Questions tagged [superalgebra]
The superalgebra tag has no usage guidance.
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Looking for a source to explain the Process of Topological twisting
as the title suggests I am looking for papers or other material that explains the notion of Topological twisting as it appears in the context of certain SUSY algebras. Concretely I am interested in ...
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Literature on Representation Theory of Graded Lie Algebras
I am currently studying 'advanced' representation theory, including topics like super-Lie algebras. I've come across various gradings (excluding the $\mathbb{Z}_2$ grading), such as how to select odd ...
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Berezin Integration, confirming an measure is invariant
I am working through the Mirror Symmetry book, available here.
I already had a question about an earlier part of the same Exercise 9.2.1 on page 157:
We are given the following action with one boson ...
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Confirming an action is invariant under a supersymmetric transformation
I am studying chapter 9 of the book Mirror Symmetry, available here. My question is relating to page 156/157 where Supersymmetry is being introduced for the first time in QFT in 0-dimensions.
We are ...
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Difference between $\mathcal{N}=2$ and $\mathcal{N}=(1,1)$ SUSY
In supersymmetry algebra, $\mathcal{N}$ refers to $I=1,2,.. \mathcal{N} $ in $Q^{I}_{\alpha}$. My question is what does it mean to write $\mathcal{N}=(1,1)$ superalgebra?
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SUSY Algebra Anti-commutation Relation
I am trying to understand one of the anti-commutation relations of SUSY algebra. The lecture notes "Supersymmetry and Extra Dimensions" (PDF) taken by Flip Tanedo, says on p.29, due to the ...
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Grassmann parameter in supersymmetry
Let's consider a free Wess-Zumino Lagrangian given by
$$\mathcal{L} = \partial^{\mu}\overline{\phi}\partial_{\mu}\phi + i\psi^{\dagger}\overline{\sigma}^{\mu}\partial_{\mu}\psi\tag{1}$$
Whose ...
user366054
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Bosonic representation of delta function for Grassmann-even quantity
Suppose I have 2 Grassmann scalars $\theta$ and $\bar{\theta}$ and form the bosonic quantity $X = \bar{\theta}\theta$. Is there a purely bosonic representation of the delta function $\delta(X - \...
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Why must Lie superalgebras always contain both $Q$ and $\bar{Q}$?
In four dimensions Lie superalgebras naturally arise by relaxing the presupposition in the Coleman-Mandula theorem that the symmetry is not classified by a Lie algebra. It is then typically stated ...
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How to construct irreducible representations of supergroup $SU(N|N)$?
In physics, it is quite common to use Young tableaux to denote the irreducible representations of Lie groups, such as $SU(N)$. There are many excellent textbooks explaining how to construct them.
...
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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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Matrix representation of Grassmann variables and Berezin Integrals
In this question, the problem of finding matrix representations for a set of Grassmann variables is discussed.
How can this representation be used in Berezin integrals or Grassmann derivatives?
Can ...
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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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Contradictory results for Berezin integral
Say $$u = (u_1, \dots, u_{2n}) = (\xi_1, \eta_1, \dots \xi_n, \eta_n)\tag{1}$$ is a vector of Grassmann variables. For an antisymmetric bosonic matrix $A$ we know that
$$
\int e^{\frac{1}{2} \sum_{a,...
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What does it exactly mean by right and left functional derivatives?
In BV formalism of the gauge theory, we need to compute the right / left functional derivatives of the actions that include fermions.
I do not quite see what it means by that.
For example, let us ...
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Complex valued Grassmann variables $(\theta \eta)^* $, $(\theta \eta)^T$ and $(\theta \eta)^\dagger$
Since hermitian conjugation and complex conjugation are serious issues in a QFT lagrangian with Grassmann variables, see here and here. Let us try to go to the bottom.
We start by accepting the ...
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Anti-de Sitter superalgebra from preserved symmetry
Following Freedman & Van Proeyen's "Supergravity", the $\mathcal{N}=1,\, D=4$ AdS supergravity action is
$$ \dfrac{1}{2\kappa^2}\int d^4x \; e \left[R(\omega) - \bar{\psi}_\mu \gamma^{\...
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Interaction with odd number of fermionic fields?
Can there exist an interacting part of the Hamiltonian with odd number of fermionic operators? In other words, can we have a vertex which couples an odd number of fermions (there can also be 1, 2, or ...
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Jacobi identity of the anti-bracket
I'm currently reading a volume 2 of Weinberg's QFT, and am puzzled by the Jacobi identities of the anti-bracket.
The anti-bracket is defined using the anti-field $\chi^n$ and $\chi_n^{‡}$ as follows
$...
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Berezin integral of a Grassmann field
Consider a time dependent Grassmann field i.e. $\theta(t)$. Now, consider the following Berezin integral $$\int [\mathcal{D}\theta] ~\prod_{t}\dot{\theta}\tag{1}$$
where $\dot{\theta}$ time derivative ...
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Conventions for graded wedge product in supergeometry
There are two conventions for the graded exterior product on superspace (see https://ncatlab.org/nlab/show/signs+in+supergeometry):
$$\alpha \wedge \beta = (-1)^{pq+|\alpha||\beta|}\beta\wedge\alpha \;...
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Odd or Even symplectic structure in BV formalism
I am studying Batalin-Vilkovisky formalism. I am a little bit confused on what an odd (or even) symplectic structure is (i know what the degree of the underlying 2-form is). I can not find a clear ...
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Logarithm of Grassmann numbers
What is $\log \theta$ where $\theta$ is a Grassmann number such that $\theta^2 = 0$. How does one then look at its logarithm? Does it even make sense?
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Integration with complex Grassmann numbers
I have a question about a convention from Peskin & Schroeder, namely that
$$\int d\theta^{*}\, d\theta \, (\theta \theta^*) = 1,$$
where $\theta$ and $\theta^*$ are independent Grassmann numbers. ...
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Superbundles and spin in Dirac equation
The Dirac operator $D$ of a Clifford bundle $S$ is a first order differential operator on $C^{\infty}(S)$.
The Clifford bundles considered are $\mathbb{Z} / 2 \text { -graded }$ (superbundles). This ...
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How do I prove that the product of chiral superfields is itself a chiral superfield?
I am currently learning about $\mathcal{N}=(2,2)$ supersymmetry and have come up against what is probably a really silly question.
The $\mathcal{}N=(2,2)$ superspace consists of bosonic coordinates $x^...
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Derivation of anti-commutation relations of massive supermultiplet generators [closed]
In almost all intro to supersymmetry notes the commutation relations are given between the generators and their conjugates however, I can not find any proofs of them anywhere and am struggling to ...
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Right derivative of Grassmann number and associated anti-commutation relation
I am reading chapter 3 of Anomalies in quantum field theory by Reinhold Bertlmann and I found one statement that I don't know how to prove. First of all he defined the right derivative on the ...
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Question for ${\cal N}=1$ supersymmetry representations
Please see this lecture note: https://arxiv.org/abs/1011.1491.
In section "2.2.5 Massless supermultiplet"
the author defines a Casimir and says it is zero.
How can we confirm it?
We take the ...
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Precise zero energy bound for supersymmetry
Usually we can shift the energy $E$ by any amount $\delta$ to redefine the lowest energy as
$$
E + \delta.
$$
However, in supersymmetry, there is a precise $E=0$ must be true, so that the supercharge $...
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Supersymmetry infinitesimal variation
In Wess & Bagger, chapter 3, the infinitesimal supersymmetric transformation is defined as: $$ \delta_\xi \psi = i\sqrt{2} \ \sigma^m \bar{\xi}\partial_m A + \sqrt{2} \ \xi F$$ and $$\delta_\xi A =...
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How do fermionic operators transform?
In quantum mechanics, if we have an operator $\Omega$, then under the transformation $T$, with infinitesimal generator $G$ (i.e. $T(\epsilon)=1-i\epsilon G + \ldots$), then operator transforms as
$$\...
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Ward identity of QED - whether the fields are all $c$-number fields
I am following Sidney Coleman's lectures of Quantum Field Theory.
At the end of ch.32, he derived the Ward identity for the 1PI generating functional $\Gamma[\psi,\bar{\psi},A_{\mu}]$ for QED:
\begin{...
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Right vs Left Derivatives
Let $\theta$ be a fermionic quantity and $f(\theta)=f(0)+\theta\frac{\partial f}{\partial\theta}=f(0)+\frac{\partial_r f}{\partial\theta}\theta$. Under a variation $\theta\mapsto\theta+\delta\theta$ ...
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Jacobi Identity in de Sitter Superalgebra
In the book "Supergravity" (by D. Freedman and A. Van Proeyen) they talk about why $\mathcal{N}=1$ de-sitter superalgebra is impossible to construct (Section 12.6.1).
Basically de-Sitter algebra is a ...
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SUSY $\mathcal{N}=1$ algebra
Given the definitions
$$
P_\mu= -i\partial_\mu
$$
$$
Q_\alpha=-i(\partial_\alpha-(\sigma^\mu\bar{\theta})_\alpha\partial_\mu)
$$
$$
\bar{Q_\dot{\alpha}}=+i(\bar{\partial}_\dot{\alpha}-({\theta}\sigma^...
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$\mathcal{N} \ge 2$ Supersymmetry massive supermultiplets
In Bertolinis SUSY notes [https://people.sissa.it/~bertmat/susycourse.pdf] we have defined:
$$
\{Q^I_\alpha,\bar{Q}_\dot{\beta}^J\}=2m\delta_{\alpha\dot{\beta}}\delta^{IJ}\tag{3.24}
$$
$$
\{Q^I_\alpha,...
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What is the meaning of a Grassmann variable?
I don't seem to understand the concept of a Grassmann-variable.
When studying superspaces and superfields I am told that the coordinates being used are $$(x^\mu, \theta_\alpha, \bar{\theta}_\alpha)$$ ...
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Are there cases where the use of the Grassmann variables simplifies computations in the usual bosonic analysis?
When one introduces complex numbers and complex analysis one can then use the new machinery to solve some real-analysis problems. A lamppost example is computing integrals via residues.
I think I've ...
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Is the expectation value of a Fermi field operator a Grassmann number?
It's often noted that Bosonic fields result from quantizing classical field theories defined on a regular numbers, whereas Fermionic fields arise when quantizing a classical field theory defined on ...
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Unitarity/Hermiticity condition for $osp(m,n|\mathbb{C})$ superalgebra
According to Dictionary on Lie Superalgebras (page 82), the compact form of $OSP(m,n|\mathbb C)$ Lie superalgebra must satisfy $M^{\text st}H\,M=1$ and $M^{\ddagger}M=1$ (is this the unitarity ...
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What exactly are "Grassmann-valued fields"?
Peskin & Schroeder define a Grassmann field $\psi(x)$ as a function whose values are anticommuting numbers, that can be written as : [p.301 eq. 9.71]
$$\psi(x) = \sum\psi_i \phi_i(x),\tag{9.71}$$
...
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Expand superspace function into component form
In 2D (1,1) superconformal field theory, the invariant "distance" between two points $Z_1=(z_1,\theta_1)$ and $Z_1=(z_1,\theta_1)$ in superspace is $$Z_{12}=z_1-z_2-\theta_1\theta_2.$$
My question ...
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One question about BRST symmetry in reading Srednicki’s book: Why should the BRST charge $Q_B$ be nilpotent?
In p.453, Srednicki claims that since the BRST transformation of a BRST transformation is zero, $Q_B$, the BRST charge, must be nilpotent:
$$Q_{B}^{2}=0.\tag{74.32}$$
I don't know why.
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What's the space of eigenvalues/field configurations for a fermion?
In the Schrödinger picture of quantum field theory, the field eigenstates of a real scalar field $\hat\phi(\mathbf x)$ with $\mathbf x \in\mathbb R^3$ are the states $\hat\phi(\mathbf x)|\phi\rangle=\...
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Expanding superfields: inconsistency of notation?
If I have a wavefunction of a fermion field $\Psi[\psi]$ I can expand it like so about some vacuum:
$$\Psi[\psi] = \Psi_0[\psi]( a + \int a(x)\psi(x)dx+\int a(x,y)\psi(x)\psi(y)dxdy+...)$$
Now all ...
user84158
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Representing quaternionic algebra with creation and annihilation operators?
The paper "Quantized Grassmann variables and unified theories" says given creation and annihilation operators $b$ and $b^\dagger$ one can represent quaternionic imaginary units $q_1$, $q_2$ and $q_3$ ...
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Why is the Jacobian factor for fermionic variables different from that for bosonic ones?
In Srednicki's textbook Quantum Field Theory, Section 77 discusses anomalies and the path integral for fermions. The path integral over the Dirac field is defined to be
\begin{equation}
Z(A) \equiv \...
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What does a supercharge physically conserve? [duplicate]
What is actually being conserved? I've calculated it for the Wess-Zumino model but I still have no idea what is actually being conserved due to Noether's Theorem.
There is already a similar question, ...
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