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Differential cohomology




A dd-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 dd propagate.

The operation of T-duality is a map that interchanges pairs of such geometric data for 2-dimensional conformal field theory sigma-models, such that the induced QFTs are equivalent.

More specifically the space of differential geometric data consisting of

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 XX turn out to be equivalent as quantum field theories for T-dual backgrounds (X,g,G)(X,g,G) and (X,g,G)(X',g',G') (at least to the approximate degree to which these are realized as full CFTs in the first place).

Further generalisations let XX be a nontrivial torus bundle, but the T-dual is then generically a bundle of non-commutative tori. (cite Mathai, Rosenberg and Hannabus)

Path integral heuristics deriving T-duality

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 n\mathbb{R}^n 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 S 1S^1, 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 2πR2 \pi R 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 2π1/R2 \pi 1/R.

A first rough look

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 RR has quantized momentum p=Rp = \frac{\ell \in \mathbb{Z}}{R}. In a state in which the string winds around the circle mm times and has \ell quanta of kinetic momentum for propagation around the circle, its energy is

E 0=p 2+M 2=(/R) 2+(Rm) 2, E_0 = \sqrt{p^2 + M^2} = \sqrt{(\ell/R)^2 + (R m)^2 } \,,

This energy is clearly invariant under exchanging

(R,(,m))(1R,(m,)). (R, (\ell, m)) \mapsto (\frac{1}{R}, (m,\ell)) \,.

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

X:ΣS R 1 X : \Sigma \to S^1_R

the σ\sigma-model field on the worldsheet Σ= 2\Sigma = \mathbb{R}^2 with values in target space S R 1S^1_R then T-duality with respect to this circle may be thought of as exchanging worldsheet momentum tX\partial_t X with worldsheet winding σX\partial_\sigma X.

This then also means that for the open string it exchanges von Neumann boundary conditions σX| σ=0=0\partial_\sigma X|_{\sigma = 0} = 0 with Dirichlet boundary conditions tX| σ=0=0\partial_t X|_{\sigma = 0} = 0. 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.

The path integral

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 Σ\Sigma be labeled by Σ (1)\partial \Sigma_{(1)}.

We consider the following fields on the worldsheet:

  • X˜:Σ/(2π/R)=S R 1\tilde X : \Sigma \to \mathbb{R}/(2\pi/R)\mathbb{Z} = S^1_R – a circle-valued function; this is the standard σ\sigma-model field describing propagation of the string on the circle;

  • X i:Σ (i)S R 1X_{i} : \partial \Sigma_{(i)} \to S^1_R – the boundary values of this field;

  • bΩ 1(Σ,)b \in \Omega^1(\Sigma, \mathbb{R}) – 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 following action functional on this collection of fields given by the assignment

S E(X˜,b)=12π Σ(12bbbdX˜)i2π i=1 s Σ (i)(X˜a i)dX i, S'_E(\tilde X,b) = \frac{1}{2 \pi} \int_\Sigma \left( \frac{1}{2} b \wedge \star b - b \wedge d \tilde X \right) - \frac{i}{2 \pi} \sum_{i = 1}^s \int_{\partial \Sigma_{(i)}} (\tilde X -a_i) d X_i \,,

where the (a i)(a_i) 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 bb 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

S˜ E=14π ΣdX˜dX˜+i2π Σ (i)(X˜a i)dX i \tilde S_E = \frac{1}{4 \pi} \int_\Sigma d \tilde X \wedge \star d \tilde X + \frac{i}{2\pi} \int_{\partial \Sigma_{(i)}} (\tilde X - a_i) d X_i

then doing the integral over the boundary values X iX_i yields

X˜| Σ i=a i \tilde X|_{\partial \Sigma_i} = a_i

This is the action functional for a σ\sigma-model on S 1/R 1S^1_{1/R} with a D-brane at X˜=a i\tilde X = a_i.

Now we evaluate the original path integral in a different way, this way first integrating over components of X˜\tilde X. To do so, we imagine that we may re-encode the field X˜\tilde X in terms of its de Rham differential

dX˜=df+2πR Aη Aω A d \tilde X = d f + \frac{2\pi}{R}\sum_A \eta_A \omega_A

where η A\eta_A are integers and

{ω A}H 1(Σ,) 2g+s1. \{\omega_A\} \subset H^1(\Sigma, \mathbb{Z}) \simeq \mathbb{Z}^{\oplus 2g + s - 1} \,.

Then formally performing the path integral over ff yields db=0d b = 0 and b| Σ=dX ib|_{\partial \Sigma} = d X_i. It follows that b=dXb = d X for some other field X:ΣS R 1X : \Sigma \to S^1_R.

So we get the action

S E=14π ΣdXdX+i2π i=1 s Σ ia idX S_E = \frac{1}{4\pi} \int_\Sigma d X \wedge \star d X + \frac{i}{2 \pi} \sum_{i = 1}^s \int_{\partial \Sigma_i} a_i d X

in terms of the field XX. This is the σ\sigma-model for string propagation on S R 1S^1_R. with D-brane wrapped on S R 1S^1_R that carries on its worldvolume a gauge field given by a constant connection 1-form a ia_i.

Topological T-duality

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 XX 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.

Geometric T-duality

In terms of generalized differential cohomology

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 c(n)\mathbf{c}(n) in the fiber sequences

BU(1)BU(n)BPU(n)c nB 2U(1) \mathbf{B}U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n) \stackrel{\mathbf{c}_n}{\to} \mathbf{B}^2 U(1)

is induced a notion of twisted cohomology which makes the Kalb-Ramond field act as a twist for twisted K-theory.

In these terms, the setup of T-duality is a correspondence of Kalb-Ramond fields over spacetime torus-bundles PXP \to X and P^X\hat P \to X that induces an integral transform

K diff +τ(P)K diff +τ1(P^) K_{diff}^{\bullet + \tau}(P) \to K_{diff}^{\bullet + \tau -1}(\hat P)

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.

See (KahleValentino).

In generalized complex geometry

Another approach to the study of T-duality takes a somewhat complementary point of view and ignores the integral cohomology class in H 3(X,)H^3(X,\mathbb{Z}) of the gerbe but does consider the Riemannian metric.

In this context T-duality is described as an isomorphism of standard Courant algebroids. This picture emerged in the study of generalized complex geometry.

Examples of T-dual pairs

In Mirror symmetry

One special cases of T-duality is mirror symmetry.

In Langlands duality

In some cases the passage to the Langlands dual group in the geometric Langlands correspondence can be understood as T-duality. (Daenzer-vanErp)

duality in physics, duality in string theory

The geometry of the fiber product of two torus fiber bundles with a circle 2-bundle on it is sometimes referred to as Bn-geometry.


Textbook accounts include


A review of T-duality from the worldsheet perspective is in

  • Jnanadeva Maharana, The Worldsheet Perspective of T-duality Symmetry in String Theory (arXiv:1302.1719)

Target space

Without supergeometry

Geometric T-duality in terms of differential cohomology as an operation on twisted differential K-theory is discussed in

More physically oriented discussion of this is in

  • Katrin Becker, Aaron Bergman, Geometric Aspects of D-branes and T-duality (arXiv:0908.2249)

Geometric T-duality is identified as an isomorphism of standard Courant algebroids (generalized complex geometry) in section 4 of

Discussion of the sigma-model description of T-duality in this context includes

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)

Discussion of the infinitesimal T-duality geometry, replacing gerbes on torus-fiber bundles with the corresponding dg-manifolds is in

For references on topological T-duality see there.

The relation to Langlands dual groups is discussed in

With supergeometry

Discussion of T-duality that takes into account the super p-brane charges (i.e. the fermionic components of the RR-fields) on super spacetime, hence also of Green-Schwarz action functionals, includes the following:

Revised on March 28, 2017 08:08:04 by Urs Schreiber (