An uplifting discussion of T-duality

Abstract

It is well known that string theory has a T-duality symmetry relating circle compactifications of large and small radius. This symmetry plays a foundational role in string theory. We note here that while T-duality is order two acting on the moduli space of compactifications, it is order four in its action on the conformal field theory state space. More generally, involutions in the Weyl group W (G) which act at points of enhanced G symmetry have canonical lifts to order four elements of G, a phenomenon first investigated by J. Tits in the mathematical literature on Lie groups and generalized here to conformal field theory. This simple fact has a number of interesting consequences. One consequence is a reevaluation of a mod two condition appearing in asymmetric orbifold constructions. We also briefly discuss the implications for the idea that T-duality and its generalizations should be thought of as discrete gauge symmetries in spacetime. May 22, 20181 We are actually being somewhat sloppy here from a mathematical viewpoint. (Most physicists will want to skip this footnote.) The "Narain moduli space" is an orbifold, and is more properly regarded as a global stack where the automorphism group of objects is always a finite group. However, it is not really the moduli stack of toroidal conformal field theories. In the latter stack, the automorphism group of an object will include continuous groups at, for example, the points of enhanced A-D-E symmetry, while in the Narain moduli stack the automorphism group of the A-D-E points is a finite group F (Γ(g)) discussed at length below. The moduli stack of conformal field theories maps to the Narain moduli space.

Anomalies of duality groups and extended conformal manifolds

“…It acts as Z 2 on the exactly marginal operators, but at the self-dual radius it acts as a Z 4 symmetry; its generator squares to minus one on the operators with half-integer SU (2) quantum numbers (see e.g. [15,16] for recent discussions).…”

“…They are parameterized by M g , the moduli space of genus-g Riemann surfaces Σ g . 16 The fact that these 4d fermions came from 6d means that they couple to a natural connection on M g . In particular, its partition function on a closed 4-manifold will be a section of a nontrivial line bundle over M g .…”

Conway subgroup symmetric compactifications of heterotic string

“…(This is the left-moving part of the group denoted F (Γ) in [41].) As noted above, the symmetry group Aut(F ⊥ L ) ⊂ Co 0 (which is, in general, only a subgroup of the full group of automorphisms of F ⊥ L ) is G 222 = U 6 (2) and the subgroup of the monomial group G 224 = 2 10 .M 22 .…”

“…The orbifold models A, B, C all satisfy standard level-matching constraints required for modular invariance. In addition one should address the "DTF" criterion of [41] (which extends earlier treatments in [55,60,61].) Namely if there is a vector p ∈ Γ such that (p, g · p) = 1 mod 2 .…”

“…(3.11) then, as described under equation (2.17) of [41] eitherĝ|p = |g · p for some p ∈ Γ g or the lifted automorphismĝ on the CFT will be order four. When the DTF criterion (3.11) holds we will use the canonical lift of equations (6.35) and (6.36) of [41]. After some work one can show that for the involutions of type A and B the DTF condition (3.11) does indeed hold for involutions of type A,B for both Q 222 and Q 224 .…”