Aspects of strings without spacetime supersymmetry

This review paper surveys the challenges and recent developments in non-supersymmetric string theory, focusing on the mechanisms of tachyon emergence, the implications of worldsheet tadpoles, and the exploration of stable, tachyon-free vacua within the non-supersymmetric string landscape.

The Gist

Imagine the universe as a giant, cosmic orchestra. For decades, physicists have tried to write the ultimate sheet music for this orchestra using String Theory. The theory suggests that everything we see—stars, atoms, you, and me—is made of tiny, vibrating strings.

For a long time, the physicists thought this music only sounded beautiful and stable if the orchestra had a perfect, hidden symmetry called Supersymmetry. It's like saying the orchestra only works if every violinist has a twin cello player perfectly mirroring their moves. This symmetry guarantees that the music doesn't fall apart into chaos.

But here's the problem: We don't see these twins in our universe. Our universe is messy, asymmetrical, and clearly not supersymmetric. So, what happens if we try to play the music without the twins?

This paper, written by Giorgio Leone and Salvatore Raucci, is a guide to the "rough patches" of that music. They explore what happens when we try to build a universe without Supersymmetry, and they find two main monsters hiding in the sheet music: Tachyons and Tadpoles.

1. The Tachyons: The Ball on the Hill

Think of a Tachyon not as a fast particle, but as a ball sitting precariously on the very top of a hill.

• In a stable universe (Supersymmetry): The ball is in a valley. It's happy. It stays put.
• In a non-supersymmetric universe: The ball is on the peak. It wants to roll down.

In physics, this "rolling down" means the universe is unstable. It's trying to change into something else. If you have a Tachyon in your string theory, it's like building a house of cards on a shaking table. The theory screams, "Wake up! You're in the wrong vacuum! Roll down to a stable spot!"

The authors explain that for a long time, we didn't know how to fix this. We didn't know what the "valley" looked like. However, they found some specific string models where the ball doesn't roll immediately. These are the "Tachyon-free" models. It's like finding a flat plateau where the ball can sit, even if it's not in a deep valley.

2. The Tadpoles: The Leaky Boat

Even if you manage to find a flat plateau (no Tachyons), there's a second problem: Tadpoles.

Imagine your universe is a boat. In a perfect, supersymmetric universe, the boat is perfectly balanced. But in a non-supersymmetric one, the boat has a hole in the bottom. Water (energy) is leaking in.

• The Leak: This "leak" is a mathematical divergence. It means the vacuum energy is trying to push the universe apart or collapse it.
• The Consequence: If you don't plug the hole, the boat sinks. In physics terms, the "vacuum" (the empty space) becomes unstable and starts to change shape violently.

The paper discusses a clever fix called the Fischler-Susskind mechanism. Think of this as a self-repairing boat. As the water leaks in, the boat automatically reshapes its hull to plug the hole. But here's the catch: reshaping the hull changes the shape of the ocean itself.

3. The New Reality: Runaway Hills

When you plug the leak using this mechanism, something strange happens to the landscape of the universe.

• Old View: We thought the universe was a flat, calm ocean (Minkowski space).
• New View: Without Supersymmetry, the ocean isn't flat. It's a runaway slope.

The authors explain that the "vacuum" of a non-supersymmetric universe isn't a place you can just sit still. It's a slide. The universe is constantly trying to expand or contract, and the "dilaton" (a field that controls the strength of gravity and the size of the strings) is sliding down this slope.

This leads to two wild possibilities for what our universe might look like:

1. The Dudas-Mourad Vacua: Imagine the universe as a long, stretched-out tube. At one end, the universe is huge and calm. At the other end, it crashes into a singularity (a point of infinite curvature). It's like a cosmic canyon where the laws of physics get weird at the edges.
2. The Climbing Scalar: Imagine a ball trying to roll up a steep hill. The energy of the universe forces the "dilaton" to climb this hill, creating a dynamic, changing universe rather than a static one.

The Big Picture: Why Should We Care?

The authors are essentially saying: "We can't just ignore the fact that our universe isn't supersymmetric."

If we try to force the universe to be supersymmetric when it clearly isn't, we miss the real, messy, beautiful physics of how gravity actually works.

• Supersymmetry is like a safety net. It catches us when we fall.
• Non-Supersymmetry is like walking a tightrope without a net. It's dangerous (unstable), but it's also where the real action is.

The paper concludes that while we have found some "safe spots" (tachyon-free models), the universe is still a dangerous place to live without that safety net. The "leaky boat" (tadpoles) forces the universe to constantly reshape itself.

In simple terms:
This paper is a map of the "danger zones" in string theory. It tells us that if we want to understand our actual, non-supersymmetric universe, we have to stop looking for a perfect, stable paradise and start learning how to navigate a universe that is constantly shifting, sliding, and trying to fix itself. It's a messy, unstable, but fascinating place where the laws of gravity are written in real-time.

Technical Summary

1. Problem Statement

String theory has historically relied on spacetime supersymmetry (SUSY) to guarantee the existence of stable vacua and the consistency of the theory (e.g., cancellation of anomalies and divergences). However, since supersymmetry is not observed in nature, understanding non-supersymmetric string theories is crucial for phenomenology and cosmology.

The absence of SUSY introduces two primary sources of instability:

1. Tachyons: States with $m^2 < 0$ appearing in the string spectrum, indicating that the background is a local maximum of a potential rather than a minimum.
2. Tadpole Divergences: Even in tachyon-free models, the absence of SUSY leads to non-vanishing vacuum amplitudes (tadpoles) for massless scalars (specifically the dilaton). These generate a runaway scalar potential, threatening the stability of the vacuum and the consistency of the perturbative expansion.

The paper addresses the challenge of characterizing these instabilities, identifying tachyon-free non-supersymmetric models, and analyzing the spacetime consequences of the resulting tadpole potentials.

2. Methodology

The authors employ a combination of worldsheet conformal field theory (CFT) techniques and spacetime effective field theory analysis.

• Worldsheet Analysis:

  • Modular Invariance & Partition Functions: The authors analyze one-loop vacuum amplitudes (Torus, Klein bottle, Annulus, Möbius strip) on Riemann surfaces. They utilize the GSO projection to construct modular-invariant partition functions.
  • Large-Mass Asymptotics: A key methodological tool is the sector-averaged sum approach. By analyzing the large-mass ($n \to \infty$) behavior of the partition function using Rademacher's exact formula and the Circle Method, they derive criteria to distinguish between physical tachyons (level-matched) and unphysical ones.
  • Off-Shell Formulations: The paper discusses sigma-model approaches and string field theory to define off-shell effective actions and potentials, acknowledging the difficulty in defining a consistent off-shell formulation for closed strings.

• Spacetime Analysis:

  • Fischler-Susskind Mechanism: The authors utilize this mechanism to interpret tadpole divergences. Instead of viewing them as inconsistencies, they are treated as signals requiring a shift in the background fields (renormalization of the dilaton) to cancel the divergences.
  • Effective Potentials: They derive the resulting scalar potentials in the Einstein and string frames, which typically take the form of exponential runaway potentials ($V \sim e^{\gamma \phi}$).
  • Vacuum Solutions: The paper solves the coupled Einstein-dilaton equations of motion with these potentials to find vacuum solutions, focusing on codimension-one geometries and flux compactifications.

3. Key Contributions

A. Characterization of Tachyons

The paper provides a rigorous characterization of tachyons in non-supersymmetric strings using the sector-averaged sum $\langle d(n) \rangle$.

• Result: For a theory to be tachyon-free, the sector-averaged sum must vanish in the large-mass limit.
• Misaligned Supersymmetry: In tachyon-free non-supersymmetric theories (like the $Spin(16) \times Spin(16)$ heterotic string), bosonic and fermionic degrees of freedom do not cancel at each mass level (as in SUSY) but cancel in an averaged sense across different mass levels. This "misaligned supersymmetry" ensures the absence of physical tachyons despite the lack of spacetime SUSY.
• Orientifolds: The authors extend this analysis to orientifold projections (Klein bottle, Annulus, Möbius strip), showing that vanishing growth in the transverse channel is a necessary but not sufficient condition for the absence of closed-string tachyons.

B. Classification of 10D Non-Supersymmetric Models

The authors review and classify the known ten-dimensional non-supersymmetric string theories:

1. Heterotic $Spin(16) \times Spin(16) \rtimes \mathbb{Z}_2$: The only known tachyon-free non-supersymmetric heterotic string. It possesses a non-vanishing one-loop vacuum amplitude but no tachyons.
2. Sugimoto Model (USp(32)): An orientifold of Type IIB with projection $\Omega(-1)^{f_L}$. It breaks SUSY in the open-string sector (brane supersymmetry breaking) and contains a tachyon-free spectrum but has uncancelled NS-NS tadpoles.
3. Type 0'B String: An orientifold of Type 0B. It is tachyon-free but exhibits non-trivial growth in the sector-averaged sum of the transverse Klein bottle amplitude, suggesting that the absence of tachyons in orientifolds is subtle.

C. Tadpole Divergences and the Fischler-Susskind Mechanism

The paper details how tadpole divergences arise in tachyon-free models due to the lack of SUSY protection.

• Mechanism: The divergence in the vacuum amplitude (interpreted as an IR divergence from massless propagators) is cancelled by renormalizing the background fields (specifically the dilaton).
• Consequence: This generates a scalar tadpole potential $V(\phi)$. In the Einstein frame, this potential is a runaway exponential (e.g., $V \sim e^{5/2 \phi}$ for heterotic, $V \sim e^{3/2 \phi}$ for orientifolds).
• Significance: This replaces the cosmological constant problem of standard field theory with a dilaton runaway problem, fundamentally altering the vacuum structure.

D. Vacuum Solutions and Stability

The authors analyze the spacetime consequences of these potentials:

• Codimension-One Vacua (Dudas-Mourad): Since maximally symmetric spaces (Minkowski/AdS) are ruled out by the runaway potential, the only viable vacuum solutions are codimension-one geometries (e.g., $ds^2 = e^{2A(y)}dx_9^2 + dy^2$). These solutions feature curvature singularities at the boundaries, interpreted as spontaneous compactification driven by SUSY breaking.
• Compactifications: The paper discusses flux compactifications (Freund-Rubin type) and warped geometries. A no-go theorem is presented, ruling out Minkowski and de Sitter vacua in the presence of leading-order tadpole potentials for these specific string theories.
• Stability: While some codimension-one solutions are perturbatively stable, the paper highlights the threat of non-perturbative instabilities (e.g., "bubbles of nothing") which are generic in the absence of SUSY.

4. Results

• Tachyon-Free Existence: Confirmed that tachyon-free non-supersymmetric strings exist in 10D, characterized by "misaligned supersymmetry" where the partition function grows slower than the exponential of the tachyonic mass.
• Inevitable Instability: Even in tachyon-free models, the absence of SUSY leads to non-vanishing tadpoles. These cannot be ignored; they force a deformation of the background.
• Runaway Potentials: The resulting spacetime potentials are runaway, preventing static Minkowski vacua. The only consistent vacua are time-dependent or involve singular spatial compactifications (Dudas-Mourad vacua).
• No-Go Theorems: Standard warped compactifications to Minkowski or de Sitter space are forbidden for these specific models when tadpole potentials are included.

5. Significance

This review is significant for several reasons:

1. Phenomenological Relevance: It provides the theoretical framework for constructing string models that do not rely on unobserved supersymmetry, offering potential pathways to realistic particle physics and cosmology.
2. Conceptual Clarity: It clarifies the distinction between tachyonic instabilities (spectrum issues) and tadpole instabilities (vacuum energy issues), showing that removing tachyons is not sufficient for stability.
3. New Vacuum Classes: It highlights a new class of string vacua (codimension-one, singular boundaries) that arise naturally from the interplay of gravity and SUSY breaking, challenging the traditional view that string vacua must be smooth, supersymmetric manifolds.
4. Open Problems: The paper underscores the lack of a complete, systematic off-shell formulation for non-supersymmetric strings and the difficulty in determining the true endpoint of the Fischler-Susskind flow (i.e., whether a stable, non-supersymmetric vacuum truly exists or if the theory decays).

In summary, the paper argues that while non-supersymmetric strings are mathematically consistent at the perturbative level (if tachyons are removed), they face profound challenges regarding vacuum stability and the existence of static, flat spacetime solutions, pointing toward a landscape dominated by dynamical, time-dependent, or singular geometries.