First post! Today I’d like to work out an example of T-duality, a crucial piece of the mirror symmetry puzzle. Along the way I’ll show you the appearance of a curious thing called the B-field, which serves as the extra bit of information in a “complexified” Kahler structure, and we’ll see some hints of the modular invariance of string theory…
A sigma model is a quantum field theory with a scalar field
at center stage. Here, when is the real numbers and is spacetime, then we have the ordinary notion of a real scalar field. Often, however, we think of as the worldspace of some perhaps-extended object such as a string and either as spacetime or as some moduli space. In the first case, this describes how the worldsheet of the string is embedded in spacetime via a field on the worldsheet itself. Scalar, by the way, means that under an isometry of , .
We will consider the case . We give this a metric with signature + – so that the circle has length 1, and we think of the factor as the time direction.
The action which is easiest to consider is the free one:
If we let be a circle of radius , we find that the partition function is invariant under the exchange . This is called T-duality and is beautifully derived in chapter 11 of Clay Math’s Mirror Symmetry. I want to consider the case when is a torus , where we have the lattice , where is real. (This disagrees with the convention in the Clay Math Mirror Symmetry text, where they would use .) When the torus is rectangular, ie. is imaginary, we can think of as a product of two circles of radius and , so the complex structure is given by the modulus . The Kahler structure, meanwhile, is just the area . We see that T-duality on one of the circle factors exchanges the complex and Kahler structures of the target space. However, when is not rectangular, the complex structure is complex, and it seems the duality breaks down unless we turn the Kahler structure also into a complex parameter.
The way we do this is by adding a “B-field” to the action. This is a background field, meaning we just pick some cohomology class and the new action is
Now we can consider the complexified Kahler parameter . We will find by a calculation that the partition function is invariant under the exchange (we replace with a representative in a standard fundamental domain of the modular group ). This does not prove the duality for this theory. It only gives an indication. The partition function is the most basic observable of a quantum field theory. If we wanted to prove the duality outright, we would have to show invariance of all observables under this duality transformation. This is best shown by a path integral manipulation, examples of which I will likely discuss some other time. Anyway…
The partition function is defined to be
where is the Hamiltonian. This is equivalent to evaluating the path integral with periodic temporal boundary conditions over a time , ie. over a rectangular torus worldsheet, of temporal length (it is easy to confuse this with the target torus, ). We can enrich this by considering a nonrectangular worldsheet, say one with modulus . Such a torus has temporal length and spacial twist . The worldsheet momentum operator, generates spacial translation, so this corresponds to the partition function
We can define right moving and left moving Hamiltonians
as well as the modular parameter , so
This technique allows us to continue thinking about the worldsheet as a rectangular tube. An aside : it turns out that and are the generators of time translation for the two chiral factors of the Virasoro algebra, though I do not yet understand this. Now I’ll move on to quantizing the classical theory just described. (I haven’t yet told you what is or even what Hilbert space it acts on!)
Fixing , describes a loop in . Time translation defines a homotopy of loops (without basepoint), so the theory will have a conserved charge in , defining so-called topological sectors into which the Hilbert space splits as a direct sum.
As for the Hilbert space of each topological sector, every in the sector of the Hilbert space can be written as a product for some “zero mode” in the contractible sector and any representative of the sector. It therefore suffices to understand those maps which can be unwound. We can think of this theory of zero modes as a sigma model with target . This is because the topologically trivial maps are precisely those that lift to the universal cover of . This lifting is unique up to the action of . States in such a theory are labelled by their momenta (they are real scalars, hence have no spin or parity, etc.), so it remains to determine the momentum spectrum.
Write in terms of the period 1 coordinates along the basis for . The canonical momenta for and are obtained from the Lagrangian by . We can write
From this, we find
where we are in the topological sector . Thus, the spectrum of the target space momentum operators for the theory with the B-field can be obtained from the spectrum without the B-field by a shift. If we were calculating the partition function within a particular topological sector, this would not matter. It would contribute an overall constant factor to the partition function, which is unphysical. However, when we consider the whole partition function, the relative shifts matter.
Consider the sigma model with target and no B-field. If is in the sector, we can take the Fourier expansion . We calculate
With this in hand, the action decomposes into a sum of harmonic oscillators , a massless mode , and a constant factor from the winding numbers. Here, takes values in and takes values in . We can find the classical Hamitonian by the Noether procedure:
We can think of each oscillator as two identical uncoupled oscillators and handle each of these in the ordinary way by introducing creation and annihilation operators and and a lowest energy state annihilated by and . This will be a short hand for us. Later we will have to square the contribution from a single oscillator to account for the complete pair. The Hilbert space is spanned by the descendants of this state, ie. it’s Verma module. On it, the Hamiltonian is
The half-Hilbert space of all these single oscillators is the tensor product of these Verma modules for with the Hamiltonian
where I use zeta function regularization . This regularization scheme is necessary for modular invariance to be preserved. I suspect it has much to do with why modular forms keep popping up in conformal field theory.
The massless mode, meanwhile, has a Hilbert space spanned by a basis of target space momentum eigenfunctions. To get the correct spectrum for these modes, we need to think about the quotient again. Target space translations are generated by the target space momentum operators, (not to be confused with the worldsheet momentum above). For any vector , we should have
since this just corresponds to circling around the torus a few times and ending up in the same spot. Thus, the eigenvalues of are contained in the dual lattice . Here I use the shorthand . I will use similar shorthand to talk about eigenvalues (an “eigenvalue” of will be a vector in ). This is just to emphasize the fact that the quantization does not depend on the coordinates of the target space. Note that and are the quantized versions of considered above. In other words, is canonically conjugate to the target space position operator . is the operator whose spectrum is shifted by the addition of a B-field.
It is easy to come up with eigenfunctions for every element of the dual lattice, so is precisely the target space momentum lattice of the system.
The Hamiltonian for the massless mode is just
We have a copy of the ground state defined above for each target space momentum –call it –such that
To finish off the construction of the quantum Hamiltonian, we need to handle the winding number term. We introduce operators that count the winding number. Here I use the same shorthand as . We have a ground state as defined above for every winding number , which we call . Then we want
There is clearly some analogy between and . Indeed, we should include in our discussion a canonically conjugate operator analogous to the target space position operator . We should have
In summary, the quantum Hilbert space has a unique ground state and other states are obtained by applying and . Then we have, for each its Verma module for the creation and annihilation operators for each . We can write down the quantum Hamiltonian for an arbitrary sector of the Hilbert space:
To find the left-moving and right-moving Hamiltonians, we need to also find the worldsheet momentum for this model. We can obtain this from the spacetime translations using the Noether procedure as we did for the Hamiltonian. The Hamiltonian is a diagonal element of the stress-energy, and the momentum is an off-diagonal one. This determines them up to signs as the stress-energy for these theories is antisymmetric (a hint of their conformal invariance!). We find that the classical worldsheet momentum is
The evident quantized operator, using the definitions of the creation and annihilation operators, is
Here I note that and commute since they are simultaneously diagonalized by . Then we see
We see from this that the only terms affected by the B-field is the zero mode terms . The oscillator terms are unaffected, so we can calculate them outright. Recall our short hand that cuts down the number of oscillators by half, so we must square the contribution to the trace. An easy calculation shows that they contribute a factor of
where is the Dedekind eta modular form. This is where the factor of is crucial to modular invariance. Go figure.
Without the B-field, the contribution from the zero modes is
With the B-field, the entire partition function becomes
where is the matrix
A quick inspection shows that this is indeed invariant under the change
is defined above.
What does this symmetry represent purely in terms of the lattices involved? What is this business with the B-field really about from a geometric standpoint?