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Igor Kriz

Publications and source records attributed to Igor Kriz.

At least 37 records · Page 2Linked to original sources
arXiv2015

D-structures and derived Koszul duality for unital operad algebras

Generalizing a concept of Lipshitz, Ozsváth and Thurs-ton from Bordered Floer homology, we define $D$-structures on algebras of unital operads, which can also be interpreted as a generalization of a seemingly unrelated concept of Getzler and Jones. This construction gives rise to an equivalence of derived categories, which can be thought of as a unital version of Koszul duality using non-unital Quillen homology. We also discuss a multi-sorted version of the construction, which provides a framework for unifying the known algebraic contexts of Koszul duality.

math.KT↗
arXiv2014

Equivariant K-theory of compact Lie groups with involution

For a compact simply connected simple Lie group $G$ with an involution $α$, we compute the $G\rtimes \Z/2$-equivariant K-theory of $G$ where $G$ acts by conjugation and $\Z/2$ acts either by $α$ or by $g\mapsto α(g)^{-1}$. We also give a representation-theoretic interpretation of those groups, as well as of $K_G(G)$.

math.KT↗
arXiv2013

A spanning tree cohomology theory for links

In their recent preprint, Baldwin, Ozsváth and Szabó defined a twisted version (with coefficients in a Novikov ring) of a spectral sequence, previously defined by Ozsváth and Szabó, from Khovanov homology to Heegaard-Floer homology of the branched double cover along a link. In their preprint, they give a combinatorial interpretation of the $E_3$-term of their spectral sequence. The main purpose of the present paper is to prove directly that this $E_3$-term is a link invariant. We also give some concrete examples of computation of this invariant.

math.GT↗
arXiv2012

Categorical Geometry and Integration Without Points

The theory of integration over infinite-dimensional spaces is known to encounter serious difficulties. Categorical ideas seem to arise naturally on the path to a remedy. Such an approach was suggested and initiated by Segal in his pioneering article \cite{segal}. In our paper we follow his ideas from a different perspective, slightly more categorical, and strongly inspired by the point-free topology. First, we develop a general (point-free) concept of measurability (extending the standard Lebesgue integration when applying to the classical $σ$-algebra). Second (and here we have a major difference from the classical theory), we prove that every finite-additive function $μ$ with values in $[0,1]$ can be extended to a measure on an abstract $σ$-algebra; this correspondence is functorial and yields uniqueness. As an example we show that the Segal space can be characterized by completely canonical data. Furthermore, from our results it follows that a satisfactory point-free integration arises everywhere where we have a finite-additive probability function on a Boolean algebra.

math.PR↗
arXiv2011

Topological Hermitian Cobordism

Extending our method for investigating Real cobordism (which was recently used by Hill, Hopkins and Ravenel in their solution of the Kervaire invariant 1 problem), we investigate the $RO(G)$-graded homotopy groups of a (non-complete) $\Z/2\times \Z/2$-equivariant spectrum called topological Hermitian cobordism. The methods of this paper may be useful in computing the homotopy groups of other $G$-equivariant spectra where $G\neq \Z/2$.

math.AT↗
arXiv2011

Tree Algebras: An algebraic axiomatization of intertwining vertex operators

We describe a completely algebraic axiom system for intertwining operators of vertex algebra modules, using algebraic flat connections, thus formulating the concept of a {\em tree algebra}. Using the Riemann-Hilbert correspondence, we further prove that a vertex tensor category in the sense of Huang and Lepowsky gives rise to a tree algebra over $\C$. We also show that the chiral WZW model of a simply connected simple compact Lie group gives rise to a tree algebra over $\Q$.

math.QA↗
arXiv2010

A Universal Approach to Vertex Algebras

We characterize vertex algebras (in a suitable sense) as algebras over a certain graded co-operad. We also discuss some examples and categorical implications of this characterization.

math.RA↗
arXiv2009

Completing Verlinde Algebras

We compute the completion of the Verlinde algebra of a simply connected simple compact Lie group $G$ at the augmentation ideal of the representation ring. By results of Freed, Hopkins, Teleman and C.Dwyer and Lahtinen, this gives a computation of (non-equivariant) twisted $K$-theory of the free loop space of $BG$.

math.AT↗
arXiv2009

Perturbative deformations of conformal field theories revisited

We investigate the moduli space of conformal field theories by setting up a canonical mathematical process for exponentiating perturbations corresponding to critical fields. We apply this process to the free field theory and the Gepner models of the Fermat quintic and quartic. We find algebraic obstructions to exponentiating purely perturbative deformations in the case of the quintic, while in the case of the quartic the obstructions vanish. While this result may seem surprising at first, we find an explanation of these effects via the renormalization analysis of Nemeschansky-Sen.

hep-th↗
arXiv2009

The symplectic Verlinde algebras and string K-theory

We construct string topology operations in twisted K-theory. We study the examples given by symplectic Grassmannians, computing the twisted K-theory of the loop spaces of quaternionic projective spaces in detail. Via the work of Freed-Hopkins-Teleman, these computations are related to completions of the Verlinde algebras of Sp(n). We compute these completions, and other relevant information about the Verlinde algebras. We also identify the completions with the twisted K-theory of the Gruher-Salvatore pro-spectra. Further comments on the field theoretic nature of these constructions are made.

math.AT↗
arXiv2008

What is the Jacobian of a Riemann surface with boundary?

We define the Jacobian of a Riemann surface with analytically parametrized boundary components. These Jacobians belong to a moduli space of ``open abelian varieties'' which satisfies gluing axioms similar to those of Riemann surfaces, and therefore allows a notion of ``conformal field theory'' to be defined on this space. We further prove that chiral conformal field theories corresponding to even lattices factor through this moduli space of open abelian varieties.

math.AG↗
arXiv2008

Laplaza Sets, or How to Select Coherence Diagrams for Pseudo Algebras

We define a general concept of pseudo algebras over theories and 2-theories. A more restrictive such notion was introduced by Hu and Kriz, but as noticed by M. Gould, did not capture the desired examples. The approach taken in this paper corrects the mistake by introducing a more general concept, allowing more flexibility in selecting coherence diagrams for pseudo algebras.

math.CT↗
arXiv2007

Comments on D-branes on Orbifolds and K-theory

We systematically revisit the description of $D$-branes on orbifolds and the classification of their charges via K-theory. We include enough details to make the results accessible to both physicists and mathematicians interested in these topics. The minimally charged branes predicted by K-theory in Z_N orbifolds with $N$ odd are only BPS. We confirm this result using the boundary state formalism for Z_3. For Z_N x Z_N orbifolds with and without discrete torsion, we show that the K-theory classification of charges agrees with the boundary state approach, largely developed by Gaberdiel and collaborators, including the types of representation on the Chan-Paton factors.

hep-th↗
arXiv2006

On effective F-theory action in type IIA compactifications

Diaconescu, Moore and Witten proved that the partition function of type IIA string theory coincides (to the extent checked) with the partition function of M-theory. One of us (Kriz) and Sati proposed in a previous paper a refinement of the IIA partition function using elliptic cohomology and conjectured that it coincides with a partition function coming from F-theory. In this paper, we define the geometric term of the F-theoretical effective action on type IIA compactifications. In the special case when the first Pontrjagin class of spacetime vanishes, we also prove a version of the Kriz-Sati conjecture by extending the arguments of Diaconescu-Moore-Witten. We also briefly discuss why even this special case contains interesting examples.

hep-th↗
arXiv2006

The RO(G)-graded coefficients of (Z/2)^n-equivariant K-theory

In this note, we calculate all untwisted and twisted (Z/2)^n-equivariant K-groups with compact supports of real finite-dimensional linear representations of (Z/2)^n. The question was motivated by the question of D-brane charges for orbifold type II string vacua.

math.KT↗
arXiv2006

A mathematical formalism for the Kondo effect in WZW branes

In this paper, we show how to adapt our rigorous mathematical formalism for closed/open conformal field theory so that it captures the known physical theory of branes in the WZW model. This includes a mathematically precise approach to the Kondo effect, which is an example of evolution of one conformally invariant boundary condition into another through boundary conditions which can break conformal invariance, and a proposed mathematical statement of the Kondo effect conjecture. We also review some of the known physical results on WZW boundary conditions from a mathematical perspective.

hep-th↗
arXiv2005

Some remarks on fundamental physical F-theory

The purpose of this paper is to investigate the possibility of a physical 12-dimensional F-theory. We study the question of geometric interaction terms in the F-theory Lagrangians. We also introduce a new supergravity multiplet in dimension $(9,3)$ which is based on a particle with 3-dimensional timelike worldvolume. A construction of signature $(9,3)$ F-theory is given using dualities analogous to those considered by Hull, and possible matches of F-theory's low energy fields with the $(9,3)$-supergravity field content is given. Finally, preliminary suggestions are made regarding a possible phenomenological compactificaton of F-theory from dimension $(9,3)$ to $(3,1)$.

hep-th↗