New Crosscap States

This paper argues for the existence of new crosscap states in two-dimensional rational conformal field theories labeled by Verlinde lines, supporting this claim with a generalized Cardy condition and exploring their transformation properties and connections to mixed anomalies.

The Gist

The Big Picture: Unfolding the Fabric of Reality

Imagine the universe as a giant, invisible piece of fabric. In physics, specifically in Conformal Field Theory (CFT), this fabric represents the rules of how particles and energy behave at the smallest scales.

Usually, physicists study this fabric as a flat sheet (like a table) or a tube (like a toilet paper roll). But sometimes, to understand the deepest secrets of the universe, they need to fold the fabric in weird ways. They create shapes like Klein bottles (a bottle with no inside or outside) or Möbius strips (a loop with a twist).

To make these shapes, you have to cut a hole in the fabric and glue the edges together in a tricky way. This "gluing" point is called a Crosscap. Think of a crosscap as a magical "glitch" in the fabric where the left side becomes the right side, and time or space flips upside down.

The Old Story: Simple Glitches

For a long time, physicists knew how to create these crosscap glitches, but only using "simple" tools. Imagine you have a set of Lego bricks. You could only build these special glitches using specific, standard bricks (called Simple Currents). These bricks were predictable: if you put one in, you knew exactly what would happen.

However, modern physics has discovered a new, more complex type of symmetry called Non-Invertible Symmetries.

• The Analogy: Imagine you have a magic wand.
  • Old Symmetry (Invertible): If you wave the wand, you turn a cat into a dog. If you wave it again, you turn the dog back into a cat. It's reversible.
  • New Symmetry (Non-Invertible): If you wave the wand, you turn a cat into a dog. But if you wave it again, the dog doesn't turn back into a cat; it turns into a pile of confetti. You can't undo it. It's a one-way street.

The New Discovery: Glitches with Magic Wands

This paper asks a big question: "Can we build these Crosscap glitches using the new, non-reversible magic wands?"

The authors, Wataru Harada and his team, say: Yes!

They propose that for every single type of "magic wand" (which they call a Verlinde line) in the theory, there is a corresponding new Crosscap state.

• The Metaphor: Imagine you have a library of different colored magic wands. Before, you could only make a "glitch" in reality using the Red wand. Now, they are saying you can make a unique, stable glitch using the Blue wand, the Green wand, or even a wand that turns things into confetti.

How They Proved It: The "Cardy" Test

How do you know if a new glitch is real and stable? You have to test it.

In physics, there is a famous rule called the Cardy Condition. Think of it like a "stress test" for a bridge. If you put a heavy truck (energy) on the bridge, does it hold together, or does it collapse?

• The authors derived a Generalized Cardy Condition. This is a new, more complex stress test designed specifically for these new, non-reversible glitches.
• They checked their new "Crosscap states" against this test. In several specific examples (like the Fibonacci and Ising models, which are like specific rulebooks for how particles interact), the new states passed the test perfectly. They are stable.

The "Parity" Puzzle: Mirrors and Anomalies

One of the most interesting parts of the paper involves Parity.

• The Analogy: Imagine looking in a mirror. Your left hand becomes your right hand. This is a "Parity" flip.
• Usually, if you have a symmetry (like a rule that says "all cats are cute"), it plays nicely with the mirror.
• But sometimes, the symmetry and the mirror don't get along. This is called an Anomaly. It's like trying to wear a left-handed glove on your right hand; it just doesn't fit right.

The authors show that these new Crosscap states are the perfect place to detect these "anomalies." By wrapping a non-reversible magic wand around the Crosscap, they can see if the symmetry is "broken" or "twisted" by the mirror. If the math works out, it tells us exactly how the universe handles these weird, one-way symmetries when space is flipped.

Why Does This Matter?

1. Expanding the Toolkit: It gives physicists a new way to build and study the "fabric" of the universe. We now know there are many more ways to fold reality than we thought.
2. Understanding the Unfolding: It helps us understand Non-Invertible Symmetries, which are currently a hot topic in physics because they might explain things like why the universe has more matter than antimatter, or how certain materials conduct electricity in weird ways.
3. Connecting the Dots: It connects the math of "glitches" (Crosscaps) with the math of "magic wands" (Verlinde lines), showing that the universe is more interconnected and flexible than our old textbooks suggested.

Summary in One Sentence

The authors discovered that you can create stable "reality glitches" (Crosscaps) using any type of symmetry, even the weird, one-way kind, and they proved it works by building a new mathematical stress test that these glitches pass with flying colors.

Technical Summary

1. Problem Statement

In the study of two-dimensional Rational Conformal Field Theories (RCFTs), crosscap states represent boundary conditions on non-orientable manifolds (specifically the real projective plane, $\mathbb{RP}^2$). Historically, crosscap states have been constructed primarily using simple currents (invertible symmetries).

However, the modern understanding of symmetry in quantum many-body systems has expanded to include non-invertible symmetries, represented by topological defect lines (specifically Verlinde lines in RCFTs). A significant gap in the literature exists regarding the interplay between crosscaps and these non-invertible symmetries. Specifically:

• Do crosscap states exist for arbitrary Verlinde lines, not just simple currents?
• How do these states behave under the action of non-invertible symmetries?
• What are the consistency conditions (generalized Cardy conditions) for such states?

This paper aims to fill this gap by proposing the existence of crosscap states labeled by arbitrary Verlinde lines and deriving the necessary consistency conditions.

2. Methodology

The authors employ the formalism of topological defect lines and fusion categories to generalize the construction of crosscap states. Their methodology involves:

• Topological Defect Formalism: Utilizing the algebra of topological lines, including fusion rules, $F$-symbols (associativity), and $G$-symbols (braiding/commutativity), to manipulate diagrams on non-orientable surfaces.
• Parity Action on Junctions: Analyzing the action of the parity operator $P$ (spacetime reflection) on trivalent junctions of topological lines. This introduces coefficients $P^c_{ab}$ which capture mixed anomalies between parity and internal symmetries.
• Diagrammatic Manipulation: Using "lasso" actions and $F$-moves to relate crosscap states to twisted Klein bottle partition functions.
• Generalized Cardy Conditions: Deriving consistency equations by equating the inner product of crosscap states (in the direct channel) with twisted partition functions (in the dual channel).
• Explicit Verification: Testing the derived formulas against specific examples: Fibonacci and Ising Unitary Modular Tensor Categories (UMTCs).

3. Key Contributions

A. Existence of New Crosscap States

The authors argue for the existence of a new class of crosscap states, denoted $|C_a\rangle$, labeled by any Verlinde line $a$ in the RCFT, not just simple currents.

• Construction: The state $|C_a\rangle$ is defined pictorially as a basic crosscap state $|C_1\rangle$ with the Verlinde line $a$ wrapped around the orientation-reversing cycle of the crosscap.
• Formula: The explicit form is given by:
  $$|C_a\rangle = \eta_a \sum_b \frac{P_{ab}}{\sqrt{S_{1b}}} |b\rangle\rangle_C$$
  where $|b\rangle\rangle_C$ are crosscap Ishibashi states, $S$ is the modular $S$-matrix, $P$ is the modular matrix relating twisted characters, and $\eta_a$ are sign factors determined by consistency.

B. Generalized Cardy Condition

The paper derives a generalized Cardy condition that relates the inner product of two crosscap states $|C_a\rangle$ and $|C_b\rangle$ to a sum over twisted Klein bottle partition functions:
$$\langle C_a | e^{-\frac{\pi i}{2\tau}(L_0+\bar{L}_0 - c/12)} | C_b \rangle = \sum_c \sqrt{\frac{d_c}{d_a d_b}} \sum_{\mu, \nu} (P^c_{ab})_{\bar{\mu}\bar{\nu}} \, \text{Tr}_{H_c} \left( P \hat{b} \, e^{2\pi i \tau (L_0+\bar{L}_0 - c/12)} \right) \Big|_{(b, \mu, \bar{\nu})}$$

• In the invertible case, this reduces to known results involving a single twisted Klein bottle.
• In the non-invertible case, the right-hand side becomes a sum of twisted Klein bottle partition functions, weighted by the trace of the parity action on the fusion space.

C. Action of Verlinde Lines

The authors derive the action of a Verlinde line $\hat{b}$ on a crosscap state $|C_a\rangle$. This action is non-trivial and involves the Frobenius-Schur (FS) indicators and the parity coefficients $P^c_{ab}$:
$$\hat{b} |C_a\rangle = \sum_{x,y} N^x_{ab} \sum_{\mu,\rho} \sum_{\nu,\sigma} (P^x_{ab})_{\mu\rho} (P^y_{xa})_{\nu\sigma} (F^y_{aba})_{(x\rho\sigma)(x\nu\mu)} |C_y\rangle$$
This formula demonstrates how non-invertible symmetries mix different crosscap states.

D. Connection to Parity Anomalies

The paper clarifies the relationship between these crosscap states and mixed anomalies between parity ($P$) and internal symmetries.

• The coefficients $P^c_{ab}$ (specifically the trace $\text{Tr} P^a_{cc}$) act as gauge-invariant measures of these anomalies.
• For a $\mathbb{Z}_2$ symmetry $g$, if the FS indicator $\kappa_g = -1$, the symmetry is anomalous. The action of $g$ on the basic crosscap $|C_1\rangle$ yields $g|C_1\rangle = \kappa_g |C_{g^2}\rangle = -|C_1\rangle$, indicating a projective realization of the symmetry.
• The authors show that redefining the parity operator (e.g., $P \to \hat{g}P$) can remove the anomaly, shifting the crosscap state to $|C_g\rangle$, where the symmetry is realized linearly.

4. Results and Examples

The authors validate their general formulas using two concrete examples:

1. Fibonacci UMTC:

   • Contains objects $\{1, W\}$ with fusion $W \times W = 1 + W$.
   • Has trivial FS indicators ($\kappa_W = 1$).
   • The derived crosscap states $|C_1\rangle$ and $|C_W\rangle$ satisfy the generalized Cardy condition.
   • The action of $W$ on the crosscap states depends on the parity eigenvalue $P^W_{WW} = \pm 1$. If $+1$, fusion rules are realized linearly; if $-1$, they are realized projectively.

2. Ising UMTC:

   • Contains objects $\{1, \sigma, \epsilon\}$ with non-trivial FS indicators (specifically $\kappa_\sigma = \pm 1$ depending on the central charge).
   • The authors explicitly compute the crosscap states $|C_1\rangle, |C_\epsilon\rangle, |C_\sigma\rangle$.
   • They confirm that the action of $\sigma$ on $|C_1\rangle$ yields a linear combination of $|C_1\rangle$ and $|C_\epsilon\rangle$, correcting a recent suggestion that it creates a "new" independent state.
   • The results match the twisted Klein bottle partition functions perfectly, validating the generalized Cardy condition.

5. Significance

• Extension of Symmetry Concepts: This work successfully extends the concept of crosscap states from invertible symmetries (simple currents) to the full landscape of non-invertible symmetries in RCFTs.
• Anomaly Detection: It provides a concrete, calculable framework for detecting mixed anomalies between parity and non-invertible symmetries using the action of topological lines on crosscap states.
• Consistency Conditions: The derivation of the generalized Cardy condition provides a necessary consistency check for any proposed crosscap state in a theory with non-invertible symmetries.
• Foundation for Future Work: By establishing the existence of these states and their transformation properties, the paper lays the groundwork for studying non-orientable manifolds in the context of modern topological phases of matter and non-invertible symmetry protected topological (SPT) phases.

In summary, the paper establishes that every Verlinde line in an RCFT labels a distinct crosscap state, governed by a generalized Cardy condition that encodes the interplay between non-invertible symmetries and spacetime parity.