text:
THE IMPACT OF CONTAMINATED RR LYRAE/GLOBULAR CLUSTER PHOTOMETRY ON THE DISTANCE SCALE

# The Impact of Contaminated Rr Lyrae/globular Cluster Photometry on the Distance Scale

D. Majaess, D. G. Turner, W. Gieren,D. J. Lane Saint Mary’s University, Halifax, NS, Canada. Universidad de Concepción, Concepción, Chile.
###### Abstract

RR Lyrae variables and the stellar constituents of globular clusters are employed to establish the cosmic distance scale and age of the universe. However, photometry for RR Lyrae variables in the globular clusters M3, M15, M54, M92, NGC2419, and NGC6441 exhibit a dependence on the clustercentric distance. For example, variables and stars positioned near the crowded high-surface brightness cores of the clusters may suffer from photometric contamination, which invariably affects a suite of inferred parameters (e.g., distance, color excess, absolute magnitude, etc.). The impetus for this study is to mitigate the propagation of systematic uncertainties by increasing awareness of the pernicious impact of contaminated and radial-dependent photometry.

globular clusters: general—stars: distances, Hertzsprung-Russell and C-M diagrams, variables: RR Lyrae—techniques: photometric

## 1 Introduction

Extragalactic photometry for classical Cepheids in M33, M81, M101, and M106 implies that at a fixed pulsation period the variables brighten () with decreasing galactocentric distance (e.g., Kennicutt et al., 1998). A debate persists as to whether the trend is tied to a metallicity gradient, or arises from photometric contamination (Majaess, 2010; Majaess et al., 2009, 2011b; Kudritzki et al., 2012, and references therein). The degeneracy exists because the surface brightness, density, and metal abundance (Andrievsky et al., 2004, their Fig. 3) all increase with decreasing galactocentric distance. A similar behavior is observed in photometric data for RR Lyrae variables in the globular cluster M15 (Majaess et al., 2011a), whereby distance moduli established for the variables likewise exhibit a radial dependence (Fig. 1). However, no linear metallicity gradient exists across the globular cluster, and thus the degeneracy plaguing the extragalactic Cepheids is absent. Consequently, observations of variables near the core of M15 suffer from photometric contamination (see also Yanny et al., 1994; Stetson et al., 2005). A dependence of the photometry on the clustercentric distance may complicate conclusions drawn from Bailey (period-amplitude) diagrams for globular clusters (Corwin et al., 2008; Smith et al., 2011, and see also Kunder et al. 2011 who describe the impact of the Blazhko effect on that diagram).

In this study, photometry for RR Lyrae variables in the globular clusters M3, M54, M92, NGC2419, NGC6441, and M15 are shown to exhibit a dependence on clustercentric distance (Figs. 12). It follows that the color-magnitude diagrams are compromised (Fig. 3). Clustercentric photometry has a direct impact on the establishment of Galactic/extragalactic distances (e.g., brightness of the Horizontal Branch), the Hubble constant (), the age of the universe, and cosmological models (e.g., , Macri & Riess, 2009).

## 2 Analysis

### 2.1 Rr Lyrae Variables

Distance moduli for RR Lyrae variables may be inferred from a reddening-free Wesenheit function (). Soszyński et al. (2003) cite the following Wesenheit function for LMC RR Lyrae (ab) variables:

 WVIc=(−2.75±0.04)logP+(17.217±0.008)

The empirical slope of the function is consistent with that predicted by models (Di Criscienzo et al., 2004). The distance modulus for each RR Lyrae variable was evaluated via:

 m−M=5logd−5 WVIc−WVIc,0=5logd−5 V−RVIc(V−Ic)−αlogP0−β=μ0 V−2.55(V−Ic)+2.73logP0−β=μ0 (1)

is the ratio of total to selective extinction (Di Criscienzo et al., 2004; Fouqué et al., 2007, and references therein), is the slope of the Wesenheit function, and is chosen to be arbitrary (relative distance). A mean of two estimates for the Wesenheit slope (LMC/IC4499) was adopted (). Only RR Lyrae variables sampled beyond the core of IC4499 were analyzed (Walker & Nemec, 1996).

Relative distances were evaluated for RR Lyrae variables discovered in M15 (Corwin et al., 2008), M3 (Benkő et al., 2006), NGC2419 (Di Criscienzo et al., 2011), M92 (Kopacki, 2001), M54 (Layden & Sarajedini, 2000), and NGC6441 (Pritzl et al., 2003). The periods of RR Lyrae (c) variables pulsating in the first overtone were fundamentalized via , where is the fundamental mode period. RA/DEC-distance diagrams were constructed for variables in M15, M3, and NGC2419 (Fig. 1). As expected the distances become particularly discrepant as the cluster core is approached. The stellar density and surface brightness increase markedly as a function of decreasing clustercentric distance. For M15 and NGC2419 a similar trend is observed, namely that distances for stars near the core appear underestimated. A significant fraction of the variables in M15 are contaminated, and numerous stars are likewise affected in M3. Four stars in NGC2419 appear particularly compromised (i.e., ), and less-pronounced contamination persists to near the periphery. NGC2419 exhibits a low stellar density111Globular Cluster Database, W. E. Harris, McMaster U.: http://physwww.mcmaster.ca/~harris/mwgc.dat relative to the other globular clusters analyzed, however, the density of stars per cubic arcminute is large since the cluster is distant ( kpc, §2.3).

Fig. 2 features parameters for RR Lyrae variables in M3, M92, M54, and NGC6441 plotted as a function of clustercentric distance. RR Lyrae variables in NGC6441 and M92 display an analogous trend to that noted for NGC2419 and M15. Stars near the core exhibit smaller distances than objects occupying the cluster periphery. Seven stars within (NGC6441) appear particularly contaminated (Fig. 2). NGC6441 is projected along a line of sight where the field density is high (). RR Lyrae variables in M54 display a similar trend to those in M3, whereby the moduli and reddenings computed appear nearer and larger with increasing clustercentric distance, respectively. It is unclear whether effects associated with the edge of CCD detectors play a role. Thus photometric contamination is not the only source of concern that imposes a clustercentric dependence (further research is necessary). Stars in M3 located appear closer than those near the core (Fig. 2). A period-reddening relation indicates the variables are redder than objects located near the core, which are anomalously blue (Fig. 2). The period-color relation does not account for the width of the instability strip, thus a symmetric distribution consisting of negative reddenings is expected for a population exhibiting . However, deviations from symmetry are observed for stars (Fig. 2).

Table 1 highlights the difference between the distance computed using all RR Lyrae variables from those located near the periphery (). The offset is largest for M15 () and smallest for M92. M3 exhibits a positive offset, and the average value for the five clusters featuring a negative offset is . Admittedly, the differences evaluated are subjective and uncertain owing partly to the neglect of superposed trends (M54, Fig. 2), and an inability to overcome poor statistics and assess the extent of the compromised photometry (NGC6441, M92, Fig. 2). Nevertheless, the results imply a potential systematic uncertainty in age. The clustercentric dependence therefore hinders efforts to determine reliable parameters for RR Lyrae variables, and age estimates for the host clusters. The clusters and their RR Lyrae variable constituents are likewise employed to establish the distance scale, from which estimates of and the age of the universe stem. Globular clusters themselves set a crucial lower limit to the age of the universe. All potential biases (regardless of their relative impact) must be assessed in the pursuit of precision cosmology.

### 2.2 Globular Cluster Cmds

Color-magnitude diagrams for globular clusters are used to determine their distance (e.g., horizontal branch, subdwarfs) and age (isochrone fitting). The radial dependencies noted for the RR Lyrae variables (Figs. 12) promulgate into the color-magnitude diagrams. Color-magnitude diagrams are tabulated for M15 (Yanny et al., 1994), M2 (Lee & Carney, 1999), M4 (Ibata et al., 1999; Mochejska et al., 2002), Cen (Rey et al., 2004), and NGC2808 (Fig. 3). Groups of stars sampling near the core and periphery of the clusters are overplotted. The severity of the contamination depends on the distance and stellar density (stars per cubic arcminute), data-processing software, and contribution to the seeing owing to the instrumentation (CCD, telescope, wavelength) and environment (e.g., ground or space-based observations), etc. (see also Yanny et al., 1994; Stetson et al., 2005). PSF photometry does not correct for multiple stars coincident well within the FWHM of the system. The clustercentric effects (Fig. 12) impose a degeneracy which complicates the analysis of multiple populations and potential radial trends which are population-dependent (e.g., Cen), although for the latter mixing over such a large timescale and mass segregation likewise require consideration. The signatures of multiple populations may be mimicked by photometric contamination.

Photometry acquired for M13 from the Abbey Ridge Observatory (ARO222The ARO is a 0.3-m robotic optical facility used to detect supernovae and conduct variable star research (e.g., Lane & Gray, 2005; Lane, 2008).) confirms the trends observed in Fig. 3. For example, as expected photometry near the core is artificially brighter and less complete owing to the increased sky brightness. The ARO’s comparatively small aperture telescope, in tandem with typically large local seeing, conspire to create a sizable FWHM (). HST seeing is more than an order of magnitude smaller, and for that reason the ARO data were acquired to provide an empirical means to evaluate contamination effects directly. The trends observed are not unique to a given dataset cited, but may be present (to an extent) in most globular cluster photometry.

### 2.3 Distance to Ngc2419

A distance to NGC2419 may be derived which is less sensitive to the trends highlighted in Fig. 1. Absolute distances may be secured by anchoring the Soszyński et al. (2003) RR Lyrae Wesenheit function to the LMC distance established by Majaess et al. (2011a, )333A mean computed from LMC distance estimates tabulated in the NASA/IPAC Extragalactic Database of galaxy Distances (NED-D) supports that result (Steer & Madore, 2011, see also Fig. 2 in Freedman & Madore 2010).:

 WVIc(RRL,LMC)=(−2.75±0.04)logP +(17.217±0.008) WVIc,0=αlogP0+β WVIc−WVIc,0=5logd−5 (−2.75±0.04)logP+(17.217±0.008)− (αlogP0+β)=μ0

can be solved by re-arranging the aforementioned equation. The absolute Wesenheit function characterizing RR Lyrae variables is thus:

 WVIc,0≃(−2.73±0.03)logP0−(1.21±0.03)

Majaess et al. (2011a) derived the LMC distance via a universal Wesenheit template, which leverages the statistical weight of the entire variable star demographic to establish precise (%) distances. The template capitalizes upon HST (Benedict et al., 2007), VLBA, and HIP (van Leeuwen, 2007) geometric distances for SX Phe, Scuti, RR Lyrae, Type II and classical Cepheid variables. Specifically, the template is anchored to 8 SX Phe and Sct variables (HIP, van Leeuwen, 2007), 4 RR Lyrae variables (HIP and HST, Benedict et al., 2002; van Leeuwen, 2007), 2 Type II Cepheids (HIP, van Leeuwen, 2007), and 10 classical Cepheids (HST, Benedict et al., 2007). The template may be refined by employing new HST parallaxes for RZ Cep, XZ Cyg, SU Dra, UV Oct, VY Pyx, and Pav (Benedict et al., 2011). However, certain of the aforementioned stars lack reliable mean photometry, which is necessary for tabulating Wesenheit magnitudes. The desired observations are presently being acquired via facilities operated by the AAVSO (T. Krajci & A. Henden/BSM). Type II Cepheids detected in the less-crowded outer region of M106 (Macri et al., 2006; Majaess et al., 2009) were likewise incorporated into the Wesenheit template owing to the availability of a precise maser distance (Macri et al., 2006, and references therein). The Wesenheit approach obviates the need for uncertain reddening corrections, and thus the maser distance is used directly. Admittedly, the approach supersedes one of the authors (DM) prior first-order (yet ultimately incorrect) effort to unify variables of the instability strip in order to establish reliable distances (Majaess et al., 2011a, and references therein).

Equation 2.3 may be employed to establish the distance to NGC2419, however, a preferred approach is to simultaneously use the SX Phe, RR Lyrae, and Type II Cepheid variables discovered by Di Criscienzo et al. (2011). An advantage of the universal Wesenheit template is that known pulsation modes for RR Lyrae (ab) constituents may be employed to constrain that (potentially ambiguous) parameter for the SX Phe variables. The universal Wesenheit template is shown as Fig. 4. The template features Galactic (Majaess et al., 2011a, and references therein) and M106 data, in addition to LMC variables cataloged by OGLE (e.g., Udalski et al., 1999; Soszyński et al., 2008) which were incorporated into the template using the LMC distance derived by Majaess et al. (2011a). Designations for several variables tabulated in the OGLE survey are inconsistent with their Wesenheit magnitudes (e.g., overtone classical Cepheids on the fundamental mode ridge, etc.), yet these objects constitute the minority (see also the break between Scuti and overtone classical Cepheids).

To mitigate the effects of contaminated photometry stars (subjective) from the core of NGC2419 were utilized to deduce the distance. Stars exhibiting anomalous Wesenheit positions were not included in the determination. The offset from the Wesenheit template implies for NGC2419, which is consistent with estimates tabulated by Harris and Di Criscienzo et al. (2011). The distance is merely evaluated for illustrative purposes since the maximum extent of the contamination is unclear (Fig. 1). The analysis reveals that the Type II Cepheid (V18) is either contaminated, a binary system, anomalous, or not a cluster member (Fig. 4). The binary hypothesis is supported by the discovery of numerous binary classical and Type II Cepheids444No bona fide binary RR Lyrae variables have been discovered/established in the LMC (Prša et al., 2008, see also Pietrzyński et al. 2012). in the Galaxy and LMC (Evans, 1991, 1992; Szabados, 2003; Soszyński et al., 2008). The three SX Phe variables lie on the Wesenheit ridge tied to fundamental mode pulsators (Fig. 4). The stars are coincident with the absolute Wesenheit magnitude for the prototype of the class (SX Phe, see also Majaess et al., 2011a). SX Phe and other metal-poor population II Scutis lie toward the short-period extension of the Wesenheit ridge characterizing population I Scutis.

### 2.4 Conclusion

The analysis reaffirms that a (deleterious) clustercentric distance dependence exists in photometry associated with the globular clusters M15, M2, M3, M54, M92, NGC2419, NGC6441, Cen, M4, M13, and NGC2808 (Figs. 123). Distances inferred from RR Lyrae variables and cluster stars are aversely affected. The impact may be reduced by remaining cognisant of the trends exhibited in Figs. 123 (see also Yanny et al., 1994; Stetson et al., 2005). The principal impetus of the present work is to increase awareness concerning clustercentric photometry in order to mitigate the propagation of biases into research on RR Lyrae variables, and to serve as a catalyst for additional discussion and continued research. Photometric contamination, in harmony with the challenges of establishing precise, standardized, multi-epoch, multi-band photometry: are foremost among sources of uncertainty associated with the distance scale (see also Majaess, 2010; Majaess et al., 2011b). Admittedly, the radial dependencies in the globular cluster photometry were partly overlooked by one of the authors (DM) previously (Majaess et al., 2011a, and references therein).

The conclusions do not mitigate the broader importance of the studies discussed. For example, the datasets contain sizable numbers of newly discovered variable stars (e.g., SX Phe, RR Lyrae, and Type II Cepheid variables, Clement et al., 2001; Samus et al., 2009). The pulsation periods determined place invaluable constraints on models since period-change is a proxy for stellar evolution (). Such analyses have hitherto yielded seminal results for RR Lyrae and Cepheid variables (Lee & Carney, 1999; Turner et al., 2006; Kunder et al., 2011; Jurcsik et al., 2012; Neilson et al., 2012).

### Acknowledgements

DM is grateful to the following individuals and consortia whose efforts lie at the foundation of the research: OGLE (A. Udalski, I. Soszyński), H. Smith (RR Lyrae Stars), the Padova Globular Cluster Group, CDS, arXiv, and NASA ADS. WG is grateful for support from the Chilean Center for Astrophysics FONDAP 15010003 and the BASAL Centro de Astrofisica y Tecnologias Afines (CATA) PFB-06/2007.

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# Factorization and vertex algebra cohomology

A chiral algebra on a smooth curve $$X$$, in the sense of Beilinson-Drinfeld, is a right $$D_{X}$$-module with a chiral bracket, which is a map $$\mathcal{V}^{\boxtimes 2}(\infty\Delta)\rightarrow \Delta_{*}\mathcal{V}$$ satisfying certain conditions making it look like a Lie bracket. In fact $$\mathcal{V}$$ is a Lie algebra with respect to a certain pseudo-tensor structure on $$D_{X}$$-modules. A pseudo-tensor structure is what one obtains when one forgets the tensor product but remembers the associated multilinear maps. Multi-linear maps from a collection $$\{\mathcal{V}_{i\in I}\}$$ to $$\mathcal{W}$$ are defined to be $$\mathrm{Hom}_{D_{X^{I}}}(\boxtimes_{i\in I}\mathcal{V}_{i}(\infty\Delta_{\mathrm{big}}),\Delta_{*}\mathcal{W})$$. One thus obtains a complex of vector spaces $$C^{*}_{CE}(\mathcal{V})$$ via the appropriate construction of Chevalley-Eilenberg cochains, which makes sense in a pseudo-tensor category. Notably this formalism does not produce CE chains.

If $$X$$ is taken to a be a formal disc $$D$$, then we obtain a vertex algebra. I believe in this case the above construction produces the vertex algebra cohomology (with coefficients in the adjoint module) studied in the work of Bakalov, de Sole, Heluani and Kac (arXiv link). Is this true? If not what is the precise relation?

Now if $$\mathcal{V}$$ is a chiral algebra, then BD construct its factorization homology. This is constructed as de Rham homology of the associated $$D$$-module on the Ran space of $$X$$. In chiral terms I believe it is gotten by taking (the colimit over finite sets $$I$$) of the Chevalley Cousins complexes on $$X^{I}$$, cf. these notes by Rozenblyum (on Gaitsgory's webpage).

This can also be performed on a formal disc $$D$$*, in which case one obtains what one might call Vertex Homology. Has this been studied anywhere? Is there a relation with the work of BdSHK cited above, in particular does the BdSHK complex act on it?

For example, if I'm not mistaken we compute $$H^{\mathrm{vtx}}_{0}(V)=V/\operatorname{Res}(V)$$, where $$\operatorname{Res}(V)$$ is the image of all the residues $$\operatorname{res}_{z=0}(v(z))$$ of the fields, which has a very similar flavour to the complex of BdSH.

*Edit, in accordance with David Ben-Zvi's comments it seems that vertex algebra homology as defined in this question would be better defined over a punctured disc $$D^{*}$$.

• Factorization homology of a vertex algebra on the disc recovers the vertex algebra itself. Its factorization algebra on the punctured disc recovers the universal enveloping algebra of the vertex algebra (the Lie algebra of Fourier coefficients), and the former is a module for the latter (in fact modules for the latter are the same as vertex/chiral modules for the vertex algebra). So it sounds like the vertex algebra cohomology you're asking about is a form of coinvariants for the enveloping algebra? – David Ben-Zvi Feb 7 at 15:48
• @ David, perhaps I'm using non standard terminology but I would have thought that factorization homology is already a sort of coinvariants construction as it is the derived functor of conformal blocks, which is fairly clearly a sort of coinvariants. In particular I don't think I understand your first sentence, do you have a reference handy? – EBz Feb 7 at 17:25
• Eg if one computes H^{fact}_{0}(X, V) for a curve X one can compute it by first choosing a point x and then computing coinv for the action of DR(X\x, V) on the fibre V|_{x}, this isn't immediate but it's not too hard. – EBz Feb 7 at 17:28
• I don't remember exact references but this is in Beilinson-Drinfeld Chiral Algebras (and in my book with Frenkel in the underived case). I find it much easier to understand in the topological context (cf Ayala-Francis, Factorization Homology), where fact.homology is DEFINED by the property that it attaches the (here $E_2$ rather than vertex) algebra to the disc and you extend by colimits over disc embeddings. – David Ben-Zvi Feb 7 at 20:55
• The key property of factorization homology is the Mayer-Vietoris formula, explaining how to glue over a cylinder. In the conformal case that's exactly the formula you mention: blocks on surface is given by relative tensor product of the punctured curve and the disc over the punctured disc, which becomes coinvariants for the "out" fields on the disc. This is consistent, in that if you apply your formula to the disc, you get relative tensor product of punctured disc (ie enveloping algebra) with the disc over the punctured disc, ie the vertex algebra again (hooray for circular reasoning :-) ) – David Ben-Zvi Feb 7 at 20:58

## 1 Answer

If $$X$$ is taken to a be a formal disc $$D$$, then we obtain a vertex algebra. I believe in this case the above construction produces the vertex algebra cohomology (with coefficients in the adjoint module) studied in the work of Bakalov, de Sole, Heluani and Kac. Is this true? If not what is the precise relation?

This is not exactly so. If you take $$X=\mathbb{A}^1$$ and your D-modules are translation equivariant and you consider the translation equivariant CE complex you will get the same complex that we studied.

Chiral algebras on the disc are not the same thing as vertex algebras. If you consider chiral algebras equivariant with respect to the action of the group $$Aut_\mathcal{O}$$ of changes of coordinates then they are the same as quasi-conformal vertex algebras (notation of Frenkel Ben-Zvi). The $$Aut_\mathcal{O}$$-equivariant complex as you describe on the punctured disk will be equivalent to ours in the quasi-conformal setting.

This can also be performed on a formal disc $$D^*$$, in which case one obtains what one might call Vertex Homology. Has this been studied anywhere? Is there a relation with the work of BdSHK cited above, in particular does the BdSHK complex act on it?

Again if you want something that should be called "vertex homology" then you should consider either $$Aut_{\mathcal O}$$ equivariant objects on the disk or translation equivariant objects on the line.

There are a few statements in this answer that are plain wrong cause I had in mind translation invariants sections on the disk. I'll strike them through.

Factorization homology of a unital chiral algebra vanishes on an open proper subset of a curve. Factorization homology of the restriction of a chiral algebra to an affine proper subset of a curve vanishes. This is the content of Lemma 4.3.4 in Beilinson and Drinfeld's book.

I also believe that David's comments about Factorization homology on the disk are wrong. There's a big difference between the topological setting and the algebraic setting and between $$E_2$$ algebras and the chiral/factorization algebras of Beilinson and Drinfeld. In particular, the chiral homology of the disk with coefficient on any unital chiral algebra vanishes. This is because chiral homology is not just the CE complex of a Lie algebra on the Ran space, but also it's de Rham complex. In particular as you point out the degree zero homology of this complex is $$V/V_{(-1)}V = 0$$ and this is because we have the vacuum vector to play around. Therefore the image of $$\mathbb{1}$$ in chiral homology vanishes and then by the last comment before the mentioned Lemma 4.3.4 the whole complex vanishes.

Chiral homology of a curve is really a global object and you cannot detect any of it by looking at formal disks (see however 2.4.12 in BD in agreement with David's comments). The complex studied in BdSHK is purely a local object, or rather a translation equivariant complex on the line. I am not aware of any direct connection between them.

There are however a couple of instances where you can compute the global chiral homology by local considerations: the case of $$X=\mathbb{P}^1$$ and the case of an elliptic curve. Both cases because you have global coordinates. In degree $$0$$ this is the work of Zhu way before chiral homology was defined. In the elliptic curve case we have a preprint with J. van Ekeren describing the nodal curve limit of the first chiral homology group and we are supposed to post a continuation describing the general elliptic curve soon.

Edit: To reflect David's comments. Indeed by $$U\subset X$$ open they really mean an open affine, so that the example of a disconnected curve is not allowed. The point needed is the vanishing of the de Rham cohomology. They use this lemma heavily in proving the quasi-isomorphism of the complex with supports in 4.4.2--4.4.3.

Commutative vertex algebras are not a counter-example to this statement as the functor of coinvariants on the disk is canonically zero as long as your algebras are unital. Here the very naive notion of coinvariants is clear, the Cousin complex on $$D^2$$ is already enough to compute the homology in degree $$0$$ and this group is $$V/V_{(-1)}V$$ (think of homologies with support at the origin).

As for the $$E_2$$ algebras business: here is where things get sloppy. As far as I know the folklore is that $$E_2$$ algebras are essentially the same as topological vertex algebras. With this I have no objection, however notice that the latter is a vertex algebra together with a BRST differential, and that this equivalence is inherently derived. Since the BRST cohomology of a topological vertex algebra is typically stupid (it lies in conformal weight zero and is a commutative algebra) then it's no wonder that the chiral homology viewed as a complex with this extra BRST differential is silly. However the complex without taking this differential into account is still very interesting, you capture irreducible modules and Jacobi elliptic functions on genus $$1$$ (at least in rational cases) and so forth.

The second point is more subtle: I have seen in many places the analogy drawn between Factorization homology in the topological world and chiral homology in the algebraic setting. I may be wrong and very much outdated in the literature that I can cope with, but I have not seen anything written to prove an actual dictionary. There is no formal way of constructing a factorization algebra (in the topological meaning) coming from a factorization algebra in the algebraic world. Folklore expected that if one takes the complex that computes chiral homology on the Ran space of $$X$$ á la BD and one takes an analitification of this and then sections with compact support, this is something that looks like a Factorization algebra in the topological setting. As far as I know the problem is that the process of analytification (or tensoring with the analytic de Rham complex) does not commute with tensor products and this breaks down being a Lie/Commutative algebra.

I don't claim that having a topological intuition is wrong, just that at least when trying to translate that intuition into an actual theorem in BD's setting the situation is not so transparent even in the simplest of cases (for example the zeoro-th Hochshild Homology of higher Zhu algebras equals zero-th chiral homology of vertex algebras, this is a theorem that requires a lot of theory and a bunch of further assumptions that have no topological counterparts).

• This is fantastic, and really clarifies things for me, many thanks! – EBz Feb 10 at 14:00
• Thanks Reimundo! I'm clearly profoundly rusty on this, but I think there's some disconnect here (maybe even just a shift). In the special case of a commutative chiral algebra (not so far from the special case of $E_2$ algebras), ie functions on the D-scheme of jets of maps into a target, BD explain that chiral homology is just functions on the mapping space - so we're not getting zero on the disc (except for the issue of left vs right D-modules, i.e. I may be off by a shift). Certainly in this case chiral homology makes sense locally as well as globally and agrees with the $E_2$ notion – David Ben-Zvi Feb 10 at 16:11
• I'm also not sure I agree with the interpretation of their Lemma 4.3.4, which is about chiral homology of a proper curve with coefficients in $j_*j^*$. To take an extreme example, I could ask for factorization homology (say in topology) of a disjoint union of surfaces where I take zero on one and whatever you want on the other, and will always get zero.. – David Ben-Zvi Feb 10 at 16:13
• The point I was making about a disjoint union is that factorization homology is a version of a "restricted tensor product" over points of the curve, so it's not surprising that if you put 0 at one point you get 0. That's different from taking restricted tensor product over a subset of the curve, which is what I think of fact.homology on an open curve (or disc) as being. – David Ben-Zvi Feb 11 at 2:35
• I don't understand your claim in the commutative case. Do you disagree with the statement that chiral homology/conformal blocks of an induced commutative fact.algebra - ie functions on jets of maps into some target - is functions on the mapping space? (that's a theorem in BD - or am I misinterpreting something?). So you get something nonzero on the disc -it's just shifted into a different cohomological degree because we're thinking of right not left D-modules. – David Ben-Zvi Feb 11 at 2:38

format: plain


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# Math Help - sum of squares

1. ## sum of squares

Suppose we want to find a closed formula for the sum of the $N$ squares. Start with the formula $\sum_{n=0}^{N} x^n = \frac{x^{N+1}-1}{x-1}$. Then do we do the following:

$\sum_{n=0}^{N} (xD)^n = \frac{(xD)^{N+1}-1}{(xD)-1}$

e.g. apply the $(xD)^2$ operator?

2. What do you call an (xD)² operator ?

And yes the second formula is correct too...


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# Since few days: Trouble with mhchem using \listoftables or \listoffigures

I'm having trouble with the mhchem package. When I have a figure with a separate TOC entry that involves a \ce{} environment, I get error messages saying that the control sequence \aaaa is not defined. Few days ago another one have had the same problems, solved by reinstalling MiKTeX, but this does not work for me. At home, with an older version of the packages the MWE works, but in the new version it does not. Here's my MWE:

\documentclass{scrreprt}
\usepackage[T1]{fontenc}
\usepackage[version=3]{mhchem}
\begin{document}
\tableofcontents
\listoffigures
\listoftables
\begin{figure}[htbp]
\centering
test
\caption[\ce{SiO2}]{\ce{SiO2}}
\label{fig:test}
\end{figure}
\end{document}

Thanks for any help! The same Version of the document was running without any problems a few days ago!

• Welcome to TeX.sx! You probably need to protect \ce: \caption[\protect\ce{SiO2}]{\ce{SiO2}} – clemens Mar 27 '14 at 10:43
• Doesn´t work for me... The problem is related to a really long document with many many formulas ... It would be a mess, if i would have to edit every single caption. – KaLi Mar 27 '14 at 10:55
• works for your MWE (the *.lof file needs to be rewritten of course, so two compilations are necessary before the error disappears) – clemens Mar 27 '14 at 11:03
• Did you have the error for the non-edited MWE and then no error with \protect? Really? What am I doing wrong? – KaLi Mar 27 '14 at 12:10
• yes I had the error with your MWE and it vanished with \protect and compiling twice. In the first compilation the old lof file is used and thus the error still occurs. In the second compilation the new lof file is used which compiles correctly. – clemens Mar 27 '14 at 12:13

My guess is that there is some mistyping in a new version of the package. I enclose a patch of what I think should be used there.

%! *latex malmhchem.tex
%   two runs
\documentclass{scrreprt}
\usepackage[T1]{fontenc}
\usepackage[version=3]{mhchem}
\begin{document}
\def\aaaa#1{\ce{#1}}
\chapter{Chemistry}
\section{My mhchem}
Some text.
\tableofcontents
\listoffigures
\listoftables
\begin{figure}[!htbp]
\centering Testing phase.
\caption[\ce{SiO2}]{\ce{SiO2}}
\label{fig:test}
\end{figure}
\end{document}

• Wow, you´re great - it works!! Thanks a lot! (It is the really first time that I had to post a question by myself in 5 years of using LateX...) – KaLi Mar 27 '14 at 13:33

Ooooops! Version 3.17 fixes that and will become available soon.

2014-03-27 mhchem v3.17
• fix of the 3.16 expansion fix – sorry, my quality control failed


text:
# Proof - Limit of a function

• Aug 25th 2010, 10:18 AM
tarheelborn
Proof - Limit of a function
Suppose that f(x) = mx + b with m > 0. Let c be a real number. Prove that
lim x-> c (f(x)) = mc + b.
• Aug 25th 2010, 10:21 AM
Ackbeet
What ideas have you had so far?
• Aug 25th 2010, 10:36 AM
tarheelborn
I know that |mx + b -(mc + b)| < \epsilon. So |mx + b -mc -b| < \epsilon. Then |mx -mc| < \epsilon. This can be factored as |m||x-c|<\epsilon. But I am not sure where to go from h ere.
• Aug 25th 2010, 10:57 AM
Ackbeet
All that looks good so far. In looking at the definition of limit, you've got

$\displaystyle{\lim_{x\to c}f(x)=L}$ if for every $\epsilon>0$ there exists a $\delta>0$ such that if $0<|x-c|<\delta$, then $|f(x)-L|<\epsilon.$

Overall, your goal is to find a $\delta(\epsilon)$ such that these conditions are satisfied. So, what do you want $\delta$ to be?
• Aug 25th 2010, 11:53 AM
tarheelborn
Ah, so if I let \delta = \epsilon/m, then |m(x-c)|<\epsilon which is what I am trying to prove. Right? Thank you!
• Aug 25th 2010, 11:54 AM
Ackbeet
Close. Note that $\delta>0.$ So that requires what modification?
• Aug 25th 2010, 12:08 PM
tarheelborn
\epsilon/m > 0.
• Aug 25th 2010, 12:51 PM
Ackbeet
Oh, you're right. The problem assumed positive slopes. I was led astray by your unnecessary absolute values in post # 3. If the problem allowed negative slopes, you could just slap on absolute values and you'd be good to go.

So... a final proof looks like what?
• Aug 25th 2010, 12:59 PM
tarheelborn
Let \epsilon > 0. Choose \delta = \epsilon/m. Now if 0<|x-c|<\delta, then -\delta < x-c < \delta.
-\epsilon/m < x-c < \epsilon
-\epsilon < m(x-c) < \epsilon
|m(x-c)| < \epsilon
|mx - mc| < \epsilon
|mx + b - mc - b| < \epsilon
|mx + b - (mc + b)| < \epsilon
==> Lim x->c (mx + b) = mc + b.
Q.E.D.

I probably have too many of my initial calculations in there, but I want to make sure it flows from start to finish.
• Aug 25th 2010, 01:01 PM
Ackbeet
Looks good to me. Your initial calculations in post # 3 are the ones that you have to reverse in the actual proof. So this is a very typical delta-epsilon proof.

I'd say you're done!


text:
# Math Help - Basic Probablility

1. ## Basic Probablility

Hi everyone. I am a math tutor at a college and we don't have any tutors here today that have done stats, so I was wondering if you guys could take a look at this and give me a thorough explanation of how to do the following problem:

The data in the table are from the voyage of the Titanic and show the survivor rates broken down by class of passenger. Use the table to answer the questions below.

$\begin{tabular}{c|c|c|c|c|c}
\null &\text{first} &\text{second}&\text{third}&\text{crew}&\text{Tota l}\\ \hline
\text{Survived}&199 &119&174&214&706\\\text{Died}&130&166&536&685&1517 \\\text{Total}&359&285&710&899&2223
\end{tabular}$

Find the following:

1. $P(\text{surviving})$

2. $P(\text{surviving }|\text{ \null }1^{st}\text{ class passenger})$

3. $P(\text{surviving }|\text{ \null }3^{rd}\text{ class passenger})$

4. Are the events of Surviving and class of passenger independent? Justify your answer.

So, for 1., would it be $P(\text{surviving})=\frac{\text{number of survivors}}{\text{total number of passengers}}$;

For 2., $P(\text{surviving }|\text{ \null }1^{st}\text{ class passenger})=\frac{\text{number of first class survivors}}{\text{total number of passengers}}$;

for 3., $P(\text{surviving }|\text{ \null }3^{rd}\text{ class passenger})=\frac{\text{number of third class survivors}}{\text{total number of passengers}}$;

and, finally for 4., The events are dependent because those passengers in 1st class were (at that time in history) given the first option to board the lifeboats.

Okay, these answers are what I've come up with. Could someone please do the problem professionally and give a really good explanation along the way? I'm trying to learn stats so that I can better help the students enrolled in the course.

Thanks.

2. Originally Posted by VonNemo19
Hi everyone. I am a math tutor at a college and we don't have any tutors here today that have done stats, so I was wondering if you guys could take a look at this and give me a thorough explanation of how to do the following problem:

The data in the table are from the voyage of the Titanic and show the survivor rates broken down by class of passenger. Use the table to answer the questions below.

$\begin{tabular}{c|c|c|c|c|c}
\null &\text{first} &\text{second}&\text{third}&\text{crew}&\text{Tota l}\\ \hline
\text{Survived}&199 &119&174&214&706\\\text{Died}&130&166&536&685&1517 \\\text{Total}&359&285&710&899&2223
\end{tabular}$

Find the following:

1. $P(\text{surviving})$

2. $P(\text{surviving }|\text{ \null }1^{st}\text{ class passenger})$

3. $P(\text{surviving }|\text{ \null }3^{rd}\text{ class passenger})$

4. Are the events of Surviving and class of passenger independent? Justify your answer.

So, for 1., would it be $P(\text{surviving})=\frac{\text{number of survivors}}{\text{total number of passengers}}$;

For 2., $P(\text{surviving }|\text{ \null }1^{st}\text{ class passenger})=\frac{\text{number of first class survivors}}{\text{total number of passengers}}$;

for 3., $P(\text{surviving }|\text{ \null }3^{rd}\text{ class passenger})=\frac{\text{number of third class survivors}}{\text{total number of passengers}}$;

and, finally for 4., The events are dependent because those passengers in 1st class were (at that time in history) given the first option to board the lifeboats.

Okay, these answers are what I've come up with. Could someone please do the problem professionally and give a really good explanation along the way? I'm trying to learn stats so that I can better help the students enrolled in the course.

Thanks.
1. Correct

2. Denominator should be Number of 1st class passengers - because the conditional probability P(A|B)= $\frac{P(A and B)}{P(B)}$

3. Denominator should be Number of 3rd class passengers -same reason as at 2.

4. Test for independent events: A and B are independent if P(A and B) = P(A) * P(B) - apply the test for this case and see if the 2 events in your question pass it or not.


text:
# CAT Quant Practice Problems

Question: Let x and y be positive integers such that x is prime and y is composite. Then,

 y – x cannot be an even integer xy cannot be an even integer $\frac{{\left( {x + y} \right)}}{x}$ cannot be an even integer None of these

#### CAT Quant Online Course

• 1000+ Practice Problems
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text:
Contents

### Context

#### Duality in string theory

duality in string theory

general mechanisms

string-fivebrane duality

string-string dualities

M-theory

F-theory

string-QFT duality

QFT-QFT duality:

#### Quantum field theory

functorial quantum field theory

# Contents

## Idea

### General idea

The AdS-CFT correspondence at its heart is the observation (Witten 98, Section 2.4) that the classical action functionals for various fields coupled to Einstein gravity on anti de Sitter spacetime are, when expressed as functions of the asymptotic boundary-values of the fields, of the form of generating functions for correlators/n-point functions of a conformal field theory on that asymptotic boundary, in a large N limit.

This is traditionally interpreted as a concrete realization of a vague “holographic principle” according to which quantum gravity in bulk spacetimes is controlled, in one way or other, by “boundary field theories” on effective spacetime boundaries, such as event horizons. The original and main motivation for the holographic principle itself was the fact that the apparent black hole entropy in Einstein gravity scales with the area of the event horizon instead of the black hole’s bulk volume (which is not even well-defined), suggesting that gravity encodes or is encoded by some boundary field theory associated with horizons; an idea that, in turn, seems to find a concrete realization in open/closed string duality in the vicinity of, more generally, black branes. The original intuition about holographic black hole entropy has meanwhile found remarkably detailed reflection in (mathematically fairly rigorous) analysis of holographic entanglement entropy, specifically via holographic tensor networks, which turn out to embody key principles of the AdS/CFT correspondence in the guise of quantum information theory, with concrete applications such as to quantum error correcting codes.

The AdS/CFT correspondence itself crucially involves the exceptional isomorphism between the isometry group of anti de Sitter spacetime $AdS_{d+1}$ (the anti de Sitter group) and the conformal group of Minkowski spacetime of dimension $d$: the connected component of both is the special orthogonal group $SO(d,2)$. But the AdS/CFT correspondence is deeper and more subtle than this group theory underlying it, in particular in how it puts fields and states on the gravity side in correspondence with sources and correlators on the field theory side, respectively.

In extrapolation of these elementary computations, the AdS/CFT correspondence conjecturally extends to a more general identification of states of gravity (quantum gravity) on asymptotically anti de Sitter spacetimes of dimension $d+1$ with correlators/n-point functions of conformal field theories on the asymptotic boundary of dimension $d$ (Gubser-Klebanov-Polyakov 98 (12), Witten 98, (2.11)), such that perturbation theory on one side of the correspondence relates to non-perturbation on the other side.

While this works to some extent quite generally (see e.g. Natsuume 15 for review), allowing applications such as AdS/CFT in condensed matter physics and AdS/CFT in quantum chromodynamics, the tightest form of the correspondence relates the 1/N expansion of superconformal field theories (super Yang-Mills theories) on the asymptotic boundaries of near-horizon limits of $N$ coincident black M2-branes/D3-branes/M5-branes to corresponding sectors of the string theory/M-theory quantum gravity in the bulk spacetime away from the brane.

Before the proposal for the actual matching rule of ADS/CFT (Gubser-Klebanov-Polyakov 98 (12), Witten 98, (2.11)) it was by matching of BPS-states in these situations that the existence of an AdS/CFT correspondence was proposed in Maldacena 97a, Maldacena 97b; these articles are now widely regarded as the origin of the idea of the AdS/CFT correspondence.

A quick way to see that the supersymmetric-cases of AdS/CFT for near horizon geometries of M2-branes, D3-branes and M5-branes must be special is to observe that these are the only dimensions in which there are super anti de Sitter spacetime-enhancements of anti de Sitter spacetime, matching the classification of simple superconformal symmetries, see there:

$d$$N$superconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
$\phantom{A}3\phantom{A}$$\phantom{A}2k+1\phantom{A}$$\phantom{A}B(k,2) \simeq$ osp$(2k+1 \vert 4)\phantom{A}$$\phantom{A}SO(2k+1)\phantom{A}$
$\phantom{A}3\phantom{A}$$\phantom{A}2k\phantom{A}$$\phantom{A}D(k,2)\simeq$ osp$(2k \vert 4)\phantom{A}$$\phantom{A}SO(2k)\phantom{A}$M2-brane
D=3 SYM
BLG model
ABJM model
$\phantom{A}4\phantom{A}$$\phantom{A}k+1\phantom{A}$$\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}$$\phantom{A}U(k+1)\phantom{A}$D3-brane
D=4 N=4 SYM
D=4 N=2 SYM
D=4 N=1 SYM
$\phantom{A}5\phantom{A}$$\phantom{A}1\phantom{A}$$\phantom{A}F(4)\phantom{A}$$\phantom{A}SO(3)\phantom{A}$D4-brane
D=5 SYM
$\phantom{A}6\phantom{A}$$\phantom{A}k\phantom{A}$$\phantom{A}D(4,k) \simeq$ osp$(8 \vert 2k)\phantom{A}$$\phantom{A}Sp(k)\phantom{A}$M5-brane
D=6 N=(2,0) SCFT
D=6 N=(1,0) SCFT

(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)

It had already been observed in (Duff-Sutton 88, see Duff 98, Duff 99) that the field theory of small perturbation of a Green-Schwarz sigma-model for a fundamental brane stretched over the asymptotic boundary of the AdS near horizon geometry of its own black brane-incarnation is, after diffeomorphism gauge fixing, a conformal field theory. This was further developed in Claus-Kallosh-Proeyen 97, DGGGTT 98, Claus-Kallosh-Kumar-Townsend 98, Pasti-Sorokin-Tonin 99. See also at super p-brane – As part of the AdS-CFT correspondence

More recently, for the archetypical case of AdS/CFT relating N=4 D=4 super Yang-Mills theory to type IIB string theory on super anti de Sitter spacetime $AdS_5 \times S^5$, fine detailed checks of the correspondence have been performed (Beisert et al. 10, Escobedo 12), see the section Checks below.

Thus regarded as a duality in string theory, the AdS/CFT correspondence is an incarnation of open/closed string duality, reflecting the fact that the physics on D-branes has two equivalent descriptions:

1) as a Yang-Mills-gauge theory coming from open strings attached the brane

2) as a gravity theory coming from closed strings emitted/absorbed by the brane.

graphics grabbed from Schomerus 07, Figure 4, see also e.g. Peschanskia 09, Figure 1

This gives a vivid intuitive picture of the mechanism underlying the correspondence: An excitation of the gauge field on the brane goes along with an excitation of the field of gravity around the brane, and either is faithfully reflected in the other; at least in the suitable limits.

### Small-$N$ corrections

The AdS/CFT correspondence has been widely discussed and is mostly understood by default only in the large $N$ limit and for large 't Hooft coupling, where the given gauge theory is dual to plain classical supergravity, which stands out as being particularly tractable and well-understood.

But it is expected (AGMOO99, p. 60) that the duality still applies in the opposite large 1/N limit, now involving on the gravity-side corrections

1. from perturbative string theory (for small 't Hooft coupling, there are some checks of such stringy corrections) and

2. from putative M-theory (for the full non-perturbative large 1/N limit, which remains largely unexplored):

$\,$

(graphics from SS22)

Notice that for real-world applications such as to the confinement/mass gap-problem of quantum chromodynamics, the value of $N$ typically is indeed small (the number of colors in quantum chromodynamics is $N_c = 3$) so that the string theory/M-theory-corrections to the AdS/QCD correspondence are going to be crucial for the full discussion of these applications:

In lack of a full formulation of M-theory (see M-theory – The open problem) approximate forms of the AdS/CFT correspondence away from the case of conformal invariance, supersymmetry, large N limit and/or exact anti de Sitter geometry are being argued to be of use for understanding quantum chromodynamics (for instance the quark-gluon plasma (Policastro-Son-Starinets 01, but most notably confined hadron-spectra – the AdS/QCD correspondence) and for various models in solid state physics (the AdS-CFT in condensed matter physics, see e.g. Hartnoll-Lucas-Sachdev 16).

More in detail, since the near horizon geometry of BPS black branes is conformal to the Cartesian product of anti de Sitter spaces with the unit $n$-sphere around the brane, the cosmology of intersecting D-brane models realizes the observable universe on the asymptotic boundary of an approximately anti de Sitter spacetime (see for instance Kaloper 04, Flachi-Minamitsuji 09). The basic structure is hence that of Randall-Sundrum models, but details differ, such as notably in warped throat geometries, see Uranga 05, section 18.

These warped throat models go back to Klebanov-Strassler 00 which discusses aspects of confinement in Yang-Mills theory on conincident ordinary and fractional D3-branes at the singularity of a warped conifold. See also Klebanov-Witten 98

snippet grabbed from Uranga 05, section 18

here: “RS”=Randall-Sundrum model; “KS”=Klebanov-Strassler 00

In particular this means that AdS-CFT duality applies in some approximation to intersecting D-brane models (e.g. Soda 10, GHMO 16), thus allowing to compute, to some approximation, non-perturbative effects in the Yang-Mills theory on the intersecting branes in terms of gravity on the ambient warped throat $\sim$ AdS (Klebanov-Strassler 00, section 6)

### Matching single trace observables to string excitation

The single trace operators/observables in conformal field theories such as super Yang-Mills theories play a special role in the AdS-CFT correspondence: They correspond to single string excitations on the AdS-supergravity side of the correspondence, where, curiously, the “string of characters/letters” in the argument of the trace gets literally mapped to a superstring in spacetime (see the references below).

From Polyakov 02, referring to gauge fields and their single trace operators as letter and words, respectively:

The picture which slowly arises from the above considerations is that of the space-time gradually disappearing in the regions of large curvature. The natural description in this case is provided by a gauge theory in which the basic objects are the texts formed from the gauge-invariant words. The theory provides us with the expectation values assigned to the various texts, words and sentences.

These expectation values can be calculated either from the gauge theory or from the strongly coupled 2d sigma model. The coupling in this model is proportional to the target space curvature. This target space can be interpreted as a usual continuous space-time only when the curvature is small. As we increase the coupling, this interpretation becomes more and more fuzzy and finally completely meaningless.

From Berenstein-Maldacena-Nastase 02, who write $Z$ for the elementary field observables (“letters”) $\mathbf{\Phi}$ above:

In summary, the “string of $Z$s” becomes the physical string and that each $Z$ carries one unit of $J$ which is one unit of $p_+$. Locality along the worldsheet of the string comes from the fact that planar diagrams allow only contractions of neighboring operators. So the Yang Mills theory gives a string bit model where each bit is a $Z$ operator.

On the CFT side these BMN operators of fixed length (of “letters”) are usefully identified as spin chains which, with the dilatation operator regarded as their Hamiltonian, are integrable systems (Minahan-Zarembo 02, Beisert-Staudacher 03).

This integrability allows a detailed matching between

under AdS/CFT duality (Beisert-Frolov-Staudacher-Tseytlin 03, …). For review see BBGK 04, Beisert et al. 10.

(…)

## Checks

At the heart of the duality is the observation that the classical action functionals for various fields coupled to Einstein gravity on anti de Sitter spacetime are, when expressed as functions of the asymptotic boundary values of the fields, equal to the generating functions for the correlators/n-point functions of a conformal field theory on that asymptotic boundary.

These computations were laid out in Witten 98, section 2.4 “Some sample computation”. These follow from elementary manipulation in differential geometry (involving neither supersymmetry nor string theory). A good exposition is in Hartnoll-Lucas-Sachdev 16, Section 1.6

For the more ambitious matching of the spectrum of the dilatation operator of N=4 D=4 super Yang-Mills theory to the corresponding spectrum of the Green-Schwarz superstring on the super anti de Sitter spacetime $AdS_5 \times S^5$ detailed checks are summarized in Beisert et al. 10, Escobedo 12

graphics grabbed from Escobedo 12

Comparison to string scattering amplitudes beyond the planar SCFT limit: ABP 18.

Numerical checks using lattice gauge theory are reviewed in Joseph 15.

Exact duality checks pertaining to the full stringy regime for $AdS_3/CFT_2$: Eberhardt-Gaberdiel 19a, Eberhardt-Gaberdiel 19b, Eberhardt-Gaberdiel-Gopakumar 19. For more see the references there.

## Examples

The solutions to supergravity that preserve the maximum of 32 supersymmetries are (e.g. HEGKS 08 (1.1))

as well as their Minkowski spacetime and plane wave limits. These are the main KK-compactifications for the following examples-

### $AdS_5 / CFT_4$ – Horizon limit of D3-branes

type II string theory on 5d anti de Sitter spacetime (times a 5-sphere) is dual to N=4 D=4 super Yang-Mills theory on the worldvolume of a D3-brane at the asymptotic boundary

### $AdS_7 / CFT_6$ – Horizon limit of M5-branes

We list some of the conjectured statements and their evidence concerning the case of $AdS_7/CFT_6$-duality.

The hypothesis (Maldacena 97, section 3.1) (see (Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.1) for a review) is that

is holographically related to

In

effectively this relation was already used to computed the 5-brane partition function in the abelian case from the states of abelian 7d Chern-Simons theory. (The quadratic refinement of the supergravity C-field necessary to make this come out right is what led to Hopkins-Singer 02 and hence to the further mathematical development of differential cohomology and its application in physics.)

In (Witten 98, section 4) this construction is argued for from within the framework of AdS/CFT, explicitly identifying the 7d Chern-Simons theory here with the compactification of the 11-dimensional Chern-Simons term of the supergravity C-field in 11-dimensional supergravity, which locally is

\begin{aligned} S_{11d SUGRA, CS}(C_3) &= \int_{AdS_7} \int_{S^4} C_3 \wedge G_4 \wedge G_4 \\ & = N \, \int_{AdS_7} C_3 \wedge G_4 \end{aligned} \,.

But in fact the quantum anomaly cancellation (GS-type mechanism) for 11d sugra introduces a quantum correction to this Chern-Simons term (DLM, equation (3.14)), making it locally become

\begin{aligned} S(\omega,C_3) &= \int_{AdS_7} \int_{S^4} C_3 \wedge G_4 \wedge (G_4 + I_8(\omega)) \\ & = N \, \int_{AdS_7} \left( C_3 \wedge G_4 + \frac{1}{48} CS_{p_2}(\omega) - \frac{1}{12} CS_{\frac{1}{2}p_1}(\omega) \wedge tr(F_\omega \wedge \omega) \right) \end{aligned} \,,

where now $\omega$ is the local 1-form representative of a spin connection and where $CS_{p_2}$ is a Chern-Simons form for the second Pontryagin class and $CS_{\frac{1}{2}p_1}$ for the first.

That therefore not an abelian, but this nonabelian higher dimensional Chern-Simons theory should be dual to the nonabelian 6d (2,0)-superconformal QFT was maybe first said explicitly in (LuWang 2010).

Its gauge field is hence locally and ignoring the flux quantization subtleties a pair consisting of the abelian 3-form field $C$ and a Spin group $Spin(6,1)$-valued connection (see supergravity C-field for global descriptions of such pairs). Or maybe rather $Spin(6,2)$ to account for the constraint that the configurations are to be asymptotic anti de Sitter spacetimes (in analogy to the well-understood situation in 3d quantum gravity, see there for more details).

Indeed, in (SezginSundell 2002, section 7) more detailed arguments are given that the 7-dimensional dual to the free 6d theory is a higher spin gauge theory for a higher spin gauge group extending the (super) conformal group $SO(6,2)$.

A non-perturbative description of this nonabelian 7d Chern-Simons theory as a local prequantum field theory (hence defined non-perturbatively on the global moduli stack of fields (twisted differential string structures, in fact)) was discussed in (FSS 12a, FSS 12b).

General discussion of boundary local prequantum field theories relating higher Chern-Simons-type and higher WZW-type theories is in (dcct 13, section 3.9.14). Specifically, a characterization along these lines of the Green-Schwarz action functional of the M5-brane as a holographic higher WZW-type boundary theory of a 7d Chern-Simons theory is found in (FSS 13).

Analogous discussion of the 6d theory as a higher WZW analog of a 7d Chern-Simons theory phrased in terms of extended quantum field theory is (Freed 12).

### $AdS_4 / CFT_3$ –Horizon limit of M2-branes

11d supergravity/M-theory on the asymptotic $AdS_4$

spacetime of an M2-brane.

### $AdS_3 / CFT_2$ – Horizon limit of D$p$-D($p+k$) brane bound states

(for more see at AdS3-CFT2 and CS-WZW correspondence)

(Dibitetto-Petri 17, …)

(Dibitetto-Petri 18, …)

### Non-conformal duals

#### QCD models

While all of the above horizon limits product super Yang-Mills theory, one can consider certain limits of these in which they look like plain QCD, at least in certain sectors. This leads to a discussion of holographic description of QCD properties that are actually experimentally observed.

### Further gauge theories induced by compactification and twisting

gauge theory induced via AdS-CFT correspondence

11d supergravity/M-theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
$\;\;\;\;\downarrow$ topological sector
7-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invarianceM5-brane worldvolume theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-brane worldvolume theory with type IIB S-duality
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$, Donaldson theory

$\,$

type II string theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$
$\;\;\;\; \downarrow$ topological sector
5-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence

## Formalizations

The full formalization of AdS/CFT is still very much out of reach, but maybe mostly for lack of trying.

But see Anderson 04.

One proposal for a formalization of a toy version in the context of AQFT is Rehren duality. However, it does not seem that this actually formalizes AdS-CFT, but something else.

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
$(D = 2n)$type IIA$\,$$\,$
D(-2)-brane$\,$$\,$
D0-brane$\,$$\,$BFSS matrix model
D2-brane$\,$$\,$$\,$
D4-brane$\,$$\,$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\,$$\,$D=7 super Yang-Mills theory
D8-brane$\,$$\,$
$(D = 2n+1)$type IIB$\,$$\,$
D(-1)-brane$\,$$\,$$\,$
D1-brane$\,$$\,$2d CFT with BH entropy
D3-brane$\,$$\,$N=4 D=4 super Yang-Mills theory
D5-brane$\,$$\,$$\,$
D7-brane$\,$$\,$$\,$
D9-brane$\,$$\,$$\,$
(p,q)-string$\,$$\,$$\,$
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection$\,$
string$\,$B2-field2d SCFT
NS5-brane$\,$B6-fieldlittle string theory
D-brane for topological string$\,$
A-brane$\,$
B-brane$\,$
M-brane11D SuGra/M-theorycircle n-connection$\,$
M2-brane$\,$C3-fieldABJM theory, BLG model
M5-brane$\,$C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
M-wave
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\,$C6-field on G2-manifold
S-brane
SM2-brane,
membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

## References

### Original articles

The rough conjecture originates in:

• Juan Maldacena, The Large $N$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2:231, 1998 (hep-th/9711200)

• Juan Maldacena, Wilson loops in Large $N$ field theories, Phys. Rev. Lett. 80 (1998) 4859 (hep-th/9803002)

The actual rule for matching bulk states to generating functions for boundary correlators/n-point functions is due to

Discussion of how Green-Schwarz action functionals for super $p$-branes in AdS target spaces induce, after diffeomorphism gauge fixing, superconformal field theory on the worldvolumes (see singleton representation) goes back to

• Mike Duff, C. Sutton, The Membrane at the End of the Universe, New Sci. 118 (1988) 67-71 (spire:268230)

and was further developed in

Review is in

The resulting super-conformal brane scan:

Sketch of a derivation of AdS/CFT:

Further references include:

• Oscar Varela, Super-Chern-Simons spectra from Exceptional Field Theory (arXiv:2010.09743)

### Introductions and surveys

Surveys and introductions include

Review of Yangian symmetry:

### Lattice gauge theory computations

Review of lattice gauge theory-numerics for the AdS-CFT correspondence:

• Anosh Joseph, Review of Lattice Supersymmetry and Gauge-Gravity Duality (arXiv:1509.01440)

Using the KK-compactification of D=4 N=4 super Yang-Mills theory to the BMN matrix model for lattice gauge theory-computations in D=4 N=4 SYM and for numerical checks of the AdS-CFT correspondence:

• Masazumi Honda, Goro Ishiki, Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, Direct test of the AdS/CFT correspondence by Monte Carlo studies of N=4 super Yang-Mills theory, JHEP 1311 (2013) 200 (arXiv:1308.3525)

### On single trace operators

The correspondence of single trace operators to superstring excitations under the AdS-CFT correspondence originates with these articles:

The identification of the relevant single trace operators with integrable spin chains is due to

which led to more detailed matching of single trace operators to rotating string excitations in

Review includes

### SYK-model in $AdS_2/CFT_1$

Discussion of the SYK-model as the AdS/CFT dual of JT-gravity in nearly AdS2/CFT1 and AdS-CFT in condensed matter physics:

Original articles:

Review:

Relation to black holes in terms of Majorana dimer states:

• Ioanna Kourkoulou, Juan Maldacena, Pure states in the SYK model and nearly-$AdS_2$ gravity (arXiv:1707.02325)

Relation to black holes in string theory and random matrix theory:

• Yuri D. Lensky, Xiao-Liang Qi, Pengfei Zhang, Size of bulk fermions in the SYK model (arXiv:2002.01961)

• Xiao-Liang Qi, Pengfei Zhang, The Coupled SYK model at Finite Temperature (arXiv:2003.03916)

• Akash Goel, Herman Verlinde, Towards a String Dual of SYK (arXiv:2103.03187)

• Tarek Anous, Felix M. Haehl, The quantum $p$-spin glass model: A user manual for holographers (arXiv:2106.03838)

Discussion of small N corrections via a lattice QFT-Ansatz on the AdS side:

• Richard C. Brower, Cameron V. Cogburn, A. Liam Fitzpatrick, Dean Howarth, Chung-I Tan, Lattice Setup for Quantum Field Theory in $AdS_2$ (arXiv:1912.07606)

• Gregory J. Galloway, Melanie Graf, Eric Ling, A conformal infinity approach to asymptotically $AdS_2 \times S^{n-1}$ spacetimes (arXiv:2003.00093)

### BFSS matrix model in $AdS_2/CFT_1$

On AdS2/CFT1 with the BFSS matrix model on the CFT side and black hole-like solutions in type IIA supergravity on the AdS side:

and on its analog of holographic entanglement entropy:

• Takeshi Morita, Hiroki Yoshida, A Critical Dimension in One-dimensional Large-N Reduced Models (arXiv:2001.02109)

### On $AdS_3 / CFT_2$

(For more see the references at AdS3/CFT2.)

An exact correspondence of the symmetric orbifold CFT of Liouville theory with a string theory on $AdS_3$ is claimed in:

based on

On black$\;$D6-D8-brane bound states in massive type IIA string theory, with defect D2-D4-brane bound states inside them realizing AdS3-CFT2 “inside” AdS7-CFT6:

On black$\;$D4-D8-brane bound states in massive type IIA string theory, with defect D2-D6-brane bound states inside them realizing AdS3-CFT2 “inside” AdS7-CFT6:

### On $AdS_5 / CFT_4$

Computing dual string scattering amplitudes by AdS/CFT beyond the planar limit:

### On $AdS_7 / CFT_6$

We list references specific to $AdS_7/CFT_6$.

In

it is argued that the conformal blocks of the 6d (2,0)-superconformal QFT are entirely controled just by the effective 7d Chern-Simons theory inside 11-dimensional supergravity, but only the abelian piece is discussed explicitly.

The fact that this Chern-Simons term is in fact a nonabelian higher dimensional Chern-Simons theory in $d = 7$, due the quantum anomaly cancellation, is clear from the original source, equation (3.14) of

but seems not to be noted explicitly in the context of $AdS_7/CFT_6$ before the references

• H. Lü, Yi Pang, Seven-Dimensional Gravity with Topological Terms Phys.Rev.D81:085016 (2010) (arXiv:1001.0042)

• H. Lu, Zhao-Long Wang, On M-Theory Embedding of Topologically Massive Gravity Int.J.Mod.Phys.D19:1197 (2010) (arXiv:1001.2349)

More on the relation between the M5-brane and supergravity on $AdS_7 \times S^4$ and arguments for the $SO(5)$ R-symmetry group on the 6d theory from the 7d theory are given in

• M. Nishimura, Y. Tanii, Local Symmetries in the $AdS_7/CFT_6$ Correspondence, Mod. Phys. Lett. A14 (1999) 2709-2720 (arXiv:hep-th/9910192)

Discussion of the $CFT_6$ in $AdS_7/CFT_6$ via conformal bootstrap:

In

arguments are given that the 7d theory is a higher spin gauge theory extension of $SO(6,2)$.

### Generalization beyond exact AdS / exact CFT

Discussion for cosmology of intersecting D-brane models (ambient $\sim$ anti de Sitter spacetimes with the $\sim$ conformal intersecting branes at the asymptotic boundary) includes the following (see also at Randall-Sundrum model):

• Igor Klebanov, Matthew Strassler, Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi^{SB}$-Resolution of Naked Singularities, JHEP 0008:052, 2000 (arXiv:hep-th/0007191)

• Igor Klebanov, Edward Witten, Superconformal Field Theory on Threebranes at a Calabi-Yau Singularity, Nucl.Phys.B536:199-218, 1998 (arXiv:hep-th/9807080)

• Nemanja Kaloper, Origami World, JHEP 0405 (2004) 061 (arXiv:hep-th/0403208)

• Angel Uranga, section 18 of TASI lectures on String Compactification, Model Building, and Fluxes, 2005 (pdf)

• Antonino Flachi, Masato Minamitsuji, Field localization on a brane intersection in anti-de Sitter spacetime, Phys.Rev.D79:104021, 2009 (arXiv:0903.0133)

• Jiro Soda, AdS/CFT on the brane, Lect.Notes Phys.828:235-270, 2011 (arXiv:1001.1011)

• Shunsuke Teraguchi, around slide 21 String theory and its relation to particle physics, 2007 (pdf)

• Gianluca Grignani, Troels Harmark, Andrea Marini, Marta Orselli, The Born-Infeld/Gravity Correspondence, Phys. Rev. D 94, 066009 (2016) (arXiv:1602.01640)

### pp-Waves as Penrose limits of $AdS_p \times S^q$ spacetimes

Discussion of pp-wave spacetimes as Penrose limits (Inönü-Wigner contractions) of AdSp x S^q spacetimes and of the corresponding limit of AdS-CFT duality:

Review:

• Michael Gutperle, Nicholas Klein, A Penrose limit for type IIB $AdS_6$ solutions (arXiv:2105.10824)

### Applications to physics

#### To gravity

Discussion of event horizons of black holes in terms of AdS/CFT (the “firewall problem”) is in

To black hole interiors:

The SYK model gives us a glimpse into the interior of an extremal black hole…That’s the feature of SYK that I find most interesting…It is a feature this model has, that I think no other model has

To symmetries in gravity:

#### To the quark-gluon plasma

Applications of AdS-CFT to the quark-gluon plasma of QCD:

Expositions and reviews include

Holographic discussion of the shear viscosity of the quark-gluon plasema goes back to

Other original articles include:

• Hovhannes R. Grigoryan, Paul M. Hohler, Mikhail A. Stephanov, Towards the Gravity Dual of Quarkonium in the Strongly Coupled QCD Plasma (arXiv:1003.1138)

• Brett McInnes, Holography of the Quark Matter Triple Point (arXiv:0910.4456)

#### To particle physics

For more see at AdS/QCD correspondence.

#### To condensed matter physics

Textbook account

Further reviews include the following:

### Applications in mathematics

#### To the volume conjecture

Suggestion that the statement of the volume conjecture is really AdS-CFT duality combined with the 3d-3d correspondence for M5-branes wrapped on hyperbolic 3-manifolds:

• Dongmin Gang, Nakwoo Kim, Sangmin Lee, Section 3.2_Holography of 3d-3d correspondence at Large $N$, JHEP04(2015) 091 (arXiv:1409.6206)

• Dongmin Gang, Nakwoo Kim, around (21) of: Large $N$ twisted partition functions in 3d-3d correspondence and Holography, Phys. Rev. D 99, 021901 (2019) (arXiv:1808.02797)

#### To deep learning in neural networks

On the deep learning algorithm on neural networks as analogous to the AdS/CFT correspondence:

• Yi-Zhuang You, Zhao Yang, Xiao-Liang Qi, Machine Learning Spatial Geometry from Entanglement Features, Phys. Rev. B 97, 045153 (2018) (arxiv:1709.01223)

• W. C. Gan and F. W. Shu, Holography as deep learning, Int. J. Mod. Phys. D 26, no. 12, 1743020 (2017) (arXiv:1705.05750)

• J. W. Lee, Quantum fields as deep learning (arXiv:1708.07408)

• Koji Hashimoto, Sotaro Sugishita, Akinori Tanaka, Akio Tomiya, Deep Learning and AdS/CFT, Phys. Rev. D 98, 046019 (2018) (arxiv:1802.08313)

Last revised on September 18, 2022 at 14:40:05. See the history of this page for a list of all contributions to it.


text:
# One or Two Components? The Scattering Transform Answers

With the aim of constructing a biologically plausible model of machine listening, we study the representation of a multicomponent stationary signal by a wavelet scattering network. First, we show that renormalizing second-order nodes by their first-order parents gives a simple numerical criterion to assess whether two neighboring components will interfere psychoacoustically. Secondly, we run a manifold learning algorithm (Isomap) on scattering coefficients to visualize the similarity space underlying parametric additive synthesis. Thirdly, we generalize the "one or two components" framework to three sine waves or more, and prove that the effective scattering depth of a Fourier series grows in logarithmic proportion to its bandwidth.

## Authors

• 21 publications
• 4 publications
• 13 publications
• ### Une ou deux composantes ? La réponse de la diffusion en ondelettes

With the aim of constructing a biologically plausible model of machine l...
05/21/2019 ∙ by Vincent Lostanlen, et al. ∙ 0

• ### Geometric Scattering on Manifolds

We present a mathematical model for geometric deep learning based upon a...
12/15/2018 ∙ by Michael Perlmutter, et al. ∙ 0

• ### Wavelet Scattering on the Pitch Spiral

We present a new representation of harmonic sounds that linearizes the d...
01/03/2016 ∙ by Vincent Lostanlen, et al. ∙ 0

• ### Group Invariant Scattering

This paper constructs translation invariant operators on L2(R^d), which ...
01/12/2011 ∙ by Stéphane Mallat, et al. ∙ 0

• ### Fractional Wavelet Scattering Network and Applications

Objective: The present study introduces a fractional wavelet scattering ...
06/30/2018 ∙ by Li Liu, et al. ∙ 0

• ### wav2shape: Hearing the Shape of a Drum Machine

Disentangling and recovering physical attributes, such as shape and mate...
07/20/2020 ∙ by Han Han, et al. ∙ 0

• ### A PolSAR Scattering Power Factorization Framework and Novel Roll-Invariant Parameters Based Unsupervised Classification Scheme Using a Geodesic Distance

We propose a generic Scattering Power Factorization Framework (SPFF) for...
06/27/2019 ∙ by Debanshu Ratha, et al. ∙ 1

##### This week in AI

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## I Introduction

In the mammalian auditory system, cochlear hair cells operate like band-pass filters whose equivalent rectangular bandwidth (ERB) grows in proportion to their center frequency. Given two sine waves and of respective frequencies and , we perceive their mixture as a musical chord insofar as and belong to disjoint critical bands. However, if or , then the tone is said to be masked by . In lieu of two pure tones, we hear a “beating tone”: i.e., a locally sinusoidal wave whose carrier frequency is and whose modulation frequency is . In humans, the resolution of beating tones involves physiological processes beyond the cochlea, i.e., in the primary auditory cortex.

The scattering transform () is a deep convolutional operator which alternates constant- wavelet decompositions and the application of pointwise complex modulus, up to some time scale . Broadly speaking, its first two layers ( and ) resemble the functioning of the cochlea and the primary auditory cortex, respectively. In the context of audio classification, scattering transforms have been succesfully employed to represent speech [anden2014deep], environmental sounds [lostanlen2018jasmp], urban sounds [salamon2015eusipco], musical instruments [lostanlen2018dlfm], rhythms [haider2019cmmr], and playing techniques [wang2019ismir]. Therefore, the scattering transform simultaneously enjoys a diverse range of practical motivations, a firm rooting in wavelet theory, and a plausible correspondence with neurophysiology.

This article discusses the response of the scattering transform operator to a complex tone input , depending on the sinusoidal parameters of and . In this respect, we follow a well-established methodology in nonstationary signal processing, colloquially known as: “One or two frequencies? The X Answers”, where X is the nonlinear operator of interest. The key idea is to identify transitional regimes in the response of X with respect to variations in relative amplitude (), relative frequency (), and relative phase (). Prior publications have done so for X being the empirical mode decomposition [rilling2008tsp], the synchrosqueezing transform [wu2011aada], and the singular spectrum analysis operator [harmouche2015gretsi]. We extend this line of research to the case where X is the scattering transform in dimension one.

## Ii Wavelet-based recursive interferometry

Let a Hilbert-analytic filter with null average, unit center frequency, and an ERB equal to . We define a constant- wavelet filterbank as the family . Each wavelet has a center frequency of , an ERB of , and an effective receptive field of in the time domain. In practice, the frequency variable get discretized according to a geometric progression of common ratio . Consequently, every continuous signal that is bandlimited to activates a number of wavelets at most.

We define the scalogram of as the squared complex modulus of its constant- transform (CQT):

 U1y:(t,λ1)⟼∣∣∣∫+∞−∞y(t′)ψλ1(t−t′)dt′∣∣∣2. (1)

Likewise, we define a second layer of nonlinear transformation for as the “scalogram of its scalogram”:

 U2y:(t,λ1,λ2)⟼∣∣∣∣y∗ψλ1∣∣2∗ψλ2∣∣2(t), (2)

where the asterisk denotes a convolution product. This construct may be iterated for every integer by “scattering” the multivariate signal into all wavelet subbands :

 Um+1y:(t,λ1…λm+1)⟼∣∣Umy(t,λ1…λm)∗ψλm∣∣2(t,λ1…λm). (3)

Note that the original definition of the scattering transform adopts the complex modulus () rather its square (

) as its activation function. This is to ensure that

is a non-expansive map in terms of Lipschitz regularity. However, to simplify our calculation and spare an intermediate stage of linearization of the square root, we choose to employ a measure of power rather than amplitude. This idea was initially proposed by [balestriero2017arxiv] in the context of marine bioacoustics.

Every layer in this deep convolutional network composes an invariant linear system (namely, the CQT) and a pointwise operation (the squared complex modulus). Thus, by recurrence over the depth variable

, every tensor

is equivariant to the action of delay operators. In order to replace this equivariance property by an invariant property, we integrate each over some predefined time scale , yielding the invariant scattering transform:

 Smy:(t,p)⟼∫+∞−∞Um(t′,p)ϕT(t−t′)dt′, (4)

where the -tuple is a scattering path and the signal is a real-valued low-pass filter of time scale .

## Iii Auditory masking in a scattering network

Given , the convolution between every sine wave and every wavelet writes as a multiplication in the Fourier domain. Because is Hilbert-analytic, only the analytic part of the real signal is preserved in the CQT:

 (yn∗ψλ1)(t)=12ˆψλ1(fn)yan(t). (5)

By linearity of the CQT, we expand the interference between and by heterodyning:

 ∣∣(y1+y2)∗ψλ1∣∣2(t)=12∣∣ˆψ(f1λ1)∣∣2a21+12∣∣ˆψ(f2λ1)∣∣2a22 +R(ˆψ(f1λ1)ˆψ∗(f2λ1))a1a2cos((f2−f1)t+(φ2−φ1)). (6)

Because the wavelet has a null average, the two constant terms in the equation above are absorbed by the first layer of the scattering network, and disappear at deeper layers. However, the cross term, proportional to , is a “difference tone” of fundamental frequency .

The authors of a previous publication [anden2012dafx] have remarked that this difference tone elicits a peak in second-order scattering coefficients for the path . In the following, we generalize their study to include the effect of the relative amplitude , the wavelet shape , the quality factor , and the time scale of local stationarity .

Equation 6 illustrates how the scalogram operator converts a complex tone (two frequencies and ) into a simple tone (one frequency ). For this simple tone to carry a nonnegligible amplitude in , three conditions must be satisfied. First, the rectangular term must be nonnegligible in comparison to the square terms and . Secondly, there must exist a wavelet whose spectrum encompasses both frequencies and . Said otherwise, must satisfy the inequalities , both for and for . Thirdly, the frequency difference must belong to the passband of some second-order wavelet . Yet, in practice, to guarantee the temporal localization of scattering coefficients and restrict the filterbank to a finite number of octaves, the scaling factor of every is upper-bounded by the temporal constant . Therefore, the period of the difference tone should be under the pseudo-period of the wavelet with support ; i.e., a pseudo-period of . Hence the third condition: .

One simple way of quantifying the amount of mutual interference between signals and is to renormalize second-order coefficients by their first-order “parent” coefficients:

 ˜S2y(t,λ1,λ2)=S2y(t,λ1,λ2)S1y(t,λ1) (7)

This operation, initially proposed by [anden2014deep], is conceptually analogous to classical methods in adaptive gain control, notably per-channel energy normalization (PCEN) [lostanlen2019spl].

In accordance with the “one or two frequencies” methodology, Figure 1 illustrates the value of this ratio of energies in the subband , for different values of relative amplitude and relative frequency difference . We fixed without loss of generality. As expected, we observe that, for and a relative frequency difference between and , second-layer wavelets resonate with the difference tone as a result of the interference between signals and .

## Iv Application to manifold learning

To demonstrate the ability of the scattering transform to characterize auditory masking, we build a dataset of complex tones according to the following additive synthesis model:

 yα,r(t)=N∑n=11+(−1)nrnαcos(nf1t)ϕT(t), (8)

where is a Hann window of duration . This additive synthesis model depends upon two parameters: the Fourier decay

and the relative odd-to-even amplitude difference

. Figure 2 displays the CQT log-magnitude spectrum of for different values of and . In practice, we set to samples, to harmonics, and between and cycles.

Our synthetic dataset comprises audio signals in total, corresponding to values of between and and values of between and , while is an integer chosen uniformly at random between and . We extract the scattering transform of each signal up to order , with and , by means of the Kymatio Python package [andreux2020jmlr]. Concatenating first-order second-order coefficients yields a representation in dimension .

For visualization purposes, we bring the -dimensional space of scattering coefficients to the dimension three by means of the Isomap algorithm for unsupervised manifold learning [tenenbaum2000science]. The appeal behind Isomap is that pairwise Euclidean distances in the 3-D point cloud approximate the corresponding geodesic distances over the -nearest neighbor graph associated to the dataset. Throughout this paper, we set the number of neighbors to and measure neighboring relationships by comparing high-dimensional distances. Crucially, in the case of the scattering transform, these distances are provably stable (i.e., Lipschitz-continuous) to the action of diffeomorphisms [mallat2012cpam, Theorem 2.12].

Figure 3 (top) illustrates our findings. We observe that, after scattering transform and Isomap dimensionality reduction, the dataset appears as a 3-D Cartesian mesh whose principal components align with , , and respectively. This result demonstrates that the scattering transform is capable of disentangling and linearizing multiple factors of variability in the spectral envelope of periodic signals, even if those factors are not directly amenable to diffeomorphisms in the time domain.

As a point of comparison, Figure 3 presents the outcome of Isomap on alternative feature representations: Open-L3 embedding (center) and mel-frequency cepstral coefficients (MFCC, bottom). The former results from training a deep convolutional network (convnet) on a self-supervised task of audiovisual correspondence, and yields coefficients [cramer2019icassp]. The latter resuts from a log-mel-spectrogram representation, followed by a discrete cosine transform (DCT) over the mel-frequency axis, and yields coefficients. We compute MFCC with librosa v0.7 [mcfee2020librosa] default parameters.

We observe that Open-L3 embeddings correctly disentangles boundary conditions () from fundamental frequency (), but fails to disentangle Fourier decay () from . Instead, correlations between and are positive for low-pitched sounds ( to ) cycles) and negative for high-pitched sounds ( to cycles). Although this failure deserves a more formal inquiry, we hypothesize that this it stems from the small convolutional receptive field of the -Net: mel subbands, i.e., roughly half an octave around kHz.

Moreover, in the case of MFCC, we find that the variability in fundamental frequency () dominates the variability in spectral shape parameters ( and ), thus yielding a rectilinear embedding (top). This observation is in line with a previous publication [lostanlen2016ismir], which showed statistically that MFCCs are overly sensitive to frequency transposition in complex tones.

From this qualitative benchmark, it appears that the scattering transform is a more interpretable representation of periodic signals than Open-L3, while incurring a smaller computational cost. However, in the presence of aperiodic signals such as environmental sounds, Open-L3 outperforms the scattering transform in terms of classification accuracy with linear support vector machines

[arandjelovic2017look]. To remain competitive, the scattering transform must not only capture heterodyne interference, but also joint spectrotemporal modulations [anden2019joint]

. In this context, future work will strive to combine insights from multiresolution analysis and deep self-supervised learning.

## V Beyond pairwise interference: full-depth scattering networks

In speech and music processing, pitched sounds are rarely approximable as a mixture of merely two components. More often than not, they contain ten components or more, and span across multiple octaves in the Fourier domain. Thus, computing the masking coefficient at the second layer only provides a crude description of the timbral content within each critical band. Indeed, encodes pairwise interference between sinusoidal components but fail to characterize more intricate structures in the spectral envelope of .

To address this issue, we propose to study the scattering transform beyond order two, thus encompassing heterodyne structures of greater multiplicity. For the sake of mathematical tractability, we consider the following mother wavelet, hereafter called “complex Shannon wavelet” after [mallat2008book, Section 7.2.2]:

 (9)

The definition of a scattering transform with complex Shannon wavelets requires to resort to the theory of tempered distributions. We refer to [strichartz2003book] for further mathematical details.

The following theorem, proven in the Appendix, describes the response of a deep scattering network in the important particular case of a periodic signal with finite bandwidth.

###### Theorem V.1.

Let a periodic signal of fundamental frequency . Let the complex Shannon wavelet as in Equation 9 and its associated scalogram operator as in Equation 1. If has a finite bandwidth of octaves, then its scattering coefficients are zero for any .

This result is in agreement with the theorem of exponential decay of scattering coefficients [waldspurger2017exponential]. Note, however, that [waldspurger2017exponential] expresses an upper bound on the energy at fixed depth for integrable signals, while we express an upper bound on the depth at fixed bandwidth for periodic signals.

We apply the theorem above to the case of a signal containing components of equal amplitudes, equal phases, and evenly spaced frequencies: . Figure 4 illustrates the decay of scatterered energy as a function of depth. The conceptual analogy between depth and scale was originally proposed by [mallat2016understanding] in a theoretical effort to clarify the role of hierarchical symmetries in convnets.

Although our findings support this analogy, we note that computing a scattering transform with layers is often impractical. However, if the Fourier series in satisfies a self-similarity assumption, it is possible to match the representational capacity of a full-depth scattering network while keeping the depth to . Indeed, spiral scattering performs wavelet convolutions over time, over log-frequency, and across octaves, thereby capturing the spectrotemporal periodicity of Shepard tones and Shepard-Risset glissandos [lostanlen2015dafx]. Further research is needed to integrate broadband demodulation into deep convolutional architectures for machine listening.

## Vi Conclusion

In this article, we have studied the role of every layer in a scattering network by means of a well-established methodology, colloquially known as “one or two components” [rilling2008tsp]. We have come up with a numerical criterion of psychoacoustic masking; demonstrated that the scattering transform disentangles multiple factors of variability in the spectral envelope; and proven that the effective scattered depth of Fourier series is bounded by the logarithm of its bandwidth, thus emphasizing the importance of capturing geometric regularity across temporal scales.

## Appendix: proof of Theorem v.1

###### Proof.

We reason by induction over the depth variable . The base case () leads to if and zero otherwise. Because

has one vanishing moment, it follows that

is zero, and likewise at deeper layers. To prove the induction step at depth , to decompose into a low-pass approximation spanning the subband and a high-pass detail spanning the subband . Denoting by the complex-valued Fourier coefficients of , we have at every time :

 y(t) =(y∗gM)(t) + (y∗hM)(t) =∑|n|≤2Mcnexp(inf1t) + ∑|n|>2Mcnexp(inf1t) (10)

On one hand, the coarse term has a bandwidth of octaves. Therefore, by the induction hypothesis, we have for , and a fortiori for . On the other hand, we consider the complex Shannon scalogram of in some subband :

 (11)

In the double sum above, all integer differences of the form range between and . Thus, is a periodic signal of fundamental frequency spanning octaves. Furthermore, because , has a smaller bandwidth than ; i.e., octaves or less. By the induction hypothesis, we have:

 ∀λ,UM+1(|y∗hM∗ψλ|2)(λ1,…,λm+1)=0. (12)

In the equation above, we recognize the scattering path of . Finally, because the scattering transform is a contractive operator [mallat2012cpam], we have the inequality:

 ∥∥UM+2(y)∥∥≤∥∥UM+2(y∗gM)∥∥+∥∥UM+2(y∗hM)∥∥=0, (13)

which implies , and likewise at deeper layers. We conclude by induction that the theorem holds for any . ∎


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# Meromorphic function at a point.

i'm study meromorphic function at the complex plane extended, but i have a trouble. I know if $$f \colon D \subseteq \mathbb{C} \to \mathbb{C}$$ $$f$$ is meromorphic on D if the singularities of $$f$$ are poles in $$D$$ But whats mean $$f$$ is meromorphic at a point $$a \in \mathbb {C}$$?

Other quickly question, anyone knows why $$\infty$$ is branched point of $$log(z)$$?

Thanks

• In answer to your second question: $\infty$ is a branch point of $\log z$ because $0$ is a branch point of $\log \frac{1}{z} = -\log z$. – mjw Jun 12 '19 at 4:47
• We usually don't define holomorphicity, meromorphicity, etc. at single points because singletons are closed and we need open sets for almost all the nice theorems. I would thus assume that when someone writes that "$f$ is meromorphic at $a$" the author is saying "$f$ is meromorphic in a neighborhood of $a$". However, without knowing the context you found this it is hard to say – Brevan Ellefsen Jun 12 '19 at 5:14
• I've found this in "Complex functions: An algebraic and geometric viewpoint" - Singerman. The point of this is extend the notions of analytic and meromorphic functions at complex plane extend, autor says $f(z)$ are analytic, meromorphic, etc... at infinity if $f(1/z)$ are analytic, meromorphic, etc... At $0$. – Davis We Jun 12 '19 at 12:31
• @DavisWe Oh, then yes - the author means analytic in a neighborhood. This is very standard terminology. You are essentially looking at some small neighborhood of the north pole on the Riemann sphere and seeing if you get holo/mero-morphicity. (For some sort of reference, here's a post on this site from a quick google search. Let me know if you still have questions on this) – Brevan Ellefsen Jun 13 '19 at 5:14