Partition function for nonlinear sigma model
$begingroup$
Let $(M,g)$ and $(N,h)$ be Riemannian manifolds. The action for the nonlinear sigma model with base $M$ and target $N$ is simply the Dirichlet energy (i.e. harmonic map energy) for maps $phi:Mto N$:
$$
S(phi)=int_M|dphi|^2;dV_g.
$$
Critical points of $S$ are harmonic maps. For example, when the target $(N,h)$ is just $mathbb{R}$ with the standard metric, then critical points of $S$ are harmonic functions, i.e. those maps $phi:Mto mathbb{R}$ satisfying $Delta_gphi=0$.
The partition function for the nonlinear sigma model is formally given by the path integral
$$
Z=int e^{-S(phi)}; Dphi.
$$
Question: How can we rigorously define $Z$ for general target manifolds $(N,h)$?
Note that there are cases when $Z$ is well defined:
Special case: $(N,h)=(mathbb{R},g_{mathrm{std}})$ In this case, we can use zeta function regularization. The Minakshisundaram–Pleijel zeta function $zeta:mathbb{C}to mathbb{C}$ defined by
$$
zeta(s)=sum_{lambdainmathrm{Spec}(Delta_g)setminus {0}} |lambda|^{-s} qquadqquad text{for}qquad mathrm{Re}(s)>dim M/2
$$ is holomorphic for $mathrm{Re}(s)>dim M/2$ and analytically extends to a meromorphic function on all of $mathbb{C}$ which is regular at $s=0$. We then define the partition function by
$$
Z=det Delta_g^{-1}=mathrm{exp}(zeta'(0)).
$$
Can this method be extended to general target manifolds?
functional-analysis differential-geometry mathematical-physics
asked Jan 23 at 0:14
rpfrpf
1,100513
$endgroup$
add a comment |
$begingroup$
Let $(M,g)$ and $(N,h)$ be Riemannian manifolds. The action for the nonlinear sigma model with base $M$ and target $N$ is simply the Dirichlet energy (i.e. harmonic map energy) for maps $phi:Mto N$:
$$
S(phi)=int_M|dphi|^2;dV_g.
$$
Critical points of $S$ are harmonic maps. For example, when the target $(N,h)$ is just $mathbb{R}$ with the standard metric, then critical points of $S$ are harmonic functions, i.e. those maps $phi:Mto mathbb{R}$ satisfying $Delta_gphi=0$.
The partition function for the nonlinear sigma model is formally given by the path integral
$$
Z=int e^{-S(phi)}; Dphi.
$$
Question: How can we rigorously define $Z$ for general target manifolds $(N,h)$?
Note that there are cases when $Z$ is well defined:
Special case: $(N,h)=(mathbb{R},g_{mathrm{std}})$ In this case, we can use zeta function regularization. The Minakshisundaram–Pleijel zeta function $zeta:mathbb{C}to mathbb{C}$ defined by
$$
zeta(s)=sum_{lambdainmathrm{Spec}(Delta_g)setminus {0}} |lambda|^{-s} qquadqquad text{for}qquad mathrm{Re}(s)>dim M/2
$$ is holomorphic for $mathrm{Re}(s)>dim M/2$ and analytically extends to a meromorphic function on all of $mathbb{C}$ which is regular at $s=0$. We then define the partition function by
$$
Z=det Delta_g^{-1}=mathrm{exp}(zeta'(0)).
$$
Can this method be extended to general target manifolds?
functional-analysis differential-geometry mathematical-physics
asked Jan 23 at 0:14
rpfrpf
1,100513
$endgroup$
add a comment |
$begingroup$
Let $(M,g)$ and $(N,h)$ be Riemannian manifolds. The action for the nonlinear sigma model with base $M$ and target $N$ is simply the Dirichlet energy (i.e. harmonic map energy) for maps $phi:Mto N$:
$$
S(phi)=int_M|dphi|^2;dV_g.
$$
Critical points of $S$ are harmonic maps. For example, when the target $(N,h)$ is just $mathbb{R}$ with the standard metric, then critical points of $S$ are harmonic functions, i.e. those maps $phi:Mto mathbb{R}$ satisfying $Delta_gphi=0$.
The partition function for the nonlinear sigma model is formally given by the path integral
$$
Z=int e^{-S(phi)}; Dphi.
$$
Question: How can we rigorously define $Z$ for general target manifolds $(N,h)$?
Note that there are cases when $Z$ is well defined:
Special case: $(N,h)=(mathbb{R},g_{mathrm{std}})$ In this case, we can use zeta function regularization. The Minakshisundaram–Pleijel zeta function $zeta:mathbb{C}to mathbb{C}$ defined by
$$
zeta(s)=sum_{lambdainmathrm{Spec}(Delta_g)setminus {0}} |lambda|^{-s} qquadqquad text{for}qquad mathrm{Re}(s)>dim M/2
$$ is holomorphic for $mathrm{Re}(s)>dim M/2$ and analytically extends to a meromorphic function on all of $mathbb{C}$ which is regular at $s=0$. We then define the partition function by
$$
Z=det Delta_g^{-1}=mathrm{exp}(zeta'(0)).
$$
Can this method be extended to general target manifolds?
functional-analysis differential-geometry mathematical-physics
asked Jan 23 at 0:14
rpfrpf
1,100513
$endgroup$
Let $(M,g)$ and $(N,h)$ be Riemannian manifolds. The action for the nonlinear sigma model with base $M$ and target $N$ is simply the Dirichlet energy (i.e. harmonic map energy) for maps $phi:Mto N$:
$$
S(phi)=int_M|dphi|^2;dV_g.
$$
Critical points of $S$ are harmonic maps. For example, when the target $(N,h)$ is just $mathbb{R}$ with the standard metric, then critical points of $S$ are harmonic functions, i.e. those maps $phi:Mto mathbb{R}$ satisfying $Delta_gphi=0$.
The partition function for the nonlinear sigma model is formally given by the path integral
$$
Z=int e^{-S(phi)}; Dphi.
$$
Question: How can we rigorously define $Z$ for general target manifolds $(N,h)$?
Note that there are cases when $Z$ is well defined:
Special case: $(N,h)=(mathbb{R},g_{mathrm{std}})$ In this case, we can use zeta function regularization. The Minakshisundaram–Pleijel zeta function $zeta:mathbb{C}to mathbb{C}$ defined by
$$
zeta(s)=sum_{lambdainmathrm{Spec}(Delta_g)setminus {0}} |lambda|^{-s} qquadqquad text{for}qquad mathrm{Re}(s)>dim M/2
$$ is holomorphic for $mathrm{Re}(s)>dim M/2$ and analytically extends to a meromorphic function on all of $mathbb{C}$ which is regular at $s=0$. We then define the partition function by
$$
Z=det Delta_g^{-1}=mathrm{exp}(zeta'(0)).
$$
Can this method be extended to general target manifolds?
functional-analysis differential-geometry mathematical-physics
functional-analysis differential-geometry mathematical-physics
asked Jan 23 at 0:14
rpfrpf
1,100513
asked Jan 23 at 0:14
rpfrpf
1,100513
edited Jan 23 at 20:32
rpf
asked Jan 23 at 0:14
rpfrpf
1,100513
asked Jan 23 at 0:14
rpfrpf
1,100513
asked Jan 23 at 0:14
rpfrpf
1,100513
1,100513
add a comment |
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