We study (2,2) supersymmetric field theories on two-dimensional worldsheet with boundaries. We determine D-branes (boundary conditions and boundary interactions) that preserve half of the bulk supercharges in nonlinear sigma models, gauged linear sigma models, and Landau-Ginzburg models. We identify a mechanism for brane creation in LG theories and provide a new derivation of a link between soliton numbers of the massive theories and R-charges of vacua at the UV fixed point. Moreover we identify Lagrangian submanifolds that arise as the mirror of certain D-branes wrapped around � �

D-Branes And Mirror Symmetry

Numerous correlated electron systems exhibit a strongly scale-dependent behavior. Upon lowering the energy scale, collective phenomena, bound states, and new effective degrees of freedom emerge. Typical examples include (i) competing magnetic, charge, and pairing instabilities in two-dimensional electron systems; (ii) the interplay of electronic excitations and order parameter fluctuations near thermal and quantum phase transitions in metals; and (iii) correlation effects such as Luttinger liquid behavior and the Kondo effect showing up in linear and nonequilibrium transport through quantum wires and quantum dots. The functional renormalization group is a flexible and unbiased tool for dealing with such scale-dependent behavior. Its starting point is an exact functional flow equation, which yields the gradual evolution from a microscopic model action to the final effective action as a function of a continuously decreasing energy scale. Expanding in powers of the fields one obtains an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes. This review is a comprehensive introduction to the functional renormalization group method for interacting Fermi systems. A self-contained derivation of the exact flow equations is presented and frequently used truncation schemes are described. Reviewing selected applications it is shown how approximations based on the functional renormalization group can be fruitfully used to improve our understanding of correlated fermion systems.

Functional renormalization group approach to correlated fermion systems

For general compact Kahler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of teh authors and Klimek and Lesniewski obtained for the tours and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finitedimensional matrix algebrasgl(N), N→∞.

/pdf/toeplitz-quantization-of-kahler-manifolds-ang-gl-n-n-limits-3qzxhkeh5e.pdf

Toeplitz quantization of Kähler manifolds ang gl(N), N→∞ limits

The subject of this book is equilibrium statistical mechanics, in particular the theory of critical phenomena, and quantum field theory. The central theme is the use of random-walk representations as a tool to derive correlation inequalities. The consequences of these inequalities for critical-exponent theory are expounded in detail. The book contains some previously unpublished results. It addresses both the researcher and the graduate student in modern statistical mechanics and quantum field theory.

/pdf/random-walks-critical-phenomena-and-triviality-in-quantum-47uq48dng3.pdf

Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory

We show that the existence of a finitely summable unbounded Fredholm module (h, D) on a C* algebra A implies the existence of a trace state on A and that no such module exists on the C* algebra of a non amenable discrete group. Both for the needs of non commutative differential geometry and of analysis in infinite dimension we are led to the better notion of the θ-summable Fredholm module.

/pdf/compact-metric-spaces-fredholm-modules-and-hyperfiniteness-35bec2z3m9.pdf

Compact metric spaces, Fredholm modules, and hyperfiniteness

We construct a cocycle on an infinite dimensional generalization of ap-summable Fredholm module. Our framework is related to Connes' cyclic cohomology and is motivated by our work on index theory on infinite dimensional manifolds. Thep-summability condition is characteristic of dimensionO(p). We replace this assumption by the requirement that there exists an underlying heat kernel which is trace class. Then we use the heat kernel to regularize states in dimension-independent fashion. Our cocycle may be interpreted as an infinite dimensional Chern character.

/pdf/quantum-k-theory-i-the-chern-character-58roavfpl9.pdf

Quantum $K$-theory. I. The Chern character

We construct a non-commutative ℂ*-algebra$C_\mu (\bar U)$ which is a quantum deformation of the algebra of continuous functions on the closed unit disc$\bar U$.$C_\mu (\bar U)$ is generated by the Toeplitz operators on a suitable Hilbert space of holomorphic functions onU.

/pdf/quantum-riemann-surfaces-i-the-unit-disc-1podfday7b.pdf

Quantum Riemann surfaces I. The unit disc

A two-parameter quantum deformation of the unit disc

Using a rigorous version of the renormalization group we construct the effective action for theY2 model. The construction starts with integrating out the bosonic field which eliminates the large fields problem. Studying the soobtained purely fermionic theory proceeds by a series of convergent perturbation expansions. We show that the continuum limit of the effective action exists and its perturbation expansion is Borel summable.

/pdf/effective-action-for-the-yukawa-2-quantum-field-theory-2p6cbemgwr.pdf

Effective action for the Yukawa 2 quantum field theory

https://www.lesniewski.us/papers/papers_2/GroundStateStructure.pdf

Andrzej Lesniewski

Papers

Quantum $K$-theory. I. The Chern character

Quantum Riemann surfaces I. The unit disc

A two-parameter quantum deformation of the unit disc

Effective action for the Yukawa 2 quantum field theory

Ground state structure in supersymmetric quantum mechanics