scholarly journals Odd dimensional analogue of the Euler characteristic

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
L. Borsten ◽  
M. J. Duff ◽  
S. Nagy
Keyword(s):  
Partition Function ◽  
Linear Combination ◽  
Betti Numbers ◽  
Gauge Fields ◽  
Euler Characteristics ◽  
Kunneth Formula ◽  
Analogous Formula ◽  
Mirror Map ◽  
M Theory ◽  
Odd Dimensions

Abstract When compact manifolds X and Y are both even dimensional, their Euler characteristics obey the Künneth formula χ(X × Y) = χ(X)χ(Y). In terms of the Betti numbers bp(X), χ(X) = Σp(−1)pbp(X), implying that χ(X) = 0 when X is odd dimensional. We seek a linear combination of Betti numbers, called ρ, that obeys an analogous formula ρ(X × Y) = χ(X)ρ(Y) when Y is odd dimensional. The unique solution is ρ(Y) = − Σp(−1)ppbp(Y). Physical applications include: (1) ρ → (−1)mρ under a generalized mirror map in d = 2m + 1 dimensions, in analogy with χ → (−1)mχ in d = 2m; (2) ρ appears naturally in compactifications of M-theory. For example, the 4-dimensional Weyl anomaly for M-theory on X4× Y7 is given by χ(X4)ρ(Y7) = ρ(X4× Y7) and hence vanishes when Y7 is self-mirror. Since, in particular, ρ(Y × S1) = χ(Y), this is consistent with the corresponding anomaly for Type IIA on X4× Y6, given by χ(X4)χ(Y6) = χ(X4× Y6), which vanishes when Y6 is self-mirror; (3) In the partition function of p-form gauge fields, ρ appears in odd dimensions as χ does in even.

scholarly journals A non-relativistic limit of M-theory and 11-dimensional membrane Newton-Cartan geometry

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Chris D. A. Blair ◽  
Domingo Gallegos ◽  
Natale Zinnato
Keyword(s):  
Poisson Equation ◽  
Longitudinal Direction ◽  
Gauge Fields ◽  
Finite Limit ◽  
Relativistic Limit ◽  
Finite Action ◽  
Symmetric Formulation ◽  
M Theory ◽  
Chern Simons ◽  
Local Tangent

Abstract We consider a non-relativistic limit of the bosonic sector of eleven-dimensional supergravity, leading to a theory based on a covariant ‘membrane Newton-Cartan’ (MNC) geometry. The local tangent space is split into three ‘longitudinal’ and eight ‘transverse’ directions, related only by Galilean rather than Lorentzian symmetries. This generalises the ten-dimensional stringy Newton-Cartan (SNC) theory. In order to obtain a finite limit, the field strength of the eleven-dimensional four-form is required to obey a transverse self-duality constraint, ultimately due to the presence of the Chern-Simons term in eleven dimensions. The finite action then gives a set of equations that is invariant under longitudinal and transverse rotations, Galilean boosts and local dilatations. We supplement these equations with an extra Poisson equation, coming from the subleading action. Reduction along a longitudinal direction gives the known SNC theory with the addition of RR gauge fields, while reducing along a transverse direction yields a new non-relativistic theory associated to D2 branes. We further show that the MNC theory can be embedded in the U-duality symmetric formulation of exceptional field theory, demonstrating that it shares the same exceptional Lie algebraic symmetries as the relativistic supergravity, and providing an alternative derivation of the extra Poisson equation.

scholarly journals "SCHWINGER MODEL" ON THE FUZZY SPHERE

2010 ◽  
Vol 25 (37) ◽  
pp. 3151-3167 ◽  
Author(s):  
E. HARIKUMAR
Keyword(s):  
Partition Function ◽  
Field Strength ◽  
Dirac Operator ◽  
Path Integral ◽  
Integral Method ◽  
Gauge Fields ◽  
Fuzzy Sphere ◽  
Schwinger Model ◽  
Spinor Fields ◽  
Path Integral Method

In this paper, we construct a model of spinor fields interacting with specific gauge fields on the fuzzy sphere and analyze the chiral symmetry of this "Schwinger model". In constructing the theory of gauge fields interacting with spinors on the fuzzy sphere, we take the approach that the Dirac operator Dq on the q-deformed fuzzy sphere [Formula: see text] is the gauged Dirac operator on the fuzzy sphere. This introduces interaction between spinors and specific one-parameter family of gauge fields. We also show how to express the field strength for this gauge field in terms of the Dirac operators Dq and D alone. Using the path integral method, we have calculated the 2n-point functions of this model and show that, in general, they do not vanish, reflecting the chiral non-invariance of the partition function.

MASSLESS SCALAR FIELDS IN NON-ABELIAN KALUZA-KLEIN THEORY

1994 ◽  
Vol 09 (04) ◽  
pp. 507-515 ◽  
Author(s):  
M. ARIK ◽  
V. GABAY
Keyword(s):  
Cosmological Constant ◽  
Direct Product ◽  
Scalar Fields ◽  
Gauge Fields ◽  
Internal Space ◽  
Massless Scalar ◽  
Klein Theory ◽  
Euler Form ◽  
Odd Dimensions ◽  
Kaluza Klein

We investigate the presence of massless scalar fields in a Kaluza—Klein theory based on a dimensionally continued Euler-form action. We show that massless scalar fields exist provided that the internal space is a direct product of two irreducible manifolds. The condition of a vanishing effective four-dimensional cosmological constant and the presence of a graviton, gauge fields and massless scalar fields can be satisfied if both irreducible manifolds have odd dimensions and the sum of these dimensions is equal to the dimension of the Euler form.

CHERN–SIMONS TERMS AND ANOMALIES IN GAUGE THEORIES

1991 ◽  
Vol 06 (21) ◽  
pp. 1915-1921 ◽  
Author(s):  
R. BANERJEE
Keyword(s):  
Effective Action ◽  
Gauge Theories ◽  
Gauge Fields ◽  
Chern Simons ◽  
External Gauge ◽  
Odd Dimensions

The parity-violating effective action for theories of fermions coupled to external gauge fields in arbitrary odd dimensions is calculated exactly by a perturbative technique. This effective action is used to obtain the structures for the Chern–Simons terms in odd dimensions and anomalies in even dimensions. Our analysis clearly elucidates the connection of the Chern–Simons terms with the covariant and consistent anomalies.

scholarly journals GAUGE FIELDS ON TORUS AND PARTITION FUNCTION OF STRINGS

1989 ◽  
Vol 04 (02) ◽  
pp. 389-400 ◽  
Author(s):  
A. NAKAMURA ◽  
K. SHIRAISHI
Keyword(s):  
Partition Function ◽  
Wilson Loop ◽  
Open String ◽  
Closed String ◽  
Point Of View ◽  
Gauge Fields ◽  
Partition Functions ◽  
Zero Modes ◽  
Background Fields ◽  
String Theories

In this paper we consider the interrelation between compactified string theories on torus and gauge fields on it. We start from open string theories with background gauge fields and derive partition functions by path integral. Since the effects of background fields and compactification correlate only through string zero modes, we investigate these zero modes. From this point of view, we discuss the Wilson loop mechanism at finite temperature. For the closed string, only a few comments are mentioned.

scholarly journals A correspondence between Ricci-flat Kerr and Kaluza-Klein AdS black hole

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Liang Ma ◽  
H. Lü
Keyword(s):  
Domain Walls ◽  
Einstein Gravity ◽  
Gravitational Instantons ◽  
Four Dimensions ◽  
Ricci Flat ◽  
Angular Momenta ◽  
Ads Black Holes ◽  
M Theory ◽  
Odd Dimensions ◽  
Kaluza Klein

Abstract We establish an explicit correspondence of Einstein gravity on the squashed spheres that are the U(1) bundles over ℂℙm to the Kaluza-Klein AdS gravity on the tori. This allows us to map the Ricci-flat Kerr metrics in odd dimensions with all equal angular momenta to charged Kaluza-Klein AdS black holes that can be lifted to become singly rotating M-branes and D3-branes. Furthermore, we find maps between Ricci-flat gravitational instantons to the AdS domain walls. In particular the supersymmetric bolt instantons correspond to domain walls that can be interpreted as distributed M-branes and D3-branes, whilst the non-supersymmetric Taub-NUT solutions yield new domain walls that can be lifted to become solutions in M-theory or type IIB supergravity. The correspondence also inspires us to obtain a new superpotential in the Kaluza-Klein AdS gravity in four dimensions.

scholarly journals Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions

10.37236/3741 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Martin Loebl ◽  
Petr Somberg
Keyword(s):  
Riemann Surface ◽  
Partition Function ◽  
Linear Combination ◽  
Dirac Operators ◽  
Finite Graph ◽  
Discrete Analogue ◽  
Square Roots ◽  
Closed Riemann Surface ◽  
Discrete Analogues ◽  
Discrete Dirac Operators

The aim of the paper is to formulate a discrete analogue of the claim made by Alvarez-Gaume et al., realizing the partition function of the free fermion on a closed Riemann surface of genus $g$ as a linear combination of $2^{2g}$ Pfaffians of Dirac operators. Let $G=(V,E)$ be a finite graph embedded in a closed Riemann surface $X$ of genus $g$, $x_e$ the collection of independent variables associated with each edge $e$ of $G$ (collected in one vector variable $x$) and $\S$ the set of all $2^{2g}$ spin-structures on $X$.  We introduce $2^{2g}$ rotations $rot_s$ and $(2|E|\times 2|E|)$ matrices $\Delta(s)(x)$, $s\in \Sigma$, of the transitions between the oriented edges of $G$ determined by rotations $rot_s$. We show that the generating function of the even sets of edges of $G$, i.e., the Ising partition function, is a linear combination of the square roots of $2^{2g}$ Ihara-Selberg functions $I(\Delta(s)(x))$ also called Feynman functions. By a result of Foata and Zeilberger $I(\Delta(s)(x))=\det(I-\Delta'(s)(x))$, where $\Delta'(s)(x)$ is obtained from $\Delta(s)(x)$ by replacing some entries by $0$. Thus each Feynman function is computable in a polynomial time. We suggest that in the case of critical embedding of a bipartite graph $G$, the Feynman functions provide suitable discrete analogues for the Pfaffians of Dirac operators.

scholarly journals Determination of features based on the formation of persistent spectra of eigenvalues of Laplace matrices

2020 ◽  
pp. 49-64
Author(s):  
S. V. Lejhter ◽  
S. N. Chukanov
Keyword(s):  
Persistent Homology ◽  
Betti Numbers ◽  
Simplicial Complexes ◽  
Wasserstein Distance ◽  
Computer Hardware ◽  
Euler Characteristics ◽  
Topological Features ◽  
Laplace Matrix ◽  
Definition Of

An algorithm for determining the spectrum of eigenvalues of the Laplace matrix for simplicial complexes has been developed in the paper. The spectrum of eigenvalues of the Laplace matrix is used as features in the data structure for image analysis. Similarly to the method of persistent homology, the filtering of embedded simplicial complexes is formed, approximating the image of the object, but the topological features at each stage of filtration is the spectrum of eigenvalues of the Laplace matrix of simplicial complexes. The spectrum of eigenvalues of the Laplace matrix allows to determine the Betti numbers and Euler characteristics of the image. Based on the method of finding the spectrum of eigenvalues of the Laplace matrix, an algorithm is formed that allows you to obtain topological features of images of objects and quantitative estimates of the results of image comparison. Software has been developed that implements this algorithm on computer hardware. The method of determining the spectrum of eigenvalues of the Laplace matrix has the following advantages: the method does not require a bijective correspondence between the elements of the structures of objects; the method is invariant with respect to the Euclidean transformations of the forms of objects. Determining the spectrum of eigenvalues of the Laplace matrix for simplicial complexes allows you to expand the number of features for machine learning, which allows you to increase the diversity of information obtained by the methods of computational topology, while maintaining topological invariants. When comparing the shapes of objects, a modified Wasserstein distance can be constructed based on the eigenvalues of the Laplace matrix of the compared shapes. Using the definition of the spectrum of eigenvalues of the Laplace matrix to compare the shapes of objects can improve the accuracy of determining the distance between images.

scholarly journals THE RAMOND–RAMOND SELF-DUAL FIVE-FORM'S PARTITION FUNCTION ON T10

2000 ◽  
Vol 15 (19) ◽  
pp. 1261-1273 ◽  
Author(s):  
LOUISE DOLAN ◽  
CHIARA R. NAPPI
Keyword(s):  
Quantum Theory ◽  
Partition Function ◽  
Automorphic Forms ◽  
Explicit Calculation ◽  
Time Dimension ◽  
Modular Invariant ◽  
Recent Interest ◽  
Type Iib String Theory ◽  
M Theory ◽  
Dual Quantum

In view of the recent interest in formulating a quantum theory of Ramond–Ramond p-forms, we exhibit an [Formula: see text] invariant partition function for the chiral four-form of Type IIB string theory on the ten-torus. We follow the strategy used to derive a modular invariant partition function for the chiral two-form of the M-theory five-brane. We also generalize the calculation to self-dual quantum fields in space–time dimension 2p = 2 + 4k, and display the [Formula: see text] automorphic forms for odd p > 1. We relate our explicit calculation to a computation of the B-cycle periods, which are discussed in the work of Witten.

Sign in / Sign up

Export Citation Format

Share Document