Consider the cotangent bundle $T^\ast X$ as a symplectic manifold with canonical symplectic form $\omega$. Consider the Hamilton-Jacobi equation
\[ \frac{\partial S}{\partial t} + H(x, \nabla S) = 0, \]
for the classical Hamilton function $S(x,t)$. Setting $x=x(t), p(t) = (\nabla S)(x(t), t)$ one sees immediately from the method of characteristics that this PDE is solved by the classical action
\[ S(x,t) = \int_0^t (p \dot{x} - H) ds, \]
where the integral is taken over the solution $(x(s),p(s))$ of Hamilton's equations with $x(0)=x_0$ and $x(t) = x$. The choice of basepoint $x_0$ involves an overall additive constant of $S$, and really this solution is only valid in some neighbourhood $U$ of $x_0$. (Reason: $S$ is in general multivalued, as the differential "$dS$" is closed but not necessarily exact.)
Now consider the case where $X$ is Riemannian, with Hamiltonian $H(x,p) = \frac{1}{2} |p|^2$. The solutions to Hamilton's equations are affinely parametrized geodesics, and by a simple Legendre transform we have
\[ S(x, t) = \frac{1}{2} \int_0^t |\dot x|^2 ds \]
where the integral is along the affine geodesic with $x(0) = x_0$ and $x(t) = x$. Since $x(s)$ is a geodesic, $|\dot x(s)|$ is a constant (in $s$) and therefore
\[ S(x, t) = \frac{t}{2} |\dot x(0)|^2. \]
Now consider the path $\gamma(s) = x($|\dot x(0)|^{-1}$s)$. This is an affine geodesic with $\gamma(0) = x_0$, $\gamma(|\dot x(0)|t) = x$ and $|\dot \gamma| = 1$. Therefore, the Riemannian distance between $x_0$ and $x$ (provided $x$ is sufficiently close to $x_0$) is
\[ d(x_0, x) = |\dot x(0)| t. \]
Combining this with the previous calculation, we see that
\[ S(x, t) = \frac{1}{2t} d(x_0, x)^2. \]
Now insert this back into the Hamilton-Jacobi equation above. With a bit of rearranging, we have the following.
Theorem. Let $x_0$ denote a fixed basepoint of $X$. Then for all $x$ in a sufficiently small neighborhood $U$ of $x_0$, the Riemannian distance function satisfies the Eikonal equation
\[ |\nabla_x d(x_0, x)|^2 = 1. \]
Now, for convenience set $r(x) = d(x_0, x)$. Then $|\nabla r|^2 = 1$, from which we obtain (by differentiating twice and contracting)
\[ g^{ij} g^{kl}\left(\nabla_{lki} r \nabla_j r + \nabla_{ki}r \nabla_{lj} r\right) = 0.\]
Quick calculation shows that
\[ \nabla_{lki} r = \nabla_{ilk} r - \left.R_{li}\right.^{b}_k \nabla_b r \]
Therefore, tracing over $l$ and $k$ we obtain
\[ g^{lk} \nabla_{lki} r = \nabla_i ( \Delta r) + Rc(\nabla r, -) \]
Plugging this back into the equation derived above, we have
\[ \nabla r \cdot \nabla(\Delta r) + Rc(\nabla r, \nabla r) + |Hr|^2 = 0, \]
where $Hr$ denotes the Hessian of $r$ regarded as a 2-tensor. Now, using $r$ as a local coordinate, it is easy to see that $\partial_r = \nabla r$ (as vector fields). So we can rewrite this identity as
\[ \partial_r (\Delta r) + Rc(\partial_r, \partial_r) + |Hr|^2 = 0. \]
Now, we can get a nice result out of this. First, note that the Hessian $Hr$ always has at least one eigenvalue equal to zero, because the Eikonal equation implies that $Hr(\partial_r, -)=0$. Let $\lambda_2, \dots, \lambda_n$ denote the non-zero eigenvalues of $Hr$. We have
\[ |Hr|^2 = \lambda_2^2 + \dots + \lambda_n^2, \]
while on the other hand
\[ |\Delta r|^2 = (\lambda_2 + \dots + \lambda_n)^2 \]
By Cauchy-Schwarz, we have
\[ |\Delta r|^2 \leq (n-1)|Hr|^2 \]
Proposition. Suppose that the Ricci curvature of $X$ satisfies $Rc \geq (n-1)\kappa$, and let $u = (n-1)(\Delta r)^{-1}$. Then
\[ u' \geq 1 + \kappa u^2. \]
Proof. From preceding formulas, $|Hr|^2$ can be expressed in terms of the Ricci curvature and the radial derivative of $\Delta r$. On the other hand, $|\Delta|^2$ is bounded above by $(n-1) |Hr|^2$. The claimed inequality then follows from simple rearrangement.
Now, the amazing thing is that this deceptively simple inequality is the main ingredient of the Bishop-Gromov comparison theorem. The Bishop-Gromov comparison theorem, in turn, is the main ingredient of the proof of Gromov(-Cheeger) precompactness. I hope to discuss these topics in a future post.
Monday, August 31, 2015
Tuesday, August 18, 2015
The Classical Partition Function
Let $(M, \omega)$ be a symplectic manifold of dimension $2n$, and let $H: M \to \mathbf{R}$ be a classical Hamiltonian. The symplectic form $\omega$ allows us to define a measure on $M$, given by integration against the top form $\omega^n / n!$. We will denote this measure by $d\mu$.
We imagine that $(M, \omega, H)$ represents some classical mechanical system. We suppose that the dynamics of this dynamical system are very complicated, e.g. some system of $10^{23}$ particles. The system is so complicated that not only can we not solve the equations of motion exactly, and even if we could, their solutions might be so complicated that we can't expect to learn very much from them.
So instead, we ask statistical questions. Imagine that we cannot measure the state of the system exactly (e.g. particles in a box), so we try to guess a probability distribution $\rho(x,p,t)$ on $M$ indicating that at time $t$ the system has probability $\rho(x,p,t) d\mu$ of being in the state $(x,p)$. Obviously, $\rho$ should satisfy the constraint $\int_M \rho d\mu = 1$.
How does $\rho$ evolve in time? We know that the system obeys Hamilton's equations,
\[ (\dot x, \dot p) = X_H = (\partial H / \partial p, -\partial H / \partial x) \]
in local Darboux coordinates. Therefore, a particle located at $(x,p)$ in phase space at time $t$ will be located at $(x,p)+X_H dt$ in phase space at time $t+dt$. Therefore, the probability that a particle is at point $(x,p)$ at time $t+dt$, should be equal to the probability that the particle is at point $(x,p)-X_H dt$ at time $t$. Therefore, we have
\[ \frac{\partial \rho}{\partial t} = \frac{\partial H}{\partial x} \frac{\partial \rho}{\partial p} - \frac{\partial H}{\partial p} \frac{\partial \rho}{\partial x} = \{H, \rho\} \]
Given a probability distribution $\rho$, the entropy is defined to be
\[ S[\rho] = -\int_M \rho \log \rho d\mu. \]
(A version of) the second law of thermodynamics. For a given average energy $U$, the system assumes a distribution of maximal possible entropy at thermodynamic equilibrium.
The goal now, is to determine what distribution $\rho$ will maximize the entropy, subject to the constraints (for fixed $U$)
\begin{align*} \int_M H \rho d\mu &= U \\\
\int_M \rho d\mu &= 1 \end{align*}
Setting aside technical issues of convergence, etc., this variational problem is easily solved using the method of Lagrange multipliers. Introducing parameters $\lambda_1, \lambda_2$, we consider the modified functional
\[ S[\rho, \lambda_1, \lambda_2, U] = \int_M\left(-\rho \log \rho +\lambda_1\rho +\lambda_2(H\rho)\right)d\mu -\lambda_1-\lambda_2 U. \]
Note that $\partial S / \partial U = -\lambda_2$, and this is conventionally identified with (minus) the inverse temperature.
Taking the variation with respect to $\rho$, we find
\[ 0= \frac{\delta S}{\delta \rho} = -\log \rho-1+\lambda_1+H\lambda_2\]
Therefore, $rho$ is proportional to $e^{-\beta H}$ where we have set $\beta=-\lambda_2$. Define the partition function $Z$ to be
\[ Z = \int_M e^{-\beta H} d\mu. \]
We therefore have proved (formally and heuristically only!):
Theorem. The probability distribution $\rho$ assumed by the system at thermodynamic equilibrium is given by
\[ \rho = \frac{e^{-\beta H}}{Z} \]
where $\beta > 0$ is a real parameter, called the inverse temperature.
Corollary. At thermodynamic equilibrium, the average energy is given by
\[ U = -\frac{\partial \log Z}{\partial \beta} , \]
and the entropy is given by
\[ S = \beta U + \log Z.\]
We imagine that $(M, \omega, H)$ represents some classical mechanical system. We suppose that the dynamics of this dynamical system are very complicated, e.g. some system of $10^{23}$ particles. The system is so complicated that not only can we not solve the equations of motion exactly, and even if we could, their solutions might be so complicated that we can't expect to learn very much from them.
So instead, we ask statistical questions. Imagine that we cannot measure the state of the system exactly (e.g. particles in a box), so we try to guess a probability distribution $\rho(x,p,t)$ on $M$ indicating that at time $t$ the system has probability $\rho(x,p,t) d\mu$ of being in the state $(x,p)$. Obviously, $\rho$ should satisfy the constraint $\int_M \rho d\mu = 1$.
How does $\rho$ evolve in time? We know that the system obeys Hamilton's equations,
\[ (\dot x, \dot p) = X_H = (\partial H / \partial p, -\partial H / \partial x) \]
in local Darboux coordinates. Therefore, a particle located at $(x,p)$ in phase space at time $t$ will be located at $(x,p)+X_H dt$ in phase space at time $t+dt$. Therefore, the probability that a particle is at point $(x,p)$ at time $t+dt$, should be equal to the probability that the particle is at point $(x,p)-X_H dt$ at time $t$. Therefore, we have
\[ \frac{\partial \rho}{\partial t} = \frac{\partial H}{\partial x} \frac{\partial \rho}{\partial p} - \frac{\partial H}{\partial p} \frac{\partial \rho}{\partial x} = \{H, \rho\} \]
Given a probability distribution $\rho$, the entropy is defined to be
\[ S[\rho] = -\int_M \rho \log \rho d\mu. \]
(A version of) the second law of thermodynamics. For a given average energy $U$, the system assumes a distribution of maximal possible entropy at thermodynamic equilibrium.
The goal now, is to determine what distribution $\rho$ will maximize the entropy, subject to the constraints (for fixed $U$)
\begin{align*} \int_M H \rho d\mu &= U \\\
\int_M \rho d\mu &= 1 \end{align*}
Setting aside technical issues of convergence, etc., this variational problem is easily solved using the method of Lagrange multipliers. Introducing parameters $\lambda_1, \lambda_2$, we consider the modified functional
\[ S[\rho, \lambda_1, \lambda_2, U] = \int_M\left(-\rho \log \rho +\lambda_1\rho +\lambda_2(H\rho)\right)d\mu -\lambda_1-\lambda_2 U. \]
Note that $\partial S / \partial U = -\lambda_2$, and this is conventionally identified with (minus) the inverse temperature.
Taking the variation with respect to $\rho$, we find
\[ 0= \frac{\delta S}{\delta \rho} = -\log \rho-1+\lambda_1+H\lambda_2\]
Therefore, $rho$ is proportional to $e^{-\beta H}$ where we have set $\beta=-\lambda_2$. Define the partition function $Z$ to be
\[ Z = \int_M e^{-\beta H} d\mu. \]
We therefore have proved (formally and heuristically only!):
Theorem. The probability distribution $\rho$ assumed by the system at thermodynamic equilibrium is given by
\[ \rho = \frac{e^{-\beta H}}{Z} \]
where $\beta > 0$ is a real parameter, called the inverse temperature.
Corollary. At thermodynamic equilibrium, the average energy is given by
\[ U = -\frac{\partial \log Z}{\partial \beta} , \]
and the entropy is given by
\[ S = \beta U + \log Z.\]
Thursday, January 29, 2015
What is generalized geometry?
The following are my notes for a short introductory talk. References below are not intended to be comprehensive!
Math references:
Physics references:
What is geometry?
Before trying to define generalized geometry, we should first decide what we mean by ordinary geometry. Of course, this question doesn't have a unique answer, so there are many ways to generalize the classical notions of manifolds and varieties. The viewpoint taken in generalized geometry is the following: the distinguishing feature of smooth manifolds is the existence of a tangent bundle
\[ TM \to M \]
which satisfies some nice axioms. The basic idea of generalized geometry is to replace the tangent bundle with some other vector bundle $L \to M$, again satisfying some nice axioms. Different generalized geometries on $M$ will correspond to different choices of bundle $L \to M$, as well as auxiliary data compatible with $L$ in some appropriate sense.
Definition. A Lie algebroid over $M$ is a smooth vector bundle $L \to M$ together with a vector bundle map $a: L \to TM$ called the anchor map and a bracket $[\cdot, \cdot]: H^0(M, L) \otimes H^0(M, L) \to H^0(M, L)$ satisfying the following axioms:
Example 1. We can take $L$ to be $TM$ with anchor map the identity.
Example 2. Let $\sigma$ be a Poisson tensor on $M$. Then we define a bracket by $[X,Y] = \sigma(X,Y)$ and an anchor by $X \mapsto \sigma(X, \cdot)$. This makes $T^\ast M$ into a Lie algebroid.
Example 3. Let $M$ be a complex manifold of and let $L \subset TM \otimes \mathbf{C}$ be the sub-bundle of vectors spanned by $\{\partial / \partial z_1, \dots, \partial / \partial z_n\}$ in local holomorphic coordinates. Then $L \to M$ is a (complex) Lie algebroid.
Courant Bracket
We'd like to try to fit the preceding examples into a common framework. Let $\mathbf{T}M = TM \oplus T^\ast M$. This bundle has a natural symmetric bilinear pairing given by
\[ \langle X \oplus \alpha, Y \oplus \beta \rangle = \frac{1}{2} \alpha(Y) + \frac{1}{2} \beta(X) \]
Note that this bilinear form is of split signature $(n,n)$. We define a bracket on sections of $\mathbf{T}M$ by
\[ [X\oplus \alpha, Y\oplus \beta] = [X,Y] \oplus \left(L_X \beta + \frac{1}{2}(d \alpha(Y))- L_Y \alpha -\frac{1}{2} d( \beta(X)) \right ) \]
Note that this bracket is not a Lie bracket. We also have an anchor map $a: \mathbf{T}M \to TM$ which is just the projection.
Let $B$ be a 2-form on $M$. Define an action of $B$ on sections of $\mathbf TM$ by
\[ X + \alpha \mapsto X + \alpha + i_X B \]
Proposition. This action preserves the Courant bracket if and only if $B$ is closed.
This shows that the diffeomorphisms of $M$ as a generalized manifold are large than the ordinary diffeomorphisms of $M$. In fact is is the semidirect product of the diffeomorphism group of $M$ with the vector space of closed 2-forms.
Dirac Structures
Definition. A Dirac structure on $M$ is an Lagrangian sub-bundle $L \subset \mathbf{T}M$ which is closed under the Courant bracket.
Theorem (Courant). A Lagrangian sub-bundle $L \subset \mathbf{T} M$ is a Dirac structure if and only if $L \to M$ is a Lie algebroid over $M$, with bracket induced by the Courant bracket and anchor given by projection.
Example 1. $TM \subset \mathbf{T}M$.
Example 2. Take $L$ to be the graph of a Poisson tensor.
Example 3. Take $L$ to be the graph of a closed 2-form.
Admissible Functions
We now let $L \to M$ be a Dirac structure on $M$.
Definition. A smooth function $f$ on $M$ is called admissible if there exists a vector field $X_f$ such that $(X_f, df)$ is a section of $L$.
The Poisson bracket is defined as follows. If $f,g$ are admissible, then define
\[ \{f, g\} = X_f g. \]
It is easy to check from the definitions that the bracket on admissible functions is well-defined (independent of choice of $X_f$) and skew-symmetric. With a little bit of calculation, we find the following.
Proposition. The vector space of admissible functions is naturally a Poisson algebra, and moreover the natural bracket satisfies the Leibniz rule.
Generalized Complex Structures
Definition. A generalized complex structure is a skew endomorphism $J$ of $\mathbf T M$ such that $J^2 = -1$ and such that the $+i$-eigenbundle is involutive under the Courant bracket.
Equivalently: A generalized complex structure is a (complex) Dirac structure $L \subset \mathbf TM$ satisfying the condition $L \cap \overline L = 0$.
Example 1. Let $J$ be an ordinary complex structure on $M$. Then the endomorphism
\[ \begin{bmatrix} -J & 0 \\ 0 & J^\ast \end{bmatrix} \]
defines a generalized complex structure on $M$.
Example 2. Let $\omega$ be a symplectic form on $M$. Then the endomorphism
\[ \begin{bmatrix} 0 & -\omega^{-1} \\ \omega & 0 \end{bmatrix} \]
defines a generalized complex structure on $M$.
Thus, generalized geometry gives a common framework for both complex geometry and symplectic geometry. Such a connection is exactly what is conjectured by mirror symmetry.
Example 3. Let $J$ be a complex structure on $M$ and let $\sigma$ be a holomorphic Poisson tensor. Consider the subbundle $L \subset \mathbf TM$ defined as the span of
\[ \frac{\partial}{\partial \bar z_1}, \dots, \frac{\partial}{\partial \bar z_n}, dz_1 - \sigma(dz_1), \dots, dz_n - \sigma(dz_n) \]
Then $L$ defines a generalized complex structure on $M$.
The last example shows that deformations of $M$ as a generalized complex manifold contain non-commutative deformations of the structure sheaf. We also have the following theorem, which shows that there is an intimate relation between generalized complex geometry and holomorphic Poisson geometry.
Theorem (Bailey). Near any point of a generalized complex manifold, $M$ is locally isomorphic to the product of a holomorphic Poisson manifold with a symplectic manifold.
Generalized Kähler Manifolds
Let $(g, J, \omega)$ be a Kähler triple. The Kähler property requires that
\[ \omega = g J. \]
Let $I_1$ denote the generalized complex structure induced by $J$, and let $I_1$ denote the generalized complex structure induced by the symplectic form $\omega$. We have
\[ I_1 I_2 = \begin{bmatrix} - J & 0 \\ 0 & J^\ast \end{bmatrix} \begin{bmatrix} 0 & -\omega^{-1} \\ \omega & 0 \end{bmatrix} = \begin{bmatrix} 0 & g^{-1} \\ g & 0 \end{bmatrix} = I_2 I_1 \]
Definition. A generalized Kähler manifold is a manifold with two commuting generalized complex structure $I_1, I_2$ such that the bilinear pairing $(I_1 I_2 u, v)$ is positive definite.
Theorem (Gualtieri). A generalized Kähler structure on $M$ induces a Riemannian metric $g$, two integrable almost complex structures $J_\pm$ Hermitian with respect to $g$, and two affine connections $\nabla_\pm$ with skew-torsion $\pm H$ which preserve the metric and complex structure $J_\pm$. Conversely, these data determine a generalized Kähler structure which is unique up to a B-field transformation.
Thus the notion of generalized Kähler manifold recovers the bihermitian geometry investigated by physicists in the context of susy non-linear $\sigma$-models.
Generalized Calabi-Yau Manifolds
Definition. A generalized Calabi-Yau manifold is a manifold $M$ together with a complex-valued differential form $\phi$, which is either purely even or purely odd, which is a pure spinor for the action of $Cl(\mathbf TM)$ and satisfies the non-degeneracy condition $(\phi, \bar \phi) \neq 0$.
Note that (by definition) $\phi$ is pure if its annihilator is a maximal isotropic subspace. Let $L \subset \mathbf TM$ be its annihilator. Then it is not hard to see that $L$ defines a generalized complex structure on $M$, so indeed a generalized Calabi-Yau manifold is in particular a generalized complex manifold.
Example. If $M$ is a complex manifold with a nowhere vanishing holomorphic $(n,0)$ form, then it is generalized Calabi-Yau.
Example. If $M$ is symplectic with symplectic form $\omega$, then $\phi = \exp(i\omega)$ gives $M$ the structure of a generalized Calabi-Yau manifold.
If $(M, \phi)$ is generalized Calabi-Yau, then so is $(M, \exp(B) \phi)$ for any closed real 2-form $B$. In the symplectic case, we obtain
\[ \phi = \exp(B+i\omega) \]
This explains the appearance of the $B$-field (or "complexified Kähler form") in discussions of mirror symmetry.
Math references:
- Courant, Dirac manifolds
- Hitchin, Generalized Calabi-Yau manifolds
- Gualtieri, Generalized complex geometry
- Cavalcanti, New aspects of the ddc-lemma
- Cavalcanti and Gualtieri, Generalized complex geometry and T-duality
- Bailey and Gualtieri, Local analytic geometry of generalized complex structures
Physics references:
- Dijkgraaf, Gukov, Neitzke, Vafa, Topological M-theory as unification of form theories of gravity
- Grana, Flux compactifications in string theory: a comprehensive review
What is geometry?
Before trying to define generalized geometry, we should first decide what we mean by ordinary geometry. Of course, this question doesn't have a unique answer, so there are many ways to generalize the classical notions of manifolds and varieties. The viewpoint taken in generalized geometry is the following: the distinguishing feature of smooth manifolds is the existence of a tangent bundle
\[ TM \to M \]
which satisfies some nice axioms. The basic idea of generalized geometry is to replace the tangent bundle with some other vector bundle $L \to M$, again satisfying some nice axioms. Different generalized geometries on $M$ will correspond to different choices of bundle $L \to M$, as well as auxiliary data compatible with $L$ in some appropriate sense.
Definition. A Lie algebroid over $M$ is a smooth vector bundle $L \to M$ together with a vector bundle map $a: L \to TM$ called the anchor map and a bracket $[\cdot, \cdot]: H^0(M, L) \otimes H^0(M, L) \to H^0(M, L)$ satisfying the following axioms:
- $[\cdot,\cdot]$ is a Lie bracket on $H^0(M, L)$
- $[X, fY] = f[X,Y] + a(X)f \cdot Y$ for $X,Y \in H^0(M,L)$ and $f \in H^0(M, \mathcal{O}_M)$
Example 1. We can take $L$ to be $TM$ with anchor map the identity.
Example 2. Let $\sigma$ be a Poisson tensor on $M$. Then we define a bracket by $[X,Y] = \sigma(X,Y)$ and an anchor by $X \mapsto \sigma(X, \cdot)$. This makes $T^\ast M$ into a Lie algebroid.
Example 3. Let $M$ be a complex manifold of and let $L \subset TM \otimes \mathbf{C}$ be the sub-bundle of vectors spanned by $\{\partial / \partial z_1, \dots, \partial / \partial z_n\}$ in local holomorphic coordinates. Then $L \to M$ is a (complex) Lie algebroid.
Courant Bracket
We'd like to try to fit the preceding examples into a common framework. Let $\mathbf{T}M = TM \oplus T^\ast M$. This bundle has a natural symmetric bilinear pairing given by
\[ \langle X \oplus \alpha, Y \oplus \beta \rangle = \frac{1}{2} \alpha(Y) + \frac{1}{2} \beta(X) \]
Note that this bilinear form is of split signature $(n,n)$. We define a bracket on sections of $\mathbf{T}M$ by
\[ [X\oplus \alpha, Y\oplus \beta] = [X,Y] \oplus \left(L_X \beta + \frac{1}{2}(d \alpha(Y))- L_Y \alpha -\frac{1}{2} d( \beta(X)) \right ) \]
Note that this bracket is not a Lie bracket. We also have an anchor map $a: \mathbf{T}M \to TM$ which is just the projection.
Let $B$ be a 2-form on $M$. Define an action of $B$ on sections of $\mathbf TM$ by
\[ X + \alpha \mapsto X + \alpha + i_X B \]
Proposition. This action preserves the Courant bracket if and only if $B$ is closed.
This shows that the diffeomorphisms of $M$ as a generalized manifold are large than the ordinary diffeomorphisms of $M$. In fact is is the semidirect product of the diffeomorphism group of $M$ with the vector space of closed 2-forms.
Dirac Structures
Definition. A Dirac structure on $M$ is an Lagrangian sub-bundle $L \subset \mathbf{T}M$ which is closed under the Courant bracket.
Theorem (Courant). A Lagrangian sub-bundle $L \subset \mathbf{T} M$ is a Dirac structure if and only if $L \to M$ is a Lie algebroid over $M$, with bracket induced by the Courant bracket and anchor given by projection.
Example 1. $TM \subset \mathbf{T}M$.
Example 2. Take $L$ to be the graph of a Poisson tensor.
Example 3. Take $L$ to be the graph of a closed 2-form.
Admissible Functions
We now let $L \to M$ be a Dirac structure on $M$.
Definition. A smooth function $f$ on $M$ is called admissible if there exists a vector field $X_f$ such that $(X_f, df)$ is a section of $L$.
The Poisson bracket is defined as follows. If $f,g$ are admissible, then define
\[ \{f, g\} = X_f g. \]
It is easy to check from the definitions that the bracket on admissible functions is well-defined (independent of choice of $X_f$) and skew-symmetric. With a little bit of calculation, we find the following.
Proposition. The vector space of admissible functions is naturally a Poisson algebra, and moreover the natural bracket satisfies the Leibniz rule.
Generalized Complex Structures
Definition. A generalized complex structure is a skew endomorphism $J$ of $\mathbf T M$ such that $J^2 = -1$ and such that the $+i$-eigenbundle is involutive under the Courant bracket.
Equivalently: A generalized complex structure is a (complex) Dirac structure $L \subset \mathbf TM$ satisfying the condition $L \cap \overline L = 0$.
Example 1. Let $J$ be an ordinary complex structure on $M$. Then the endomorphism
\[ \begin{bmatrix} -J & 0 \\ 0 & J^\ast \end{bmatrix} \]
defines a generalized complex structure on $M$.
Example 2. Let $\omega$ be a symplectic form on $M$. Then the endomorphism
\[ \begin{bmatrix} 0 & -\omega^{-1} \\ \omega & 0 \end{bmatrix} \]
defines a generalized complex structure on $M$.
Thus, generalized geometry gives a common framework for both complex geometry and symplectic geometry. Such a connection is exactly what is conjectured by mirror symmetry.
Example 3. Let $J$ be a complex structure on $M$ and let $\sigma$ be a holomorphic Poisson tensor. Consider the subbundle $L \subset \mathbf TM$ defined as the span of
\[ \frac{\partial}{\partial \bar z_1}, \dots, \frac{\partial}{\partial \bar z_n}, dz_1 - \sigma(dz_1), \dots, dz_n - \sigma(dz_n) \]
Then $L$ defines a generalized complex structure on $M$.
The last example shows that deformations of $M$ as a generalized complex manifold contain non-commutative deformations of the structure sheaf. We also have the following theorem, which shows that there is an intimate relation between generalized complex geometry and holomorphic Poisson geometry.
Theorem (Bailey). Near any point of a generalized complex manifold, $M$ is locally isomorphic to the product of a holomorphic Poisson manifold with a symplectic manifold.
Generalized Kähler Manifolds
Let $(g, J, \omega)$ be a Kähler triple. The Kähler property requires that
\[ \omega = g J. \]
Let $I_1$ denote the generalized complex structure induced by $J$, and let $I_1$ denote the generalized complex structure induced by the symplectic form $\omega$. We have
\[ I_1 I_2 = \begin{bmatrix} - J & 0 \\ 0 & J^\ast \end{bmatrix} \begin{bmatrix} 0 & -\omega^{-1} \\ \omega & 0 \end{bmatrix} = \begin{bmatrix} 0 & g^{-1} \\ g & 0 \end{bmatrix} = I_2 I_1 \]
Definition. A generalized Kähler manifold is a manifold with two commuting generalized complex structure $I_1, I_2$ such that the bilinear pairing $(I_1 I_2 u, v)$ is positive definite.
Theorem (Gualtieri). A generalized Kähler structure on $M$ induces a Riemannian metric $g$, two integrable almost complex structures $J_\pm$ Hermitian with respect to $g$, and two affine connections $\nabla_\pm$ with skew-torsion $\pm H$ which preserve the metric and complex structure $J_\pm$. Conversely, these data determine a generalized Kähler structure which is unique up to a B-field transformation.
Thus the notion of generalized Kähler manifold recovers the bihermitian geometry investigated by physicists in the context of susy non-linear $\sigma$-models.
Generalized Calabi-Yau Manifolds
Definition. A generalized Calabi-Yau manifold is a manifold $M$ together with a complex-valued differential form $\phi$, which is either purely even or purely odd, which is a pure spinor for the action of $Cl(\mathbf TM)$ and satisfies the non-degeneracy condition $(\phi, \bar \phi) \neq 0$.
Note that (by definition) $\phi$ is pure if its annihilator is a maximal isotropic subspace. Let $L \subset \mathbf TM$ be its annihilator. Then it is not hard to see that $L$ defines a generalized complex structure on $M$, so indeed a generalized Calabi-Yau manifold is in particular a generalized complex manifold.
Example. If $M$ is a complex manifold with a nowhere vanishing holomorphic $(n,0)$ form, then it is generalized Calabi-Yau.
Example. If $M$ is symplectic with symplectic form $\omega$, then $\phi = \exp(i\omega)$ gives $M$ the structure of a generalized Calabi-Yau manifold.
If $(M, \phi)$ is generalized Calabi-Yau, then so is $(M, \exp(B) \phi)$ for any closed real 2-form $B$. In the symplectic case, we obtain
\[ \phi = \exp(B+i\omega) \]
This explains the appearance of the $B$-field (or "complexified Kähler form") in discussions of mirror symmetry.
Thursday, December 11, 2014
Virasoro Algebra from the Free Boson without Regularization
Usual physics derivations of the Virasoro algebra from the free boson in two dimensions usually use some sort of regularization procedure to compute the central charge. Following these notes I'd like to give a purely algebraic calculation of the central charge.
Let $A = \mathbf C[x_1, x_2, \dots]$ be the polynomial algebra in countably many generators. For an integer $k > 0$, define a $k$-linear operator $a_k$ on $A$ by
\[ a_k = \frac{\partial}{\partial x_k}, \ k > 0 \]
Similarly, for $k < 0$ we define $a_k$ by multiplication:
\[ a_{-k} = k x_k, k > 0 \]
For $k = 0$, we define $a_0$ to be multiplication by some fixed complex number (which by abuse of notation we also denote by $a_0$).
Lemma. We have the commutation relation $[a_m, a_n] = m \delta_{m+n}$ as linear operators on $A$.
For any monomial in the $a_k$, we define normal ordering $::$ to be the monomial obtained by reordering the terms so that the indices are increasing. (Mathematical interpretation: it is a section of the quotient map from the tensor algebra in the $a_k$ to the symmetric algebra, defined by lexicographic order.) For example,
\[ :a_j a_k:\ = \left\{ \begin{array}{rr} a_j a_k, & j \leq k \\ a_k a_j, & j > k \end{array} \right. \]
Next we formally define a set of operators $L_k$ by
\[ L_k = \frac{1}{2}\sum_j :a_j a_{k-j}: \]
Proposition. The $L_k$ are well-defined as linear operators on $A$.
Proof. For sufficiently large $|j|$, at least one of $j$ or $k-j$ is positive, and hence $:a_j a_{k-j}:$ contains a differentiation (on the right). Since any element $f \in A$ is annihilated by all but finitely many of the differentiation operators $\partial_j$, the formal expression $L_k f$ contains only finitely many non-zero terms, and hence is well-defined.
Lemma. As operators on $A$, we have $[a_k, L_n] = k a_{k+n}$.
Theorem. As operators on $A$, we have
\[ [L_m, L_n] = (m-n) L_{m+n} + \frac{1}{12} (m^3-m) \delta_{m+n} \]
Proof. Fix $m,n$. For the sake of simplicity we will assume $m \neq n$ and $mn \neq 0$. (The other special cases can be treated by similar arguments.) By the same argument as the proof of the preceding proposition, for any fixed element $f \in A$ there exists some $N \gg 0$ such that
\[ [L_m, L_n]f = [L_m^N, L_n] f \]
where $L_m^N$ is the truncated operator
\[ L_m^N = \frac{1}{2}\sum_{|j| < N} :a_j a_{m-j}: \]
Let us compute (noting that since $m \neq 0$, $:a_j a_{m-j}: = a_j a_{m-j}$)
\begin{align}
[L_m^N, L_n] &= \frac{1}{2} \sum_{|j| < N} [a_j a_{m-j}, L_n] \\
&= \frac{1}{2} \sum_{|j| < N} (m-j) a_j a_{m+n-j} + \frac{1}{2} \sum_{|j| < N}j a_{n+j} a_{m-j}
\end{align}
Denote the two sums above by $S_1$ and $S_2$. It is clear that these should be related to the operator $L_{m+n}$, but to see the exact relation we will have to normal order the terms. Let's start with $S_1$. Note that $a_j a_{m+n-j}$ is already normal ordered, unless $j > m+n-j$. Hence
\begin{align}
S_1 &= \frac{1}{2} \sum_{|j| < N} (m-j) :a_j a_{m+n-j}: + \frac{1}{2} \sum_{m+n\lt2j\lt2N} (m-j) [a_j, a_{m+n-j}] \\
&= \frac{1}{2} \sum_{|j| < N} (m-j) :a_j a_{m+n-j}: + \frac{\delta_{m+n}}{2} \sum_{m+n\lt2j\lt2N} j(m-j) \\
&= \frac{1}{2} \sum_{|j| < N} (m-j) :a_j a_{m+n-j}: + \frac{\delta_{m+n}}{2} \sum_{0\lt j\lt N} j(m-j)
\end{align}
Similarly, we normal order the terms in $S_2$:
\begin{align}
S_2 &= \frac{1}{2} \sum_{|j| < N}j :a_{n+j} a_{m-j}: -\frac{1}{2} \sum_{m-n\lt2j\lt2N}j [a_{m-j}, a_{n+j}] \\
&= \frac{1}{2} \sum_{|j| < N}j :a_{n+j} a_{m-j}: -\frac{\delta_{m+n}}{2} \sum_{m\lt j\lt N}j (m-j) \\
&= \frac{1}{2} \sum_{|j| < N}j :a_{n+j} a_{m-j}: -\frac{\delta_{m+n}}{2} \sum_{m\lt j\lt N}j (m-j) \\
&= \frac{1}{2} \sum_{|j| < N}j :a_{n+j} a_{m-j}: -\frac{\delta_{m+n}}{2} \sum_{m\lt j\lt N}j (m-j)
\end{align}
Hence we have
\[ S_1 + S_2 = \frac{1}{2} \sum_{|j| < N} (m-j) :a_j a_{m+n-j}: + \frac{1}{2} \sum_{|j| < N}j :a_{n+j} a_{m-j}: + \frac{\delta_{m+n}}{2} \sum_{0\lt j\leq m} j(m-j) \]
Now that everything is normal ordered, we can take the limit $N \to \infty$ without fear. After a simple cancellation, and explicitly summing the last term using well-known formulas for sums of powers of integers, we obtain:
\[ [L_m, L_n] = (m-n) L_{m+n} + \frac{1}{12} (m^3-m) \delta_{m+n} \]
For the usual conventions of the Virasoro algebra, this shows that this representation corresponds to central charge $c=1$.
Remark. If we tried to take the limit $N \to \infty$ in each of the terms $S_1, S_2$ separately, before taking their sum, we would obtain a formal infinite constant $\sum_{j} j(m-j)$. In any physics textbook, the author will simply zeta-regularize this sum to obtain a fininte result. However, the above calculation shows that this is not necessary. By taking care that each term in the expression $S_1+S_2$ was normal-ordered, before taking the limit, we obtain only finite constants with no need to regularize. Zeta regularization certainly has its uses (for example in rigorous definitions of functional determinants), but as the above calculation shows, it can also be an unnecessary crutch that obscures the underlying mathematical phenomena.
Remark. There is an analogous calculation which shows that one obtains a Virasoro representation from affine Lie algebras. Physically, this corresponds to the WZW model. Roughly, the generators of the affine Lie algebra behave as an infinite set of harmonic oscillators, similar to the Heisenberg algebra above. Sometime in the future I may write a sequel to this post giving the details of this calculation.
Let $A = \mathbf C[x_1, x_2, \dots]$ be the polynomial algebra in countably many generators. For an integer $k > 0$, define a $k$-linear operator $a_k$ on $A$ by
\[ a_k = \frac{\partial}{\partial x_k}, \ k > 0 \]
Similarly, for $k < 0$ we define $a_k$ by multiplication:
\[ a_{-k} = k x_k, k > 0 \]
For $k = 0$, we define $a_0$ to be multiplication by some fixed complex number (which by abuse of notation we also denote by $a_0$).
Lemma. We have the commutation relation $[a_m, a_n] = m \delta_{m+n}$ as linear operators on $A$.
For any monomial in the $a_k$, we define normal ordering $::$ to be the monomial obtained by reordering the terms so that the indices are increasing. (Mathematical interpretation: it is a section of the quotient map from the tensor algebra in the $a_k$ to the symmetric algebra, defined by lexicographic order.) For example,
\[ :a_j a_k:\ = \left\{ \begin{array}{rr} a_j a_k, & j \leq k \\ a_k a_j, & j > k \end{array} \right. \]
Next we formally define a set of operators $L_k$ by
\[ L_k = \frac{1}{2}\sum_j :a_j a_{k-j}: \]
Proposition. The $L_k$ are well-defined as linear operators on $A$.
Proof. For sufficiently large $|j|$, at least one of $j$ or $k-j$ is positive, and hence $:a_j a_{k-j}:$ contains a differentiation (on the right). Since any element $f \in A$ is annihilated by all but finitely many of the differentiation operators $\partial_j$, the formal expression $L_k f$ contains only finitely many non-zero terms, and hence is well-defined.
Lemma. As operators on $A$, we have $[a_k, L_n] = k a_{k+n}$.
Theorem. As operators on $A$, we have
\[ [L_m, L_n] = (m-n) L_{m+n} + \frac{1}{12} (m^3-m) \delta_{m+n} \]
Proof. Fix $m,n$. For the sake of simplicity we will assume $m \neq n$ and $mn \neq 0$. (The other special cases can be treated by similar arguments.) By the same argument as the proof of the preceding proposition, for any fixed element $f \in A$ there exists some $N \gg 0$ such that
\[ [L_m, L_n]f = [L_m^N, L_n] f \]
where $L_m^N$ is the truncated operator
\[ L_m^N = \frac{1}{2}\sum_{|j| < N} :a_j a_{m-j}: \]
Let us compute (noting that since $m \neq 0$, $:a_j a_{m-j}: = a_j a_{m-j}$)
\begin{align}
[L_m^N, L_n] &= \frac{1}{2} \sum_{|j| < N} [a_j a_{m-j}, L_n] \\
&= \frac{1}{2} \sum_{|j| < N} (m-j) a_j a_{m+n-j} + \frac{1}{2} \sum_{|j| < N}j a_{n+j} a_{m-j}
\end{align}
Denote the two sums above by $S_1$ and $S_2$. It is clear that these should be related to the operator $L_{m+n}$, but to see the exact relation we will have to normal order the terms. Let's start with $S_1$. Note that $a_j a_{m+n-j}$ is already normal ordered, unless $j > m+n-j$. Hence
\begin{align}
S_1 &= \frac{1}{2} \sum_{|j| < N} (m-j) :a_j a_{m+n-j}: + \frac{1}{2} \sum_{m+n\lt2j\lt2N} (m-j) [a_j, a_{m+n-j}] \\
&= \frac{1}{2} \sum_{|j| < N} (m-j) :a_j a_{m+n-j}: + \frac{\delta_{m+n}}{2} \sum_{m+n\lt2j\lt2N} j(m-j) \\
&= \frac{1}{2} \sum_{|j| < N} (m-j) :a_j a_{m+n-j}: + \frac{\delta_{m+n}}{2} \sum_{0\lt j\lt N} j(m-j)
\end{align}
Similarly, we normal order the terms in $S_2$:
\begin{align}
S_2 &= \frac{1}{2} \sum_{|j| < N}j :a_{n+j} a_{m-j}: -\frac{1}{2} \sum_{m-n\lt2j\lt2N}j [a_{m-j}, a_{n+j}] \\
&= \frac{1}{2} \sum_{|j| < N}j :a_{n+j} a_{m-j}: -\frac{\delta_{m+n}}{2} \sum_{m\lt j\lt N}j (m-j) \\
&= \frac{1}{2} \sum_{|j| < N}j :a_{n+j} a_{m-j}: -\frac{\delta_{m+n}}{2} \sum_{m\lt j\lt N}j (m-j) \\
&= \frac{1}{2} \sum_{|j| < N}j :a_{n+j} a_{m-j}: -\frac{\delta_{m+n}}{2} \sum_{m\lt j\lt N}j (m-j)
\end{align}
Hence we have
\[ S_1 + S_2 = \frac{1}{2} \sum_{|j| < N} (m-j) :a_j a_{m+n-j}: + \frac{1}{2} \sum_{|j| < N}j :a_{n+j} a_{m-j}: + \frac{\delta_{m+n}}{2} \sum_{0\lt j\leq m} j(m-j) \]
Now that everything is normal ordered, we can take the limit $N \to \infty$ without fear. After a simple cancellation, and explicitly summing the last term using well-known formulas for sums of powers of integers, we obtain:
\[ [L_m, L_n] = (m-n) L_{m+n} + \frac{1}{12} (m^3-m) \delta_{m+n} \]
For the usual conventions of the Virasoro algebra, this shows that this representation corresponds to central charge $c=1$.
Remark. If we tried to take the limit $N \to \infty$ in each of the terms $S_1, S_2$ separately, before taking their sum, we would obtain a formal infinite constant $\sum_{j} j(m-j)$. In any physics textbook, the author will simply zeta-regularize this sum to obtain a fininte result. However, the above calculation shows that this is not necessary. By taking care that each term in the expression $S_1+S_2$ was normal-ordered, before taking the limit, we obtain only finite constants with no need to regularize. Zeta regularization certainly has its uses (for example in rigorous definitions of functional determinants), but as the above calculation shows, it can also be an unnecessary crutch that obscures the underlying mathematical phenomena.
Remark. There is an analogous calculation which shows that one obtains a Virasoro representation from affine Lie algebras. Physically, this corresponds to the WZW model. Roughly, the generators of the affine Lie algebra behave as an infinite set of harmonic oscillators, similar to the Heisenberg algebra above. Sometime in the future I may write a sequel to this post giving the details of this calculation.
Wednesday, May 14, 2014
The Basic Idea of the Quantum BV Complex
$
\newcommand{\h}{\hbar}
\newcommand{\X}{\mathfrak{X}}
\newcommand{\PV}{\text{PV}}
\newcommand{\K}{{\mathbb K}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\too}{\xrightarrow}
\newcommand{\im}{\text{Im }}
$
Let $M$ be an oriented $n$-dimensional manifold and $\X^\bullet(M):=\Gamma(M,\wedge^{-\bullet} TM)$, so that $\X^\bullet(M)$ is concentrated in non-positive degree. Let $\mu\in \Omega^n(M)$ a volume form on $M$. Then interior product with $\mu$ gives an isomorphism $\vee\mu: \X^k(M)\too{\cong}\Omega^{n-k}(M)$. From this, we induce a degree 1 differential $\Delta_\mu$ on $\X^\bullet(M)$ from the de Rham differential $d$ on $\Omega^\bullet(M)$, defined by $\Delta_\mu= (\vee\mu)^{-1}\circ d\circ\vee\mu$, making $\X^\bullet(M)$ into a cochain complex isomorphic to $\Omega^\bullet(M)[k]$; in particular $H^k(\X)=H^{n+k}(\Omega)$.
If $M$ is compact and connected, then $H^0(\X)=H^n(\Omega)$ is one dimensional, and upon fixing a basis to identify $H^0(\X)\too{\cong}\R$, the quotient map gives a map $\pi:C^\infty(M)\to \R$. We have the following:
Proposition Let $1\in C^\infty(M)$ be the constant function taking the value $1$. Then $[1]\in H^0(\X)$ is nontrivial and after choosing it as a basis the resulting map $\pi:C^\infty(M)\to \R$ is given by $$ f\mapsto\frac{ \int_M f\mu}{\int_M \mu}$$ Proof Let $f\in C^\infty(M)$ and suppose $f=\Delta_\mu(X)$ for $X\in\X^1(M)$. Then $$f\mu = f\vee \mu = \Delta_\mu(X)\vee\mu= d(X\vee \mu)$$ so that $f\mu$ is exact and by Stokes' theorem integrates to $0$ on $M$. Thus all $f\in \im \Delta_\mu$ integrate to zero against $\mu$ on $M$, so that the above map indeed descends to the quotient.
The above arguement also implies that were $1\in\im\Delta_\mu$ then $\mu$ would integrate to zero, contradicting that $\mu$ is a volume form, so that $[1]\in H^0(\X)$ is indeed nontrivial. The map is thus well-defined, linear, and has the appropriate action on a basis so that it is correct as claimed. $\square$
This recovers the standard definition of the expectation of an observable in the path integral picture of quantum field theory. The upshot is that having formulated the integration homologically, we can hope to extend this homological definition of expectation to situations where the integral itself is not well defined.
Consider the case $M=V$ a vector space, which in particular described the situation in coordinates on $M$. We are interested in measures of the form $\mu=e^{-S/\h}\mu_0$ where $\mu_0$ is the Lesbesgue measure on $V$, given by $\mu_0=dx^1\wedge ...\wedge dx^n$. Let $X\in \X^k(M)$, we have: \begin{align*} d(X\vee \mu) & = d(e^{-S/\h}(X\vee\mu_0)) \\ & = de^{-S/\h}\wedge (X\vee \mu_0)+e^{-S/\h}d(X\vee \mu_0)\\ & = -\frac{1}{\h}e^{-S/\h} dS\wedge (X\vee \mu_0) + e^{-S/\h} (\Delta_{\mu_0}X)\vee \mu_0 \\ & = -\frac{1}{\h}e^{-S/\h} (dS\vee X)\vee \mu_0 + (\Delta_{\mu_0}X)\vee \mu\\ & = \left(-\frac{1}{\h}dS\vee X + \Delta_{\mu_0}X \right)\vee \mu \end{align*} so that $$\Delta_\mu=\Delta_{\mu_0}-\frac{1}{\h}\iota_{dS}$$ Further, we can explicitly compute $\Delta_{\mu_0}$: for $X\in\X^k(M)$ we have that $X=\sum_I X^I \partial_I$, where the sum is over increasing $k$-tuples $I\subset\{1,...,n\}$. We have \begin{align*} d(X\vee \mu_0) & = d\left( \sum_I X^I \partial_I\vee(dx^1\wedge ...\wedge dx^n)\right)\\ & = d\left( \sum_I X^I (-1)^?dx^1\wedge ...\wedge \hat{dx^I} \wedge ...\wedge dx^n \right)\\ & = \sum_I \sum_{i\in I} \partial_{x^i} X^I (-1)^?dx^i\wedge dx^1\wedge ...\wedge \hat{ dx^I}\wedge ...\wedge dx^n\\ & = \left( \sum_i \partial_{x^i} (dx^i\vee X) \right) \vee \mu \end{align*} so that $$\Delta_{\mu_0} = \sum_i \partial_{x^i} \iota_{dx^i}$$ To be slightly more careful about the formal variable $\h$ and allow the $\h\to 0$ limit to be more clear, we refine our complex to be: $$ \X^\bullet(M)[[\h]] \quad\quad\text{equipped with}\quad\quad \h \Delta_\mu= \h\Delta_{\mu_0} -\iota_{dS}$$ Next, we restrict consideration only to polynomial observables (functions), and vector fields with polynomial coefficients, denoting the resulting complex $\PV^\bullet$. One can check that $H^0(\PV)$ is still one dimensional, so that the above proposition holds and we maintain the integral intepretation of this cohomology.
Now, we can identify $\PV^\bullet[[\h]]$ with the graded-commutative graded algebra $\K[[x^1,...,x^n,\xi_1,...,\xi_n,\h]]$ where $x^1,...,x^n,\h$ are in degree 0 and $\xi_1,...,\xi_n$ are in degree -1 as follows: $$ x^i \mapsto x^i \quad\quad \partial_i\mapsto \xi_i\quad\quad \partial_i\wedge\partial_j\mapsto \xi_i\xi_j \quad\quad \h\mapsto \h$$ Under this identification, the map $\iota_{dx^i}:PV^k(M)\to \PV^{k-1}(M)$ is identified with $\partial_{\xi_i}$. Now, we require our action function $S:M\to \R$ also be polynomial, and further, that $$S(x)=\frac{1}{2}\sum_{i,j} a_{ij}x^ix^j- b(x)$$ for $a_{ij}$ symmetric and non-degenerate and for $b\in I^3$ where $I=(x^1,...,x^n)\subset \K[[x^1,...,x^n]]$, that is, $b(x)$ a polynomial with no terms of degree less than 3. This implies that \begin{align*} \h\Delta_\mu & = \h\Delta_{\mu_0} -\iota_{dS} \\ & = \h \sum_i \partial_{x^i} \iota_{dx^i} - \sum_i (\partial_{x^i}S) \iota_{dx^i} \\ & = \h \sum_i \partial_{x^i} \iota_{dx^i} + \sum_i (\partial_{x^i}b) \iota_{dx^i} - \sum_{i,j}a_{ij}x^i\iota_{dx^j}\\ \end{align*} and under our identification this becomes $$\h\Delta_\mu = \h \sum_i \partial_{x^i}\partial_{\xi_i} + \sum_i (\partial_{x^i}b)\partial_{\xi_i} - \sum_{i,j}a_{ij}x^i\partial_{\xi_j}$$ The computation of the degree 0 cohomology of a given polynomial $f\in\K[[x_1,...,x_n]]$ under this differential is taken up in Gwilliam, Johnson-Freyd where it is shown the answer is precisely the Feynman diagram expansion for the expectation of $f$ which we expect.
If $M$ is compact and connected, then $H^0(\X)=H^n(\Omega)$ is one dimensional, and upon fixing a basis to identify $H^0(\X)\too{\cong}\R$, the quotient map gives a map $\pi:C^\infty(M)\to \R$. We have the following:
Proposition Let $1\in C^\infty(M)$ be the constant function taking the value $1$. Then $[1]\in H^0(\X)$ is nontrivial and after choosing it as a basis the resulting map $\pi:C^\infty(M)\to \R$ is given by $$ f\mapsto\frac{ \int_M f\mu}{\int_M \mu}$$ Proof Let $f\in C^\infty(M)$ and suppose $f=\Delta_\mu(X)$ for $X\in\X^1(M)$. Then $$f\mu = f\vee \mu = \Delta_\mu(X)\vee\mu= d(X\vee \mu)$$ so that $f\mu$ is exact and by Stokes' theorem integrates to $0$ on $M$. Thus all $f\in \im \Delta_\mu$ integrate to zero against $\mu$ on $M$, so that the above map indeed descends to the quotient.
The above arguement also implies that were $1\in\im\Delta_\mu$ then $\mu$ would integrate to zero, contradicting that $\mu$ is a volume form, so that $[1]\in H^0(\X)$ is indeed nontrivial. The map is thus well-defined, linear, and has the appropriate action on a basis so that it is correct as claimed. $\square$
This recovers the standard definition of the expectation of an observable in the path integral picture of quantum field theory. The upshot is that having formulated the integration homologically, we can hope to extend this homological definition of expectation to situations where the integral itself is not well defined.
Consider the case $M=V$ a vector space, which in particular described the situation in coordinates on $M$. We are interested in measures of the form $\mu=e^{-S/\h}\mu_0$ where $\mu_0$ is the Lesbesgue measure on $V$, given by $\mu_0=dx^1\wedge ...\wedge dx^n$. Let $X\in \X^k(M)$, we have: \begin{align*} d(X\vee \mu) & = d(e^{-S/\h}(X\vee\mu_0)) \\ & = de^{-S/\h}\wedge (X\vee \mu_0)+e^{-S/\h}d(X\vee \mu_0)\\ & = -\frac{1}{\h}e^{-S/\h} dS\wedge (X\vee \mu_0) + e^{-S/\h} (\Delta_{\mu_0}X)\vee \mu_0 \\ & = -\frac{1}{\h}e^{-S/\h} (dS\vee X)\vee \mu_0 + (\Delta_{\mu_0}X)\vee \mu\\ & = \left(-\frac{1}{\h}dS\vee X + \Delta_{\mu_0}X \right)\vee \mu \end{align*} so that $$\Delta_\mu=\Delta_{\mu_0}-\frac{1}{\h}\iota_{dS}$$ Further, we can explicitly compute $\Delta_{\mu_0}$: for $X\in\X^k(M)$ we have that $X=\sum_I X^I \partial_I$, where the sum is over increasing $k$-tuples $I\subset\{1,...,n\}$. We have \begin{align*} d(X\vee \mu_0) & = d\left( \sum_I X^I \partial_I\vee(dx^1\wedge ...\wedge dx^n)\right)\\ & = d\left( \sum_I X^I (-1)^?dx^1\wedge ...\wedge \hat{dx^I} \wedge ...\wedge dx^n \right)\\ & = \sum_I \sum_{i\in I} \partial_{x^i} X^I (-1)^?dx^i\wedge dx^1\wedge ...\wedge \hat{ dx^I}\wedge ...\wedge dx^n\\ & = \left( \sum_i \partial_{x^i} (dx^i\vee X) \right) \vee \mu \end{align*} so that $$\Delta_{\mu_0} = \sum_i \partial_{x^i} \iota_{dx^i}$$ To be slightly more careful about the formal variable $\h$ and allow the $\h\to 0$ limit to be more clear, we refine our complex to be: $$ \X^\bullet(M)[[\h]] \quad\quad\text{equipped with}\quad\quad \h \Delta_\mu= \h\Delta_{\mu_0} -\iota_{dS}$$ Next, we restrict consideration only to polynomial observables (functions), and vector fields with polynomial coefficients, denoting the resulting complex $\PV^\bullet$. One can check that $H^0(\PV)$ is still one dimensional, so that the above proposition holds and we maintain the integral intepretation of this cohomology.
Now, we can identify $\PV^\bullet[[\h]]$ with the graded-commutative graded algebra $\K[[x^1,...,x^n,\xi_1,...,\xi_n,\h]]$ where $x^1,...,x^n,\h$ are in degree 0 and $\xi_1,...,\xi_n$ are in degree -1 as follows: $$ x^i \mapsto x^i \quad\quad \partial_i\mapsto \xi_i\quad\quad \partial_i\wedge\partial_j\mapsto \xi_i\xi_j \quad\quad \h\mapsto \h$$ Under this identification, the map $\iota_{dx^i}:PV^k(M)\to \PV^{k-1}(M)$ is identified with $\partial_{\xi_i}$. Now, we require our action function $S:M\to \R$ also be polynomial, and further, that $$S(x)=\frac{1}{2}\sum_{i,j} a_{ij}x^ix^j- b(x)$$ for $a_{ij}$ symmetric and non-degenerate and for $b\in I^3$ where $I=(x^1,...,x^n)\subset \K[[x^1,...,x^n]]$, that is, $b(x)$ a polynomial with no terms of degree less than 3. This implies that \begin{align*} \h\Delta_\mu & = \h\Delta_{\mu_0} -\iota_{dS} \\ & = \h \sum_i \partial_{x^i} \iota_{dx^i} - \sum_i (\partial_{x^i}S) \iota_{dx^i} \\ & = \h \sum_i \partial_{x^i} \iota_{dx^i} + \sum_i (\partial_{x^i}b) \iota_{dx^i} - \sum_{i,j}a_{ij}x^i\iota_{dx^j}\\ \end{align*} and under our identification this becomes $$\h\Delta_\mu = \h \sum_i \partial_{x^i}\partial_{\xi_i} + \sum_i (\partial_{x^i}b)\partial_{\xi_i} - \sum_{i,j}a_{ij}x^i\partial_{\xi_j}$$ The computation of the degree 0 cohomology of a given polynomial $f\in\K[[x_1,...,x_n]]$ under this differential is taken up in Gwilliam, Johnson-Freyd where it is shown the answer is precisely the Feynman diagram expansion for the expectation of $f$ which we expect.
Tuesday, March 18, 2014
Clifford Algebras and Spinors III: Bochner identity
Let $M$ be a Riemannian manifold, and let $Cl(M)$ be its Clifford bundle. Let $E \to M$ be any vector bundle with connection, and assume that $C^\infty(M, E)$ is a $Cl(M)$-module. We can define a Dirac operator $\mathcal{D}$ acting on sections of $E$ via the formula
\[ \mathcal{D} \sigma = \sum_{i=1}^n e_i \cdot \nabla_i \sigma \]
for any orthonormal frame $\{e_1, \dots, e_n\}$ on $M$, and where $\cdot$ denotes the Clifford module action. We demand that the connection on $E$ is compatible with Clifford multiplication in the following sense:
\[ \nabla_j (e_i \cdot \sigma) = (\nabla_j e_i) \cdot \sigma + e_i \cdot \nabla_j \sigma. \]
Let $R$ denote the curvature of $E$, i.e. we have
\[ [\nabla_i, \nabla_j] \sigma = R(e_i, e_j) \sigma+ \nabla_{[e_i, e_j]} \sigma \]
We can define an endomorphism $\mathcal{R}$ on $E$ by
\[ \mathcal{R} = \frac{1}{2} \sum_{ij} R(e_i, e_j). \]
Theorem. We have the identity $\mathcal{D}^2 = -\Delta + \mathcal{R}$.
Proof. We compute
\begin{align}
\mathcal{D}^2 \sigma &= \sum_{ij} e_i \nabla_i \left( e_j \nabla_j \sigma \right) \\
&= \sum_{ij} e_i e_j \nabla_i \nabla_j \sigma + e_i ( \nabla_i e_j ) \nabla_j \sigma \\
&= -\Delta \sigma + \frac{1}{2}\sum_{ij}e_i e_j [\nabla_i, \nabla_j] \sigma + \sum_{ij} e_i ( \nabla_i e_j) \nabla_j \sigma \\
&= -\Delta \sigma + \frac{1}{2}\sum_{ij}e_i e_j R(e_i, e_j) \sigma + \frac{1}{2}\sum_{ij} e_i e_j \nabla_{[e_i, e_j]} \sigma+ \sum_{ij} e_i ( \nabla_i e_j) \nabla_j \sigma \\
&= (-\Delta + \mathcal{R})\sigma + \frac{1}{2} \sum_{ij} \left( e_i e_j \nabla_{[e_i, e_j]}\sigma + e_i (\nabla_i e_j) \nabla_j + e_j (\nabla_j e_i) \nabla_i \right)\sigma
\end{align}
We will be done provided we can show that the last term vanishes. Notice that it is fully tensorial, since it can be expressed as $\mathcal{D}^2 + \Delta - \mathcal{R}$. On the other hand, the terms $[e_i, e_j]$ and $\nabla_j e_i$ are (by definition!) proportional to Christoffel symbols. Since we can always choose a frame so that these vanish at a point, these terms must vanish identically. Hence we have $0 = \mathcal{D}^2 + \Delta - \mathcal{R}$, as desired.
\[ \mathcal{D} \sigma = \sum_{i=1}^n e_i \cdot \nabla_i \sigma \]
for any orthonormal frame $\{e_1, \dots, e_n\}$ on $M$, and where $\cdot$ denotes the Clifford module action. We demand that the connection on $E$ is compatible with Clifford multiplication in the following sense:
\[ \nabla_j (e_i \cdot \sigma) = (\nabla_j e_i) \cdot \sigma + e_i \cdot \nabla_j \sigma. \]
Let $R$ denote the curvature of $E$, i.e. we have
\[ [\nabla_i, \nabla_j] \sigma = R(e_i, e_j) \sigma+ \nabla_{[e_i, e_j]} \sigma \]
We can define an endomorphism $\mathcal{R}$ on $E$ by
\[ \mathcal{R} = \frac{1}{2} \sum_{ij} R(e_i, e_j). \]
Theorem. We have the identity $\mathcal{D}^2 = -\Delta + \mathcal{R}$.
Proof. We compute
\begin{align}
\mathcal{D}^2 \sigma &= \sum_{ij} e_i \nabla_i \left( e_j \nabla_j \sigma \right) \\
&= \sum_{ij} e_i e_j \nabla_i \nabla_j \sigma + e_i ( \nabla_i e_j ) \nabla_j \sigma \\
&= -\Delta \sigma + \frac{1}{2}\sum_{ij}e_i e_j [\nabla_i, \nabla_j] \sigma + \sum_{ij} e_i ( \nabla_i e_j) \nabla_j \sigma \\
&= -\Delta \sigma + \frac{1}{2}\sum_{ij}e_i e_j R(e_i, e_j) \sigma + \frac{1}{2}\sum_{ij} e_i e_j \nabla_{[e_i, e_j]} \sigma+ \sum_{ij} e_i ( \nabla_i e_j) \nabla_j \sigma \\
&= (-\Delta + \mathcal{R})\sigma + \frac{1}{2} \sum_{ij} \left( e_i e_j \nabla_{[e_i, e_j]}\sigma + e_i (\nabla_i e_j) \nabla_j + e_j (\nabla_j e_i) \nabla_i \right)\sigma
\end{align}
We will be done provided we can show that the last term vanishes. Notice that it is fully tensorial, since it can be expressed as $\mathcal{D}^2 + \Delta - \mathcal{R}$. On the other hand, the terms $[e_i, e_j]$ and $\nabla_j e_i$ are (by definition!) proportional to Christoffel symbols. Since we can always choose a frame so that these vanish at a point, these terms must vanish identically. Hence we have $0 = \mathcal{D}^2 + \Delta - \mathcal{R}$, as desired.
Thursday, March 6, 2014
Clifford Algebras and Spinors, Part II: Spin Structures and Dirac Operators
A very good reference for today's material is Dan Freed's (unpublished) notes on Dirac operators, available here.
\[ (e_1 \cdots e_k)^t = e_k \cdots e_1, \ \beta(e_1 \dots e_k) = (-1)^k e_k \dots e_2 e_1 \]
There is a natural inclusion \(\mathbb E^n \hookrightarrow Cl(\mathbb E^n)\). Given \(x \in Cl(\mathbb E^n)\) and \(v \in \mathbb E^n\), we can consider the product \(x v x^t\). In general, this might not be contained in \(\mathbb E^n \subset Cl(\mathbb E^n)\).
Definition. We define the group \(Pin(n)\) to consist of all those \(g \in Cl(\mathbb E^n)\) such that
\[ g \beta(g) = 1, \ \ g v \beta(g) \subset \mathbb E^n \ \forall\ v \in \mathbb E^n. \]
Similarly, we define the group \(Spin(n)\) to be the subgroup of \(Pin(n)\) such that \(gg^t = 1\).
Theorem. The natural action of \(Pin(n)\) on \(\mathbb E^n\) is by othogonal transformations, giving a natural map \(Pin(n) \to O(n)\). This map is a double cover. Similarly, \(Spin(n)\) is a double cover of \(SO(n)\). If \(n \geq 2\), \(Spin(n)\) is simply connected.
The importance of the spin groups is due to the following basic fact. Suppose that \(G\) is a Lie group with Lie algebra \(\mathfrak{g}\). Any representation of \(G\) induces a representation of \(\mathfrak{g}\). However, given a representation of \(\mathfrak{g}\), it is not always possible to integrate it to a representation of \(G\). But it is always possible to integrate a representation of \(\mathfrak{g}\) to produce a representation of the universal cover of \(G\). For \(n \geq 2\), \(Spin(n)\) is the universal cover of \(SO(n)\).
Suppose that \(V\) is a representation of \(SO(n)\). Then we may form the associated bundle \(SO(M) \times_{SO(n)} V\), which is a vector bundle over \(M\) with structure group \(SO(n)\). If we take the defining representation then we obtain the tangent bundle, but of course there are many others. Unfortunately, since \(SO(n)\) is not simply connected, not every representation of \(\mathfrak{so}_n\) can be integrated to a representation of \(SO(n)\). At the level of geometry, this means that in a certain sense there are certain vector bundles over \(M\) that are "missing"! Even more disturbing, is that these "missing" bundles appear to be necessary to describe many of the fundamental particles that appear in the standard model--so this has real world consequences. The solution is to equip \(M\) with a spin structure.
Definition. A spin structure on \(M\) is a principal \(Spin(n)\)-bundle \(Spin(M)\) over \(M\) together with a bundle morphism \(Spin(M) \to SO(M)\) which is a reduction of structure (i.e., satisfies the obvious axioms).
As you might expect, not every manifold admits a spin structure, and spin structures may not be unique. Loosely speaking, a spin structure is a slightly stronger notion of orientability. Spin structures may always be chosen locally, and the obstruction to consistent gluing is not too difficult to characters as a certain \(\mathbb Z_2\) cohomology class, called the second Stiefel-Whitney class.
\[ S = Spin(M) \times_{Spin(n)} S_0 \]
which is called the spinor bundle. Moreover, since \(S_0\) is a Clifford module, there is well-defined notion of Clifford multiplication on sections of \(S\). We may then define the Dirac operator \(\mathcal{D}\) by
\[ \mathcal{D} = \sum_{a=1}^n c(e_a) \nabla_{e_a} \]
where \(\{e_a\}\) is any orthonormal frame, \(\nabla\) is the spin connection, and \(c\) denotes Clifford multiplication.
Next time: the Weitzenböck formula, and maybe a vanishing theorem.
Spin(n)
Consider the Clifford algebra \(Cl(\mathbb E^n)\) as constructed in yesterday's post. Define maps \(t, \beta: Cl(\mathbb E^n) \to Cl(\mathbb E^n)\) via\[ (e_1 \cdots e_k)^t = e_k \cdots e_1, \ \beta(e_1 \dots e_k) = (-1)^k e_k \dots e_2 e_1 \]
There is a natural inclusion \(\mathbb E^n \hookrightarrow Cl(\mathbb E^n)\). Given \(x \in Cl(\mathbb E^n)\) and \(v \in \mathbb E^n\), we can consider the product \(x v x^t\). In general, this might not be contained in \(\mathbb E^n \subset Cl(\mathbb E^n)\).
Definition. We define the group \(Pin(n)\) to consist of all those \(g \in Cl(\mathbb E^n)\) such that
\[ g \beta(g) = 1, \ \ g v \beta(g) \subset \mathbb E^n \ \forall\ v \in \mathbb E^n. \]
Similarly, we define the group \(Spin(n)\) to be the subgroup of \(Pin(n)\) such that \(gg^t = 1\).
Theorem. The natural action of \(Pin(n)\) on \(\mathbb E^n\) is by othogonal transformations, giving a natural map \(Pin(n) \to O(n)\). This map is a double cover. Similarly, \(Spin(n)\) is a double cover of \(SO(n)\). If \(n \geq 2\), \(Spin(n)\) is simply connected.
The importance of the spin groups is due to the following basic fact. Suppose that \(G\) is a Lie group with Lie algebra \(\mathfrak{g}\). Any representation of \(G\) induces a representation of \(\mathfrak{g}\). However, given a representation of \(\mathfrak{g}\), it is not always possible to integrate it to a representation of \(G\). But it is always possible to integrate a representation of \(\mathfrak{g}\) to produce a representation of the universal cover of \(G\). For \(n \geq 2\), \(Spin(n)\) is the universal cover of \(SO(n)\).
Spin Structures
Let \((M^n, g)\) be a Riemannian manifold. Recall that the frame bundle \(O(M)\) is the manifold consisting of pairs \((x, \mathbb{e})\) where \(x \in M\) and \(\mathbb{e} = \{e_1, \dots, e_n\}\) is an orthonormal frame in \(T_x M\). Since the orthogonal group \(O(n)\) acts on the set of orthonormal frames, this makes \(F(M)\) into a principal \(O(n)\) bundle over \(M\). Let us assume that \(M\) is oriented, so that we may reduce its structure group to \(SO(n)\).Suppose that \(V\) is a representation of \(SO(n)\). Then we may form the associated bundle \(SO(M) \times_{SO(n)} V\), which is a vector bundle over \(M\) with structure group \(SO(n)\). If we take the defining representation then we obtain the tangent bundle, but of course there are many others. Unfortunately, since \(SO(n)\) is not simply connected, not every representation of \(\mathfrak{so}_n\) can be integrated to a representation of \(SO(n)\). At the level of geometry, this means that in a certain sense there are certain vector bundles over \(M\) that are "missing"! Even more disturbing, is that these "missing" bundles appear to be necessary to describe many of the fundamental particles that appear in the standard model--so this has real world consequences. The solution is to equip \(M\) with a spin structure.
Definition. A spin structure on \(M\) is a principal \(Spin(n)\)-bundle \(Spin(M)\) over \(M\) together with a bundle morphism \(Spin(M) \to SO(M)\) which is a reduction of structure (i.e., satisfies the obvious axioms).
As you might expect, not every manifold admits a spin structure, and spin structures may not be unique. Loosely speaking, a spin structure is a slightly stronger notion of orientability. Spin structures may always be chosen locally, and the obstruction to consistent gluing is not too difficult to characters as a certain \(\mathbb Z_2\) cohomology class, called the second Stiefel-Whitney class.
Spin Connection and Dirac Operators
The reduction of structure \(Spin(M) \to SO(M)\) allows us to pull back the Levi-Civita connection on \(SO(M)\) to obtain a connection on \(Spin(M)\), called the spin connection. Let \(S_0\) be the spinor module described in the previous post. Then we may construct the associated bundle\[ S = Spin(M) \times_{Spin(n)} S_0 \]
which is called the spinor bundle. Moreover, since \(S_0\) is a Clifford module, there is well-defined notion of Clifford multiplication on sections of \(S\). We may then define the Dirac operator \(\mathcal{D}\) by
\[ \mathcal{D} = \sum_{a=1}^n c(e_a) \nabla_{e_a} \]
where \(\{e_a\}\) is any orthonormal frame, \(\nabla\) is the spin connection, and \(c\) denotes Clifford multiplication.
Next time: the Weitzenböck formula, and maybe a vanishing theorem.
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