sequence

$$\cdots \to H^{n-k}(X_{n-k},\mathbb{Z})\to H^{n-k}(X_{n-k},\mathbb{R}) \to H^{n-k}(X_{n-k},U(1))\to 0.$$

Since $X_{n-k}$ is a closed oriented manifold, we have $H^{n-k}(X_{n-k},\mathbb{Z})=\mathbb{Z}$, $H^{n-k}(X_{n-k},\mathbb{R})=\mathbb{R}$, and the map $H^{n-k}(X_{n-k},\mathbb{Z})\to H^{n-k}(X_{n-k},\mathbb{R})$ is the inclusion of $\mathbb{Z}$ into $\mathbb{R}$. Hence $H^{n-k}(X_{n-k},U(1))\cong \mathbb{R}/\mathbb{Z}\cong U(1)$.

This means that the transgression of the Dijkgraaf-Witten background field

$$\alpha : \mathbf{B}G \to \mathbf{B}^n U(1)$$

to the space of field configurations $[\Pi(X_{n-k}), \mathbf{B}G]$ over $X_{n-k}$ is a cocycle of the form

$$[\Pi(X_{n-k}), \alpha] : [\Pi(X_{n-k}), \mathbf{B}G] \to \mathbf{B}^k U(1) \,.$$

This classifies a $\mathbf{B}^{k-1} U(1)$-principal ∞-bundle $P$ over the space of field configurations, given by the pullback

$$\array{ P &\to & \mathbf{E} \mathbf{B}^{k-1} U(1) \\ \downarrow && \downarrow \\ [\Pi(X_{n-k}), \mathbf{B}G] &\stackrel{[\Pi(X_{n-k}), \rho]}{\to}& \mathbf{B}^k U(1) } \,.$$

(Here $\mathbf{E} \mathbf{B}^{k-1} U(1)$ is as described at universal principal ∞-bundle.)

By the canonical $k$-representation $\rho : \mathbf{B}^k U(1) \to k Vect_{\mathbb{C}}$ of $\mathbf{B}^{k-1}U(1)$ on complex k-vector spaces, we have associated to this canonically a $k$-vector bundle $E$, which may be realized as the pullback

$$\array{ E &\to & k Vect_* \\ \downarrow && \downarrow \\ [\Pi(X_{n-k}), \mathbf{B}G] &\stackrel{\rho \circ [\Pi(X_{n-k}), \rho]}{\to}& k Vect } \,.$$

Here $k Vect_*$ is the k-category of pointed $k$-vector bundles, see again generalized universal bundle for more.

If $X_{n-k}$ is closed then the $k$-vector spaces associated by the TFT to $X_{n-k}$ is the (k-1)-category of sections of this bundle $E$.

…

Relation to Chern-Simons theory

Dijkgraaf-Witten theory is to be thought of as the finite group version of Chern-Simons theory. Chern-Simons theory looks formally just as the above, only that all finite $n$-groupoids appearing here are replaced by Lie ∞-groupoids (∞-stacks on CartSp).

Related concepts

• transgression in group cohomology

• ∞-Chern-Simons theory

• higher dimensional Chern-Simons theory

  • 1d Chern-Simons theory

    • 1d Dijkgraaf-Witten theory

  • 2d Chern-Simons theory

  • 3d Chern-Simons theory

    • Dijkgraaf-Witten theory

    • 2d Wess-Zumino-Witten theory

  • 4d Chern-Simons theory

  • 5d Chern-Simons theory

  • 6d Chern-Simons theory

  • 7d Chern-Simons theory

  • infinite-dimensional Chern-Simons theory

  • AKSZ sigma-model

References

The idea originates, of course, in

• Robbert Dijkgraaf, Edward Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990), 393 (euclid:cmp/1104180750)

The discussion of the quasi-Hopf algebra associated with a group cohomology 3-cocycle $c \colon \mathbf{B}G \to \mathbf{B}^3 U(1)$ originates in

• Robbert Dijkgraaf, V. Pasquier, P. Roche, QuasiHopf algebras, group cohomology and orbifold models, Nucl. Phys. B Proc. Suppl. 18B (1990), 60-72; Quasi-quantum groups related to orbifold models, Modern quantum field theory (Bombay, 1990), 375–383, World Sci. 1991

A review is in

• A. Coste, J-M. Maillard, Representation Theory of Twisted Group Double, Annales Fond.Broglie 29 (2004) 681-694, (arXiv:hep-th/0309257)

and conceptual clarifications were established in

• Simon Willerton, The twisted Drinfeld double of a finite group via gerbes and finite groupoids, arXiv/math.QA/0503266

and, earlier, in an unpublished manuscript of Paul Bressler (2002-2004). See at Drinfeld double for more on this.

A first comprehensive structural account of DW theory as a functorial QFT was given in

• Daniel Freed, Frank Quinn, Chern-Simons theory with finite gauge group Commun.Math.Phys. 156:435-472, 1993, (arXiv:hep-th/9111004)

A review is given on p. 68 of

• Bruce Bartlett, Categorical aspects of topological quantum field theory, arXiv/math.QA/0512103

First steps towards understand DW theory as an extended TQFT appear in

• Daniel Freed, Higher Algebraic Structures and Quantization Commun.Math.Phys. 159 (1994) 343-398 (arXiv:arXiv:hep-th/9212115)

Discussion aiming towards a refinement of DW theory to an extended TQFT is in

• Kevin Wray, Extended topological gauge theories in codimension 0 and higher (pdf)

Further conceptual refinement of this is indicated in section 3 and section 8 of

• Dan Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups (2010)

This proposes a general abstract way to construct path integral quantizations for finite group theories such as DW, see also at prequantum field theory. More along these lines is in

• Jacob Lurie, Finiteness and ambidexterity in $K(n)$-local stable homotopy theory, talk at Notre Dame Graduate Summer School on Topology and Field Theories and Harvard lecture 2012 (video part 1, part 2, part 3 part 4, pdf lecture notes by Chris Elliott)

See also

• Thomas Nikolaus, Equivariant Dijkgraaf-Witten theory, a talk at Muenster (pdf)

For more on this see the discussion on the n-Forum.

Dijkgraaf–Witten theory via Kan extensions

• Jeffrey C. Morton, Cohomological Twisting of 2-Linearization and Extended TQFT, arXiv:1003.5603.

• Section 2 and 3 of Gijs Heuts and Jacob Lurie‘s Ambidexterity, in: Topology and Field Theories, doi.

• Section 3 of Daniel Freed, Michael Hopkins, Jacob Lurie, Constantin Teleman, Topological quantum field theories from compact Lie groups, arXiv:0905.0731.