Spin-Cobordisms, Surgeries and Fermionic Modular Bootstrap

Grigoletto, Andrea; Putrov, Pavel

doi:10.1007/s00220-023-04710-z

Cited by 6 publications

(6 citation statements)

References 59 publications

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“…Many authors have studied the anomalies of the continuum 1+1-dimensional theory of Majorana fermions. See e.g., the recent discussions in [37,38,42,43,[51][52][53][54]. And many authors have studied the lattice Majorana chain [15,51,[55][56][57][58].…”

Section: Majorana Chainmentioning

confidence: 99%

“…For a single Majorana fermion, the chiral fermion parity (−1) F L realizes the ν = 1 ∈ 8 anomaly. It was argued in [42] (see also [36,43]) that the minus sign in (39) is a consequence of the anomaly when ν ∈ 8 is odd.…”

Section: Rr Hilbert Space and The Projective Algebramentioning

confidence: 99%

“…The anomaly of 2 × f 2 labeled by ν ∈ Hom(Tors Ω Spin 3 (B 2 ), U( 1)) = 8 can be detected from the eigenvalue of the momentum operator P in the NSR Hilbert space. More specifically, the eigenvalue of P in the 2 -twisted Hilbert space (from the NSNS Hilbert space) takes values in [42,43]…”

Section: Nsr Hilbert Space and The Mod 8 Anomalymentioning

confidence: 99%

“…This matches with the bosonization relation (A.24) in the continuum. 43 Finally, we discuss the action of the translation operator T in (196), which comes from the square of the Majorana translation operators. We first work in the bigger Hilbert space H. Using…”

Section: Ising Modelmentioning

confidence: 99%

“…It is known that the description of symmetries and anomalies in fermionic systems is more subtle than in bosonic systems [33][34][35][36][37][38][39][40][41][42][43][44]. Our goal here is to present the analog of the analysis in [32] for fermionic systems.…”

Section: Introductionmentioning

confidence: 99%

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Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries

Seiberg,

Shao

2024

SciPost Phys.

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We study the symmetries of closed Majorana chains in 1+1d, including the translation, fermion parity, spatial parity, and time-reversal symmetries. The algebra of the symmetry operators is realized projectively on the Hilbert space, signaling anomalies on the lattice, and constraining the long-distance behavior. In the special case of the free Hamiltonian (and small deformations thereof), the continuum limit is the 1+1d free Majorana CFT. Its continuum chiral fermion parity (-1)^{F_L}(−1)FL emanates from the lattice translation symmetry. We find a lattice precursor of its mod 8 ’t Hooft anomaly. Using a Jordan-Wigner transformation, we sum over the spin structures of the lattice model (a procedure known as the GSO projection), while carefully tracking the global symmetries. In the resulting bosonic model of Ising spins, the Majorana translation operator leads to a non-invertible lattice translation symmetry at the critical point. The non-invertible Kramers-Wannier duality operator of the continuum Ising CFT emanates from this non-invertible lattice translation of the transverse-field Ising model.

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Section: Majorana Chainmentioning

confidence: 99%

Section: Rr Hilbert Space and The Projective Algebramentioning

confidence: 99%

Section: Nsr Hilbert Space and The Mod 8 Anomalymentioning

confidence: 99%

Section: Ising Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries

Seiberg,

Shao

2024

SciPost Phys.

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The geometry of the modular bootstrap

Chiang,

Huang,

Huang

et al. 2024

J. High Energ. Phys.

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We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment curves on R+⊗ℤ, with boundaries characterized by the total positivity of generalized Hankel matrices. We phrase the Hankel constraints as a semi-definite program, which has several advantages, such as the validity of bounds irrespective of spin truncation. We derive bounds on the gap, twist-gap, and the space of Taylor coefficients themselves. We find that if the gap is above $$ {\Delta }_{\textrm{gap}}^{\ast } $$ ∆ gap ∗ , where $$ \frac{c-1}{12}<{\Delta}_{\textrm{gap}}^{\ast }<\frac{c}{12} $$ c − 1 12 < Δ gap ∗ < c 12 , all coefficients become bounded on both sides and kinks develop in the space. In the second part, we propose an analytic method of imposing the integrality condition for the degeneracy number in the spinless bootstrap, which leads to a non-convex geometry. We find that even at very low derivative order this condition rules out regions otherwise allowed by bootstraps at high derivative order.

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Constraints on the topology of Type IIB string theory

Yonekura

2024

J. High Energ. Phys.

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We discuss some topological constraints on Type IIB string theory that cannot be described by elementary characteristic classes. Anomaly cancellation of the worldvolume theory of a D3-brane requires a shifted Dirac quantization condition of the Ramond-Ramond 5-form flux. However, the required shift is sometimes inconsistent depending on the topology of spacetime backgrounds. The obstruction to the existence of a shifted quantization is described by a degree-6 cohomology class whose definition involves spin structure of spacetime manifolds. The cohomology class is interpreted as a background D3-charge, and the Gauss law constraint requires inclusion of additional D3-branes to cancel it.

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Spin-Cobordisms, Surgeries and Fermionic Modular Bootstrap

Cited by 6 publications

References 59 publications

Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries

Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries

The geometry of the modular bootstrap

Constraints on the topology of Type IIB string theory

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