Sebastian Franco, Daniele Galloni, Alberto Mariotti
We review recent progress in Bipartite Field Theories. We cover topics such as their gauge dynamics, emergence of toric Calabi-Yau manifolds as master and moduli spaces, string theory embedding, relationships to on-shell diagrams, connections to cluster algebras and the Grassmannian, and applications to graph equivalence and stratification of the Grassmannian.
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Bipartite field theories (BFTs) are a new class of 4d N=1 quantum field theories defined by bipartite graphs on bordered Riemann surfaces. In this paper we derive, purely in terms of the gauge theory, the cluster transformations of face weights under square moves in the graph. In this context, we obtain them by connecting regular parametrizations of the master space of the associated BFTs. For BFTs on a disk, these transformations follow from the properties of coordinates in the Grassmannian. This represents a new addition to the list of combinatorial objects for the Grassmannian, such as matc…
The fundamental role of on-shell diagrams in quantum field theory has been recently recognized. On-shell diagrams, or equivalently bipartite graphs, provide a natural bridge connecting gauge theory to powerful mathematical structures such as the Grassmannian. We perform a detailed investigation of the combinatorial and geometric objects associated to these graphs. We mainly focus on their relation to polytopes and toric geometry, the Grassmannian and its stratification. Our work extends the current understanding of these connections along several important fronts, most notably eliminating rest…
We give a new, fully mathematical, construction of the space of local operators in the holomorphic-topological twist of 4d $\mathcal N=2$ gauge theories. It is based on computations of morphism spaces in the DG category of line operators, which (by work of Kapustin-Saulina and Cautis-Williams) may be represented as ind-coherent sheaves on the affine Grassmannian and, more generally, on the $\mathcal R_{G,N}$ spaces of Braverman-Finkelberg-Nakajima. We prove that characters of our morphisms spaces reproduce the Schur indices of 4d $\mathcal N=2$ theories, and that the spaces themselves agree wi…
This is a companion paper of arXiv:1601.03586. We study Coulomb branches of unframed and framed quiver gauge theories of type $ADE$. In the unframed case they are isomorphic to the moduli space of based rational maps from ${\mathbb C}P^1$ to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian.
We perform a detailed investigation of Bipartite Field Theories (BFTs), a general class of 4d N=1 gauge theories which are defined by bipartite graphs. This class of theories is considerably expanded by identifying a new way of assigning gauge symmetries to graphs. A new procedure is introduced in order to determine the toric Calabi-Yau moduli spaces of BFTs. For graphs on a disk, we show that the matroid polytope for the corresponding cell in the Grassmannian coincides with the toric diagram of the BFT moduli space. A systematic BFT prescription for determining graph reductions is presented. …
Hilbert Series of Bipartite Field Theories
本文利用 Hilbert 级数分析二部场论的介子模空间，确定其生成元与关系，识别增强对称性，并给出两类圆柱面上的完全交 toric CY 3-fold 单参数理论族。
We present bipartiteSUSY, a Mathematica package designed to perform calculations for physical theories based on bipartite graphs. In particular, the package can employ the recently developed arsenal of techniques surrounding on-shell diagrams in N=4 SYM scattering amplitudes, including those for non-planar diagrams, with particular attention to computational speed. It also contains a host of tools for computations in N=1 Bipartite Field Theories, which utilize the same bipartite graphs. Through the use of an interactive graphical tool, it is possible to draw the desired diagrams on the screen …
We introduce and initiate the investigation of a general class of 4d, N=1 quiver gauge theories whose Lagrangian is defined by a bipartite graph on a Riemann surface, with or without boundaries. We refer to such class of theories as Bipartite Field Theories (BFTs). BFTs underlie a wide spectrum of interesting physical systems, including: D3-branes probing toric Calabi-Yau 3-folds, their mirror configurations of D6-branes, cluster integrable systems in (0+1) dimensions and leading singularities in scattering amplitudes for N=4 SYM. While our discussion is fully general, we focus on models that …
We derive superconformal partial waves for all scalar four-point functions on a super Grassmannian space Gr(m|n,2m|2n) for all m,n. This family of four-point functions includes those of all (arbitrary weight) half BPS operators in both N=4 SYM (m=n=2) and in N=2 superconformal field theories in four dimensions (m=2,n=1) on analytic superspace. It also includes four-point functions of all (arbitrary dimension) scalar fields in non-supersymmetric conformal field theories (m=2,n=0) on Minkowski space, as well as those of a certain class of representations of the compact SU(2n) coset spaces. As an…
Brane systems provide a large class of gauge theories that arise in string theory. This paper demonstrates how such brane systems fit with a somewhat exotic geometric object, called the affine Grassmannian. This gives a strong motivation to study physical aspects of the affine Grassmannian. Explicit quivers are presented throughout the paper, and a quiver addition algorithm to generate the affine Grassmannian is introduced. An important outcome of this study is a set of quivers for new elementary slices.
We study the algebra of Wilson line operators in three-dimensional N=2 supersymmetric U(M) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M,N), and its connection to K-theoretic Gromov-Witten invariants for Gr(M,N). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M,N), isomorphic to the Verlinde algebra for U(M), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.
We develop tools for determining the gauge theory resulting from a configuration of Type IIB D3-branes probing a non-compact, toric Calabi-Yau 3-fold, in the presence of additional flavor D7-branes with general embeddings. Two main ingredients of our approach are dimer models and mirror symmetry. D7-branes with general embeddings are obtained by recombination of elementary D7-brane constituents. These tools are then used to engineer a large set of Bipartite Field Theories, a class of 4d, N=1 quantum field theories defined by bipartite graphs on bordered Riemann surfaces. Several explicit examp…
We study a correspondence between 3d $\mathcal{N}=2$ topologically twisted Chern-Simons-matter theories on $S^1 \times Σ_g$ and quantum $K$-theory of Grassmannians. Our starting point is a Frobenius algebra depending on a parameter $β$ associated with an algebraic Bethe ansatz introduced by Gorbounov-Korff. They showed that the Frobenius algebra with $β=-1$ is isomorphic to the (equivariant) small quantum $K$-ring of the Grassmannian, and the Frobenius algebra with $β=0$ is isomorphic to the equivariant small quantum cohomology of the Grassmannian. We apply supersymmetric localization formulas…
We derive exact formulas for bipartite von Neumann entanglement entropy after partial projective local measurement in $1+1$ dimensional conformal field theories with periodic and open boundary conditions. After defining the set up we will check numerically the validity of our results in the case of Klein-Gordon field theory (coupled harmonic oscillators) and spin-$1/2$ XX chain in a magnetic field. The agreement between analytical results and the numerical calculations is very good. We also find a lower bound for localizable entanglement in coupled harmonic oscillators.
We provide a string theory embedding for N = 1 superconformal field theories defined by bipartite graphs inscribed on a disk. We realize these theories by exploiting the close connection with related N = 2 generalized (A_(k-1), A_(n-1)) Argyres-Douglas theories. The N = 1 theory is obtained from spacetime filling D5-branes wrapped on an algebraic curve and NS5-branes wrapped on special Lagrangians of C^2 which intersect along the BPS flow lines of the corresponding N = 2 Argyres-Douglas theory. Dualities of the N = 1 field theory follow from geometric deformations of the brane configuration wh…
We study D-brane instantons in systems of D3-branes at toric CY 3-fold singularities. The instanton effect can be described as a backreaction modifying the geometry of the mirror configuration, in which the breaking of $U(1)$ symmetries by the instanton translates into the recombination of gauge D-branes, which also directly generates the instanton-induced charged field theory operator. In this paper we describe the D-brane instanton backreaction in terms of a combinatorial operation in the bipartite dimer diagram of the original theory. Interestingly, the resulting theory is a general Biparti…
Dual conformal invariant kinematics and folding of Grassmannian cluster algebras
本文针对一般$n$情形，通过推导$D=3$运动学约束、构造可折叠种子，建立了$D=3$运动学与格拉斯曼簇代数折叠之间的联系。
We define and study a new class of 4d N=1 superconformal quiver gauge theories associated with a planar bipartite network. While UV description is not unique due to Seiberg duality, we can classify the IR fixed points of the theory by a permutation, or equivalently a cell of the totally non-negative Grassmannian. The story is similar to a bipartite network on the torus classified by a Newton polygon. We then generalize the network to a general bordered Riemann surface and define IR SCFT from the geometric data of a Riemann surface. We also comment on IR R-charges and superconformal indices of …
We obtain the entanglement negativity for various bipartite zero and finite temperature pure and mixed state configurations in a class of $(1+1)$-dimensional Galilean conformal field theories. In this context we establish a construction for computing the entanglement negativity for such bipartite states involving a suitable replica technique. Our construction exactly reproduces certain universal features observed for entanglement negativity of corresponding states in relativistic $(1+1)$-dimensional conformal field theories.
We consider the construction of operator bases for massless, relativistic quantum field theories, and show this is equivalent to obtaining the harmonic modes of a physical manifold (the kinematic Grassmannian), upon which observables have support. This enables us to recast the approach of effective field theory (EFT) through the lens of harmonic analysis. We explicitly construct harmonics corresponding to low mass dimension EFT operators.
Diagnosing Inconsistencies in 4d N=1 Gauge Theories with Explainable AI
本研究利用可解释AI（XAI）从Kasteleyn矩阵训练CNN，高准确率识别4d N=1规范理论膜平铺的几何不一致性，并通过梯度显著性分析定位需Higgsing的手性场，展示了XAI在理论物理缺陷诊断中的潜力。
Solvable Models in Two-Dimensional N=2 Theories
二维N=2超对称场论在过去几年中得到了广泛的研究。它们的许多性质可以沿着整个重整化群流确定，如它们的耦合依赖性和孤子谱。本文讨论了几个模型，当超场的数目很大时，通过研究它们的拓扑-反拓扑融合方程，可以完全解决这些模型。这些模型是CPN模型、格拉斯曼流形上的西格玛模型和某些扰动的$N=2$极小模型。
We study the Faddeev-Jackiw symplectic Hamiltonian reduction for 3+1-dimensional free and Abelian gauged Rarita-Schwinger theories that comprise Grassmannian fermionic fields. We obtain the relevant fundamental brackets and find that they are in convenient forms for quantization. The brackets are independent of whether the theories contain mass or gauge fields, and the structure of constraints and symplectic potentials largely determine characteristic behaviors of the theories. We also note that, in contrast to the free massive theory, the Dirac field equations for free massless Rarita-Schwing…
The odd entanglement entropy (OEE) for bipartite states in a class of $(1+1)$-dimensional Galilean conformal field theories ($GCFT_{1+1}$) is obtained through an appropriate replica technique. In this context our results are compared with the entanglement wedge cross section (EWCS) for $(2+1)$-dimensional asymptotically flat geometries dual to the $GCFT_{1+1}$ in the framework of flat holography. We find that our results are consistent with the duality of the difference between the odd entanglement entropy and the entanglement entropy of bipartite states, with the bulk EWCS for flat holographi…
We study the algebra of BPS Wilson loops in 3d gauge theories with N=2 supersymmetry and Chern-Simons terms. We argue that new relations appear on the quantum level, and that in many cases this makes the algebra finite-dimensional. We use our results to propose the mapping of Wilson loops under Seiberg-like dualities and verify that the proposed map agrees with the exact results for expectation values of circular Wilson loops. In some cases we also relate the algebra of Wilson loops to the equivariant quantum K-ring of certain quasi projective varieties. This generalizes the connection between…
Bipartite entanglement entropy of a segment with the length $l$ in $1+1$ dimensional conformal field theories (CFT) follows the formula $S=\frac{c}{3}\ln l+γ$, where $c$ is the central charge of the CFT and $γ$ is a cut-off dependent constant which diverges in the absence of an ultraviolet cutoff. According to this formula, systems with larger central charges have {\it{more}} bipartite entanglement entropy. Using quantum Fisher information (QFI), we argue that systems with bigger central charges not only have larger bipartite entanglement entropy but also have more multipartite entanglement co…
Genuine Tripartite Entanglement Selects Gauge-Invariant Theories
该研究将极值纠缠的方法扩展至三方系统，发现真正三方纠缠可唯一识别胶子与引力子散射中对应规范不变、微分同胚不变的理论。
We obtain the reflected entropy for bipartite states in a class of $(1+1)$-dimensional Galilean conformal field theories ($GCFT_{1+1}$) through a replica technique. Furthermore we compare our results with the entanglement wedge cross section (EWCS) obtained for the dual (2+1) dimensional asymptotically flat geometries in the context of flat holography. We find that our results are consistent with the duality between the reflected entropy and the bulk EWCS for flat holographic scenarios.
We advance holographic constructions for the entanglement negativity of bipartite states in a class of $(1+1)-$dimensional Galilean conformal field theories dual to asymptotically flat three dimensional bulk geometries described by Einstein Gravity and Topologically Massive Gravity. The construction involves specific algebraic sums of the lengths of bulk extremal curves homologous to certain combinations of the intervals appropriate to such bipartite states. Our analysis exactly reproduces the corresponding replica technique results in the large central charge limit. We substantiate our constr…
A large class of two-dimensional $\mathcal{N}=(2,2)$ superconformal field theories can be understood as IR fixed-points of Landau-Ginzburg models. In particular, there are rational conformal field theories that also have a Landau-Ginzburg description. To understand better the relation between the structures in the rational conformal field theory and in the Landau-Ginzburg theory, we investigate how rational B-type boundary conditions are realised as matrix factorisations in the $SU(3)/U(2)$ Grassmannian Kazama-Suzuki model. As a tool to generate the matrix factorisations we make use of a parti…
We study 4d superconformal indices for a large class of N=1 superconformal quiver gauge theories realized combinatorially as a bipartite graph or a set of "zig-zag paths" on a two-dimensional torus T^2. An exchange of loops, which we call a "double Yang-Baxter move", gives the Seiberg duality of the gauge theory, and the invariance of the index under the duality is translated into the Yang-Baxter-type equation of a spin system defined on a "Z-invariant" lattice on T^2. When we compactify the gauge theory to 3d, Higgs the theory and then compactify further to 2d, the superconformal index reduce…
Anomalies are a powerful way to gain insight into possible lattice regularizations of a quantum field theory. In this work, we argue that the continuum anomaly for a given symmetry can be matched by a manifestly-symmetric, local, lattice regularization in the same spacetime dimensionality only if (i) the symmetry action is offsite, or (ii) if the continuum anomaly is reproduced exactly on the lattice. We consider lattice regularizations of a class of prototype models of QCD: the (1+1)-dimensional asymptotically-free Grassmannian nonlinear sigma models (NLSMs) with a $θ$ term. Using the Grassma…
We establish a construction for the entanglement wedge in asymptotically flat bulk geometries for subsystems in dual $(1+1)$-dimensional Galilean conformal field theories in the context of flat space holography. In this connection we propose a definition for the bulk entanglement wedge cross section for bipartite states of such dual non relativistic conformal field theories. Utilizing our construction for the entanglement wedge cross section we compute the entanglement negativity for such bipartite states through the generalization of an earlier proposal, in the context of the usual $AdS/CFT$ …
Superconformal field theories from crystal lattices
利用M-理论的T-对偶变换，我们提出了（2+1）d，$\mathcal{N}=2$超共形理论（CFT $_3$）的一种膜构型.我们得到的M5-膜上的三个环面，形成一个三维二分晶格的方式类似的二维二聚体模型CFT $_4$的交叉点。CFT $_3$的基本场是M2膜盘，位于交叉点附近，超势项与晶体的原子相同。该模型正确地再现了介子和重子的完整BPS谱。
We argue that the entanglement of purification for two dimensional holographic CFT can be obtained from conformal blocks with internal twist operators. First, we explain our formula from the view point of tensor network model of holography. Then, we apply it to bipartite mixed states dual to subregion of AdS${}_3$ and the static BTZ blackhole geometries. The formula in CFT agrees with the entanglement wedge cross section in the bulk, which has been recently conjectured to be equivalent to the entanglement of purification.
This is the second part of two papers where we study the effect of integrable line defects on bipartite entanglement entropy in integrable field theories. In this paper, we consider non-topological line defects in Ising field theory. We derive an infinite series expression for the entanglement entropy and show that both the UV and IR limits of the bulk entanglement entropy are modified by the line defect. In the UV limit, we give an infinite series expression for the coefficient in front of the logarithmic divergence and the exact defect $g$-function. By tuning the defect to be purely transmis…
Graded Geometry and Tensor Gauge Theories
本文在分级几何框架下构造混合对称性高阶自旋场的普适拉格朗日量，识别出标量、微分形式和二分张量的Galileon型高阶导数相互作用，并论证自旋高于2时规范不变性与局域性不可兼得。
We propose a covariant holographic conjecture for the entanglement negativity of mixed states in bipartite systems described by $d$-dimensional conformal field theories dual to bulk non static $AdS_{d+1}$ configurations. Application of our conjecture to $(1+1)$-dimensional conformal field theories dual to bulk rotating BTZ black holes exactly reproduces the corresponding entanglement negativity in the large central charge limit and characterizes the distillable entanglement. We further demonstrate that our conjecture applied to the case of bulk extremal rotating BTZ black holes also characteri…
Motivated by recent studies of symmetry protected topological (SPT) phases, we explore the possible gapless quantum disordered phases in the $(2+1)d$ nonlinear sigma model defined on the Grassmannian manifold $\frac{U(N)}{U(n)\times U(N - n)}$ with a Wess-Zumino-Witten (WZW) term at level $k$, which is the effective low energy field theory of the boundary of certain $(3+1)d$ SPT states. With $k = 0$, this model has a well-controlled large-$N$ limit, $i.e.$ its renormalization group equations can be computed exactly with large-$N$. However, with the WZW term, the large-$N$ and large-$k$ limit a…
We calculate numerically the Rényi bipartite entanglement entropy of the ground state of Klein-Gordon field theory (coupled harmonic oscillators) after fixing the position (partial measurement) of some of the oscillators in $d=1,2$ and $3$ dimensions. We show that after partial measurement as far as the two parts are connected the area-law is valid in generic dimensions except in $d=1$ conformal field theories. When due to the measured region the two parts are disconnected, the entanglement entropy decreases exponentially with respect to the minimum distance between the regions. In the massles…
We investigate the application of our recent holographic entanglement negativity conjecture for higher dimensional conformal field theories to specific examples which serve as crucial consistency checks. In this context we compute the holographic entanglement negativity for bipartite pure and finite temperature mixed state configurations in $d$-dimensional conformal field theories dual to bulk pure $AdS_{d+1}$ geometry and $AdS_{d+1}$-Schwarzschild black holes respectively. It is observed that the holographic entanglement negativity characterizes the distillable entanglement for the finite tem…
The purpose of this review is to bridge the gap between a standard course in quantum field theory and recent fascinating developments in the studies of on-shell scattering amplitudes. We build up the subject from basic quantum field theory, starting with Feynman rules for simple processes in Yukawa theory and QED. The material covered includes spinor helicity formalism, on-shell recursion relations, superamplitudes and their symmetries, twistors and momentum twistors, loops and integrands, Grassmannians, polytopes, and amplitudes in perturbative supergravity as well as 3d Chern-Simons-matter t…
We calculate analytically the Rényi bipartite entanglement entropy $S_α$ of the ground state of $1+1$ dimensional conformal field theories (CFT) after performing a projective measurement in a part of the system. We show that the entanglement entropy in this setup is dependent on the central charge and the operator content of the system. When the measurement region $A$ separates the two parts $B$ and $\bar{B}$, the entanglement entropy between $B$ and $\bar{B}$ decreases like a power-law with respect to the characteristic distance between the two regions with an exponent which is dependent on t…
Odd entanglement entropy in boundary conformal field theories and holographic moving mirrors
本文通过复制技巧研究（1+1）维边界共形场论中二分混合态的奇纠缠熵，并验证其与AdS$_3$几何中体纠缠楔截面的全息对偶，同时扩展到全息移动镜模型以模拟霍金辐射。
The field content and interactions of almost all known gauge theories in AdS_5/CFT_4 can be expressed in terms of dimer models or bipartite graphs drawn on a torus. Associated with the fundamental cell is a complex structure parameter tau_R. Based on the brane realization of these theories, we can specify a special Lagrangian (SLag) torus fibration that is the natural candidate to be identified as the torus on which the dimer lives. Using the metrics known in the literature, we compute the complex structure tau_G of this torus. For the theories on C^3 and the conifold and for orbifolds thereof…
Holographic entanglement negativity for a single subsystem in conformal field theories with a conserved charge
该研究利用全息构造计算带守恒荷共形场论中单个子系统的全息纠缠负度，验证了领头阶热贡献消除的结论，为全息构造提供了一致性检验。
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