Showing posts with label string theory. Show all posts
Showing posts with label string theory. Show all posts

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:
  • $[\cdot,\cdot]$ 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 $f \in H^0(M, \mathcal{O}_M)$
Note that we can take $L$ to be either a real or complex vector bundle. In the latter case the anchor map should map to the complexified tangent bundle.

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.

Sunday, February 23, 2014

Virasoro Algebra

Conformal Invariance in 2D

To begin, recall that in two dimensions, the conformal transformations are generated by holomorphic and anti-holomorphic transformations. At the infinitesimal level, let \(\ell_n := -z^{n+1} \partial_z\) be a basis of holomorphic vector fields. These satisfy the Witt algebra
\[ [\ell_m, \ell_n] = (m-n)\ell_{m+n}. \]
Similarly, we can define \(\bar{\ell}_m = -\bar{z}^{n+1} \partial_{\bar{z}}\), and in addition to the Witt algebra these new generators satisfy \([\bar{\ell}_m, \ell_n]=0\).

Now, we could try to define a 2D conformal quantum field theory to be a unitary representation of the Witt algebra (or rather, of two copies of the Witt algebra, since we have both holomorphic and anti-holomorphic vector fields--but nevermind that). But this is too naive.


Central Extensions

Recall that in quantum mechanics, states are represented by vectors in some Hilbert space \(\mathcal{H}\). However, the state \(|\phi\rangle\) and \(\alpha|\phi\rangle\) are physically equivalent for any non-zero complex number \(\alpha\). The reason, of course, is that the expectation value of an operator \(\mathcal{O}\) is defined to be \(\langle \phi|\mathcal{O}|\phi\rangle / \langle \phi|\phi\rangle\), and such expressions are invariant under rescaling in \(\mathcal{H}\).

Thus,  a symmetry group \(G\) for a theory does not necessarily act via a map \(G \to U(\mathcal{H})\). It suffices to have a projective representation \(G \to PU(\mathcal{H})\). Let \(\mathfrak{g}, \mathfrak{pu}\) be the Lie algebras of \(G\) and \(PU\), respectively. A projective representation gives a map
\[ \mathfrak{g} \to \mathfrak{pu}. \]
Since \(PU\) is a quotient of \(U\), we have a short exact sequence
\[ 0 \to \mathbb{C} \to \mathfrak{u} \to \mathfrak{pu} \to 0. \]
Now let \(\hat{\mathfrak{g}}\) be defined as
\[ \hat{\mathfrak{g}} = \{ (\xi, \eta) \in \mathfrak{u}\oplus\mathfrak{g} \ | \ \pi(\xi) = \rho(\eta) \} \]
This comes with a natural projection \(\hat{\mathfrak{g}} \to \mathfrak{g}\). If we suppose that the projective representation \(\rho\) is faithful, then the kernel of this map is exactly \(\mathbb{C}\). Hence, a faithful projective representation of \(\mathfrak{g}\) yields a short exact sequence of Lie algebras
\[ 0 \to \mathbb{C} \to \hat{\mathfrak{g}} \to \mathfrak{g} \to 0. \]
We have obtained a central extension of \(\mathfrak{g}\).


Virasoro Algebra

Finally, we can define the Virasoro algebra. It has generators \(L_n\) and \(c\), with defining relations
\[ [L_m, L_n] = (m-n) L_{m+n} + \frac{c}{12}(m^3-m) \delta_{m+n,0}, [c, L_n] = 0. \]
The generator \(c\) acts as a scalar in any irreducible representation, and its value is called the central charge. The factor of \(1/12\) is entirely conventional. Now, the amazing fact is the following.

Theorem. Up to equivalence, the Virasoro algebra is the unique non-trivial central extension of the Witt algebra.

Proof sketch. This is essentially just a calculation. Any central extension has to be of the form
\[ [L_m, L_n] = (m-n) L_{m+n} + A(m,n) c \]
for some function \(A(m,n)\). If we make the replacement \(L_m \mapsto L_m + a_m c\), then we have
\[ [L_m, L_n] = (m-n) L_{m+n} + \left( A(m,n) + (m-n) a_{m+n} \right) c \]
Taking \(n = 0\), we have
\[ [L_m, L_0] = m L_{m} + \left( A(m,0) + m a_{m} \right) c \]
Hence for \(m\neq0\) we can take \(a_m = m^{-1} A(m,0)\). Having done this, we are now free to assume that \(A(m,0) = 0 \) for all \(m\). Then we may apply the Jacobi identity to deduce that \(A(m,n)=0\) except possibly for \(m=-n\), so that \(A(m,n)\) can be written in the form \(A(m,n) = A_m \delta_{m+n, 0}\). Finally, another application of the Jacobi identity yields a simple recurrence relation for the coefficients \(A_m\), and it is easily seen that every solution of this recurrence is proportional to \(m^3-m\).

Now we can take our (preliminary, and still too naive) definition of a quantum conformal field theory to be a unitary representation of the Virasoro algebra.


Stress-Energy Tensor and OPE

The operator \(L_0\) behaves like the Hamiltonian of the theory, and the Virasoro relations show that \(L_n\) for \(n>0\) act as lowering operators. Hence, in a physically sensible representation, the vacuum vector \(|\Omega\rangle\) will be annihilated by \(L_n\) for all \(n > 0\). Unitary requires \(L_n^\dagger = L_{-n}\), so additionally we have \(\langle \Omega|L_n = 0\) for \(n < 0\). Hence
\[ \langle \Omega | L_m L_n | \Omega \rangle = 0 \ \textrm{unless}\ n \leq 0, m \geq 0 \]

Now define the stress-energy tensor to be the operator-valued formal power series
\[ T(z) = \sum_n \frac{L_n}{z^{n+2}} \]
We can consider the vacuum expectation of the product \(T(z) T(w)\). By the above remarks, many terms in the expansion will vanish. In fact, it is a straightforward (but tedious!) exercise to check the following.

Theorem. The stress-energy tensor satisfies the operator product expansion
\[ T(z) T(w) \sim \frac{c/2}{(z-w)^4} + \frac{2 T(w)}{(z-w)^2} + \frac{\partial_w T(w)}{z-w} \]
where \(\sim\) denotes that the left- and right-hand sides are equal up to the addition of terms with vanishing vev and/or regular as \(z \to w\).

Monday, August 27, 2012

A Toy Model for Effective Field Theory and Extra Dimensions

I wanted to see how the Fourier transform can turn field theory into many-particle mechanics. This is just silly fooling around, so you shouldn't take what follows too seriously (there are much better models of extra dimensions, to be sure!).

Take \(\phi(t, s)\) to be a field on a cylinder of radius \(R\). We consider the action

\[ S = \frac{1}{R} \int_{-\infty}^\infty \int_0^{R} |\nabla \phi|^2 ds dt \]

Expand \(\phi(t, s)\) in Fourier series:
\[ \phi(t,s ) = \sum_n \phi_n(t) e^{2 \pi i n s / R} \]
Then in Lorentzian signature, we have
\[ \int_0^{R} |\nabla \phi|^2 d\theta = R \sum_n \dot{\phi}_n^2 - \left(\frac{2\pi n}{R}\right)^2 \phi_n^2. \]

Putting this back into the action, we find
\[ S = \sum_n \int_{-\infty}^\infty \dot{\phi}_n^2- \left(\frac{2\pi n}{R}\right)^2 \phi_n^2 dt. \]

This is the action for infinitely many harmonic oscillators, with frequencies \(\omega_n = 2\pi |n| / R\). Recall that the energy levels of the harmonic oscillator are \(k\omega\) for \(k = 0, 1, \ldots\). So supposing that only a finite energy \(E\) is accessible in some particular experiment, we can only excite those modes \(\phi_n\) for which
\[ \frac{2\pi |n|}{R} < E. \]
In particular, only finitely many \(\phi_n\) may be excited at energies below \(E\), effectively reducing the field theory on the cylinder to many-particle quantum mechanics.

Wednesday, October 5, 2011

Susskind on String Theory

Found this and thought I'd share:

Lectures on String Theory by Susskind

This is a series of introductory lectures on string theory by cofounder Leonard Susskind. Regardless of whether you think string theory has anything to say about the universe we live it, it's very cool to hear about its origins from one of the people who dreamt it up.