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Nihar Karve
Nihar Karve
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  1. Preliminary: $\mathrm{AdS_3}$ can be embedded in $\mathbb R^{2,2}$ subject to the hyperboloid constraint $-\eta_{\mu\nu}x^\mu x^\nu=L^2$, $\eta_{\mu\nu}=\mathrm{diag}(-1,-1,1,1)$. This can additionally be parameterised by $$ \chi=\frac{1}{L^2}\pmatrix{x^0-x^2 & -x^1+x^3 \\ x^1+x^3 & x^0+x^2}\overset!\in\mathrm{SL}(2,\mathbb R)\tag{1} $$ as the hyperboloid constraint imposes $\det\chi=1$.

  2. The flat spacetime Polyakov action can be converted into a $\mathrm U(1)^D$ model: $$ S\sim\int_\Sigma\mathrm d^2\sigma\ \sqrt{-h}h^{\alpha\beta}\eta_{\mu\nu}\partial_\alpha X^\mu\partial_\beta X^\nu \\\sim\int\mathrm d^2\sigma\ \mathrm{Tr}\left(\partial g\ \bar\partial g^{-1}\right) $$ $$= -\int\mathrm d^2\sigma\ \mathrm{Tr}\left(g^{-1}\partial g\ g^{-1}\bar\partial g\right)\tag{2} $$ where $\mathrm U(1)^D\ni g = \exp(t_i X^i), [t_i,t_j]=0, \mathrm{Tr}(t_i t_j)=\eta_{ij}$. Note that $g$ is not the metric, but rather the space of maps from the 2D worldsheet $\Sigma$ into the target (group) manifold.

  3. The action $(2)$ is easy to generalise to non-abelian semisimple Lie algebras, yielding the corresponding metric in the non-linear sigma model. In particular, choosing the (simple) Lie algebra $\mathfrak{sl}(2, \mathbb R)$ results in a non-linear sigma model over $\mathrm{AdS_3}$, from $(1)$. Note that $g^{-1}\partial g\in\Omega^1(M)\otimes\mathfrak{Lie}(G)$, cf. the Maurer-Cartan form. Now $\mathrm d\omega+\frac12[\omega, \omega]=0$Maurer-Cartan form. In particular, choosing the Maurer-Cartan form as the connection on $G$ leads to $R=0$, via the Cartan structure equation $\mathrm d\omega+\frac12[\omega, \omega]=0$. But this precisely encodes the constraints imposed upon worldsheet fields by the one-loop beta functions!

  4. However, unlike for the $\mathrm U(1)^D$ model, the group manifold action for the non-linear sigma model does not automatically satisfy the worldsheet beta function constraints for a general connection. Enter the Witten term: $$ S_W=\int_\mathcal{Y} W=\int_\mathcal{Y}\mathrm{Tr}(g^{-1}\mathrm dg\ g^{-1}\mathrm dg\ g^{-1}\mathrm dg),\qquad \partial\mathcal Y=\Sigma \\\Omega^3(\mathcal Y)\ni W\overset!=\mathrm dB\Rightarrow\int_\mathcal{Y} W = \int_\Sigma B $$$$ S_W=\int_\mathcal{Y} W=\int_\mathcal{Y}\mathrm{Tr}(g^{-1}\mathrm dg\ g^{-1}\mathrm dg\ g^{-1}\mathrm dg),\qquad \partial\mathcal Y=\Sigma \\\Omega^3(\mathcal Y)\ni W\overset!=\mathrm dB\Rightarrow\int_\mathcal{Y} W = \int_\Sigma B\tag{3} $$ This term corresponds to the topological NS-NS B-field.

  5. Ordinarily, this term would have a quantized prefactor in the action, to ensure conformal invariance at the quantum level. However, for $\mathrm{SL}(2, \mathbb R)$, this is unnecessary as the NS-NS three-form field vanishes.

  1. Preliminary: $\mathrm{AdS_3}$ can be embedded in $\mathbb R^{2,2}$ subject to the hyperboloid constraint $-\eta_{\mu\nu}x^\mu x^\nu=L^2$, $\eta_{\mu\nu}=\mathrm{diag}(-1,-1,1,1)$. This can additionally be parameterised by $$ \chi=\frac{1}{L^2}\pmatrix{x^0-x^2 & -x^1+x^3 \\ x^1+x^3 & x^0+x^2}\overset!\in\mathrm{SL}(2,\mathbb R)\tag{1} $$ as the hyperboloid constraint imposes $\det\chi=1$.

  2. The flat spacetime Polyakov action can be converted into a $\mathrm U(1)^D$ model: $$ S\sim\int_\Sigma\mathrm d^2\sigma\ \sqrt{-h}h^{\alpha\beta}\eta_{\mu\nu}\partial_\alpha X^\mu\partial_\beta X^\nu \\\sim\int\mathrm d^2\sigma\ \mathrm{Tr}\left(\partial g\ \bar\partial g^{-1}\right) $$ $$= -\int\mathrm d^2\sigma\ \mathrm{Tr}\left(g^{-1}\partial g\ g^{-1}\bar\partial g\right)\tag{2} $$ where $\mathrm U(1)^D\ni g = \exp(t_i X^i), [t_i,t_j]=0, \mathrm{Tr}(t_i t_j)=\eta_{ij}$.

  3. The action $(2)$ is easy to generalise to non-abelian semisimple Lie algebras, yielding the corresponding metric in the non-linear sigma model. In particular, choosing the (simple) Lie algebra $\mathfrak{sl}(2, \mathbb R)$ results in a non-linear sigma model over $\mathrm{AdS_3}$, from $(1)$. Note that $g^{-1}\partial g\in\Omega^1(M)\otimes\mathfrak{Lie}(G)$, cf. the Maurer-Cartan form. Now $\mathrm d\omega+\frac12[\omega, \omega]=0$. In particular, choosing the Maurer-Cartan form as the connection on $G$ leads to $R=0$. But this precisely encodes the constraints imposed upon worldsheet fields by the one-loop beta functions!

  4. However, unlike for the $\mathrm U(1)^D$ model, the group manifold action for the non-linear sigma model does not automatically satisfy the worldsheet beta function constraints for a general connection. Enter the Witten term: $$ S_W=\int_\mathcal{Y} W=\int_\mathcal{Y}\mathrm{Tr}(g^{-1}\mathrm dg\ g^{-1}\mathrm dg\ g^{-1}\mathrm dg),\qquad \partial\mathcal Y=\Sigma \\\Omega^3(\mathcal Y)\ni W\overset!=\mathrm dB\Rightarrow\int_\mathcal{Y} W = \int_\Sigma B $$ This term corresponds to the NS-NS B-field.

  5. Ordinarily, this term would have a quantized prefactor to ensure conformal invariance at the quantum level. However, for $\mathrm{SL}(2, \mathbb R)$, this is unnecessary as the NS-NS three-form field vanishes.

  1. Preliminary: $\mathrm{AdS_3}$ can be embedded in $\mathbb R^{2,2}$ subject to the hyperboloid constraint $-\eta_{\mu\nu}x^\mu x^\nu=L^2$, $\eta_{\mu\nu}=\mathrm{diag}(-1,-1,1,1)$. This can additionally be parameterised by $$ \chi=\frac{1}{L^2}\pmatrix{x^0-x^2 & -x^1+x^3 \\ x^1+x^3 & x^0+x^2}\overset!\in\mathrm{SL}(2,\mathbb R)\tag{1} $$ as the hyperboloid constraint imposes $\det\chi=1$.

  2. The flat spacetime Polyakov action can be converted into a $\mathrm U(1)^D$ model: $$ S\sim\int_\Sigma\mathrm d^2\sigma\ \sqrt{-h}h^{\alpha\beta}\eta_{\mu\nu}\partial_\alpha X^\mu\partial_\beta X^\nu \\\sim\int\mathrm d^2\sigma\ \mathrm{Tr}\left(\partial g\ \bar\partial g^{-1}\right) $$ $$= -\int\mathrm d^2\sigma\ \mathrm{Tr}\left(g^{-1}\partial g\ g^{-1}\bar\partial g\right)\tag{2} $$ where $\mathrm U(1)^D\ni g = \exp(t_i X^i), [t_i,t_j]=0, \mathrm{Tr}(t_i t_j)=\eta_{ij}$. Note that $g$ is not the metric, but rather the space of maps from the 2D worldsheet $\Sigma$ into the target (group) manifold.

  3. The action $(2)$ is easy to generalise to non-abelian semisimple Lie algebras, yielding the corresponding metric in the non-linear sigma model. In particular, choosing the (simple) Lie algebra $\mathfrak{sl}(2, \mathbb R)$ results in a non-linear sigma model over $\mathrm{AdS_3}$, from $(1)$. Note that $g^{-1}\partial g\in\Omega^1(M)\otimes\mathfrak{Lie}(G)$, cf. the Maurer-Cartan form. In particular, choosing the Maurer-Cartan form as the connection on $G$ leads to $R=0$, via the Cartan structure equation $\mathrm d\omega+\frac12[\omega, \omega]=0$. But this precisely encodes the constraints imposed upon worldsheet fields by the one-loop beta functions!

  4. However, unlike for the $\mathrm U(1)^D$ model, the group manifold action for the non-linear sigma model does not automatically satisfy the worldsheet beta function constraints for a general connection. Enter the Witten term: $$ S_W=\int_\mathcal{Y} W=\int_\mathcal{Y}\mathrm{Tr}(g^{-1}\mathrm dg\ g^{-1}\mathrm dg\ g^{-1}\mathrm dg),\qquad \partial\mathcal Y=\Sigma \\\Omega^3(\mathcal Y)\ni W\overset!=\mathrm dB\Rightarrow\int_\mathcal{Y} W = \int_\Sigma B\tag{3} $$ This term corresponds to the topological NS-NS B-field.

  5. Ordinarily, this term would have a quantized prefactor in the action, to ensure conformal invariance at the quantum level. However, for $\mathrm{SL}(2, \mathbb R)$, this is unnecessary as the NS-NS three-form field vanishes.

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Nihar Karve
Nihar Karve
  • 8.6k
  • 4
  • 29
  • 50

  1. Preliminary: $\mathrm{AdS_3}$ can be embedded in $\mathbb R^{2,2}$ subject to the hyperboloid constraint $-\eta_{\mu\nu}x^\mu x^\nu=L^2$, $\eta_{\mu\nu}=\mathrm{diag}(-1,-1,1,1)$. This can additionally be parameterised by $$ \chi=\frac{1}{L^2}\pmatrix{x^0-x^2 & -x^1+x^3 \\ x^1+x^3 & x^0+x^2}\overset!\in\mathrm{SL}(2,\mathbb R)\tag{1} $$ as the hyperboloid constraint imposes $\det\chi=1$.

  2. The flat spacetime Polyakov action can be converted into a $\mathrm U(1)^D$ model: $$ S\sim\int_\Sigma\mathrm d^2\sigma\ \sqrt{-h}h^{\alpha\beta}\eta_{\mu\nu}\partial_\alpha X^\mu\partial_\beta X^\nu \\\sim\int\mathrm d^2\sigma\ \mathrm{Tr}\left(\partial g\ \bar\partial g^{-1}\right) $$ $$= -\int\mathrm d^2\sigma\ \mathrm{Tr}\left(g^{-1}\partial g\ g^{-1}\bar\partial g\right)\tag{2} $$ where $\mathrm U(1)^D\ni g = \exp(t_i X^i), [t_i,t_j]=0, \mathrm{Tr}(t_i t_j)=\eta_{ij}$.

  3. The action $(2)$ is easy to generalise to non-abelian semisimple Lie algebras, yielding the corresponding metric in the non-linear sigma model. In particular, choosing the (simple) Lie algebra $\mathfrak{sl}(2, \mathbb R)$ results in a non-linear sigma model over $\mathrm{AdS_3}$, from $(1)$. Note that $g^{-1}\partial g\in\Omega^1(M)\otimes\mathfrak{Lie}(G)$, cf. the Maurer-Cartan form. Now $\mathrm d\omega+\frac12[\omega, \omega]=0$. In particular, choosing the Maurer-Cartan form as the connection on $G$ leads to $R=0$. But this precisely encodes the constraints imposed upon worldsheet fields by the one-loop beta functions!

  4. However, unlike for the $\mathrm U(1)^D$ model, the group manifold action for the non-linear sigma model does not automatically satisfy the worldsheet beta function constraints for a general connection. Enter the Witten term: $$ S_W=\int_\mathcal{Y} W=\int_\mathcal{Y}\mathrm{Tr}(g^{-1}\mathrm dg\ g^{-1}\mathrm dg\ g^{-1}\mathrm dg),\qquad \partial\mathcal Y=\Sigma \\\Omega^3(\mathcal Y)\ni W\overset!=\mathrm dB\Rightarrow\int_\mathcal{Y} W = \int_\Sigma B $$ This term corresponds to the NS-NS B-field.

  5. Ordinarily, this term would have a quantized prefactor to ensure conformal invariance at the quantum level. However, for $\mathrm{SL}(2, \mathbb R)$, this is unnecessary as the NS-NS three-form field vanishes.