this entry is going to contain one chapter of geometry of physics
In the previous chapters we have set up prequantum field theory and classical field theory in generality. Here we discuss examples of such field theories in more detail.
We introduce a list of important examples of field theories in fairly tradtional terms.
We study the above physical systems with the tools of of cohesive (∞,1)-topos-theory as developed in the previous semantics-layers.
The prequantum field theory which describes the gauge interaction of a single nonabelian charged particle – a Wilson loop – turns out to be equivalent to what in mathematics is called the orbit method. We discuss here the traditional formulation of these matters. Below in Semantics layer – Nonabelian charged particle and Wilson loops we then show how all this is naturally understood from a certain extended Lagrangian which is induced by a regular coadjoint orbit.
A useful review of the following is also in (Beasley, section 4).
Throughout, let $G$ be a semisimple compact Lie group. For some considerations below we furthermore assume it to be simply connected.
Write $\mathfrak{g}$ for its Lie algebra. Its canonical (up to scale) binary invariant polynomial we write
Since this is non-degenerate, we may equivalently think of this as an isomorphism
that identifies the vector space underlying the Lie algebra with its dual vector space $\mathfrak{g}^*$.
We discuss the coadjoint orbits of $G$ and their relation to the coset space/flag manifolds of $G$.
Write
1 $\mathfrak{t} \hookrightarrow \mathfrak{g}$ the corresponding Cartan subalgebra
In all of the following we consider an element $\langle\lambda,-\rangle \in \mathfrak{g}^*$.
For $\langle\lambda,-\rangle \in \mathfrak{g}^*$ write
for its coadjoint orbit
Write $G_\lambda \hookrightarrow G$ for the stabilizer subgroup of $\langle \lambda,-\rangle$ under the coadjoint action.
There is an equivalence
given by
An element $\langle\lambda,-\rangle \in \mathfrak{g}^*$ is regular if its coadjoint action stabilizer subgroup coincides with the maximal torus: $G_\lambda \simeq T$.
For generic values of $\lambda$ it is regular. The element in $\mathfrak{g}^*$ farthest from regularity is $\lambda = 0$ for which $G_\lambda = G$ instead.
We describe a canonical symplectic form on the coadjoint orbit/coset $\mathcal{O}_\lambda \simeq G/G_\lambda$.
Write $\theta \in \Omega^1(G, \mathfrak{g})$ for the Maurer-Cartan form on $G$.
Write
for the 1-form obtained by pairing the value of the Maurer-Cartan form at each point with the gixed element $\lambda \in \mathfrak{g}^*$.
Write
for its de Rham differential.
The 2-form $\nu_\lambda$ from def.
satisfies
it descends to a closed $G$-invariant 2-form on the coset space, to be denoted by the same symbol
this is non-degenerate and hence defines a symplectic form on $G/G_\lambda$.
We discuss the geometric prequantization of the symplectic manifold given by the coadjoint orbit $\mathcal{O}_\lambda$ equipped with its symplectic form $\nu_\lambda$ of def. .
Assume now that $G$ is simply connected.
The weight lattice $\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t}$ of the Lie group $G$ is isomorphic to the group of group characters
where the identification takes $\langle \alpha , -\rangle \in \mathfrak{t}^*$ to $\rho_\alpha : T \to U(1)$ given on $t = \exp(\xi)$ for $\xi \in \mathfrak{t}$ by
The symplectic form $\nu_\lambda \in \Omega^2_{cl}(G/T)$ of prop. is integral precisely if $\langle \lambda, - \rangle$ is in the weight lattice.
The group $G$ canonically acts on the coset space $G/G_{\lambda}$ (by multiplication from the left). We discuss a lift of this action to a Hamiltonian action with respect to the symplectic manifold structure $(G/T, \nu_\lambda)$ of prop. , equivalently a momentum map exhibiting this Hamiltonian action.
Above (…) we discussed how an irreducible unitary representation of $G$ is encoded by the prequantization of a coadjoint orbit $(\mathcal{O}_\lambda, \nu_\lambda)$. Here we discuss how to express Wilson loops/holonomy of $G$-principal connections in this representation as the path integral of a topological particle charged under this background field, whose action functional is that of a 1-dimensional Chern-Simons theory.
Let $A|_{S^1} \in \Omega^1(S^1, \mathfrak{g})$ be a Lie algebra valued 1-form on the circle, equivalently a $G$-principal connection on the circle.
For
a representation of $G$, write
for the holonomy of $A$ around the circle in this representation, which is the trace of its parallel transport around the circle (for any basepoint). If one thinks of $A$ as a background gauge field then this is alse called a Wilson loop.
Let the action functional
be given by sending $g T : S^1 \to G/T$ represented by $g : S^1 \to G$ to
where
is the gauge transformation of $A$ under $g$.
The Wilson loop of $A$ over $S^1$ in the unitarry irreducible representation $R$ is proportional to the path integral of the 1-dimensional sigma-model with
target space the coadjoint orbit $\mathcal{O}_\lambda \simeq G/T$ for $\langle \lambda, - \rangle$ the weight corresponding to $R$ under the Borel-Weil-Bott theorem
action functional the functional of def. :
See for instance (Beasley, (4.55)).
Notice that since $\mathcal{O}_\lambda$ is a manifold of finite dimension, the path integral for a point particle with this target space can be and has been defined rigorously, see at path integral.
exposition and survey is in (FSS 13).
For some $n \in \mathbb{N}$ let
be the Lie group homomorphism from the unitary group to the circle group which is given by sending a unitary matrix to its determinant.
Being a Lie group homomorphism, this induces a map of deloopings/moduli stacks
Under geometric realization of cohesive infinity-groupoids this is the universal first Chern class
Moreiver this has the evident differential refinement
given on Lie algebra valued 1-forms by taking the trace
So we get a 1d Chern-Simons theory with $\widehat{\mathbf{B}det}$ as its extended Lagrangian.
We consider now extended Lagrangians defined on fields as above in Nonabelian charged particle trajectories – Wilson loops. This provides a natural reformulation in higher geometry of the constructions in the orbit method as reviewed above in Model layer – Nonabelian charged particle.
We discuss how for $\lambda \in \mathfrak{g}$ a regular element, there is a canonical diagram of smooth moduli stacks of the form
where
$\mathbf{J}$ is the canonical 2-monomorphism;
the left square is a homotopy pullback square, hence $\mathbf{\theta}$ is the homotopy fiber of $\mathbf{J}$;
the bottom map is the extended Lagrangian for $G$-Chern-Simons theory, equivalently the universal Chern-Simons circle 3-bundle with connection;
the top map denoted $\langle \lambda,- \rangle$ is an extended Lagrangian for a 1-dimensional Chern-Simons theory;
the total top composite modulates a prequantum circle bundle which is a prequantization of the canonical symplectic manifold structure on the coadjoint orbit $\Omega_\lambda \simeq G/T$.
Write $\mathbf{H} =$ Smooth∞Grpd for the cohesive (∞,1)-topos of smooth $\infty$-groupoids.
For the following, let $\langle \lambda, - \rangle \in \mathfrak{g}^*$ be a regular element, def. , so that the stabilizer subgroup is identified with a maximal torus: $G_\lambda \simeq T$.
As usual, write
for the moduli stack of $G$-principal connections.
Write
for the canonical map, as indicated.
The map $\mathbf{J}$ is the differential refinement of the delooping $\mathbf{B}T \to \mathbf{B}G$ of the defining inclusion. By the general discussion at coset space we have a homotopy fiber sequence
By the discussion at ∞-action this exhibits the canonical action $\rho$ of $G$ on its coset space: it is the universal rho-associated bundle.
The following proposition says what happens to this statement under differential refinement
The homotopy fiber of $\mathbf{J}$ in def. is
given over a test manifold $U \in$ CartSp by the map
which sends $g \mapsto g^* \theta$, where $\theta$ is the Maurer-Cartan form on $G$.
We compute the homotopy pullback of $\mathbf{J}$ along the point inclusion by the factorization lemma as discussed at homotopy pullback – Constructions.
This says that with $\mathbf{J}$ presented canonically as a map of presheaves of groupoids via the above definitions, its homotopy fiber is presented by the presheaf of groupids $hofib(\mathbf{J})$ which is the limit cone in
Unwinding the definitions shows that $hofib(\mathbf{J})$ has
objects over a $U \in$ CartSp are equivalently morphisms $0 \stackrel{g}{\to} g^* \theta$ in $\Omega^1(U,\mathfrak{g})//C^\infty(U,G)$, hence equivalently elements $g \in C^\infty(U,G)$;
morphisms are over $U$ commuting triangles
in $\Omega^1(U,\mathfrak{g})//C^\infty(U,G)$ with $t \in C^\infty(U,T)$, hence equivalently morphisms
in $C^\infty(U,G)//C^\infty(U,T)$.
The canonical map $hofib(\mathbf{J}) \to \Omega^1(-,\mathfrak{g})//T$ picks the top horizontal part of these commuting triangles hence equivalently sends $g$ to $g^* \theta$.
If $\langle \lambda ,- \rangle \in \Gamma_{wt} \hookrightarrow \mathfrak{g}^*$ is in the weight lattice, then there is a morphism of moduli stacks
in $\mathbf{H}$ given over a test manifold $U \in$ CartSp by the functor
which is given on objects by
and which maps morphisms labeled by $\exp(\xi) \in T$, $\xi \in C^\infty(-,\mathfrak{t})$ as
That this construction defines a map $*//T \to *//U(1)$ is the statement of prop. . It remains to check that the differential 1-forms gauge-transform accordingly.
For this the key point is that since $T \simeq G_\lambda$ stabilizes $\langle \lambda , - \rangle$ under the coadjoint action, the gauge transformation law for points $A : U \to \mathbf{B}G_{conn}$, which for $g \in C^\infty(U,G)$ is
maps for $g = exp( \xi ) \in C^\infty(U,T) \hookrightarrow C^\infty(U,G)$ to the gauge transformation law in $\mathbf{B}U(1)_{conn}$:
The composite of the canonical maps of prop. and prop. modulates a canonical circle bundle with connection on the coset space/coadjoint orbit:
The curvature 2-form of the circle bundle $\langle \lambda, \mathbf{\theta}\rangle$ from remark is the symplectic form of prop. . Therefore $\langle \lambda, \mathbf{\theta}\rangle$ is a prequantization of the coadjoint orbit $(\mathcal{O}_\lambda \simeq G/T, \nu_\lambda)$.
The curvature 2-form is modulated by the composite
Unwinding the above definitions and propositions, one finds that this is given over a test manifold $U \in$ CartSp by the map
which sends
Let $\Sigma$ be an oriented closed smooth manifold of dimension 3 and let
be a submanifold inclusion of the circle: a knot in $\Sigma$.
Let $R$ be an irreducible unitary representation of $G$ and let $\langle \lambda,-\rangle$ be a weight corresponding to it by the Borel-Weil-Bott theorem.
Regarding the inclusion $C$ as an object in the arrow (∞,1)-topos $\mathbf{H}^{\Delta^1}$, say that a gauge field configuration for $G$-Chern-Simons theory on $\Sigma$ with Wilson loop $C$ and labeled by the representation $R$ is a map
in the arrow (∞,1)-topos $\mathbf{H}^{(\Delta^1)}$ of the ambient cohesive (∞,1)-topos. Such a map is equivalently by a square
in $\mathbf{H}$. In components this is
a $G$-principal connection $A$ on $\Sigma$;
a $G$-valued function $g$ on $S^1$
which fixes the field on the circle defect to be $(A|_{S^1})^g$, as indicated.
Moreover, a gauge transformation between two such fields $\kappa : \phi \Rightarrow \phi'$ is a $G$-gauge transformation of $A$ and a $T$-gauge transformation of $A|_{S^1}$ such that these intertwine the component maps $g$ and $g'$. If we keep the bulk gauge field $A$ fixed, then his means that two fields $\phi$ and $\phi'$ as above are gauge equivalent precisely if there is a function $t \;\colon\; S^1 \to T$ such that $g = g' t$, hence gauge equivalence classes of fields for fixed bulk gauge field $A$ are parameterized by their components $[g] = [g'] \in [S^1, G/T]$ with values in the coset space, hence in the coadjoint orbit.
For every such field configuration we can evaluate two action functionals:
that of 3d Chern-Simons theory, whose extended Lagrangian is $\mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$;
that of the 1-dimensional Chern-Simons theory discussed above whose extended Lagrangian is $\langle \lambda, -\rangle : \Omega^1(-,\mathfrak{g})//T \to \mathbf{B}U(1)_{conn}$, by prop. .
These are obtained by postcomposing the above square on the right by these extended Lagrangians
and then preforming the fiber integration in ordinary differential cohomology over $S^1$ and over $\Sigma$, respectively.
For the bottom map this gives the ordinary action functional of Chern-Simons theory. For the top map inspection of the proof of prop. shows that this gives the 1d Chern-Simons action whose partition function is the Wilson loop observable by prop. above.
In the context of string theory, the background gauge field for the open string sigma-model over a D-brane in bosonic string theory or type II string theory is a unitary principal bundle with connection, or rather, by the Kapustin-part of the Freed-Witten-Kapustin anomaly cancellation mechanism, a twisted unitary bundle, whose twist is the restriction of the ambient B-field to the D-brane.
We considered these fields already above. Here we discuss the corresponding action functional for the open string coupled to these fields
The first hint for the existence of such background gauge fields for the open string 2d-sigma-model comes from the fact that the open string’s endpoint can naturally be taken to carry labels $i \in \{1, \cdots n\}$. Further analysis then shows that the lowest excitations of these $(i,j)$-strings behave as the quanta of a $U(n)$-gauge field, the $(i,j)$-excitation being the given matrix element of a $U(n)$-valued connection 1-form $A$.
This original argument goes back work by Chan and Paton. Accordingly one speaks of Chan-Paton factors and Chan-Paton bundles .
We discuss the Chan-Paton gauge field and its quantum anomaly cancellation in extended prequantum field theory.
Throughout we write $\mathbf{H} =$ Smooth∞Grpd for the cohesive (∞,1)-topos of smooth ∞-groupoids.
For $X$ a type II supergravity spacetime, the B-field is a map
If $X = G$ is a Lie group, this is the prequantum 2-bundle of $G$-Chern-Simons theory. Viewed as such we are to find a canonical ∞-action of the circle 2-group $\mathbf{B}U(1)$ on some $V \in \mathbf{H}$, form the corresponding associated ∞-bundle and regard the sections of that as the prequantum 2-states? of the theory.
The Chan-Paton gauge field is such a prequantum 2-state.
We discuss the Chan-Paton gauge fields over D-branes in bosonic string theory and over $Spin^c$-D-branes in type II string theory.
We fix throughout a natural number $n \in \mathbb{N}$, the rank of the Chan-Paton gauge field.
The extension of Lie groups
exhibiting the unitary group as a circle group-extension of the projective unitary group sits in a long homotopy fiber sequence of smooth ∞-groupoids of the form
where for $G$ a Lie group $\mathbf{B}G$ is its delooping Lie groupoid, hence the moduli stack of $G$-principal bundles, and where similarly $\mathbf{B}^2 U(1)$ is the moduli 2-stack of circle 2-group principal 2-bundles (bundle gerbes).
Here
is a smooth refinement of the universal Dixmier-Douady class
in that under geometric realization of cohesive ∞-groupoids ${\vert- \vert} \colon$ Smooth∞Grpd $\to$ ∞Grpd we have
By the discussion at ∞-action the homotopy fiber sequence in prop.
in $\mathbf{H}$ exhibits a smooth∞-action of the circle 2-group on the moduli stack $\mathbf{B}U(n)$ and it exhibits an equivalence
of the moduli stack of projective unitary bundles with the ∞-quotient of this ∞-action.
For $X \in \mathbf{H}$ a smooth manifold and $\mathbf{c} \;\colon\; X \to \mathbf{B}^2 U(1)$ modulating a circle 2-group-principal 2-bundle, maps
in the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}^2 U(1)}$, hence diagrams of the form
in $\mathbf{H}$ are equivalently rank-$n$ unitary twisted bundles on $X$, with the twist being the class $[\mathbf{c}] \in H^3(X, \mathbb{Z})$.
There is a further differential refinement
where $\mathbf{B}^2 U(1)_{conn}$ is the universal moduli 2-stack of circle 2-bundles with connection (bundle gerbes with connection).
Write
for the differential smooth universal Dixmier-Douady class of prop. , regarded as an object in the slice (∞,1)-topos over $\mathbf{B}^2 U(1)_{conn}$.
Let
be an inclusion of smooth manifolds or of orbifolds, to be thought of as a D-brane worldvolume $Q$ inside an ambient spacetime $X$.
Then a field configuration of a B-field on $X$ together with a compatible rank-$n$ Chan-Paton gauge field on the D-brane is a map
in the arrow (∞,1)-topos $\mathbf{H}^{(\Delta^1)}$, hence a diagram in $\mathbf{H}$ of the form
This identifies a twisted bundle with connection on the D-brane whose twist is the class in $H^3(X, \mathbb{Z})$ of the bulk B-field.
This relation is the Kapustin-part of the Freed-Witten-Kapustin anomaly cancellation for the bosonic string or else for the type II string on $Spin^c$ D-branes. (FSS)
If we regard the B-field as a background field for the Chan-Paton gauge field, then remark determines along which maps of the B-field the Chan-Paton gauge field may be transformed.
On the local connection forms this acts as
This is the famous gauge transformation law known from the string theory literature.
The D-brane inclusion $Q \stackrel{\iota_X}{\to} X$ is the target space for an open string with worldsheet $\partial \Sigma \stackrel{\iota_\Sigma}{\hookrightarrow} \Sigma$: a field configuration of the open string sigma-model is a map
in $\mathbf{H}^{\Delta^1}$, hence a diagram of the form
For $X$ and $Q$ ordinary manifolds just says that a field configuration is a map $\phi_{bulk} \;\colon\; \Sigma \to X$ subject to the constraint that it takes the boundary of $\Sigma$ to $Q$. This means that this is a trajectory of an open string in $X$ whose endpoints are constrained to sit on the D-brane $Q \hookrightarrow X$.
If however $X$ is more generally an orbifold, then the homotopy filling the above diagram imposes this constraint only up to orbifold transformations, hence exhibits what in the physics literature are called “orbifold twisted sectors” of open string configurations.
The moduli stack $[\iota_\Sigma, \iota_X]$ of such field configurations is the homotopy pullback
For $\Sigma$ a smooth manifold with boundary $\partial \Sigma$ of dimension $n$ and for $\nabla \;\colon \; X \to \mathbf{B}^n U(1)_{conn}$ a circle n-bundle with connection on some $X \in \mathbf{H}$, then the transgression of $\nabla$ to the mapping space $[\Sigma, X]$ yields a section of the complex line bundle associated to the pullback of the ordinary transgression over the mapping space out of the boundary: we have a diagram
This is the higher parallel transport of the $n$-connection $\nabla$ over maps $\Sigma \to X$.
The operation of forming the holonomy of a twisted unitary connection around a curve fits into a diagram in $\mathbf{H}$ of the form
By the discussion at ∞-action the diagram in prop. says in particular that forming traced holonomy of twisted unitary bundles constitutes a section of the complex line bundle on the moduli stack of twisted unitary connection on the circle which is the associated bundle to the transgression $\exp(2 \pi i \int_{S^1} [S^1, \widehat\mathbf{dd}_n])$ of the universal differential Dixmier-Douady class.
It follows that on the moduli space of the open string sigma-model of prop. above there are two $\mathbb{C}//U(1)$-valued action functionals coming from the bulk field and the boundary field
Neither is a well-defined $\mathbb{C}$-valued function by itself. But by pasting the above diagrams, we see that both these constitute sections of the same complex line bundle on the moduli stack of fields:
Therefore the product action functional is a well-defined function
This is the Kapustin anomaly-free action functional of the open string.
We discuss how an extended Lagrangian for $G$-Chern-Simons theory with Wilson loop defects is naturally obtained from the above higher geometric formulation of the orbit method. In particular we discuss how the relation between Wilson loops and 1-dimensional Chern-Simons theory sigma-models with target space the coadjoint orbit, as discussed above is naturally obtained this way.
More formally, we have an extended Chern-Simons theory as follows.
The moduli stack of fields $\phi : C \to \mathbf{J}$ in $\mathbf{H}^{(\Delta^1)}$ as above is the homotopy pullback
in $\mathbf{H}$, where square brackets indicate the internal hom in $\mathbf{H}$.
Postcomposing the two projections with the two transgressions of the extended Lagrangians
and
to yield
and then forming the product yields the action functional
This is the action functional of 3d $G$-Chern-Simons theory on $\Sigma$ with Wilson loop $C$ in the representation determined by $\lambda$.
Similarly, in codimension 1 let $\Sigma_2$ now be a 2-dimensional closed manifold, thought of as a slice of $\Sigma$ above, and let $\coprod_i {*} \to \Sigma_2$ be the inclusion of points, thought of as the punctures of the Wilson line above through this slice. Then we have prequantum bundles given by transgression of the extended Lagrangians to codimension 1
and
and hence a total prequantum bundle
One checks that this is indeed the correct prequantization as considered in (Witten 98, p. 22).
(…)
Last revised on December 26, 2022 at 20:10:16. See the history of this page for a list of all contributions to it.