19

Physicist chiming in; "quantum corrections of a geometry" represents the vague idea that gravity should be quantum. Classically, gravity is a geometric theory: "reality" is modelled as a (Lorentzian) manifold, whose metric can be found -- in principle -- by solving a system of PDEs. To be more specific, the metric $g$ is such that the classical action $S[g]$...


14

You should probably read the conclusion (Section 6) of Atiyah's laudatio on Witten work during the 1990 ICM (specifically, the positive mass theorem is treated in Section 3). 6. Conclusion From this very brief summary of Witten's achievements it should be clear that he has made a profound impact on contemporary mathematics. In his hands physics is once ...


11

I presume this refers to quantum corrections to the metric tensor. These are expected to be important in general relativity when the curvature of space-time approaches the Planck scale $\sqrt{G\hbar/c^3}\simeq 10^{-33}$ cm. Some pointers to the literature: Einstein Equation with Quantum Corrections Quantum Corrections to the Schwarzschild and Kerr Metrics ...


7

The question is not clearly stated, but if appropriately interpreted it does make sense. Let $S$ be a spinor bundle over a pseudo-Riemannian manifold $(M,g)$, whose bundle of Clifford algebras is denoted by $Cl(M,g)$. Since $S$ is a spinor bundle, there exists a morphism of bundles of associative, unital algebras: $\gamma \colon Cl(M,g)\to End(S)$ to the ...


7

Summary: The integral $\int_0^\infty t^{\frac{s-1}{2}} \operatorname{Tr}[ D^u (\exp(-t D^2) ] \, dt$ has a simple pole at $s=0$ with residue $2 C_{-1/2}$, from which your expression for $\eta(0)$ follows. To compute the residue, we compute the integral explicitly (up to a holomorphic remainder) near $t=0$ for $\operatorname{Re} s$ large, then analytically ...


6

No, we cannot (completely) hear the shape of a drum, even if it is spinorial. Two metric fields with the same collection of eigenvalues are called isospectral. There exist Dirac isospectral deformations; continuous 1-parameter families of mutually non- isometric metrics with the same Dirac spectrum have been constructed. They are of the form $M_s =...


6

The assertion is supposed to be that $d(e^{-tD^2})/dt$ has the same smooth kernel as $-D^2 e^{-tD^2}$, i.e. they are the same operator. This is because $e^{-tD^2}$ is the solution operator to the heat equation: $\frac{du}{dt} = -D^2 u$. I'd be comfortable saying this is the definition of $e^{-tD^2}$, but you might prefer to construct $e^{-tD^2}$ using the ...


5

Your question is entirely covered by Section 2 of Brain–Mesland–Van Suijlekom, but the fgp case is simple enough to ultimately boil down to folklore proved by Chakraborty–Mathai. Let me summarise what happens, while incorporating some technical simplifications from Blecher–Kaad–Mesland. For convenience, let me take $A$ and $B$ to be unital $C^\ast$-algebras ...


5

The essence of "geometric algebra" (better known as Kahler-Atiyah algebra) is the classical Chevalley-Riesz isomorphism, which presents the Clifford algebra of a quadratic space $(V,h)$ as a deformation quantization of the exterior algebra of $V$. The systematic use of this presentation allows for the automatic translation of various computations ...


4

The label "geometic algebra" was William Cifford's name for the algebra he discovered (invented). This bit of history is recounted by David Hestenes, who has perhaps been most responsible for promoting this lovely mathematics for physics, and has many books and papers on this subject. There is also a Cambridge lot, including Chris Doran and ...


4

Here's a reference to a paper; Theorem 3 gives a general formula for differential forms with no constraints on the boundary values. I'm not sure if this shows up in a book yet. Raulot, S.; Savo, A. A Reilly formula and eigenvalue estimates for differential forms. J. Geom. Anal. 21 (2011), no. 3, 620--640. MR2810846. Here is a direct link to the article.


2

First some historical comments: The first sufficiently clear reference that I know for the spinorial Weierstrass is a preprint by Rob Kusner and Nick Schmitt link. From the point of view of the results, Friedrich does not add much, the value of Friedrich's article mostly lies in its very explicit presentation in invariant language. Friedrich's article was ...


2

Most of the questions raised above are answered in the article "The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles" by Christian Bär and myself published in Annals of Global Analysis and Geometry June 1998, Volume 16, Issue 3, pp 221–253. http://link.springer.com/article/10.1023/A:1006553302362. A preprint version is available on the arxive as ...


2

The map $\sigma$ is a representation of $U_q(\mathfrak{l})$ on the vector space $\Sigma_{2m}$, let's say for the moment just any such representation. To answer the second question, $\sigma(S(X))\psi$ should really be thought as $(1\otimes\sigma(S(X)))\psi$. Let $\psi$ be decomposable of the form $\psi_1\otimes\psi_2$ with $\psi_1\in \mathbb C_q[G]$ and $\...


2

This is too long for a comment. I found the article Lazaroiu, Calin Iuliu; Babalic, Elena Mirela; Coman, Ioana Alexandra, Geometric algebra techniques in flux compactifications, Adv. High Energy Phys. 2016, Article ID 7292534, 42 p. (2016). ZBL1366.83098. on the classification of Killing (s)pinors using geometric algebra with applications to $\mathcal N=...


2

The action $T\colon S\to S$ is compatible with the Clifford-multiplication, hence the decomposition into $\pm1$ eigenspaces is preserved. Moreover, the spin connection is also preserved, and by the definition of the Dirac operator this should solve question 1.


1

For question 2, the original manifold M will be a double cover of the quotient space since it's an involution, and if you have an operator that commutes with the involution on the covering space, it should descend to the quotient space. If there are "nice" fixed points one can get an orbifold or a stratified space, where you can still make sense of ...


1

The general boundary condition for the Dirac equation is a local linear restriction on the components of the spinor wave function at the boundary, $$\psi=M\psi,\;\;M=\begin{pmatrix} n_z&n_x-in_y\\ n_x+in_y&-n_z \end{pmatrix}$$ with ${\mathbf n}=(n_x,n_y,n_z)$ a unit vector. (Check that $M^2=\mathbb{1}$.) If you are interested in the generalization ...


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