Fortuity beyond counting: an explicit construction

Introduction to the show: ident: High Energy Physics: Theory. The day in high-energy theory: gravity, strings, fields and the questions left open.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Fortuity beyond counting: an explicit construction".

Mira: Here's the gist of the paper. The paper investigates how "fortuity" mechanisms modify BPS cohomologies in the D1D5 CFT when subjected to twist-two deformations, revealing non-trivial constraints on identifying monotone and fortuitous states Investigating Fortuitous States The study goes beyond counting BPS states to examine how the explicit form of BPS cohomologies is altered by twist-two deformations The authors calculate the action of supercharges in the sector (h, j) = (1, 0) for different central charges and derive explicit expressions for primary states Equipped with this information, they compare protected three-point couplings in the free and gravity regimes They argue that a map relating theories with different central charges must commute with the supercharges that define the cochain complex They find an explicit example of a non-vanishing coupling between two monotone and a fortuitous state, providing evidence that the two sectors are dynamically connected Defining Monotone vs.

Kai: Fortuitous States Monotone states form sequences whose elements are mapped one into the other by the projection operator Π and are characterized by observables, such as three-point couplings, that are smooth functions of N It is natural to assume that graviton gas states are monotone, even if this is not obvious in sectors where black holes exist The analysis involves identifying monotone and fortuitous states by looking at a cohomology problem defined by the supercharges one wishes to preserve The naive projection operator does not commute with the supercharges QA, which is a feature shared also by a two flavours SYK model Analyzing State Lifting and Multiplet Structure The paper constructs explicit sequences of states that naturally admit an interpretation as monotone, supported by the regular pattern of three-point couplings The form of such monotone states coincides with their naive gravitongas representatives only at leading order in the large-N expansion The analysis shows that for N > 2, a candidate fortuitous state becomes part of a long multiplet, consistent with its interpretation as a fortuitous state The deformation of the action through superdescendants of the twist-2 operator leads to lifted states that combine to form long supermultiplets (quartets) Resolving Ambiguities in Protected Sectors The analysis focuses on how the lifting of (1, 1, 3, 1) states changes at general N > 2 The final outcome of this procedure is that the unique protected state at this level is given by an expression involving a sum over states The results suggest that there is no superselection rule separating dynamically monotone and fortuitous states in the N = 2 case, as demonstrated by the OPE between the fortuitous state and a monotone state Connecting Free Theory to Strong Coupling The protected three-point correlators computed in the free theory are shown to agree with strong coupling results obtained from holography The comparison of the OPE limits of the strong coupling and weak coupling four-point functions suggests that there exists a protected state with (h, j) = (1, 0) in the (1, 1, 3, 1) irrep for any value of N The free theory four-point function OPE at the orbifold point is predicted to be the sum of the squares of three three-point couplings Revisiting Monotone Definitions The proposal is made that the definition of the projection map should be modified so it commutes with supercharges This modification suggests that in cases where supercharges split a multi-wound strand, the result is an arbitrary linear combination of states with the same quantum numbers The analysis shows that this modification can ensure that

Π, G˜+A−1/2: vanishes when acting on all states with (h, j) = (1, 0) The refined projection map is shown to link monotone cohomology classes in a natural way Holographic Interpretation for Black Holes The analysis suggests that the unique protected state coincides with the naive supergraviton state in the large N limit and can be interpreted holographically as a bound state of two 1/2-BPS objects The distinction between monotone and fortuitous states may be relevant beyond BPS cases, as suggested by the top elements of quartets at different values of N forming sequences related by the map Π The paper concludes that the holographic interpretation of this family of states is in terms of a single-particle stringy mode Numerical Verification for N=2 The numerical coefficients in Equation (3.11) are derived using the basis of states and bosonization relations The action of the deformed supercharge on the state 0, 1⟩ i 1 is shown to be proportional to 1/2, 1⟩ i A2 The action of δG˜−A−1/2 on this state is found to be zero Final Conclusion on State Protection The unique protected state at this level is given by an expression involving a sum of states The analysis demonstrates that the protected three-point correlators computed in the free theory are equal to the supergravity result due to the exchange of two supersymmetric states The OPE limit of the free four-point function at z → 0 reproduces exactly (3.28) The anti-symmetrized correlator is singular in the z → 0 limit, signaling a lack of flavor symmetry in the free theory The Role of Fortuity Beyond Counting The paper concludes that there is no superselection rule separating dynamically monotone and fortuitous states in the N = 2 case The mixing between monotone and fortuitous states may be exponentially suppressed in N, but it is unclear if this decouples the two classes of states in the gravity regime The distinction between primary and secondary invariants for operators constructed from hermitean matrices has many features common with the fortuity mechanism without relying on supersymmetry The paper suggests that a full matrix comprising both monotone and fortuitous states should be considered when diagnosing chaos Future Directions It is suggested that in the large N scaling, new contributions may arise that are non-linear in the coherent parameters, which could be tested at strong coupling extending the precision holography analysis The distinction between monotone and fortuitous states may be relevant beyond BPS cases where it has been developed so far It is natural to increase N by keeping fixed the curvature scale of the backreacted geometry for superstrata coherent superposition of gravitons The analysis suggests that in the large N scaling, there may be new contributions that are non-linear in the coherent parameters, something that could be tested at strong coupling extending the precision holography analysis Acknowledgements The authors thank several individuals for their discussions and collaboration on related projects The work is supported by the UK EPSRC grant “CFT and Gravity: Heavy States and Black Holes” EP/W019663/1 and by the Science and Technology Facilities Council (STFC) Consolidated Grant ST/X00063X/1 “Amplitudes, Strings & Duality” Calculating Coefficients for N=2 The numerical coefficients in Equation (3.11) are derived using the basis of states and bosonization relations The action of δG˜+A−1/2 on the state 0, 1⟩ i 1 is found to be proportional to 1/2, 1⟩ i A2 The action of δG˜−A−1/2 on this state is zero The numerical coefficients in Equation (3.

Paper summary: Kai: So to recap, this paper, "Fortuity beyond counting: an explicit construction," is taking the "fortuity" mechanism in D1D5 CFT and looking at how twist-two deformations modify the BPS cohomologies. The thesis is that going beyond just counting BPS states, they investigate perturbatively how those cohomologies change under these deformations.

Mira: They calculate the action of the supercharges in the sector (h, j) = (one zero) for different central charges to derive explicit expressions for primary states <ref:2606.20353#pg1,calculate the action of the supercharges in the sector (h, j) = (1>. They then use this to compare protected three-point couplings in the free and gravity regimes.

Lev: The key claim they make is that this agreement between those two descriptions imposes non-trivial constraints on how you identify monotone and fortuitous states, forcing them to define a map relating theories with different central charges in a very specific way.

Kai: It matters because they argue that this map must commute with the supercharges defining the cochain complex, which is a big condition for consistency. If it doesn't commute, the description breaks down under deformation.

Mira: And they find that there’s an explicit example of a non-vanishing coupling between a monotone state and a fortuitous state, suggesting these sectors are dynamically connected in some way.

Lev: That connection is significant because it means you can’t treat them as separate entities when you're trying to build something that spans both regimes. It requires modeling the interaction explicitly.

Kai: They also construct sequences of states that look monotone based on the regular pattern of three-point couplings, but these candidates become part of long supermultiplets, consistent with their interpretation as fortuitous states for N greater than two.

Mira: So they are showing that for N greater than two, a candidate fortuitous state actually becomes part of a long multiplet when you look at the deformation through superdescendants of the twist-two operator.

Lev: From an error correction perspective, seeing those lifted states form quartets tells you about the structure of the Hilbert space that you’re working with. It's not just a simple single state anymore.

Kai: They also look at how this lifting process changes for N greater than two and end up with a unique protected state defined by a sum over states.

Mira: Ultimately, they conclude that the protected three-point correlators in the free theory match the supergravity result due to the exchange of two supersymmetric states, which provides strong evidence for their construction.

Lev: So they’re not just doing an abstract calculation; they are building a structure that actually connects different theoretical descriptions.

Kai: It seems like this paper is laying down some groundwork for how we should approach the problem of distinguishing monotone and fortuitous states in these CFTs.

Conclusion: Kai: So, looking at the title "Fortuity beyond counting" and the authors Stefano Giusto a and James Inglis a and Rodolfo Russo b, it tells us that they are focusing on moving past just counting BPS states.

Mira: They are emphasizing that this is about providing an explicit construction—a concrete way to build these fortuity mechanisms, not just abstract ideas. It’s about showing the actual mechanism in action under deformation.

Lev: For the implications, it means for anyone doing quantum hardware work, you need to understand that these states aren't cleanly separated anymore. The coupling between them is real and needs accounting for when you are trying to build a system spanning different theoretical regimes.

Kai: It boils down to this: the distinction between monotone and fortuitous states might be less like a strict superselection rule in the N equals two case, with mixing that can be exponentially suppressed, but it's not entirely settled in the gravity regime.

Mira: So they’re giving us a concrete example of how these sectors are dynamically connected through explicit couplings, which is what matters for understanding the underlying physics.

Lev: And for hardware guys, it means you have to build in that interaction into your error correction models because those two states are not independent entities when you're dealing with the same underlying theory.