T-Duality in Lorentz-Breaking String Theories

• T-duality is a symmetry that interchanges momentum and winding modes, defining equivalence between distinct compactified backgrounds.
• The formulation employs standard Buscher rules and gauging techniques, preserving duality even when target space Lorentz invariance is broken.
• This robust framework facilitates analysis of modified Hamiltonian constraints and non-perturbative aspects of string theory dynamics.

T-duality is a non-perturbative symmetry of string theory that relates theories compactified on geometrically distinct, yet physically equivalent, backgrounds. At its core, T-duality interchanges the momentum and winding modes of a string on a compact direction and is implemented mathematically as a mapping between background fields, often with deep implications for the spectrum, symmetries, and geometry of the theory. In the context of Lorentz Breaking String (LBS) theories—string theories with modified Hamiltonian constraints that explicitly break Lorentz invariance—the persistence and precise formulation of T-duality is a nontrivial structural property. Despite the breaking of target space Lorentz symmetry, standard techniques permit the explicit derivation of T-duality transformations, and detailed analysis reveals that these coincide exactly with the Buscher rules of relativistic string theory (Kluson et al., 2010). This invariance underscores the universality of T-duality as a stringy symmetry.

1. Lorentz Breaking String Theories: Structure and Motivation

LBS theories are constructed by modifying the Hamiltonian constraint in the standard Polyakov string, explicitly breaking target space Lorentz invariance while preserving worldsheet symmetries. Despite the varying critical exponent $z$ in such theories, the infrared limit ($z \to 1$) asymptotically restores conformal invariance and, up to details of the worldsheet lapse function, recovers the relativistic Polyakov action. These deformations, inspired by Hořava–Lifshitz gravity, serve as laboratories to study string dynamics and dualities under altered symmetry principles.

A central question addressed in this context is the fate of fundamental string-theoretic symmetries—most notably, T-duality—when Lorentz symmetry (a backbone of relativistic string theory) is explicitly broken.

2. Formulation of T-duality in LBS Theories

The implementation of T-duality in LBS theories utilizes a formulation closely paralleling the Buscher approach. The starting point is the requirement of a continuous shift symmetry (isometry) in one target space direction, say $\varphi$, so that the background depends only on coordinates $x^M = (\varphi, x^a)$ and is invariant under $\varphi \to \varphi + \epsilon$. This symmetry induces a conserved worldsheet current $J^a$ via Noether’s theorem: $\partial_a J^a = 0.$ Gauging this isometry involves promoting the global shift parameter $\epsilon$ to a local worldsheet function and introducing gauge fields $a_\alpha$, yielding covariant derivatives: $$D_\alpha \varphi = \partial_\alpha \varphi + a_\alpha,$$ with $a_\alpha \to a_\alpha - \partial_\alpha \epsilon$ under local shifts.

Consistency with worldsheet foliation-preserving diffeomorphisms is achieved by constructing a Lagrangian invariant under both the local shift and diffeomorphism transformations, which may necessitate additional terms depending on the microscopic details of the LBS theory [(Kluson et al., 2010), eqs. (3.5)-(3.8)].

A Lagrange multiplier $\tilde{\varphi}$ is introduced to impose the flatness of $a_\alpha$, thereby ensuring that the constraint remains non-dynamical. Gauge fixing (e.g., by setting $\varphi = 0$) and integrating out $a_\alpha$ leads directly to a dualized worldsheet action (see below for its explicit structure).

3. Explicit Dualization Procedure and T-duality Rules

The dualized Lagrangian density for LBS theories (after integrating out the gauge field and gauge fixing) takes the schematic form [(Kluson et al., 2010), eq. (4.22)]: $$\mathcal{L} = - V_\omega \frac{1}{4\pi\alpha'} \partial_\tau \tilde{\varphi} N^T (\partial_\tau x^0 - n^0 \partial_\sigma x^0)^2 - \cdots,$$ with additional terms involving spatial and worldsheet fields. Here $V_\omega$ is a worldsheet volume factor, and the explicit structure mirrors that of the original (Lorentz invariant) theory—demonstrating that the core dualization mechanism applies even in the LBS setting.

The fundamental result is the explicit set of transformation rules for the background fields under T-duality. For the metric, lapse, and shift fields: $$\begin{aligned} \hat{h}_{\varphi\varphi} & = h_{\varphi\varphi} - \frac{(h_{\varphi 0})^2}{h_{00}}, \ \hat{h}_{\varphi a} &= 0, \ \hat{N} &= N, \quad \hat{N}^\sigma = 0, \quad \hat{N}^\varphi = N^0, \ \hat{g}_{00} &= -N^2 + N_i N^i, \ \hat{g}_{0\varphi} &= 0, \ \hat{g}_{\varphi\varphi} &= h_{\varphi\varphi}, \end{aligned}$$ with the inverse metric components accordingly: $$\hat{g}^{00} = -\frac{1}{N^2}, \qquad \hat{g}^{0\varphi} = 0, \qquad \hat{g}^{\varphi\varphi} = h^{\varphi\varphi} - \frac{(N^\varphi)^2}{N^2}.$$

For the antisymmetric B-field, the T-duality relations are

$$\begin{aligned} \hat{g}_{\varphi\varphi} &= \frac{1}{g_{\varphi\varphi}}, \ \hat{g}_{00} &= g_{00} - \frac{(g_{0\varphi})^2}{g_{\varphi\varphi}}, \ \hat{B}_{0\varphi} &= \frac{g_{0\varphi}}{g_{\varphi\varphi}}, \end{aligned}$$

and the full set reduces to the standard Buscher rules for relativistic strings: $$\hat{g}_{\mu\nu} = g_{\mu\nu} - \frac{g_{\mu\varphi} g_{\nu\varphi}}{g_{\varphi\varphi}}; \qquad \hat{g}_{\varphi\varphi} = \frac{1}{g_{\varphi\varphi}}; \qquad \hat{B}_{\mu\varphi} = \frac{g_{\mu\varphi}}{g_{\varphi\varphi}}; \qquad \hat{B}_{\varphi\varphi} = 0.$$ These formulas govern the dualization of the background data when an isometry is present and establish precise correspondences between the fields in the original and the dual theory.

4. Crucial Role of Target Space Isometries in T-duality

The existence of a target space isometry (invariance under a shift of a coordinate) is central to the construction. The isometry ensures:

• Existence of a conserved current crucial for gauging.
• Possibility of promoting the isometric shift to a worldsheet local symmetry, enabling the introduction of gauge fields.
• Accessibility of a consistent dual coordinate in the dual theory via a Lagrange multiplier that enforces flatness of the associated field strength.
• Consistency of integrating out the gauge field and yielding a dual sigma model.

In the absence of this isometry, the Buscher-type dualization procedure fails, as neither the necessary globally defined conservation law nor the covariantization procedure is available.

5. Equivalence with Buscher’s T-duality: Universality of Duality Rules

A foundational outcome is that, despite explicit Lorentz breaking in the target space, the LBS dualization yields exactly the Buscher rules for the transformation of the metric, B-field, and related data, as in standard relativistic string theory. This equivalence is encapsulated in the explicit formulae above and both formalizes and extends the universality of T-duality. The analysis holds even though the initial worldsheet theory lacks target space Lorentz symmetry [(Kluson et al., 2010), eqs. (4.24), (4.25), (4.28), (4.31)].

This result demonstrates that the structure of T-duality is deeper than its manifestation through target space Lorentz invariance; it is rooted in the existence of the isometry and the global properties of the string worldsheet action.

6. Technical Summary and LaTeX Formulæ

The essential elements of the symmetry in the context of LBS theories can be summarized with the following key formulas:

• Covariant derivatives along isometry:

$$D_\alpha \varphi = \partial_\alpha \varphi + a_\alpha$$

• Gauge field transformations:

$a'_\alpha = a_\alpha - \partial_\alpha \epsilon$

• T-duality rules for background fields (sample):

$$\hat{g}_{\varphi\varphi} = \frac{1}{g_{\varphi\varphi}};\quad \hat{g}_{0\varphi} = 0;\quad \hat{g}_{00} = g_{00} - \frac{(g_{0\varphi})^2}{g_{\varphi\varphi}};\quad \hat{B}_{0\varphi} = \frac{g_{0\varphi}}{g_{\varphi\varphi}}$$

These encapsulate the essence of the duality mechanism and are structurally identical to those in the relativistic setting.

7. Implications and Universality

The coincidence of the T-duality rules in LBS theories and standard string theory, under the requirement of target space isometry, has important implications:

• The presence or absence of target space Lorentz invariance does not affect the T-duality transformation laws, provided worldsheet reparameterization and gauge symmetries are properly implemented.
• T-duality is robust to deformations that break target space Lorentz invariance, as long as certain global structures (notably, isometries) are maintained.
• This universality reinforces the central role of T-duality as a fundamental symmetry of string theory, possibly extending beyond traditional symmetry-based formulations.

In summary, T-duality in Lorentz breaking string theories is structurally identical to the relativistic case, and its implementation via standard gauging, covariantization, and dualization proceeds unimpeded. The invariance of the theory under dualization is ensured solely by the presence of an isometry in the target space, rendering T-duality a robust and universal feature of string theory dynamics (Kluson et al., 2010).