Types of quantum field thories
abstract duality: opposite category,
A -dimensional sigma-model is a quantum field theory that is induced from certain differential geometric and differential cohomological data, to be thought of as encoding the background geometry on which quantum objects of dimension propagate.
More specifically the space of differential geometric data consisting of
a smooth manifold
admits a certain operation that, roughly, inverts the Riemannian circumference of the torus fibers and mixes the metric with the gerbe data, such that the induced 2-dimensional sigma-model QFTs for these backgrounds are equivalent. This is the operation called T-duality.
This was noticed originally in the study of conformal field theories in the context of string theory: the conformal field theory sigma-models with target space turn out to be equivalent as quantum field theories for T-dual backgrounds and (at least to the approximate degree to which these are realized as full CFTs in the first place).
Further generalisations let be a nontrivial torus bundle, but the T-dual is then generically a bundle of non-commutative tori?. (cite Mathai, Rosenberg and Hannabus)
We indicate how one can see T-duality from formal manipulations of the path integral for the string sigma-model. We look at the simplest situation, where the torus bundle in question is a trivial circle bundle over a Cartesian space carrying the metric induced from the standard flat metric on and where there are no other nontrivial background fields. In fact, for the purpose of the following computation we can entirely ignore the base of this bundle and consider target space to be nothing but a circle. Since the sigma-model for this is on the worldsheet just the theory of a single free field with values in , this is often also called the “free boson on the circle”.
This means that the only geometric datum determining the background geometry is the circumference of the fiber of the circle bundle. The statement of T-duality in this situation is that the 2-dimensional sigma-model on this background yields the same 2-dimensional CFT as that for this kind of background with circumference of the circle being .
A quick way to get an indication for this is to consider the center-of-mass energy of the string in such a circle-bundle background. In the simplified setup we mentioned before, a string on a circle of radius has quantized momentum . In a state in which the string winds around the circle times and has quanta of kinetic momentum for propagation around the circle, its energy is
This energy is clearly invariant under exchanging
This is of course far from being a proof that the corresponding two QFTs are equivalent, but it does already capture a good deal of the essence of what T-duality does and why it works.
In slightly more detail, but still at a very rough level, if we denote by
the -model field on the worldsheet with values in target space then T-duality with respect to this circle may be thought of as exchanging worldsheet momentum with worldsheet winding .
This then also means that for the open string it exchanges von Neumann boundary conditions with Dirichlet boundary conditions . The first boundary condition is that describing an open string whose endpoints are free to propagate in worldsheet time, whereas the second boundary condition describes a situation where the endpoint of the string is fixed at some point in target space. In terms of the language of geometric target space data, a sigma-model with such a constraint is said to describe a D-brane in target space: the locus where the endpoints of the string are fixed. This is a first indication that the T-duality operation on geometric background also involves the RR-field.
We follow Kentaro Hori’s path integral discussion of T-duality. Here the strategy is to consider a path integral over a certain space of auxiliary fields and show or argue that by “algebraically integrating out” some of these in two different ways, the path integral is equivalent to that over two different action functionals, which describe two T-dual geometric backgrounds.
Let the boundary components of the worldsheet be labeled by .
We consider the following fields on the worldsheet:
– a circle-valued function; this is the standard -model field describing propagation of the string on the circle;
– the boundary values of this field;
– a 1-form; this is the auxiliary field that will not contribute to the dynamics but serves to make the T-duality manifest.
Consider then the action functional on this collection of fields given by the assignment
where the are a collection of real numbers.
We now want to formally perform the path integral over the fields in two different orders, which should give the same quantum field theories but in terms of different effective action functionals.
If we do first the path integral over the field then by the general formal rule of “algebraically integrating out a non-dynamical field” which says that we can evaluate this path integral that formally looks like a Gaussian integral by the usual formulas for Gaussian integrals, we obtain the action functional
then doing the integral over the boundary values yields
This is the action functional for a -model on with a D-brane at .
Now we evaluate the original path integral in a different way, this way first integrating over components of . To do so, we imagine that we may re-encode the field in terms of its de Rham differential
where are integers and
Then formally performing the path integral over yields and . It follows that for some other field .
So we get the action
in terms of the field . This is the -model for string propagation on . with D-brane wrapped on that carries on its worldvolume a gauge field given by a constant connection 1-form .
It turns out to be possible and useful to discuss just the topological aspects of T-duality, meaning all the aspects that depend on the as a topological space, on the topological class of the gerbe and of its 3-form curvature, but not on the Riemannian metric and not on the precise connection on the gerbe (there may be several inequivalent one for a given curvature)!
This sub-phenomenon is discussed in more detail at topological T-duality.
Gauge fields are cocycles in differential cohomology. The Kalb-Ramond field is given by degree-3 ordinary differential cohomology, the differential refinement on degree-3 integral cohomology. The RR-field is given by differential K-theory.
Induced by the morphisms in the fiber sequences
of twisted differential K-theory classes.
This is an isomorphism – the action of the T-duality isomorphism on the Kalb-Ramond field and the RR-field.
One special cases of T-duality is mirror symmetry.
A review of T-duality from the worldsheet perspective is in
More physically oriented discussion of this is in
A discussion of the sigma-model description of T-duality in this context is in
Further references are
Willie Carl Merrell, Application of superspace techniques to effective actions, complex geometry and T-duality in String theory (pdf)
Peggy Kao, T-duality and Poisson-Lie T-duality in generalized geometry (pdf)
For references on topological T-duality see there.
The relation to Langlands dual groups is discussed in