Sergei Gukov, Po-Shen Hsin, Hiraku Nakajima, Sunghyuk Park, Du Pei, Nikita Sopenko
By studying Rozansky-Witten theory with non-compact target spaces we find new connections with knot invariants whose physical interpretation was not known. This opens up several new avenues, which include a new formulation of $q$-series invariants of 3-manifolds in terms of affine Grassmannians and a generalization of Akutsu-Deguchi-Ohtsuki knot invariants.
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We first construct the Rozansky-Witten model coupled to BF theory and Chern-Simons theory using the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) method. Then we apply the machinery developed in some earlier papers about AKSZ theories and characteristic classes to these concrete models: the BF-Rozansky-Witten model and the Chern-Simons-Rozansky-Witten model. In the former case, we obtain characteristic classes on the target hyperKahler manifold equipped with a group action as a generalization of the original Rozansky-Witten classes. We also give the prescription for similar classes associate…
We investigate the perturbative aspects of Rozansky-Witten's 3d $σ$-model using Costello's approach to the Batalin-Vilkovisky (BV) formalism. We show that the BV quantization (in Costello's sense) of the model, which produces a perturbative quantum field theory, can be obtained via the configuration space method of regularization due to Kontsevich and Axelrod-Singer. We also study the factorization algebra structure for quantum observables following Costello-Gwilliam. In particular, we show that the cohomology of local quantum observables on a genus $g$ handle body is given by $H^*(X,(\wedge^*…
We introduce symplectic structures on "Lie pairs" of (real or complex) algebroids as studied by Chen, Stienon and the second author (From Atiyah classes to homotopy Leibniz algebras, arXiv:1204.1075), encompassing homogeneous symplectic spaces, symplectic manifolds with a $\mathfrak g$-action and holomorphic symplectic manifolds. We show that to each such symplectic Lie pair are associated Rozansky-Witten-type invariants of three-manifolds and knots, given respectively by weight systems on trivalent and chord diagrams.
We show that recently constructed invariants of 3-dimensional manifolds and of hyperkaehler manifolds (L.Rozansky and E.Witten, hep-th/9612216) come from characteristic classes of foliations and from Gelfand-Fuks cohomology. In particular, any symplectic foliation gives invariants of 3-manifolds. Our preprint has many intersections with the preprint alg-geom/9704009 by M.Kapranov.
Using the AKSZ formalism, we construct the Batalin-Vilkovisky master action for the Rozansky-Witten model, which can be defined for any complex manifold with a closed (2,0)-form. We also construct the holomorphic version of Rozansky-Witten theory defined over Calabi-Yau 3-fold.
We construct and study a new topological field theory in three dimensions. It is a hybrid between Chern-Simons and Rozansky-Witten theory and can be regarded as a topologically-twisted version of the N=4 d=3 supersymmetric gauge theory recently discovered by Gaiotto and Witten. The model depends on a gauge group G and a hyper-Kahler manifold X with a tri-holomorphic action of G. In the case when X is an affine space, we show that the model is equivalent to Chern-Simons theory whose gauge group is a supergroup. This explains the role of Lie superalgebras in the construction of Gaiotto and Witte…
Let Z^{LMO} be the 3-manifold invariant of [LMO]. It is shown that Z^{LMO}(M)=1, if the first Betti number of M, b_{1}(M), is greater than 3. If b_{1}(M)=3, then Z^{LMO}(M) is completely determined by the cohomology ring of M. A relation of Z^{LMO} with the Rozansky-Witten invariants Z_{X}^{RW}[M] is established at a physical level of rigour. We show that Z_{X}^{RW}[M] satisfies appropriate connected sum properties suggesting that the generalized Casson invariant ought to be computable from the LMO invariant.
Link invariants, for 3-manifolds, are defined in the context of the Rozansky-Witten theory. To each knot in the link one associates a holomorphic bundle over a holomorphic symplectic manifold X. The invariants are evaluated for b_{1}(M) \geq 1 and X Hyper-Kaehler. To obtain invariants of Hyper-Kaehler X one finds that the holomorphic vector bundles must be hyper-holomorphic. This condition is derived and explained. Some results for X not Hyper-Kaehler are presented.
Recently, L.Rozansky and E.Witten (hep-th/9612216) associated to any hyperKaehler manifold X a system of "weights" (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a very simple cohomological definition of these weights in terms of the Atiyah class of X (the obstruction to existence of a holomorphic connection). We show that the analogy between the tensor of curvature of a hyperKaehler metric and the tensor of structure constants of a Lie algebra observed by Rozansky and Witten, holds in fact for any complex manifold, if we work …
Truncated affine Rozansky--Witten models as extended TQFTs
本文从以T*C^n为目标的Rozansky–Witten模型出发，通过截断3-范畴得到体理论、面缺陷与线缺陷同构类的2-范畴，并应用配边假设构造唯一扩展定向2维TQFT，进而计算闭曲面态空间及圆周限制对应的交换Frobenius代数。
Truncated affine Rozansky--Witten models as extended defect TQFTs
将带奇点的配边假设应用于仿射Rozansky--Witten模型，通过证明同伦2-范畴的枢轴性及运用3维图形演算，构建含线和面缺陷的扩展拓扑量子场论，可计算缺陷网络曲面态空间并给出对称缺陷与边界条件实例。
The Seiberg-Witten analysis of the low-energy effective action of d=4 N=2 SYM theories reveals the relation between the Donaldson and Seiberg-Witten (SW) monopole invariants. Here we apply analogous reasoning to d=3 N=4 theories and propose a general relationship between Rozansky-Witten (RW) and 3-dimensional Abelian monopole invariants. In particular, we deduce the equality of the SU(2) Casson invariant and the 3-dimensional SW invariant (this includes a special case of the Meng-Taubes theorem relating the SW invariant to Milnor torsion). Since there are only a finite number of basic RW invar…
We describe a globalization construction for the Rozansky-Witten model in the BV-BFV formalism for a source manifold with and without boundary in the classical and quantum case. After having introduced the necessary background, we define an AKSZ sigma model, which, upon globalization through notions of formal geometry extended appropriately to our case, is shown to reduce to the Rozansky-Witten model. The relations with other relevant constructions in the literature are discussed. Moreover, we split the model as a $BF$-like theory and we construct a perturbative quantization of the model in th…
Foams and KZ-equations in Rozansky-Witten theories
本文提出泡沫的几何描述并从代数视角探索Rozansky-Witten理论几何，通过几何框架构建KZ方程推导辫子与结合子态射，并讨论目标空间作为6d SCFT或Little String Theory的Coulomb分支的应用。
The paper has two parts, in the first part, we apply the localisation technique to the Rozansky-Witten theory on compact HyperKähler targets. We do so via first reformulating the theory as some supersymmetric sigma-model. We obtain the exact formula for the partition function with Wilson loops on $S^1\timesΣ_g$ and the lens spaces, the results match with earlier computations using Feynman diagrams on K3. The second part is motivated by a very curious preprint \cite{Gukov:2020lqm}, where the equivariant index formula for the dimension of the Hilbert space the theory is interpreted as a kind of …
We formulate a deformation of Rozansky-Witten theory analogous to the $Ω$-deformation. It is applicable when the target space $X$ is hyperkähler and the spacetime is of the form $\mathbb{R} \times Σ$, with $Σ$ being a Riemann surface. In the case that $Σ$ is a disk, the $Ω$-deformed Rozansky-Witten theory quantizes a symplectic submanifold of $X$, thereby providing a new perspective on quantization. As applications, we elucidate two phenomena in four-dimensional gauge theory from this point of view. One is a correspondence between the $Ω$-deformation and quantization of integrable systems. The…
B-twisted Gaiotto-Witten theory and topological quantum field theory
本研究发展表示论技术，基于李超代数展开量化的权模范畴相对模结构，构造三维非半单拓扑量子场论，建模含线性物质的阿贝尔Gaiotto-Witten理论的同调截断拓扑B扭曲，纳入非真实线算子，涵盖多类Chern-Simons相关理论特例。
We calculate the homological blocks for Seifert manifolds from the exact expression for the $G=SU(N)$ Witten-Reshetikhin-Turaev invariants of Seifert manifolds obtained by Lawrence, Rozansky, and Mariño. For the $G=SU(2)$ case, it is possible to express them in terms of the false theta functions and their derivatives. For $G=SU(N)$, we calculate them as a series expansion and also discuss some properties of the contributions from the abelian flat connections to the Witten-Reshetikhin-Turaev invariants for general $N$. We also provide an expected form of the $S$-matrix for general cases and the…
We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition function at or around roots of unity $q=e^{2πi \frac{1}{K}}$ with rational level $K=\frac{r}{s}$ where $r$ and $s$ are coprime integers. From the exact expression for the $G=SU(2)$ Witten-Reshetikhin-Turaev invariants of Seifert manifolds at other roots of unity obtained by Lawrence and Rozansky, we provide an expected form of the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks at other roots of unity. Also, we discuss the asymptotic expansion of knot invariants around roots …
The path integral generalization of the Casson invariant as developed by Rozansky and Witten is investigated. The path integral for various three manifolds is explicitly evaluated. A new class of topological observables is introduced that may allow for more effective invariants. Finally it is shown how the dimensional reduction of these theories corresponds to a generalization of the topological B sigma model.
In this paper, we apply recent methods of localized GLSMs to make predictions for Gromov-Witten invariants of noncommutative resolutions, as defined by e.g. Kontsevich, and use those predictions to examine the connectivity of the SCFT moduli space. Noncommutative spaces, in the present sense, are defined by their sheaves, their B-branes. Examples of abstract CFT's whose B-branes correspond with those defining noncommutative spaces arose in examples of abelian GLSMs describing branched double covers, in which the double cover structure arises nonperturbatively. This note will examine the GLSM f…
A 3-dimensional model dual to the Rozansky-Witten topological sigma-model with a hyper-Kaehler target space is considered. It is demonstrated that a Feynman diagram calculation of the classical part of its partition function yields the Milnor linking number.
We study the Coulomb branches of the rank-one 4d $\mathcal{N} = 2$ quantum field theories, including the KK theories obtained from the circle compactification of the 5d $\mathcal{N}= 1$ $E_n$ Seiberg theories. The focus is set on the relation between their Seiberg-Witten geometries and rational elliptic surfaces, with more attention being given to the modular surfaces, which we completely classify using the classification of subgroups of the modular group ${\rm PSL}(2,\mathbb{Z})$. We derive closed-form expressions for the modular functions for all congruence and some of the non-congruence sub…
The colored Jones function of a knot is a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. It has been understood in terms of representations of quantum groups and Witten gave an intrinsic quantum field theory interpretation of the colored Jones function as the expectation value of Wilson loops of a 3-dimensional gauge theory, the Chern-Simons theory. We present the colored Jones function as an evaluation of the inverse of a non-commutative fermionic partition function. This result is in the form familiar in quantum field theory, namely the inverse…
The periods of arbitrary abelian forms on hyperelliptic Riemann surfaces, in particular the periods of the meromorphic Seiberg-Witten differential, are shown to be in one-to-one correspondence with the conformal blocks of correlation functions of the rational logarithmic conformal field theory with central charge c=c(2,1)=-2. The fields of this theory precisely simulate the branched double covering picture of a hyperelliptic curve, such that generic periods can be expressed in terms of certain generalised hypergeometric functions, namely the Lauricella functions of type F_D.
It has recently been suggested that, in a large N limit, a particular four dimensional gauge theory is indistinguishable from the six dimensional CFT with (0,2) supersymmetry compactified on a torus. We give further evidence for this correspondence by studying the Seiberg-Witten curve for the "deconstructed" theory and demonstrating that along the reduced Coulomb branch of moduli space (on the intersection of the Higgs and Coulomb branches) it describes the low energy physics on a stack of M5-branes on a torus, which is the (0,2) theory on a torus as claimed. The M-theory construction helps to…
We comment on various aspects of the the dynamics of 3d N=2 Chern-Simons gauge theories and their possible phases. Depending on the matter content, real masses and FI parameters, there can be non-compact Higgs or Coulomb branches, compact Higgs or Coulomb branches, and isolated vacua. We compute the Witten index of the theories, and show that it does not change when the system undergoes a phase transition. We study aspects of monopole operators and solitons in these theories, and clarify subtleties in the soliton collective coordinate quantization. We show that solitons are compatible with a m…
We develop a systematic method to describe the moduli space of vacua of four dimensional $\mathcal{N}=2$ class ${\cal S}$ theories including Coulomb branch, Higgs branch and mixed branches. In particular, we determine the Higgs and mixed branch roots, and the dimensions of the Coulomb and Higgs components of mixed branches. They are derived by using generalized Hitchin's equations obtained from twisted compactification of 5d maximal Super-Yang-Mills, with local degrees of freedom at punctures given by (nilpotent) orbits. The crucial thing is the holomorphic factorization of the Seiberg-Witten …
Gauge Theory and Integrability: An Overview
本文综述了通过四维Chern-Simons理论的微扰分析解决可积模型在量子场论中解释的问题，为统一可积模型提供了新框架。
Even More On Twisted $A_{2n}$ Class-S Theories
本文延续扭曲$A_{2n}$类S理论库仑分支研究，通过精确定义4阶自同构、找到局部约束、利用sp(n)逆序映射及构造Seiberg-Witten曲线例子，完善了对扭曲punctures的直接分析。
$q$-Series Invariants of Three-Manifolds and Knots-Quivers Correspondence
本文验证了不同群下三维流形$\u2c6bZ$不变量的相等关系，并通过逆向工程MMR形式体系得到双扭结的箭图表示，推进了GPPV猜想与纽结-箭图对应猜想的研究。
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