On supersymmetric interface defects, brane parallel transport, order-disorder transition and homological mirror symmetry

Galakhov, Dmitry

doi:10.1007/jhep10(2022)076

Cited by 3 publications

(5 citation statements)

References 187 publications

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“…where ξ : Ω U * −→ Λ is a homomorphism of an algebraic cobordism ring to a Lazard ring. The major question we would like to pose in this note is if based on hints of (1.1) the QFT provides a basis for generalized cohomology as it does in the case of the Morse theory [12,13] or its categorical lifts [14][15][16][17][18][19][20], at least for the mentioned family of quiver models. For a recent discussion on similar questions for elliptic cohomology and BPS states in 3d N = 4 theories see [15,16,21,22].…”

Section: Summary and Discussionmentioning

confidence: 99%

“…Also we would like to translate into the current framework an idea of [14,[29][30][31]: interface defects providing Berry connections on Hilbert spaces induce morphisms on geometric structures emerging in the physical systems. For instance, soliton amplitudes in linear gauged sigma-models support Fourier-Mukai transforms on the derived coherent sheave category of D-branes [17]. In particular, we construct matrix coefficients for the resulting BPS algebra as elements of instanton amplitudes, also in the universal language of e ξ (z).…”

Section: Summary and Discussionmentioning

confidence: 99%

“…A nice appropriate tool to include such a type of moduli "dynamics" into the theory in a natural way is to consider an interface defect. Interfaces [14][15][16][17]29] introduce a variation of the moduli along some spacial directions and might preserve some supersymmetry subgroup. After applying the Wick rotation and sending the time direction along the interface one would naturally arrive to a time-dependent (moduli-dependent) Hamiltonian H(t) and a supercharge Q(t).…”

Section: Berry Connection From An Interfacementioning

confidence: 99%

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BPS states meet generalized cohomology

Galakhov

2023

J. High Energ. Phys.

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In this note we review a construction of a BPS Hilbert space in an effective supersymmetric quiver theory with 4 supercharges. We argue abstractly that this space contains elements of an equivariant generalized cohomology theory $$ {E}_G^{\ast}\left(-\right) $$ E G ∗ − of the quiver representation moduli space giving concretely Dolbeault cohomology, K-theory or elliptic cohomology depending on the spacial slice is compactified to a point, a circle or a torus respectively, and something more amorphous in other cases. Furthermore BPS instantons — basic contributors to interface defects or a Berry connection — induce a BPS algebra on the BPS Hilbert spaces representing Fourier-Mukai transforms on the quiver representation moduli spaces descending to an algebra over $$ {E}_G^{\ast}\left(-\right) $$ E G ∗ − as its representation. In the cases when the quiver describes a toric Calabi-Yau three-fold (CY3) the algebra is a respective generalization of the quiver BPS Yangian algebra discussed in the literature, in more general cases it is given by an abstract generalized cohomological Hall algebra.

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Section: Summary and Discussionmentioning

confidence: 99%

Section: Summary and Discussionmentioning

confidence: 99%

Section: Berry Connection From An Interfacementioning

confidence: 99%

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BPS states meet generalized cohomology

Galakhov

2023

J. High Energ. Phys.

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“…With these data, it makes sense to talk about elliptic cohomology classes, that is holomorphic sections of L. Precisely such a setting plays central role in the constructions of elliptic stable envelopes in mathematics [66] and physics [28,29]. See also a somewhat different in approach but similar in goals series of papers [67][68][69][70][71].…”

Section: Relation To Elliptic Cohomologymentioning

confidence: 99%

Remarks on Berry connection in QFT, anomalies, and applications

Dedushenko

2023

SciPost Phys.

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Berry connection has been recently generalized to higher-dimensional QFT, where it can be thought of as a topological term in the effective action for background couplings. Via the inflow, this term corresponds to the boundary anomaly in the space of couplings, another notion recently introduced in the literature. In this note we address the question of whether the old-fashioned Berry connection (for time-dependent couplings) still makes sense in a QFT on \Sigma^{(d)}× \mathbb{R}Σ(d)×ℝ, where \Sigma^{(d)}Σ(d) is a dd-dimensional compact space and \mathbb{R}ℝ is time. Compactness of \Sigma^{(d)}Σ(d) relieves us of the IR divergences, so we only have to address the UV issues. We describe a number of cases when the Berry connection is well defined (which includes the tt^*tt* equations), and when it is not. We also mention a relation to the boundary anomalies and boundary states on the Euclidean \Sigma^{(d)} × \mathbb{R}_{≥ 0}Σ(d)×ℝ≥0. We then work out the examples of a free 3D Dirac fermion and a 3D \mathcal{N}=2𝒩=2 chiral multiplet. Finally, we consider 3D theories on \mathbb{T}^2× \mathbb{R}𝕋2×ℝ, where the space \mathbb{T}^2𝕋2 is a two-torus, and apply our machinery to clarify some aspects of the relation between 3D SUSY vacua and elliptic cohomology. We also comment on the generalization to higher genus.

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“…The BPS soliton equations correspond to the stationary field configurations that annihilate the Q-transformations of the fermionic fields. In the GLSM model in question, the soliton equations have the form of a flow equation [58,66]:…”

Section: Jhep11(2022)119mentioning

confidence: 99%

Gauge/Bethe correspondence from quiver BPS algebras

Galakhov

Yamazaki

2022

J. High Energ. Phys.

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We study the Gauge/Bethe correspondence for two-dimensional $$ \mathcal{N} $$ N = (2, 2) supersymmetric quiver gauge theories associated with toric Calabi-Yau three-folds, whose BPS algebras have recently been identified as the quiver Yangians. We start with the crystal representations of the quiver Yangian, which are placed at each site of the spin chain. We then construct integrable models by combining the single-site crystals into crystal chains by a coproduct of the algebra, which we determine by a combination of representation-theoretical and gauge-theoretical arguments. For non-chiral quivers, we find that the Bethe ansatz equations for the crystal chain coincide with the vacuum equation of the quiver gauge theory, thus confirming the corresponding Gauge/Bethe correspondence. For more general chiral quivers, however, we find obstructions to the R-matrices satisfying the Yang-Baxter equations and the unitarity conditions, and hence to their corresponding Gauge/Bethe correspondence. We also discuss trigonometric (quantum toroidal) versions of the quiver BPS algebras, which correspond to three-dimensional $$ \mathcal{N} $$ N = 2 gauge theories and arrive at similar conclusions. Our findings demonstrate that there are important subtleties in the Gauge/Bethe correspondence, often overlooked in the literature.

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On supersymmetric interface defects, brane parallel transport, order-disorder transition and homological mirror symmetry

Cited by 3 publications

References 187 publications

BPS states meet generalized cohomology

BPS states meet generalized cohomology

Remarks on Berry connection in QFT, anomalies, and applications

Gauge/Bethe correspondence from quiver BPS algebras

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