Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants

Gukov, Sergei; Hsin, Po-Shen; Nakajima, Hiraku; Park, Sunghyuk; Pei, Du; Sopenko, Nikita

doi:10.1016/j.geomphys.2021.104311

“…For a 3D N = 4 SCFT T of non-zero rank, the topological twisted theories RW±[T ] are not genuine TQFTs since they do not obey the standard axioms of TQFT. Nevertheless, interesting 3-manifold/knot invariants can be studied using the RW±[T ][68].…”

mentioning

confidence: 99%

Non-unitary TQFTs from 3D $$ \mathcal{N} $$ = 4 rank 0 SCFTs

Gang

¹

,

Kim

²

,

Lee

³

et al. 2021

J. High Energ. Phys.

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We propose a novel procedure of assigning a pair of non-unitary topological quantum field theories (TQFTs), TFT±[$$ \mathcal{T} $$ T rank 0], to a (2+1)D interacting $$ \mathcal{N} $$ N = 4 superconformal field theory (SCFT) $$ \mathcal{T} $$ T rank 0 of rank 0, i.e. having no Coulomb and Higgs branches. The topological theories arise from particular degenerate limits of the SCFT. Modular data of the non-unitary TQFTs are extracted from the supersymmetric partition functions in the degenerate limits. As a non-trivial dictionary, we propose that F = maxα (− log|$$ {S}_{0\alpha}^{\left(+\right)} $$ S 0 α + |) = maxα (− log|$$ {S}_{0\alpha}^{\left(-\right)} $$ S 0 α − |), where F is the round three-sphere free energy of $$ \mathcal{T} $$ T rank 0 and $$ {S}_{0\alpha}^{\left(\pm \right)} $$ S 0 α ± is the first column in the modular S-matrix of TFT±. From the dictionary, we derive the lower bound on F, F ≥ − log $$ \left(\sqrt{\frac{5-\sqrt{5}}{10}}\right) $$ 5 − 5 10 ≃ 0.642965, which holds for any rank 0 SCFT. The bound is saturated by the minimal $$ \mathcal{N} $$ N = 4 SCFT proposed by Gang-Yamazaki, whose associated topological theories are both the Lee-Yang TQFT. We explicitly work out the (rank 0 SCFT)/(non-unitary TQFTs) correspondence for infinitely many examples.

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“…[48]. We expect that the theories T A G,k studied in the present paper are 3d mirrors of the Rozansky-Witten-twisted (or "B-twisted") sigma-models with targets X k described in [43], or some enhancements thereof; we expand further on the relation to [43] from a 6d perspective in Section 1.6.2.…”

mentioning

confidence: 80%

“…Such flat connections will play an important role in the VOA and QFT perspectives as well. Finally, we describe multiple physical constructions of the QFT T A G,k , and comment on their relations to analytically continued Chern-Simons theory, the 3d-3d correspondence, and the setup of [43]. In Section 1.7, we give a more precise formulation of our new results.…”

Section: Organizationmentioning

confidence: 99%

“…The search for a physical QFT that computes invariants based on the full, non-semisimple category U q (g)-mod was already instigated last year [43], motivated in part by recent developments in the 3d-3d correspondence, and in particular the discovery of logarithmic VOA characters [44] in the "homological blocks" of [45,46]. This line of reasoning was developed in [47], which in particular clarified the role of spin (c) structures in physical QFT's underlying CGP/ADO invariants.…”

mentioning

confidence: 99%

“…The latter often evaluate to zero or infinity without careful regularization. Some of these issues were addressed in [43], as well as [13] in the related context of supergroup Chern-Simons; and they were one of the principal difficulties to overcome in mathematical work on non-semisimple invariants, which we come to next. We hope that a full, cohomological TQFT can be (re)constructed directly from the physical QFT T A G,k in the future.…”

mentioning

confidence: 99%

See 2 more Smart Citations

A QFT for non-semisimple TQFT

Creutzig¹,

Dimofte²,

Garner³

et al. 2021

Preprint

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We construct a family of 3d quantum field theories T A n,k that conjecturally provide a physical realization -and derived generalization -of non-semisimple mathematical TQFT's based on the modules for the quantum group U q (sl n ) at an even root of unity q = exp(iπ/k). The theories T A n,k are defined as topological twists of certain 3d N = 4 Chern-Simons-matter theories, which also admit string/M-theory realizations. They may be thought of as SU (n) k−n Chern-Simons theories, coupled to a twisted N = 4 matter sector (the source of non-semisimplicity). We show that T A n,k admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an sl n -type Feigin-Tipunin algebra; and we conjecture that these two vertex operator algebras are related by a novel logarithmic level-rank duality. (We perform detailed computations to support the conjecture.) We thus relate the category of line operators in T A n,k to the derived category of modules for a boundary Feigin-Tipunin algebra, and -using a logarithmic Kazhdan-Lusztiglike correspondence that has been established for n = 2 and expected for general n -to the derived category of U q (sl n ) modules. We analyze many other key features of T A n,k and match them from quantum-group and VOA perspectives, including deformations by flat P GL(n, C) connections, one-form symmetries, and indices of (derived) genus-g state spaces.

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BPS indices, modularity and perturbations in quantum K-theory

Jockers

¹

,

Mayr²,

Ninad³

et al. 2022

J. High Energ. Phys.

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We study a perturbation family of $$ \mathcal{N} $$ N = 2 3d gauge theories and its relation to quantum K-theory. A 3d version of the Intriligator-Vafa formula is given for the quantum K-theory ring of Grassmannians. The 3d BPS half-index of the gauge theory is connected to the theory of bilateral hypergeometric q-series, and to modular q-characters of a class of conformal field theories in a certain massless limit. Turning on 3d Wilson lines at torsion points leads to mock modular behavior. Perturbed correlators in the IR regime are computed by determining the UV-IR map in the presence of deformations.

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Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants

Cited by 38 publications

References 79 publications

Non-unitary TQFTs from 3D $$ \mathcal{N} $$ = 4 rank 0 SCFTs

Non-unitary TQFTs from 3D $$ \mathcal{N} $$ = 4 rank 0 SCFTs

A QFT for non-semisimple TQFT

BPS indices, modularity and perturbations in quantum K-theory

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