Transverse generalized metrics and 2d sigma models

TL;DR

This paper introduces the concept of transverse generalized metrics within Courant algebroids, providing a geometric framework for string theory dynamics and gauging procedures.

Contribution

It reformulates compatibility conditions between generalized metrics and Dirac structures using Dirac geometry, extending the understanding of string phase space dynamics.

Findings

Defines transverse generalized metrics in Courant algebroids

Links these metrics to string phase space dynamics

Provides a Dirac-geometric reformulation of gauging conditions

Abstract

We reformulate the compatibility condition between a generalized metric and a small (non-maximal rank) Dirac structure in an exact Courant algebroid found in the context of the gauging of strings and formulated by means of two connections in purely Dirac-geometric terms. The resulting notion, a transverse generalized metric, is also what is needed for the dynamics on the reduced phase space of a string theory.

Equations

$$[\Gamma (D),\Gamma (W)]\subset \Gamma (W)$$

$$\bigl{[}(X,0),(u,h(u,\cdot))\bigr{]}=\bigl{(}[X,u],(\mathcal{L}_{X}h)(u,\cdot)+h([X,u],\cdot)+H(X,u,\cdot)\bigr{)}\in\Gamma(W)$$

$${\mathcal{L}}_{X}h=0,{\iota }_{X}H=0.$$

$$\mathrm{Sym}\left(\overline{\nabla }{\overline{\rho }}_D-{\varphi }^{\ast }({\alpha }_D)\right)$$

$$\mathrm{Alt}\left(\overline{\nabla }{\alpha }_D-{\varphi }^{\ast }({\overline{\rho }}_D)-\frac{1}{2}{\iota }_{{\rho }_D}H\right)$$

$$S[a]={\int }_{\Sigma }\frac{1}{2}||\mathrm{d}X-A_{TM}|{|}^2+⟨A_{T^{\ast }M}\stackrel{\wedge }{,}\mathrm{d}X-\frac{1}{2}{A}_{TM}⟩+\int H,$$

$$V_{+}:=V=\{(u,{\iota }_{u}g):=u_{+}\mid u\in TM\},V_{-}:=V^{\perp }=\{(u,-{\iota }_{u}g):=u_{-}\mid u\in TM\}$$

$$D^{\perp }\cap V_{\pm }=\{u_{\pm }\mid u\in \mathrm{Ann}(D_{\pm })\}.$$

$$⟨[(X,\alpha ),u_{+}],v_{-}⟩=0\text{whenever u+ and v− are in D⊥},$$

$$[(X,\alpha ),u_{+}]=({\mathcal{L}}_{X}u,{\iota }_{({\mathcal{L}}_{X}u)}g+{\iota }_u({\mathcal{L}}_{X}g-\mathrm{d}\alpha +{\iota }_{X}H)).$$

$$⟨[(X,\alpha ),u_{+}],v_{-}⟩=({\mathcal{L}}_{X}g+i_{X}H-\mathrm{d}\alpha )(u,v).$$

$${\mathcal{L}}_{X}g+i_{X}H-d\alpha \in \Gamma (D_{+}\otimes T^{\ast }M+T^{\ast }M\otimes D_{-}).$$

$${\mathcal{L}}_{X_a}g+{\iota }_{X_a}H-\mathrm{d}{\alpha }_a={\beta }_b^{+}\otimes ({\omega }^{+}{)}_a^b-({\omega }^{-}{)}_a^b\otimes {\beta }_b^{-}.$$

$${\mathcal{H}}_V(f)=\frac{1}{2}{\int }_{S^1}⟨f({\partial }_{\sigma }),R_{V}f({\partial }_{\sigma })⟩d\sigma ,$$

$$({\mu }^{\ast }(s))(f)={\int }_{S^1}⟨s,f⟩.$$

$${\mathcal{H}}_W([f])=\frac{1}{2}{\int }_{S^1}⟨Pf({\partial }_{\sigma }),R_{W}Pf({\partial }_{\sigma })⟩d\sigma ,$$

Full text

The condition $[\Gamma(D),\Gamma(W)]\subset\Gamma(W)$ means

[TABLE]

for every $X\in\Gamma(F)$ and every vector field $u$, and thus

[TABLE]

Together with $\iota_{X}h=0$, $\mathrm{d}H=0$, and the semi-positivity requirement on $W$, these two equations imply that $h$ and $H$ are the pullback of a Riemannian metric $g_{Q}$ and a closed 3-form $H_{Q}$, respectively. ∎

3. If $(M,g)$ is a Riemannian manifold and $\pi\colon M\to Q$ a submersion, then $\pi$ is called a Riemannian submersion iff $g$ descends to a Riemannian metric on $Q$. This means the following: for any $m\in M$ we decompose $T_{m}M$ orthogonally to $T_{m}^{\parallel}M\oplus T_{m}^{\perp}M$, where $T_{m}^{\parallel}M:=\ker d_{m}\pi$ is the subspace tangent to the fiber of $\pi$. Define $h$ as the unique symmetric bilinear form on $T_{m}M$ which agrees with $g$ on $T_{m}^{\perp}M$ and which gives zero when one of the vectors is in $T_{m}^{\parallel}$. Then $\pi$ is a Riemannian submersion iff $h=\pi^{*}g_{Q}$ for some Riemannian metric $g_{Q}$ on $Q$.

We reformulate this in a language adapted to our purposes: Take $V$ and $W$ to be the graphs of $g$ and $h$ inside the standard Courant algebroid $E_{0}:=(T\oplus T^{*})M$, respectively, and let $D:=\{(u,0)\in TM\oplus T^{*}M|\pi_{*}u=0\}$. One verifies easily that then $W=D\oplus(D^{\perp}\cap V)$. As we saw above, the relation $h=\pi^{*}g_{Q}$ for some $g_{Q}$ is equivalent to $W$ being a $D$-transverse generalized metric.

If one, more generally, has a foliation on a Riemannian manifold $(M,g)$ where there is no good quotient manifold $Q$ but where $g$ descends to the local spaces of leaves (i.e. to a transverse Riemannian metric), one talks of a (regular) Riemannian foliation [10]. This motivates the following definition:

A triple $(E,V,D)$—of an exact Courant algebroid $E$, a generalized metric $V\subset E$, and a small Dirac structure $D\subset E$—is a Dirac-Riemannian foliation iff $V_{D}:=D\oplus(D^{\perp}\cap V)$ is a $D$-transverse generalized metric.

4. In their simplest setting, sigma models are variational problems defined on the space of maps $X\colon\Sigma\to M$ from a $d$-dimensional, Lorentzian signature pseudo-Riemannian manifold $(\Sigma,\gamma)$ to a Riemannian manifold $(M,g)$. In the traditional setting of gauging such a theory, one needs to be given a group $G$ acting isometrically on $(M,g)$. The procedure is devised in such a way that when the group acts freely and properly on $M$, then the gauged sigma model is equivalent to the ungauged one with target manifold $M/G$.

The gauging is achieved by coupling to $\mathrm{Lie}(G)$-valued 1-forms $A$ on $\Sigma$ in such a way that two maps $X_{1}$ and $X_{2}$ which differ from one another only by the application of a $\Sigma$-dependent group element are related by a symmetry of the action functional depending on $X$ and $A$. If the original model is twisted by a closed $d+1$-form $H$, called a Wess-Zumino term, then gauging in this way requires that $H$ has an equivariantly closed extension [5, 4].

While every isometric $G$-action equips $(M,g)$ with an—in general only singular—Riemannian foliation, where the leaves are given by the $G$-orbits, not every such a foliation results from a $G$-action. According to [8, 9]222See also Section 2 of [3] for a possible definition of such a generalized notion of gauging., it is not necessary to restrict gauging to eventual isometries of $g$, it is sufficient that $(M,g)$ defines (a somewhat controlled form of) a singular Riemannian foliation. While the general theory of such gaugings is not worked out yet in the presence of a Wess-Zumino twist for arbitrary dimensions $d$, it was done so in [3] for $d=2$. Two-dimensional sigma models are somewhat particular since the Hodge dual of a 1-form $A$ is again a 1-form, yielding additional options for the gauging, and they are in general intimately related to Dirac geometry, see, e.g., [11, 1].

5. Let $V\subset E$ be a generalized metric and $D\subset E$ a small Dirac structure. Let us use the (unique) splitting $E\cong(T\oplus T^{*})M$ that turns $V$ into the graph of a Riemann metric. The resulting inclusion $D\to(T\oplus T^{*})M$ then gives us a section $(\rho_{D},\alpha_{D})\in\Gamma(D^{*}\otimes(T\oplus T^{*})M)$. Let us set $\overline{\rho}_{D}:=\iota_{\rho_{D}}g\in\Gamma(D^{*}\otimes T^{*}M)$.

In [3] it was shown that a two-dimensional sigma model with the above data on the target space can be gauged with respect to $D$, if $D$ can be equipped with two connections $\nabla^{\pm}:=\nabla\pm\phi$, $\phi\in\Omega^{1}(\mathrm{End}(D))$, such that $(M,g,H,D)$ or $(E,V,D)$ satisfy the following compatibility conditions:

[TABLE]

where $\overline{\nabla}$ is the extension of $\nabla$ to $T^{*}M$ by means of the Levi-Civita connection of $g$, $\mathrm{Sym}$ and $\mathrm{Alt}$ denote the symmetrization and antisymmetrization projections in $T^{*}M\otimes T^{*}M$, respectively, and $\phi^{*}\in\Omega^{1}(\mathrm{End}(D^{*}))$ is the 1-form valued map dual to $\phi$.

In this case the variational problem can be gauged by extending the fields from maps $X\colon\Sigma\to M$ to vector bundle morphisms $a\colon T\Sigma\to D$—thus adding gauge field 1-forms $A\in\Omega^{1}(\Sigma,X^{*}D)$. Using the canonical splitting given by the above data, the independent field $A$ gives rise to its projections $A_{TM}\in\Omega^{1}(\Sigma,X^{*}TM)$ and $A_{T^{*}\!M}\in\Omega^{1}(\Sigma,X^{*}T^{*}\!M)$ to $TM$ and $T^{*}\!M$, respectively. The gauged variational problem is then described symbolically by [3, 6]

[TABLE]

where for every $\nu\in\Omega^{1}(\Sigma,X^{*}TM)$ one has $||\nu||^{2}\equiv(X^{*}\!g)(\nu\stackrel{{\scriptstyle\wedge}}{{,}}\ast\nu)$ with $\ast$ denoting the Hodge dual associated to $\gamma$—symbolically, since $S$ is not really a functional due to the Wess-Zumino term, while it still defines a unique variational problem for the field $a$ in the standard manner.

6. It is one of the main purposes of the current letter to reformulate the conditions (2) and (3) in terms which are intrinsic to Courant algebroids. This is essentially achieved by means of the following statement.

[TABLE]

where we implicitly defined maps $u\mapsto u_{\pm}$ from $TM$ to $V_{\pm}$. Consider in addition the vector bundle maps $\pi_{\pm}\colon D\to T^{*}M,(X,\alpha)\mapsto\alpha\pm\iota_{X}g$. These are isomorphisms between $D_{\pm}:=\pi_{\pm}(D)\subset T^{*}M$ and $D$, since $\iota_{X}(\alpha\pm\iota_{X}g)=\pm||X||^{2}$ vanishes only for $X=0$. We have

[TABLE]

As $V_{D}^{\hphantom{D}\perp}=D\oplus(D^{\perp}\cap V_{-})$ and $[\Gamma(D),\Gamma(D)]\subset\Gamma(D)$, Condition (1) for $V_{D}\equiv D\oplus(D^{\perp}\cap V_{+})$ can be restated as that for every $(X,\alpha)\in\Gamma(D)$:

[TABLE]

i.e. whenever $u$ is annihilated by $D_{+}$ and $v$ by $D_{-}$ On the other hand, one computes

[TABLE]

giving

[TABLE]

This shows

[TABLE]

Denote by $e_{a}:=(X_{a},\alpha_{a})$ a local basis of $D$ and let $\beta_{a}^{\pm}=\pi_{\pm}(e_{a})$ be the induced bases in $D_{\pm}$. On a local level, Condition (6), and thus $V_{D}$ to be a transverse generalized metric, is equivalent to the existence of locally defined coefficient 1-forms $({\omega}^{\pm})_{a}^{b}$ such that333For the special case $H=\mathrm{d}B$ together with $D=\mathrm{graph}(-B)$, i.e. $\alpha=-\iota_{X}B$ in the description of $D$ above, the left-hand side of the following equation becomes simply ${\cal L}_{X_{a}}(g+B)$, thus reproducing Equation (2.21) in [2].

[TABLE]

This now is verified to be the local form of the equations (2) and (3), with $({\omega}^{\pm})_{a}^{b}$ being the connection coefficients of $\nabla^{\pm}$ in the chosen basis, $\nabla^{\pm}e_{a}=({\omega}^{\pm})_{a}^{b}\otimes e_{b}$. The global existence of the connections then follows by a standard argument using a partition of unity. ∎

7. An exact Courant algebroid $E\to M$ gives rise to an infinite-dimensional symplectic manifold ${\cal M}$ which is the phase space of 2-dimensional sigma models: If a splitting of $E=(T\oplus T^{*})M$ is chosen, giving rise to a closed 3-form $H\in\Omega^{3}(M)$, then we have ${\cal M}=T^{*}(LM)$ with the standard symplectic 2-form modified by the transgression of $H$. More naturally, ${\cal M}$ is the space of all vector bundle maps $f\colon TS^{1}\to E$, covering the base map $f_{0}\colon S^{1}\to M$, such that $\rho\circ f\colon TS\to TM$ agrees with the tangent map of $f_{0}$.

A generalized metric $V\subset E$ then defines a function $\mathcal{H}_{V}$ on ${\cal M}$ which is the Hamiltonian of the corresponding two-dimensional sigma model:

[TABLE]

where $\sigma$ is the coordinate on $S^{1}$ and $R_{V}\colon E\to E$ is the reflection with respect to $V$.

A small Dirac structure $D\subset E$ defines the Lie algebra $\mathfrak{g}_{D}:=C^{\infty}(S^{1})\otimes\Gamma(D)$ together with a Lie algebra map $\mu^{*}\colon\mathfrak{g}_{D}\to C^{\infty}({\cal M})$ given by [1]

[TABLE]

The reduced phase space ${\cal M}/\!/\mathfrak{g}_{D}$ is composed of maps $f\colon TS^{1}\to D^{\perp}$, i.e. $f$’s in ${\cal M}$ satisfying $(\mu^{*}(s))(f)=0$ for all $s\in\mathfrak{g}_{D}$, modulo the action of $\mathfrak{g}_{D}$. A $D$-transverse generalized metric $W\subset E$ is then precisely what is needed to provide a Hamiltonian on the reduced phase space ${\cal M}/\!/\mathfrak{g}_{D}$. Define

[TABLE]

where $P$ denotes the natural projection from $D^{\perp}$ to $D^{\perp}/D$, and $R_{W}\colon D^{\perp}/D\to D^{\perp}/D$ is the orthogonal reflection with respect to $W/D$ which corresponds to the transverse generalized metric $W$.

Acknowledgement

The work was supported by the grant MODFLAT of the European Research Council and the NCCR SwissMAP of the Swiss National Science Foundation.

T.S. is grateful for the hospitality and the wonderful working conditions in the group of Anton Alekseev during visits when this work was began.