本文研究 d 维CFT中二维球形缺陷的RG流，构造单调递减熵函数，给出缺陷RG流不可逆性的替代推导，并以微扰情形和示例说明其C函数作用。
摘要：
我们研究 d 维共形场论中的二维球形缺陷。我们论证了此类缺陷上的重整化群（RG）流允许存在一个递减的熵函数。在流的不动点处，该熵函数等于缺陷 Weyl 反常中与 Euler 密度相乘的反常系数。我们的构造展示了对二维缺陷上 RG 流不可逆性的另一种推导。此外，在微扰 RG 流情形下，该熵函数单调递减，并起到 C 函数的作用。我们提供了一个简单示例，以显式算出所提构造中的 RG 流细节。
英文摘要：
We study two-dimensional spherical defects in d-dimensional Conformal Field Theories. We argue that the Renormalization Group (RG) flows on such defects admit the existence of a decreasing entropy function. At the fixed points of the flow, the entropy function equals the anomaly coefficient which multiplies the Euler density in the defect's Weyl anomaly. Our construction demonstrates an alternative derivation of the irreversibility of RG flows on two-dimensional defects. Moreover in the case of perturbative RG flows, the entropy function decreases monotonically and plays the role of a C-function. We provide a simple example to explicitly work out the RG flow details in the proposed construction.
The computation of the effect of a simple monodromy defect in the case of a sphere with twisted boundary conditions is revisited and streamlined using earlier calculations for a similar system. Compact and explicit expressions are found for arbitrary integer dimensions. Comments on other work are made.
We deform a defect conformal field theory by an exactly marginal bulk operator and we consider the dependence on the marginal coupling of flat and spherical defect expectation values. For even dimensional spherical defects we find a logarithmic divergence which can be related to a $a$-type defect anomaly coefficient. This coefficient, for defect theories, is not invariant on the conformal manifold and its dependence on the coupling is controlled to all orders by the one-point function of the associated exactly marginal operator. For odd-dimensional defects, the flat and spherical case exhibit …
Conformal symmetry is broken by a flat or spherical defect operator $\mathcal{D}$. We show that this defect operator, may be identified as a pair of twist operators which are inserted at the tips of its causal diamond. Any $k-$point correlation function in a flat or spherical defect CFT is equivalent to a $(k+2)-$point correlation function. We reproduce one point correlation functions and also solve two point correlation functions in defect CFTs . Mutual Rényi entropy is computed and agrees with previous result in a certain limit. We conjecture there may be universal terms in general co-dimens…
Deforming a two dimensional conformal field theory on one side of a trivial defect line gives rise to a defect separating the original theory from its deformation. The Casimir force between these defects and other defect lines or boundaries is used to construct flows on bulk moduli spaces of CFTs. It turns out, that these flows are constant reparametrizations of gradient flows of the g-functions of the chosen defect or boundary condition. The special flows associated to supersymmetric boundary conditions in N=(2,2) superconformal field theories agree with the attractor flows studied in the con…
We show that topological defects in the language of Landau-Ginzburg models carry information about the RG flow between the non-compact orbifolds $\mathbb{C}/\mathbb{Z}_d$. We show that such defects correctly implement the bulk-induced RG flow on the boundary.
We study the effect of bulk perturbations of N=(2,2) superconformal minimal models on topological defects. In particular, symmetries and more general topological defects which survive the flow to the IR are identified. Our method is to consider the topological subsector and make use of the Landau-Ginzburg formulation to describe RG flows and topological defects in terms of matrix factorizations.
Using the properties of defect lines, we study boundary renormalisation group flows. We find that when there exists a flow between maximally symmetric boundary conditions "a" and "b" then there also exists a boundary flow between "c x a" and "c x b" where "x" denotes the fusion product. We also discuss applications of this simple observation.
We consider defect operators in scalar field theories in dimensions $d=4-ε$ and $d=6-ε$ with self-interactions given by a general marginal potential. In a double scaling limit, where the bulk couplings go to zero and the defect couplings go to infinity, the bulk theory becomes classical and the quantum defect theory can be solved order by order in perturbation theory. We compute the defect $β$ functions to two loops and study the Renormalization Group flows. The defect fixed points can move and merge, leading to fixed point annihilation; and they exhibit a remarkable factorization property whe…
Recently Gaiotto [1] considered conformal defects which produce an expansion of infrared local fields in terms of the ultraviolet ones for a given renormalization group flow. In this paper we propose that for a boundary RG flow in two dimensions there exist boundary condition changing fields (RG defect fields) linking the UV and the IR conformal boundary conditions which carry similar information on the expansion of boundary fields at the fixed points. We propose an expression for a pairing between IR and UV operators in terms of a four-point function with two insertions of the RG defect field…
In this paper we study a simple example of a two-parameter space of renormalisation group flows of defects in Virasoro minimal models. We use a combination of exact results, perturbation theory and the truncated conformal space approach to search for fixed points and investigate their nature. For the Ising model, we confirm the recent results of Fendley et al. In the case of central charge close to one, we find six fixed points, five of which we can identify in terms of known defects and one of which we conjecture is a new non-trivial conformal defect. We also include several new results on ex…
Global topological defects described by real scalar field in (3,1) dimensions coupled to gravity are analyzed. We consider a class of scalar potentials with explicit dependence with distance, evading Derrick's theorem and leading to defects with spherical symmetry. The analysis shows that the defects have finite energy on flat space, contrary to the observed for the global monopole. With the aim to study the gravitational field produced by such defects, after an {\it Ansatz} for the static metric with spherical symmetry, we obtain the coupled system of Einstein and field equations. On the Newt…
We study the defect groups of $D_p^b(G)$ theories using geometric engineering and BPS quivers. In the simple case when $b=h^\vee (G)$, we use the BPS quivers of the theory to see that the defect group is compatible with a known Maruyoshi-Song flow. To extend to the case where $b\neq h^\vee (G)$, we use a similar Maruyoshi-Song flow to conjecture that the defect groups of $D_p^b(G)$ theories are given by those of $G^{(b)}[k]$ theories. In the cases of $G=A_n, \;E_6, \;E_8$ we cross check our result by calculating the BPS quivers of the $G^{(b)}[k]$ theories and looking at the cokernel of their …
In the context of two-dimensional rational conformal field theories we consider topological junctions of topological defect lines with boundary conditions. We refer to such junctions as open topological defects. For a relevant boundary operator on a conformal boundary condition we consider a commutation relation with an open defect obtained by passing the junction point through the boundary operator. We show that when there is an open defect that commutes or anti-commutes with the boundary operator there are interesting implications for the boundary RG flows triggered by this operator. The end…
We consider perturbation defects obtained by perturbing a 2D conformal field theory (CFT) by a relevant operator on a half-plane. If the perturbed bulk theory flows to an infrared fixed point described by another CFT, the defect flows to a conformal defect between the ultraviolet and infrared fixed point CFTs. For short bulk renormalization group flows connecting two fixed points which are close in theory space we find a universal perturbative formula for the boundary entropy of the corresponding conformal perturbation defect. We compare the value of the boundary entropy that our formula gives…
We investigate relevant deformation and the renormalization group flow in a defect conformal field theory from the point of view of the holography. We propose a candidate of g-function in the context of the holography, and prove the g-theorem: the g-function is monotonically non-increasing along the RG flow. We apply this g-theorem to the D5-brane solution which is an asymptotically AdS_4 x S^2 brane in AdS_5 x S^5. This solution corresponds to the mass deformation of the defect CFT. We checked that the g-function is monotonically non-increasing in this solution.
Quantum field theories (QFT) in the presence of defects exhibit new types of anomalies which play an important role in constraining the defect dynamics and defect renormalization group (RG) flows. Here we study surface defects and their anomalies in conformal field theories (CFT) of general spacetime dimensions. When the defect is conformal, it is characterized by a conformal $b$-anomaly analogous to the $c$-anomaly of 2d CFTs. The $b$-theorem states that $b$ must monotonically decrease under defect RG flows and was proven by coupling to a spurious defect dilaton. We revisit the proof by deriv…
We investigate perturbatively tractable deformations of topological defects in two-dimensional conformal field theories. We perturbatively compute the change in the $g$-factor, the reflectivity, and the entanglement entropy of the conformal defect at the end of these short RG flows. We also give instances of such flows in the diagonal Virasoro and Super-Virasoro Minimal Models.
We study a number of (3+1)- and (2+1)-dimensional defect and boundary conformal field theories holographically dual to supergravity theories. In all cases the defects or boundaries are planar, and the defects are codimension-one. Using holography, we compute the entanglement entropy of a (hemi-)spherical region centered on the defect (boundary). We define defect and boundary entropies from the entanglement entropy by an appropriate background subtraction. For some (3+1)-dimensional theories we find evidence that the defect/boundary entropy changes monotonically under certain renormalization gr…
The effect of a spherical monodromy defect on the entanglement entropy and central charge $C_T$ of a free conformal scalar field propagating on an odd-dimensional sphere is investigated. As on even spheres the central charge becomes negative for a range of the flux parameter, $δ$, with possible implications for reflection positivity. The work is mostly numerical but closed forms are found for the $Z_2$ monodromy ($δ=1/2$) yielding an explicit $d\to\infty$ limit, $C_T\to -6/π$
The generalized Wilson loop operator interpolating between the supersymmetric and the ordinary Wilson loop in ${\cal N}=4$ SYM theory provides an interesting example of renormalization group flow on a line defect: the scalar coupling parameter $ζ$ has a non-trivial beta function and may be viewed as a running coupling constant in a 1d defect QFT. In this paper we continue the study of this operator, generalizing previous results for the beta function and Wilson loop expectation value to the case of an arbitrary representation of the gauge group and beyond the planar limit. Focusing on the scal…
The effect of a co--dimension--2 spherical monodromy defect on the free--field theory of a conformal scalar is investigated. The conformal anomaly is computed (in two ways) and thence the Rényi and entanglement entropies. The $C_T$ central charge is found to be negative for a range of monodromy flux.The calculations are pursued for any even dimensional sphere. A Possible conflict with existing results is raised.
We consider line defects in d-dimensional Conformal Field Theories (CFTs). The ambient CFT places nontrivial constraints on Renormalization Group (RG) flows on such line defects. We show that the flow on line defects is consequently irreversible and furthermore a canonical decreasing entropy function exists. This construction generalizes the g theorem to line defects in arbitrary dimensions. We demonstrate our results in a flow between Wilson loops in 4 dimensions.
Solutions of Einstein's equations are found for global defects in a higher-dimensional spacetime with a nonzero cosmological constant Lambda. The defect has a (p-1)-dimensional core (brane) and a `hedgehog' scalar field configuration in the n extra dimensions. For Lambda = 0 and n > 2, the solutions are characterized by a flat brane worldsheet and a solid angle deficit in the extra dimensions. For Lambda > 0, one class of solutions describes spherical branes in an inflating higher-dimensional universe. Instantons obtained by a Euclidean continuation of such solutions describe quantum nucleatio…
We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries and Verlinde lines, TDLs together with their attached defect operators provide models of fusion categories without braiding. We study the crossing relations of TDLs, discuss their relation to the 't Hooft anomaly, and use them to constrain renormalization group flows to either conformal critical points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-…
A conformal field theory (CFT) in dimension $d\geq 3$ coupled to a planar, two-dimensional, conformal defect is characterized in part by a "central charge" $b$ that multiplies the Euler density in the defect's Weyl anomaly. For defect renormalization group flows, under which the bulk remains critical, we use reflection positivity to show that $b$ must decrease or remain constant from ultraviolet to infrared. Our result applies also to a CFT in $d=3$ flat space with a planar boundary.
We explore a $C$-theorem in defect conformal field theories (DCFTs) that unify all the known conjectures and theorems until now. We examine as a candidate $C$-function the additional contributions from conformal defects to the sphere free energy and the entanglement entropy across a sphere in a number of examples including holographic models. We find the two quantities are equivalent, when suitably regularized, for codimension-one defects (or boundaries), but differ by a universal constant term otherwise. Moreover, we find in a few field theoretic examples that the sphere free energy decreases…
We investigate superconformal surface defects in four-dimensional N=2 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfeld-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group f…
We consider loop observables in gauged Wess-Zumino-Witten models, and study the action of renormalization group flows on them. In the WZW model based on a compact Lie group G, we analyze at the classical level how the space of renormalizable defects is reduced upon the imposition of global and affine symmetries. We identify families of loop observables which are invariant with respect to an affine symmetry corresponding to a subgroup H of G, and show that they descend to gauge-invariant defects in the gauged model based on G/H. We study the flows acting on these families perturbatively, and qu…
