Questions tagged [superconformality]

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Why for 6d SCFT we consider (1,0) and (2,0) only?

It is known to have a stress-energy tensor we must have the supercharges $\mathcal{N}<2$. My confusion is why in most cases we consider chiral supercharges, and what is the problem with $\mathcal{N}...
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General derivation of Supervirasoro algebra

I'm looking for a derivation derivation of the ($\mathcal{N} = 1$) Supervirasoro algebra (NS sector) that does't just apply to specific examples. Most books/papers either just cite the result, or ...
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Is there a guide for superconformal field theories?

Is there a guide for superconformal field theories for $\mathcal N=1,2,3,4…$ ? I don't understand what characteristics they have, what the motivation for each is, and how they relate to supergravity ...
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1 answer
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Is there superconformal group? (Exponentiating the superconformal algebra)

I (only superficially) know that not every Lie algebra can be exponentiated to give a Lie group. I also have only heard about the superconformal algebras, and not the superconformal groups. This is in ...
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Zamolodchikov metric and coupling-dependent field rescalings

this is my first post/question, so I apologize in advance for any mistakes in phrasing or format or anything else. Consider a conformal field theory (CFT) with exactly marginal deformations described ...
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2 votes
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Control Parameter for Coulomb Phase of $\mathcal{N}=4$ SYM?

When discussing the vacua of $\mathcal{N}=4$ super-Yang-Mills (SYM) theory (with any connected gauge group you like, such as $U(N)$ or $SU(N)$), a simple analysis of the potential tells you the vacuum ...
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Intuition for Plethystic Exponentiation

I have been studying Abjit Gadde's lecture notes on the superconformal index, and I can't seem to understand what the intuition for the plethystic exponentiation is. He motivates it with the ...
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Why study 6d SCFT, when it is 2d CFT that is well studied?

According to AGT correspondence, if we compactify 6d SCFT to a 4d Riemann surface, then we find that certain physical quantities of 4d QFT on this surface are related to certain properties of that ...
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Worldsheet SCFT on a lattice

My question is clear from the title. I'm curious whether it is possible to put the string world sheet SCFT on a lattice. I expect when the world sheet theory is chiral, then it's not possible. But I ...
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What Operator is the Superconformal Index Counting?

Given a differential operator $\mathcal{D}$ with adjoint $\mathcal{D}^\dagger$, the analytical index of $\mathcal{D}$ is usually defined by $$\text{ind }\mathcal{D}=\dim\ker\mathcal{D}-\dim\ker\...
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Superconformal transformation (Polchinski text 12.3.8)

I am reading Polchinski's text book STING THEORY. In the above of eq.(12.3.8), the differential $D_\theta = \partial_\theta + \theta\partial_z$ is defined and \begin{equation} D_\theta = D_\theta\...
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On the picture of Schur limit of the superconformal index

My goal is to understand qualitatively (hopefully quantitatively in future) the existence of the relationship of the Schur limit of the superconformal index given by \begin{align} \mathcal{I}(q)=\...
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  • 511
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1 answer
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Weyl SUSY vs Conformal SUSY

Is it possible to add the generators of dilatations to Poincare superalgebra in any dimensions with any number of supercharges without adding the full superconformal generators? I have only seen ...
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1 answer
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6D (2,0) superconformal field theory

I'm looking for a good reference book or textbook to study on 6D (2,0) superconformal field theory as a part of string theory.
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$(0,2)$ worldsheet SUSY implies ${\cal N}=1$ spacetime SUSY (reference request)

In [1] they write that "Boucher et al. [14] found quite generally that $(0, 2)$ world-sheet SUSY ensures the existence of $N = 1$ spacetime SUSY." But this reference "[14]" they cite just says: "...
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1 answer
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Supercurrent of the $bc$-$\beta\gamma$ SCFT

In Polchinksi's Sec. 10.1, the $bc$-$\beta\gamma$ SCFT is introduced with action $$S_{BC} = \frac{1}{2\pi} \int d^2z (b \bar \partial c + \beta \bar \partial \gamma)$$ and supercurrent $$T_F = -\...
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How to represent a superconformal algebra as differential operators acting on the fields in superspace?

Both super algebras and conformal algebras can be nicely represented as differential operators acting on the fields in superspace, but I've never seen one for superconformal algebras. I would ...
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Basic question about $\mathcal{N}=2$ SCFT

Consider a $\mathcal{N}=2$ SCFT radially quantized on $R^3 \times S^1$. The Lorentz group is $SU(2)_+ \times SU(2)_-$ and there are 8 independent supercharges $Q_{\alpha}^I,\tilde{Q}_{I \dot \alpha} $...
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Resource Recommendations: Extended Supersymmetry in 2+1 Dimensions

I'm interested in comprehensive resource recommendations covering extended supersymmetry (all cases) in 2+1 dimensions with the aim of understanding topics such as 2+1 dimensional SCFTs [ABJ(M), BLG ...
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1 answer
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Why doesn't the Superconformal Algebra close?

What I mean is, if I take the conformal algebra and the supersymmetry charges, how do I show that $[K, Q]$ is not in this algebra? I see this fact stated everywhere but with no proof. Any references ...
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Are there superconformal field theories in 10D?

I've heard that there is a belief that interacting conformal field theories do not exist in dimensions greater than 6, and in 6D the only known nontrivial CFTs are superconformal field theories. What ...
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Non-renormalization theorems at large N

I know of numerous examples of non-renormaliation theorems in theories with SUSY - e.g. the non-renormaliation of the superpotential in 4d theories. However, I've never seen a non-renormaliation ...
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6 votes
1 answer
459 views

Question about multiplets of 6d $\mathcal{N}=(1,0)$ SUSY

In Strathdeee's "Extended Poincare Supersymmetry", the first entry on page 16 lists the massless multiplets of 6d $\mathcal{N} = (1,0)$ supersymmetry as $2^2 = (2,1; 1) \oplus (1,1; 2)$. This is the ...
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Does scale invariance and R-invariance of Kähler potential imply superconformal symmetry?

Consider a four-dimensional $\mathcal{N} = 1$ field theory with Lagrangian: $ \mathcal{L} = \int d^4 \theta K(\Phi, \bar \Phi) $ and assume $K$ transforms well under dilations with scaling dimension ...
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5 votes
1 answer
409 views

Derivation of conformal generators in spinor helicity formalism

I've been trying for some time to find the expressions for conformal generators of Witten's paper in perturbative Yang-Mills. Given $P_{\alpha \dot{\alpha}} = \lambda_{\alpha} \overline{\lambda}_{\...
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Question about complex conjugation, and dotted/undotted indices in the 4d $\mathcal{N} = 1$ superconformal algebra

The 4-dimensional $\mathcal{N} = 1$ superconformal algebra as presented in equations (2.2, 2.3, and 2.4) of the paper, "Counting chiral primaries in N = 1 d = 4 superconformal algebras (arXiv:hep-th/...
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2 votes
1 answer
256 views

6D $(1,0)$ supersymmetry from properties of 6D spinors

It is known from string/M-theory considerations that six dimensional superconformal field theories exist with $(1,0)$ supersymmetry. But if one looks at Table 2.4 on page 47 of Sergio Cecotti's book,...
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0 votes
1 answer
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Conformal supergravity or local tensor calculus? [closed]

What are the pros and cons of the two formalisms? In particular for explicit calculations of components. Are there textbooks providing self-contained, detailed description of superconformal ...
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1 vote
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On the action of superconformal generators in maximally supersymmetric Yang-Mills

Consider maximally supersymmetric Yang-Mills theory in 3+1 dimensions. This theory has 32 supercharges: 16 ordinary ones, conventionally labeled $Q$; and 16 superconformal ones, conventionally ...
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2 answers
550 views

Question about the superconformal index

According to arXiv:1507.08553v1, the superconformal index, defined by $$I(\beta_j) = \mbox{Tr}_{\mathcal{H}}(-1)^F e^{-\gamma\{Q,Q^\dagger\}}e^{-\sum_{j}\beta_j t_j}$$ is independent of the ...
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1 vote
1 answer
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Superconformal description of supergravity action

I've been reading this paper http://arxiv.org/abs/hep-th/0110263. In section 4, he discusses the benefits of writing the supergravity action in a superconformal way. I have a few questions regarding ...
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Diiference between three squashed sphere and three sphere and number of susy

First I know that three sphere $S^3$ and squashed sphere $S_b^3$ \begin{align} S_{b}^{3} = \begin{array} & R^2 \times S_{r}, \quad r=b, \quad b\rightarrow 0 \\ R^2 \times S_{r}, \quad r=\frac{1}{b}...
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5 votes
0 answers
395 views

Connection between the M5 brane and (2, 0) superconformal field theory

I have read that the worldvolume theory of the M5 brane is a $(2, 0)$ superconformal field theory (SCFT). But I have also learnt from talks that the $(2, 0)$ theory lacks a Lagrangian description. ...
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9 votes
4 answers
5k views

What does the $I$-$V$ curve in josephson junction mean?

According to the $I$-$V$ curve for Josephson junction tunneling for S-I-S (superconductor-insulator-superconductor), Do we have any tunneling current for $0< V\leq V_c$? If yes, then why don't we ...
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  • 1,363
6 votes
1 answer
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Why the bosonic part of the superconformal group $SU(2,2|1)$ is $SO(4,2) \times U(1)_R$?

Why in $d=4$ $\mathcal{N}=1$ SCFT the bosonic part of the superconformal group $SU(2,2|1)$ is $SO(4,2) \times U(1)_R$? More generally how can I determine the such a thing in other theories? Is there ...
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4 votes
1 answer
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Questions on the $N=2$ superconformal algebra

In my understanding, mirror symmetry in physics originates from representation of the $N=2$ superconformal algebra. Why do we need precisely 2 supersymmetries (why not 1 or 4)? Moreover, a chiral (...
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7 votes
1 answer
314 views

Superconformal approach to supergravity

In the book (Supergravity - Daniel Z.Freedman & Antoine Van Proeyen - Cambridge), there is (Chapters 16-17) a presentation of pure supergravity or supergravity with matter, from a superconformal ...
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8 votes
1 answer
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About the general expression of trace anomaly and CFT partition functions

I have put up a question here, https://mathoverflow.net/questions/139685/proof-of-the-general-expression-for-anomaly-in-a-cft-and-its-partition-function Here I am putting up a slightly different ...
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  • 4,439
11 votes
1 answer
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About defining "baryons" and "mesons"

I want to understand the proof of the claims (of the construction as well as of its uniqueness) of gauge singlet states given around equation 2.13 (page 10) of this paper. Also does the listing of ...
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  • 4,439
3 votes
1 answer
282 views

Other Gross-Neveu like theories?

By "Gross-Neveu like" I mean non-supersymmetric QFTs whose partition function/beta-function (or any n-point function) is somehow exactly solvable in the large $N_c$ or $N_f$ or 't Hooft limit. (.....
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9 votes
3 answers
274 views

Embedding of $F(4)$ in $OSp(8|4)$?

Is the superconformal algebra in five dimensions, $F(4)$, a subalgebra of the (maximal) six-dimensional superconformal algebra $OSp(8|4)$?
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Boundary conditions in AdS/CFT

This question is in reference to this very famous paper of Witten. In general through the whole paper why is the author able to just focus on the scalar field propagating in the bulk and not need to ...
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2 votes
1 answer
729 views

Pedagogic reference for calculation of 2-loop anomalous dimension (supersymmetric)

I want to know of pedagogic references which teach how to compute anomalous dimensions (..wave-function renormalization..) at lets say 2-loops. I guess there might be specialized techniques for ...
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6 votes
1 answer
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About 2+1 dimensional superconformal algebra

I would like to get some help in interpreting the main equation of the superconformal algebra (in $2+1$ dimenions) as stated in equation 3.27 on page 18 of this paper. I am familiar with supersymmetry ...
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6 votes
0 answers
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The ${\cal N} = 3$ Chern-Simons matter lagrangian

This question is sort of a continuation of this previous question of mine. I would like to know of some further details about the Lagrangian discussed in this paper in equation 2.8 (page 7) and in ...
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4 votes
0 answers
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Some more questions on conformal spinors of $SO(n,2)$

This is somewhat of a continuation of my previous question. I had stated there that a conformal spinor ($V$) of $SO(n,2)$ can be created by taking a direct sum of two $SO(n-1,1)$ spinors $Q$ and $S$ ...
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  • 4,361
4 votes
1 answer
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Lorentz spinors of $SO(n,1)$ and conformal spinors of $SO(n,2)$

It would be great if someone can give me a reference (short enough!) which explains the (spinor) representation theory of the groups $SO(n,1)$ and $SO(n,2)$. I have searched through a few standard ...
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  • 4,361
8 votes
1 answer
356 views

Reference for the ${\cal N}=3$ Chern-Simons Lagrangian at general $N_c$, $N_f$

I was wondering if someone could give me a reference where someone has explicitly written the Lagrangian for ${\cal N}=3$ $SU(N_c)$ Chern-Simons theory coupled to $N_f$ fundamental hypermultiplets. ...
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  • 4,439
4 votes
0 answers
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Central charge at the fixed point of the ${\cal N}=2$ Landau-Ginzburg theory in $1+1$ dimensions

Let me first believe that the ${\cal N}=2$ Landau-Ginzburg theory does in the IR flow to a non-trivial fixed point and that if the potential is of the form $\Phi ^k$ then the central charge of the CFT ...
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  • 4,361
2 votes
1 answer
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A certain $\cal{N}=2$ superconformal theory (or is it?)

I want to look at the following theory in $1+1$ dimensions with $\Phi$ being the chiral superfield, $L = \int d^2x d^4\theta \bar{\Phi}\Phi - \int d^2x d^2\theta \frac{\Phi^{k+2}}{k+2} - \int d^2x d^...
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