2.1. T-Duality and the ’t Hooft Anomaly: Basics
Our purpose in this section is to review some basic features of T-duality in 2D scalar field theories, namely nonlinear sigma models. Our main focus will be to highlight some of the features that are usually not covered in other treatments of the subject, such as the mixed ’t Hooft anomaly between the momentum and winding symmetries and the T-duality of models whose couplings are not only functions of the scalar fields but of their derivatives as well.
It is instructive to begin with the simplest and most pedagogical case of a single compact scalar field
X(
${\sigma}^{\mu}$), where
${\sigma}^{\mu},\mu =0,1$ are 2D spacetime (worldsheet) coordinates and the target space is a circle of radius
R. In other words, we consider a map
$X:{\Sigma}_{2}\to {S}^{1}$ and denoting the compact circle coordinate by
$x\sim x+2\pi R$, the scalar field is the pullback
$X\left(\sigma \right)={X}^{*}\left(x\right)$. The simplest field theory in this setting is a kinetic term for
X, the sigma model
where we work in units where the Regge slope parameter is
${\alpha}^{\prime}=1/2\pi $. Moreover, we have employed a differential form notation where the 2D metric
${\gamma}_{\mu \nu}$ and determinant
$\gamma $ are packaged in the Hodge star operator ∗ and the associated volume form. Let us denote
$F=\mathrm{d}X$, essentially the field strength of the scalar field. At this stage it is useful to clarify that the symbol
F will be used in two different ways below. In accordance with standard nomenclature, we refer to
$F=\mathrm{d}X$ as the
constrained 1-form or field strength, otherwise (i.e., when
F is not given by definition as
$\mathrm{d}X$, but eventually becomes
$\mathrm{d}X$ dynamically) we refer to it as the
unconstrained 1-form.
The field equations obtained from the action functional (
1) accompanied by the Bianchi identity for the field strength
F read as
The theory under consideration has two global symmetries, the momentum and winding ones, with ordinary vector currents
Using the field Equations (
2), they are both conserved,
$\mathrm{d}\ast {J}_{\mathrm{mom}}=0=\mathrm{d}\ast {J}_{\mathrm{win}}$. These two currents may be used to introduce background fields in the theory in the usual way of adding to the action terms of the form
${A}^{\mu}{J}_{\mu}$ or
$A\wedge \ast J$ in differential form notation. In the present case, the background fields are Abelian 1-forms and we denote them by
A and
$\widehat{A}$, which will appear in the theory through
$A\wedge \ast {J}_{\mathrm{mom}}$ and
$\widehat{A}\wedge \ast {J}_{\mathrm{win}}$. These fields transform under background gauge transformations as
where
$\u03f5$ and
$\widehat{\u03f5}$ are the corresponding background gauge transformation parameters. Let us emphasize that being background fields,
A and
$\widehat{A}$ are nondynamical and they can be switched on and off at will without spoiling the theory and its symmetries. Nevertheless, introducing them in the theory by means of the terms
$A\wedge \ast {J}_{\mathrm{mom}}$ and
$\widehat{A}\wedge \ast {J}_{\mathrm{win}}$ requires a suitable counter term to preserve invariance under the transformation of
A. Indeed, the dynamical field
X transforms as
and is inert under the “winding” transformation given by
$\widehat{\u03f5}$, therefore the invariant 1-form is
$\mathrm{d}X-A$. This dictates that under suitable rearrangements of terms the action (
1) extended by background fields has the form
To avoid confusion, we stress once more that this is not a gauged action and the background fields may be switched off without any issues. In particular, one should not think of
$\widehat{A}$ here as a Lagrange multiplier.
One may ask whether
A and
$\widehat{A}$ could now be promoted to dynamical fields,
3 in other words whether both global symmetries can be gauged. For this to be possible one should make sure that no ’t Hooft anomaly is present in the theory. Such an anomaly exists and prevents the gauging of both global symmetries simultaneously. This is easily seen by examining how the action (
6) changes under background gauge transformations. The result, discarding boundary terms, is
where in the second term we took the liberty of integrating by parts, ignoring global issues that could however be treated in a proper way [
24], a topic that we shall not pursue further here. The main observation is that the action with background fields is not invariant but instead it includes shifts under the background gauge transformation of
A.
4 Unlike the case of ABJ anomalies, these are c-number shifts and the anomaly goes away as long as the background fields are switched off. This is typical of ’t Hooft anomalies, which represent an obstruction to gauging a global symmetry rather than a sickness in the theory. This is precisely what happens in the present case, as mentioned in footnote 11 of Ref. [
1]. There exists a mixed ’t Hooft anomaly between the momentum and winding global symmetries that prevents them from being gauged simultaneously. As discussed in [
6], one can present the anomalous term in multiple ways upon adding counter terms, however, it can never be removed and persists in all descriptions of the theory, the latter being a manifestation of ’t Hooft anomaly matching. In a more symmetric way, the shift in (
7) may be obtained via inflow from the 4-form anomaly polynomial
using the descent equations.
The above discussion can now be directly related to the gauging of global symmetries in 2D sigma models. In this simple setting the situation is very clear, since the sigma model (
1) has a global (momentum) symmetry
$\delta X=\u03f5$ with a rigid parameter
$\u03f5$. Gauging this symmetry means that
$\u03f5$ is promoted to a
$\sigma $-dependent parameter which according to the basic principles of gauge theory means that we must introduce gauge field 1-forms
$\mathcal{A}$ that transform as
$\delta \mathcal{A}=\mathrm{d}\u03f5$ and give rise to the covariant worldsheet differentials
Employing minimal coupling, the gauged action functional is
In this action,
$\mathcal{A}$ is a true gauge field and not a background, which explains the difference in notation. One may now immediately see that
In other words, the gauged action is obtained from the ungauged one coupled to background fields for all global symmetries, after one of the background fields is promoted to a genuine gauge field and the other is switched off. This is the essence of the ’t Hooft anomaly, which suggests that only one of the background fields can be promoted to a dynamical one.
Of course, one may ask whether we could switch off
A instead of
$\widehat{A}$ and gauge the other global symmetry. This is better seen in terms of dual variables, which is where T-duality comes into play. Moreover, T-duality can be better understood in terms of an action involving the unconstrained 1-from
F. To be fully general, we consider this “parent” action already coupled to background fields,
where
$\widehat{X}$ is a Lagrange multiplier which will turn out to be the dual scalar field under T-duality. The action
$\mathcal{S}$ is obtained from (
6) upon replacing the constrained 1-form
$F=\mathrm{d}X$ by an unconstrained one and adding the
$\widehat{X}$-dependent term
$F\wedge \mathrm{d}\widehat{X}$. Moreover, background gauge invariance dictates that the new field should transform according to
a sign of the winding global symmetry as we discuss shortly. One can immediately see that variation with respect to the Lagrange multiplier
$\widehat{X}$ leads to the Euler–Lagrange equation
$\mathrm{d}F=0$, which may be locally solved by the constrained
$F=\mathrm{d}X$ due to the Poincaré lemma. In that case, the field
$\widehat{X}$ is eliminated or integrated out from the theory and one returns to the original action (
6), or (
1) when the background fields are switched off. On the other hand, we can now instead eliminate the unconstrained field
F through its field equation. The latter is
This is substituted back in the action (
12) to give
This is then the action for the dual scalar field
$\widehat{X}$ that transforms with shifts under the winding symmetry of the theory. Three observations follow: firstly, switching off the background fields, we immediately see that the dual scalar propagates on a target
${S}^{1}$ with radius
$1/R$ (recall that we work in units where
${\alpha}^{\prime}$ is 1/2
$\pi $; from dimensional analysis, the dual radius is actually
${\alpha}^{\prime}/R$). Secondly, in this dual formulation of the theory, the momentum and winding currents are exchanged. Indeed, the duality relation (
14) taken with the constrained
$F=\mathrm{d}X$ says that the Bianchi identity and wave equation of the original theory are mapped to the wave equation and the Bianchi identity respectively in the dual theory, as in [
25]. Thirdly, the dual action contains a term
$A\wedge \widehat{A}$ that couples the two background fields, unlike the action (
6). The reason for this can be traced in the matching of the ’t Hooft anomaly, since
In other words, we observe that
and the ’t Hooft anomaly is precisely matched in both T-dual formulations of the theory.
Summarizing, within the simple setting of a 2D compact scalar with target space being a circle we have discussed the global momentum and winding symmetries, their ’t Hooft anomaly and the T-duality that exchanges the two symmetries along with the inversion of the radius of the target circle. This is the extension with background fields of the usual picture of equivalence between sigma models by means of a parent action functional found in string theory textbooks and can be depicted as
Here, $\mathcal{S}$ is the parent action coupled to background fields, the inner arrows lead to the two dual theories coupled to background fields once by integrating out the field $\widehat{X}$ (left) and once the unconstrained field F (right), and the outer arrows mean that the background fields are switched off and lead to the two dual second-order theories in terms of X and $\widehat{X}$, respectively. As we will see below, all the essential features of this picture and the relation of the duality to the ’t Hooft anomaly are reflected in various generalizations of this basic model.
2.2. T-Duality beyond Strings: Heisenberg’s Pion Fireball Model
The discussion in
Section 2.1 can be generalized to 2D higher-derivative single scalar theories in a straightforward way. We present this through the example of Heisenberg’s pion fireball model [
10], which is based on the Dirac–Born–Infeld type action functional
where
ℓ is a length parameter. Here, we are interested in the massless case,
$m=0$. Then, in the limit
$\ell \to \infty $ one obtains the linear scalar theory (
1) studied in the previous section. Hence we set
$R=1$ for simplicity. The field equation obtained by variation of this action with respect to
X and the Bianchi identity have the form
in terms of the constrained field strength
$F=\mathrm{d}X$. This theory has the same momentum and winding zero-form global symmetries as the linear theory (
1), with corresponding vector currents
The first is conserved (
$\mathrm{d}\ast {J}_{\mathrm{mom}}=0$) by virtue of the field equation, while the second is conserved (
$\mathrm{d}\ast {J}_{\mathrm{win}}=0$) due to the Bianchi identity.
Just like before, one can couple background fields
A and
$\widehat{A}$ to the Heisenberg model. These transform under the background gauge transformations (
4) and the dynamical field
X transforms only under the first one through a shift. Since
X is inert under the winding transformation, the current
${J}_{\mathrm{win}}$ can be coupled simply through the invariant (up to a total derivative) term
$\widehat{A}\wedge \ast {J}_{\mathrm{win}}$ and no additional counter-terms are needed. On the other hand, the term
$A\wedge \ast {J}_{\mathrm{mom}}$ is not invariant on its own. Thus, one has to add an infinite sequence of counterterms which are of higher order
$\mathcal{O}\left({A}^{2}\right)$ to ensure invariance. These are all encoded in the following action
in which the term linear in
A reads precisely as
$A\wedge \ast {J}_{\mathrm{mom}}$. To see this, one can first expand the square root in the denominator and rewrite the components of the current as
where
${F}^{2m}\equiv {\left({F}_{\mu}{F}^{\mu}\right)}^{m}$, etc. Subsequently, we can expand the first term in the action as
The terms linear in
${A}_{\mu}$, i.e., the ones with
$k=1$ in the series, are
Finally, the identity
$\left(\genfrac{}{}{0pt}{}{n-3/2}{n-1}\right)=2n{(-1)}^{n+1}\left(\genfrac{}{}{0pt}{}{1/2}{n}\right)$ reveals that
and this concludes the proof of the statement.
As we saw earlier for the case of the string sigma model, the action is not invariant under background gauge transformations, since
up to boundary terms. The above shift may be obtained via inflow from the same 4-form anomaly polynomial
${\mathcal{I}}_{4}$ given in (
8). This reflects the fact that in both the string sigma model and in this particular higher-derivative theory the momentum and winding global symmetries are the same. If one now wishes to gauge part of the global symmetry, using minimal coupling the gauged action reads as
Once more the gauged action ${S}_{H,\phantom{\rule{0.166667em}{0ex}}\mathrm{gauged}}[X,\mathcal{A}]$ is obtained from ${S}_{H}[X,A,\widehat{A}\phantom{\rule{0.166667em}{0ex}}]$ by promoting the background field A to the dynamical gauge field $\mathcal{A}$ and switching off the background field $\widehat{A}$.
To uncover T-duality in this case, let us consider the parent action
which is already coupled to the background fields
A and
$\widehat{A}$. The field
F is now an unconstrained 1-form and
$\widehat{X}$ is a Lagrange multiplier, which will be identified with the dual scalar field. The original second order theory (
22) is obtained by varying the parent action with respect to the Lagrange multiplier
$\widehat{X}$, as before. On the other hand, the equation of motion for
F is the duality relation
Multiplying both sides of this relation with
$({F}_{\mu}-{A}_{\mu})$ and
${\u03f5}_{\mu \kappa}({\partial}^{\kappa}\widehat{X}-{\widehat{A}}^{\kappa})$ respectively gives the two equations
where we have introduced the shorthand notation
for brevity. These equations imply that
Using this final equation, as well as the duality relation (
30) from which it originates, one can obtain the dual theory from the parent action by direct substitution. The result is
We observe once again that the dual action contains the term
$-{\int}_{{\Sigma}_{2}}A\wedge \widehat{A}$, which ensures the matching of the corresponding ’t Hooft anomaly. Finally, the equivalence reflected in (
18) holds for the Heisenberg model too, with the same radius inversion as in the string model.
2.3. Multiple Fields, Target Space Geometry, and Jets
A further natural generalization of the models discussed so far regards theories with multiple fields and richer target space isometries. In that case the sigma model map
$X:{\Sigma}_{2}\to M$ has a target manifold
M replacing the circle
${S}^{1}$. The components of the map
X are scalar fields
${X}^{i}$ with
$i=1,\cdots ,\mathrm{dim}\phantom{\rule{0.166667em}{0ex}}M$, not all of which need to be necessarily compact. However, when some of them are compact, dualization may be performed along these directions.
5 We consider a nonlinear, possibly higher-derivative sigma model for the scalar fields
${X}^{i}$ given by the action functional
where
${G}_{ij}$ and
${B}_{ij}$ are background fields, which play the role of couplings in the 2D theory. In usual cases, such as for closed strings in background fields, these couplings are functions of
${\sigma}^{\mu}$ only through the scalar fields
${X}^{i}$. Here we consider an extension of this picture to theories where the background fields can also depend on
${\sigma}^{\mu}$ via the worldsheet differentials (field strength)
$\mathrm{d}{X}^{i}$. We have already seen a realization of this when we discussed the Heisenberg pion fireball model.
Firstly, it is important to identify under which conditions this action has some global symmetries. In the case where
${G}_{ij}$ and
${B}_{ij}$ are either constant or strictly
X-dependent, the answer is well known, see for instance [
12,
26] and references therein. Geometrically,
${G}_{ij}$ are the pull-back components of a Riemannian metric
G on
M and
${B}_{ij}$ the ones of a 2-form
B, with field strength
$H=\mathrm{d}B$. Then, the action is invariant under the global symmetry
provided that the following invariance conditions on the background fields hold
Here,
${\u03f5}^{a}$ is the (rigid) parameter of the global symmetry and
${\rho}_{a}^{i}$ are the components of a Lie algebra valued vector field
$\rho $ in the configuration space of fields, i.e.,
$\rho ={\rho}^{i}{\partial}_{i}$ and
${\rho}^{i}={\rho}_{a}^{i}{t}^{a}$ with
${t}^{a}$ generators of the global symmetry.
6 Furthermore,
${\beta}_{a}$ is an arbitrary 1-form on
M. Recall that the Lie derivative of e.g., the metric
G in a local coordinate system reads as
This tells us that the global symmetries of the theory are isometries of the target space geometry specified by
G and
H (a generalized metric in the language of [
28]), and
${\rho}_{a}$ is thus a host of Killing vector fields. Another way to think about this is to rewrite (
38) as
where
${\alpha}_{a}={\beta}_{a}-{\iota}_{{\rho}_{a}}B$. Then, one may think of
${\rho}_{a}+{\alpha}_{a}$ as a section of the generalized tangent bundle
$TM\oplus {T}^{\ast}M$ and the two invariance conditions can be combined into
where
$E=G+B$ and the bracket is the one on an exact Courant algebroid. In other words, this represents a generalized flow for the tensor field
E, see [
29,
30] for more details. The single compact scalar is obviously a special case of this with the isometry being the one on the circle
${S}^{1}$, in which case the invariance conditions are trivially satisfied. In the following, we revisit these well-known statements for the apparently more general case of background couplings
${G}_{ij},{B}_{ij}$ that depend both on the fields
${X}^{i}$ and their field strength
$\mathrm{d}{X}^{i}$, simultaneously generalizing the closed string on a circle and the Heisenberg model. In this case, one must treat
${X}^{i}$ and
$\mathrm{d}{X}^{i}$ as independent fields. There is a natural mathematical framework that allows us to deal with the variational principle of the action where fields and their higher derivatives are independent. It goes under the name of “variational bicomplex” [
14] which is based on an extension of the configuration space to the jet space. Below we provide a basic toolkit for our purposes.
Let us consider a smooth fiber bundle
$(E,\pi ,\Sigma )$ with base
$\Sigma $, fiber
M and projection
$\pi :E\to \Sigma $, and a point
$p\in \Sigma $. Two local sections
$X,Y\in {\Gamma}_{p}\left(\pi \right)$ are called 1-equivalent
7 at
p if
$X\left(p\right)=Y\left(p\right)$ and in an adapted coordinate system
$({\sigma}^{\mu},{x}^{i})$ of the bundle around
$X\left(p\right)$
for
$i=1,\cdots ,\mathrm{dim}\left(M\right)$ and
$\mu =1,\cdots ,\mathrm{dim}(\Sigma )$ and
${X}^{i}={X}^{\ast}\left({x}^{i}\right)$. This equivalence class is called the 1-jet of
X at
p and is denoted by
${j}_{p}^{1}X$. The first jet manifold of
$\pi $ is the set
The functions
${\pi}_{1}$ and
${\pi}_{1,0}$ defined by
are called the source and target projections respectively. Now let
$(U,u)$ be an adapted coordinate system on
E, where
$u=({\sigma}^{\mu},{x}^{i})$ and
U is an open neighbourhood around
$X\left(p\right)$. The induced coordinate system
$({U}^{\prime},{u}^{\prime})$ on
${J}^{1}\pi $ is defined by
where the new functions
${x}_{\mu}^{i}:{U}^{\prime}\to \mathbb{R}$ are specified by
and are known as derivative coordinates. The triples
$({J}^{1}\pi ,{\pi}_{1},\Sigma )$ and
$({J}^{1}\pi ,{\pi}_{1,0},E)$ are also fibered manifolds. These steps can be repeated
n times to form an
n-jet bundle.
In the same manner that
$(\frac{\partial}{\partial {\sigma}^{\mu}},\frac{\partial}{\partial {x}^{i}})$ form a complete basis for a fiber bundle, one can generalize this to an
n-jet space with the basis
A general vector field in this basis can be written as
At this point its components are completely arbitrary. For
$n=1$, the vector field
${V}^{\left(1\right)}\in \mathfrak{X}\left({J}^{1}\pi \right)$ is projectable if
${V}^{\mu}$,
${V}^{i}$ are smooth functions on
U and
${V}_{\mu}^{i}$ is a smooth function on
${U}^{\prime}\equiv {J}^{1}\left(U\right)$. However, there are two ways of extending the coordinate space to the one on the jet-bundle which restricts the general form of the vector field, as we briefly explain below.
Firstly, considering the bundle
$({T}^{\ast}E,{\pi}_{E}^{\ast},E)$ the total space
${\pi}_{1,0}^{\ast}\left({T}^{\ast}E\right)$ can be interpreted as a submanifold of
${T}^{\ast}{J}^{1}\pi $. This allows us to define differential forms on
${J}^{1}\pi $. A contact 1-form
${\omega}_{{\mu}_{1}\cdots {\mu}_{k}}^{i}$ can be written in local coordinates as
${\omega}_{{\mu}_{1}\cdots {\mu}_{k}}^{i}=d{x}_{{\mu}_{1}\cdots {\mu}_{k}}^{i}-{x}_{{\mu}_{1}\cdots {\mu}_{k}\nu}^{i}d{\sigma}^{\nu}$. Then a
total vector field ${V}^{\left(n\right)}$ on an
n-jet space is a vector field of the form (
50) such that for a contact form
${\omega}_{{\mu}_{1}\cdots {\mu}_{k}}^{i}$, it satisfies
$\omega \left({V}^{\left(n\right)}\right)=0$. This restricts the form of
${V}^{\left(n\right)}$ to
with the condition that
${x}_{{\mu}_{1}{\mu}_{2}\cdots {\mu}_{k}\nu}^{i}=-{x}_{\nu {\mu}_{1}\cdots {\mu}_{k}}^{i}$. More precisely, the total vector fields are the sections of bundle of holonomic tangent vectors which is a pullback of the bundle of vertical tangent vectors in the sense that they are in the kernel of the map
${\pi}_{\ast}:TE\to T\Sigma $, and the contact forms are the sections of bundle of contact cotangent vectors which is a pullback of bundle of the horizontal cotangent vectors. These contact forms are annihilated by total vector fields. In the following, however, we will not use this extension of vector space but rather the prolongation approach described below.
Another way of extending the basis to a complete basis in the jet space is through the prolongation of the vector space spanned by
$(\frac{\partial}{\partial {\sigma}^{\mu}},\frac{\partial}{\partial {x}^{i}})$. Corresponding to each local section of the bundle
$\pi $, there is a uniquely determined local section of the bundle
${\pi}_{1}$. This new section is called first prolongation and its coordinate representation is obtained by appending to the coordinates of the original section the derivatives of those coordinates. If
$(E,\pi ,\Sigma )$ is a bundle,
$W\subset \Sigma $ is an open submanifold and
$X\in {\Gamma}_{W}\left(\pi \right)$, then the first prolongation of
X is the section
${j}^{1}X\in {\Gamma}_{W}\left({\pi}_{1}\right)$ defined by
${j}^{1}X\left(p\right)={j}_{p}^{1}X$ for
$p\in W$. Therefore, the coordinate representation of
${j}^{1}X$ is
$({X}^{i},\frac{\partial {X}^{i}}{\partial {\sigma}^{\mu}})$. A general local section
$\psi \in {\Gamma}_{W}\left({\pi}_{1}\right)$ will have coordinates
$({\psi}^{i},{\psi}_{\mu}^{i})$ where
${\psi}_{\mu}^{i}$ is completely independent of
${\psi}^{i}$. Let us consider a vector field
$V\in \mathfrak{X}\left(E\right)$ with the coordinate representation
Then, the prolongation of
V is
${V}^{\left(1\right)}\in \mathfrak{X}\left({J}^{1}\pi \right)$ and can be represented in coordinates as [
31,
32]
where
$\frac{d}{d{\sigma}^{\mu}}$ is a total derivative defined as
A local basis for the full exterior algebra
$\Omega \left({J}^{n}\pi \right)$ can be given by the differential forms
After this short review of the jet-bundle, we are ready to go back to the original problem of what is the global symmetry of the action (
35) and how it is related to isometries of the target space geometry. The independent fields of the model are
${X}^{i}$ and
$\mathrm{d}{X}^{i}$, therefore we are extending the fiber bundle to a 1-jet bundle. In the following we use
${X}_{\mu}^{i}$ instead of
$\mathrm{d}{X}^{i}$, where we consider
$\mathrm{d}{X}^{i}={X}_{\mu}^{i}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}{\sigma}^{\mu}$ and
$\mathrm{d}{\sigma}^{\mu}$ contributes to the volume-form on the worldsheet. Moreover, both
${X}^{i}$ and
${X}_{\mu}^{i}$ are functions of
${\sigma}^{\mu}$. This means that we are considering a jet space defined via the basis
8 $\left\{{\partial}_{{X}^{i}},{\partial}_{{X}_{\mu}^{i}}\right\}$. Consider now the following vector field defined using the prolongation (
53) written in this basis,
with
${\Lambda}^{\mu},{\rho}^{i},{\xi}_{\mu}^{i}$ to be Lie algebra valued functions of
${\sigma}^{\mu},{X}^{i}$ in general. The transformation of the fields is
To make contact with the problem of symmetries of the action (
35), we restrict to the case where
${\rho}^{i}$,
${\xi}_{\mu}^{i}$ do not depend explicitly on
${\sigma}^{\mu}$, therefore
${\partial}_{\mu}{\rho}^{i}=0$. Moreover, we restrict to the prolongation where
${\Lambda}^{\mu}=0$. Then the candidate global symmetry becomes
By direct calculation the action (
35) is invariant under this global symmetry, if and only if the following generalized invariance conditions hold
in terms of the usual Lie derivative on the space with coordinate basis
${\partial}_{{X}^{i}}$.
These conditions acquire a more natural geometric interpretation in the 1-jet space, using the Lie derivative of a differentiable multilinear map
T of smooth sections of the 1-jet tangent and cotangent spaces defined using
${\pi}_{1}$ and
${\pi}_{1,0}$ into
$\mathbb{R}$, i.e.,
$T:{\otimes}^{q}\Gamma ({T}^{\ast}\Sigma )\times {\otimes}^{p}\Gamma (T\Sigma )\to \mathbb{R}$. If
${\omega}_{1},...{\omega}_{p}\in \Omega \left({J}^{1}\pi \right)$ and
${V}_{1},{V}_{2},...,{V}_{q}\in \mathfrak{X}\left({J}^{1}\pi \right)$, then the Lie derivative along some vector field
V in the 1-jet space is given
9 as
This formula also holds for nonprojectable vector fields since the right hand side of this equation is still well defined [
32]. Using this definition and the discussion above, it becomes obvious that the invariance conditions (
61) and (
62) acquire the form
for
G and
B the ordinary Riemannian metric and Kalb–Ramond field on the target space
M. This is the main result of the present section. It extends the well-known result of just
X-dependent background fields to the more general case of
$(X,\mathrm{d}X)$-dependent ones, such as the Heisenberg model and generalizations thereof. Remarkably, the usual statement about the global symmetries being Killing vector isometries of the target space
M is promoted to Killing vector isometries of the 1-jet for the mapping space. As for the single compact scalar with the circle of radius
R as target space, which is the prototypical example satisfying the usual invariance conditions trivially, the massless Heisenberg model does so for these extended invariance conditions. The model itself may be written in the form of a nonlinear sigma model
which is simply a rewriting of (
19), obtained by Taylor-expanding the square root. In this example, there only exists a single scalar field and, thus, there is no coupling
B. In the above language, the vector
V in the 1-jet space reads as
The coupling is now
$G=G\left({F}_{\mu}\right)$, with
$\frac{\partial G}{\partial X}=0$ and the derivatives of
X are assumed to be independent variables, i.e.,
${\partial}_{\mu}X\equiv {F}_{\mu}$. The generalized Lie derivative along this vector is now simply given by
This vanishes if and only if
Given the global symmetry (
60), we have
${\xi}_{\nu}=\frac{\partial \rho}{\partial X}{F}_{\nu}$, from which one can simplify further (
69) to
Obviously, the only solution to this condition is
$\frac{\partial \rho}{\partial X}=0$ and
$\rho $ can be scaled away such that the global shift symmetry is given by
$\delta X=\u03f5$, as in
Section 2.2. This is, however, a somewhat degenerate example because
G is only a function of derivative coordinates and does not reflect the full power of jet-space formulation. On one hand, that would be the case for effective theories that involve couplings which are functions of both fields and their derivatives. Possible candidates for such effective theories are the so-called generalized Galileons [
33]. On the other hand, the transformation of
${X}_{\mu}^{i}$ can be written in even more general way by (i) considering two different symmetry generators, i.e., for example
$\delta {X}_{\mu}^{i}={\xi}_{\mu a}^{i}{\u03f5}^{a}+{\xi}_{\mu J}^{i}{\lambda}^{J}$ where
${\lambda}^{J}$ are parameters of another type of symmetry, (ii) allowing
${\Lambda}^{\mu}\ne 0$ or
${\partial}_{\mu}{\rho}^{i}\ne 0$. A theory with Galilean shift symmetry [
34], or more general polynomial shift symmetries [
35], is an example of this extended notion of transformation. A Galilean symmetry is described by
${X}_{\mu}^{i}\to {X}_{\mu}^{i}+{b}_{\mu}^{i}$, where
${b}_{\mu}^{i}$ are constants. In our notation, this can be written as
This extra term in the transformation of
${X}^{i}$ inspires new terms in the transformation of
${X}_{\mu}^{i}$, which would then lead to more stringent geometrical constraints on the target space geometry. We leave a complete discussion of these apparently more general theories for future work.
2.4. T-Duality and the ’t Hooft Anomaly: Multiple Fields
Having identified the global symmetry of the multifield action (
35), we can now complete the discussion by first coupling spacetime background fields to it along the same lines as in
Section 2.1 and
Section 2.2, and then stating the duality rules for the target space background fields. In this section, we focus on the case where
${G}_{ij},{B}_{ij}$ are only functions of the dynamical fields
${X}^{i}$. Therefore, we restrict to those transformations under which
${X}^{i}$ transform as
$\delta {X}^{i}={\rho}_{a}^{i}{\u03f5}^{a}$ and the background fields
${A}^{a}$ and
${\widehat{A}}_{a}$ transform as
Here, the momentum and winding parameters
${\u03f5}^{a}$ and
${\widehat{\u03f5}}^{\phantom{\rule{0.166667em}{0ex}}a}$ are functions of the spacetime coordinates
${\sigma}^{\mu}$. The action (
35) can now be coupled to these background fields as
For vanishing background fields, the field equation for the scalars reads as
where
${\Gamma}_{jk}^{i}$ are the connection coefficients obtained from the metric
G and
${H}_{ijk}$ are the components of the 3-form field strength of
B. It is then straightforward to identify the two currents in this case, which read as
in terms of the constrained
${F}^{i}$. The conservation of these currents is a direct consequence of the field equation and the Bianchi identity respectively. The action (
73) contains the usual terms that couple background fields through these currents, improved by additional counter terms. With the already stated invariance conditions on the couplings
G and
B, this action is invariant under the background gauge transformation only when
${\widehat{A}}_{a}$ is set to zero. The transformation of the full action (
73) (which involves the term
$\ast {J}_{m}\wedge {\widehat{A}}_{a}$) under the background gauge symmetry is non-vanishing, specifically
Using adapted coordinates along the isometry directions such that
${\rho}_{a}^{i}={\delta}_{a}^{i}$, this anomalous term can be obtained from an anomaly polynomial in 4 dimensions
We thus observe that more possibilities arise in the multifield case when one wishes to gauge part of the full global symmetry. Apart from electric and magnetic gaugings, one may consider dyonic ones where some of the A and $\widehat{A}$ fields are kept and promoted to true gauge fields, whereas their conjugates in the anomaly polynomial are set to zero.
The action (
73) can be now dualized along the isometric directions with the same procedure as in
Section 2.1 and
Section 2.2. As before, the dual theory will have the same ’t Hooft anomaly with the original one but it will comprise different, T-dual couplings
${\widehat{G}}_{ij},{\widehat{B}}_{ij}$. As discussed in textbooks e.g., [
36], the dual action is
where
${\widehat{X}}^{i}=({X}^{\alpha},{\widehat{X}}_{m})$ are dual dynamical fields, differing from the original ones
${X}^{i}=({X}^{\alpha},{X}^{m})$ only in the directions
$m=1,\cdots d$ that undergo duality. The procedure establishes that the dual couplings are related through the Buscher rules, which in terms of the generalized metric
can be written as
where
$m,n$ run from 1 to the number
d of directions along which the dualization is performed. These are
$O(d,d,\mathbb{R})$ transformations of the generalized metric. This set of rules holds true for all corresponding sigma models of differential
$2p$-form fields in self-dual dimensions, e.g., 2-forms in 6 dimensions, with a kinetic term and a generalized theta term that corresponds to the antisymmetric coupling
${B}_{ij}$. For more details see [
12,
20].