Questions tagged [superspace-formalism]
The Green-Schwarz formalism, or the superspace-formalism, are formalisms for supersymmetry with explicit spacetime supersymmetry.
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How do we construct an action on a superspace lattice?
I am interested in the formulation of supersymmetric theories on a discrete spacetime, such as a lattice. I know that there are some difficulties in preserving supersymmetry on a lattice, such as the ...
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References to lattice supermanifolds
Do you have any references (textbooks and/or internet links) to lattice supermanifolds or, more generally, discrete superspaces?
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Superfields over superspaces
Suppose that we have an $n$-dimensional vector space $X$ together with a basis $\{\theta_i\}^n_{i = 1}\in X$.
$$\bigwedge(X) = \mathbb{C}\oplus\bigwedge^1(X)\oplus\bigwedge^2(X)\oplus\cdots\oplus\...
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Generalized Stokes theorem in superspace
Do the generalized Stokes theorem apply in superspace? Any issues or uncommon behaviour of the gradient, divergence and rotational in superspace?
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Symmetries and equations of superspheres and other superspaces
What are the symmetries and the most studied/most standard examples of superspaces? I include exceptional superalgebras and infinite-dimensional spaces.
Bonus: Do quantum groups apply in the above ...
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Functions on Superspace
I find the definition of a function $\phi$ on a superspace $(z,\theta)$ confusing for the following reason. $\phi(z,\theta)$ can be expanded as $$\phi(z,\theta) = \phi_0(z) + \theta \psi(z)\tag{1}.$$ ...
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Is this interpretation of fermionic dimensions correct?
The number of Grassmann coordinates in ${\cal N}=1$, $3+1$ dimensional superspace is $4$. Let's call them:
$\theta_1$ $\theta_2$ $\theta_3$ $\theta_4$.
The Grassmann variables can be represented by ...
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How do you show that $DD \Phi = 4 m \Phi^{\dagger}$ yields massive field equations for the component fields of a chiral superfield $\Phi$?
Suppose $\Phi(x, \theta, \bar{\theta})$ is a chiral superfield ($\bar{D}_{\dot{\alpha}} \Phi = 0$). One can write it in components as
$$
\Phi(x, \theta, \bar{\theta}) = f(x) + \theta \phi(x) + \bar{\...
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Grassmann numbers and fermionic strings
Is it correct that by introducing Grassmann numbers as new directions of spacetime we can make strings behave like fermions (that is, 1/2-spin objects)?
And if so, is it possible to show how that ...
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Fermionic components of super-gauge transformations
Consider $4d$ $\mathcal{N}=1$ super Yang-Mills with gauge group $U(N)$, but really any supersymmetric gauge theory in any dimension will do. People have developed a formalism to quantize such theories ...
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Dimension of moduli space for SQCD
We are in $\mathcal{N}=1$ SUSY. Consider massless SQCD with gauge group $SU(N)$ and $F$ flavours. The quarks superfields $Q$ and $\tilde{Q}$ are $F\times N$ and $N\times F$ matrices respectively and ...
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Functional integral for unconstrained superfields
Context
In this paper by Srivastava (also in his book "Supersymmetry, Superfields and Supergravity"), he proposes the functional integral for a chiral superfield $\Phi$. In order to work ...
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Problem in deriving equations of motion for a free chiral superfield
Setting of the problem
We are in $\mathcal{N}=1$ SUSY and we are using superspace formalism in the set of coordinates $(x^\mu,\theta_\alpha,\bar\theta_{\dot{\alpha}})$ where $\theta_\alpha$ and $\bar\...
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Component field expansion of $(1,1)$ supersymmetric Polyakov action
I am working on the renormalizibility of two-dimensional nonlinear sigma models in string theory and particularly the $(1,1)$ supersymmetric extension of it. The following is based on the review by C. ...
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Chiral constraint on superfield
In SUSY to select the required supermultiplet from the entire set of fields in the general superfield we must impose the follow constraint
$\bar{D}_{\dot{\alpha}}\Phi=0$
And a superfield subject to ...
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Relation between the on-shell superspace and lightcone superspace
There is an on-shell superspace (introduced by V.Nair) for $\mathcal{N} = 4$.
It is introduced in the section 4.3 in this paper https://arxiv.org/abs/1308.1697 , for instance.
Four Grassmann
variables ...
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How do I prove that the product of chiral superfields is itself a chiral superfield?
I am currently learning about $\mathcal{N}=(2,2)$ supersymmetry and have come up against what is probably a really silly question.
The $\mathcal{}N=(2,2)$ superspace consists of bosonic coordinates $x^...
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Invariant of supersymmetry?
Given two vectors in 3D superspace $(x_1^\mu,\theta_1^\alpha,\overline{\theta}_1^\alpha)$ and $(x_2^\mu,\theta_2^\alpha,\overline{\theta}_2^\alpha)$ I am trying to find a polynomial invariant under ...
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Extended SUSY and superspace
I am trying to understand how to construct an action using the superspace formalism for $\mathcal N>1$. I have read that this is quite difficult to do, so let's consider a simple example. Suppose I ...
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Auxiliary fields in non-supersymmetric theory
It is well-known, that in superspace formulation of supersymmetric theories auxiliary fields appear. In present of such fields SUSY transformations are linear and independent of model.
Are some non-...
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What is the form of local supersymmetry transformations in superspace?
So the most general local superspace transformation generator I can write is:
$$\hat{L} = A^\mu(x,\theta)\frac{\partial}{\partial x^\mu} + B^\alpha(x,\theta)\frac{\partial}{\partial \theta^\alpha}$$
...
user84158
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Gauge invariant supersymmetric transformations
Given the following action
$$
\mathcal{S}=\int{d^4x\;d^2\theta\;d^2\bar{\theta}\left(\bar{Q}_+e^{2V}Q_++\bar{Q}_-e^{-2V}Q_--2\xi V\right)}+\int{d^4x\;d^2\theta\left(mQ_-Q_++\frac{\tau}{16\pi i}W^\...
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SUGRA & consistent field theories
Why must a consistent theory with a rarita schwinger field (i.e. massless gravitino in the spectrum) be supersymmetric? I was reviewing the GSO projection, Spin Structure, etc. & wasn’t able to ...
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General expression for a Scalar Superfield with ${\cal N = 1}$ in $D = 10$
I need to write explicitly the most general form of a scalar superfield in 10 dimensions up to second order in grassman variables $\theta^\alpha$. I know that up to first order one can write
\begin{...
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Expand superspace function into component form
In 2D (1,1) superconformal field theory, the invariant "distance" between two points $Z_1=(z_1,\theta_1)$ and $Z_1=(z_1,\theta_1)$ in superspace is $$Z_{12}=z_1-z_2-\theta_1\theta_2.$$
My question ...
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Expanding superfields: inconsistency of notation?
If I have a wavefunction of a fermion field $\Psi[\psi]$ I can expand it like so about some vacuum:
$$\Psi[\psi] = \Psi_0[\psi]( a + \int a(x)\psi(x)dx+\int a(x,y)\psi(x)\psi(y)dxdy+...)$$
Now all ...
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Determination of Torsion constraints in ${\cal N} = 1$, D = 10 Superspace
For the on-shell theory, containing the graviton $e_m^{\ \ \ a}$, gravitino $\psi_m^{\ \ \ \alpha}$, dilaton $\phi$, dilatino $\lambda$ and 3-form $H_{n m p}$, one has to demand that the SUSY algebra ...
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Why is an action built from superfields guaranteed to be supersymmetric?
Given a superfield (in 0+1 spacetime + 2 superspace coordinates)
$$X(t,\theta_1,\theta_2) = x(t) + \theta_i \psi_i(t) + \theta_1 \theta_2 F_{12}(t)\tag{1}$$
and given the standard supercharges ...
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Supersymmetry Generator Definition for ${\cal N }= 1$
I am studying SYM $\mathcal{N}$ = 1 in D = 10, and using the bimodular representations for the 32x32 gamma matrices $\Gamma^a$. This means that I work with the off-diagonal 16x16 matrices, which I ...
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Anti-Commutator of derivatives of Grassmann variables
How do I evaluate the anti-commutator of $\frac{\partial}{\partial\chi}$ and $\frac{\partial}{\partial\eta}$ when both $\chi$ and $\eta$ are Grassmann variables?
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What are Grassmann numbers in field theory?
I've been struggling with the use of Grassmann numbers in QFT e.g. Peskin and Schroeder. They are introduced as "numbers" whose product is antisymmetric, and associative (this isn't said, but used in ...
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${\cal N} = 1$ SUSY Non-renormalization theorem
In Ref. 1, on Page 53, the ${\cal N} = 1$ SUSY non-renormalization theorem is derived. One first specifies the symmetries of the general ${\cal N} = 1$ SUSY action in the superspace formalism, and ...
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Auxiliary Grassmann variables in supergeometry
I was reading on super geometry and how it is used to model fermions and supersymmetry in classical field theory.
In texts like [1] or [2] the authors introduced auxiliary Grassmann odd variables to ...
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Grassmann-odd extra dimensions and gravity
Take a world with $D=3+n$ space-time dimensions, where $n$ are extra space-like dimensions.
With extra-dimensional newton gravity
$$F=G_N(D)\dfrac{Mm}{r^{2+n}}$$
Can $n$ affect IF the extra ...
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Could you get real space from Grassmann numbers?
You can get a vector field from a pair of spinor fields with $A_\mu(x)=\psi(x) \gamma_\mu \overline{\psi}(x)$. Using this fact could you define a space-time vector in terms of Grasman numbers?
Say ...
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Super-gauge transformation in two dimensional $\mathcal{N}= (0,2)$ superspace
I'm trying to couple matter to $\mathcal{N}=(0,2)$ SYM in 2d using superfield formalism. There are some paper (this on Sec. 6, or this on Sec. 3 [whose notation will be used here]) that construct what ...
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2D ${\cal N}=(2,2)$ Super Yang-Mills with Superspace
I'm reading this famous paper by Witten. There is the expression of field strength for the abelian vector multiplet (eq. (2.16)):
$$\Sigma = \frac{1}{\sqrt{2}}\bar{D}_+D_- V\;.\tag{2.16}$$
I'm ...
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Are there SUSY Lagrangian terms that are not D-term nor F-term?
I've read that a way to construct supersymmetric invariant lagrangian could be either to integrate a superfield in the whole superspace, i.e. in all anticommuting coordinates (D-term), or in half of ...
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Supergravity action as a total integral, over 4 spacetime and 4 Grassmann coordinates
Wess and Bagger, in their Supersymmetry and Supergravity, give the action for a global SUSY, ${\cal N}=1$, $D=4$, Yang-Mills gauge model as an integral over the 4 spacetime coordinates and 4 Grassmann ...
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Super Field Strength Identity
I work on a introduction into Super-symmetry. In the course we define
\begin{equation}
D_{\alpha} = \frac{\partial}{\partial \theta^{\alpha}} - i \sigma^{\mu}_{\alpha \dot{\alpha}} \bar{\theta}^{\dot{\...
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Ambiguity of Time Derivative of Superfunctions
I think that there is an ambiguity for defining the time derivative of a superfunction on the phase space of pseudo-classical mechanics of Grassmann numbers.
Let $\xi$ be a Grassmann odd number. Its ...
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Questions about the Ferrara-Zumino Multiplet
These questions arose while reading the paper ``Comments on Supercurrent Multiplets, Supersymmetric Field Theories and Supergravity" by Komargodski and Seiberg (arXiv:1002.2228)
The Ferrara-Zumino ...
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What is the definition of $\delta^{m|n}$ and $\delta^{m}$?
I am reading the paper. What is the definition of $\delta^{m|m}$ and $\delta^{m+k}$ in (1.1) and (1.3) on pages 2,3? Are they some kind of delta function?
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Grassmann numbers & supermanifolds
I'm asking this question because I'm currently trying to learn about Super Symmetry but I'm having trouble understanding the concept of super-space and super-manifold.
I read that in super-spaces you ...
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Should the complex conjugate of a derivative of a Grassmann number include a sign?
Take a real Grassmann variable, by which I mean $\theta=\theta^*$. We have
$$\int d\theta~ \theta =1,\qquad \frac{\partial}{\partial\theta}\theta=1$$
If I define the conjugation of Grassmann ...
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Fayet-Iliopoulos terms
It is mentioned in first page of this paper by Seiberg and Komargodski that the Lagrangian in superspace of a $U(1)$ gauge SUSY theory with FI terms is not gauge invariant. However, the FI terms in ...
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State counting in the d = 1+2, $\cal{N} = 2$ vector multiplet
The question is from Box 8.2, page 282 of the book "Gauge Gravity Duality" by Ammon and Erdmenger. The link to the specific page from Google Books is here.
According to the authors, a $\mathcal{N} = ...
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Projective superspace: why extra bosonic coordinates
I'm studying the projective superspace formalism for N = 4 supersymmetric $\sigma$-models in two dimensions. My question is: why do we need the extra bosonic coordinates for the manifest action?
I ...
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Do the Grassmann coordinates in the superfield formalism have any physical meaning?
In the superfield formalism we consider fields in a space who has four so called bosonic coordinates $x^{\nu}$ and four so called fermionic coordinates $\theta_1$,$\theta_2$,$\bar{\theta_1}$,$\bar{\...
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What mathematical structure describes superspace and superfields?
In every book related to supersymmetry I have encountered at some point the idea of superspace is introduced. Superspace is presented as a space spanned by 4 "normal" directions and 4 Grassmannian ...
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