# Questions tagged [superalgebra]

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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 ...
1 vote
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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 ... 84 views

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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 ...
1 vote
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In Bertolinis SUSY notes [https://people.sissa.it/~bertmat/susycourse.pdf] we have defined: $$\{Q^I_\alpha,\bar{Q}_\dot{\beta}^J\}=2m\delta_{\alpha\dot{\beta}}\delta^{IJ}\tag{3.24}$$ $$\{Q^I_\alpha,... 0 votes 1 answer 1k views ### What is the meaning of a Grassmann variable? I don't seem to understand the concept of a Grassmann-variable. When studying superspaces and superfields I am told that the coordinates being used are$$(x^\mu, \theta_\alpha, \bar{\theta}_\alpha)$$... 1 vote 1 answer 90 views ### Are there cases where the use of the Grassmann variables simplifies computations in the usual bosonic analysis? When one introduces complex numbers and complex analysis one can then use the new machinery to solve some real-analysis problems. A lamppost example is computing integrals via residues. I think I've ... 2 votes 1 answer 308 views ### Is the expectation value of a Fermi field operator a Grassmann number? It's often noted that Bosonic fields result from quantizing classical field theories defined on a regular numbers, whereas Fermionic fields arise when quantizing a classical field theory defined on ... 2 votes 1 answer 64 views ### Unitarity/Hermiticity condition for osp(m,n|\mathbb{C}) superalgebra According to Dictionary on Lie Superalgebras (page 82), the compact form of OSP(m,n|\mathbb C) Lie superalgebra must satisfy M^{\text st}H\,M=1 and M^{\ddagger}M=1 (is this the unitarity ... 6 votes 1 answer 1k views ### What exactly are "Grassmann-valued fields"? Peskin & Schroeder define a Grassmann field \psi(x) as a function whose values are anticommuting numbers, that can be written as : [p.301 eq. 9.71]$$\psi(x) = \sum\psi_i \phi_i(x),\tag{9.71}$$... 2 votes 1 answer 64 views ### Expand superspace function into component form In 2D (1,1) superconformal field theory, the invariant "distance" between two points Z_1=(z_1,\theta_1) and Z_1=(z_1,\theta_1) in superspace is$$Z_{12}=z_1-z_2-\theta_1\theta_2.$$My question ... 2 votes 2 answers 505 views ### One question about BRST symmetry in reading Srednicki’s book: Why should the BRST charge Q_B be nilpotent? In p.453, Srednicki claims that since the BRST transformation of a BRST transformation is zero, Q_B, the BRST charge, must be nilpotent:$$Q_{B}^{2}=0.\tag{74.32}$$I don't know why. 7 votes 1 answer 322 views ### What's the space of eigenvalues/field configurations for a fermion? In the Schrödinger picture of quantum field theory, the field eigenstates of a real scalar field \hat\phi(\mathbf x) with \mathbf x \in\mathbb R^3 are the states \hat\phi(\mathbf x)|\phi\rangle=\... 3 votes 1 answer 221 views ### Expanding superfields: inconsistency of notation? If I have a wavefunction of a fermion field \Psi[\psi] I can expand it like so about some vacuum:$$\Psi[\psi] = \Psi_0[\psi]( a + \int a(x)\psi(x)dx+\int a(x,y)\psi(x)\psi(y)dxdy+...) Now all ... 1 vote
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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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